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i am grateful to alekos kechris for informing me of t.dyck/ ; the proof given seems to be due to alain louveau .
i thank norm levenberg for references .
hough , j.b .
, krishnapur , m. , peres , y. , and virg , b. , _ zeros of gaussian analytic functions and determinantal point processes_. university lecture series , * 51*. american mathematical society , providence , ri , 2009 .
mester , p. , invariant monotone coupling need not exist .
* 41 * ( 2013 ) , 3a , 11801190 .
morris , b. , the components of the wired spanning forest are recurrent . _
probab . theory related fields _ * 125 * ( 2003 ) , 259265 .
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we describe the fundamental constructions and properties of determinantal probability measures and point processes , giving streamlined proofs .
we illustrate these with some important examples . we pose several general questions and conjectures .
primary 60k99 , 60g55 ; secondary 42c30 , 37a15 , 37a35 , 37a50 , 68u99 .
random matrices , eigenvalues , orthogonal projections , positive contractions , exterior algebra , stochastic domination , negative association , point processes , mixtures , spanning trees , orthogonal polynomials , completeness , bernoulli processes .
determinantal point processes were originally defined by macchi @xcite in physics . starting in the 1990s ,
determinantal probability began to flourish as examples appeared in numerous parts of mathematics @xcite .
recently , applications to machine learning have appeared @xcite .
a discrete determinantal probability measure is one whose elementary cylinder probabilities are given by determinants . more specifically ,
suppose that @xmath0 is a finite or countable set and that @xmath1 is an @xmath2 matrix .
for a subset @xmath3 , let @xmath4 denote the submatrix of @xmath1 whose rows and columns are indexed by @xmath5 .
if @xmath6 is a random subset of @xmath0 with the property that for all finite @xmath7 , we have e.dpm = ( qa ) , then we call @xmath8 a . the inclusion - exclusion principle in combination with yields the probability of each elementary cylinder event .
therefore , for every @xmath1 , there is at most one probability measure , to be denoted @xmath9 , on subsets of @xmath0 that satisfies .
conversely , it is known ( see , e.g. , b.lyons:det/ ) that there is a determinantal probability measure corresponding to @xmath1 if @xmath1 is the matrix of a positive contraction on @xmath10 ( in the standard orthonormal basis ) .
technicalities are required even to define the corresponding concept of determinantal point process for @xmath0 being euclidean space or a more general space .
we present a virtually complete development of their basic properties in a way that minimizes such technicalities by adapting the approach of b.lyons:det/ from the discrete case .
in addition , we use an idea of goldman b.goldman/ to deduce properties of the general case from corresponding properties in the discrete case .
space limitations prevent mention of most of what is known in determinantal probability theory , which pertains largely to the analysis of specific examples .
we focus instead on some of the basic properties that hold for all determinantal processes and on some intriguing open questions .
let @xmath0 be a denumerable set .
we identify a subset of @xmath0 with an element of @xmath11 in the usual way .
there are several approaches to prove the basic existence results and identities for determinantal probability measures .
we sketch the one used by b.lyons : det/. this depends on understanding first the case where @xmath1 is the matrix of an orthogonal projection .
it also relies on exterior algebra so that the existence becomes immediate .
any unit vector @xmath12 in a hilbert space with orthonormal basis @xmath0 gives a probability measure @xmath13 on @xmath0 , namely , @xmath14 associated to orthogonal projections @xmath15 .
we refer to b.lyons:det/ for details not given here .
identify @xmath0 with the standard orthonormal basis of the real or complex hilbert space @xmath10 . for @xmath16 ,
let @xmath17 denote a collection of ordered @xmath18-element subsets of @xmath0 such that each @xmath18-element subset of @xmath0 appears exactly once in @xmath17 in some ordering .
define @xmath19 if @xmath20 , then @xmath21 and @xmath22 .
we also define @xmath23 to be the scalar field , @xmath24 or @xmath25 .
the elements of @xmath26 are called of @xmath18 , or for short .
we then define the ( or ) of multivectors in the usual alternating multilinear way : @xmath27 for any permutation @xmath28 , and @xmath29 for any scalars @xmath30 ( @xmath31,\ ; e \in e'$ ] ) and any finite @xmath32 .
( thus , @xmath33 unless all @xmath34 are distinct . ) the inner product on @xmath26 satisfies e.ipdet = _ i , j when @xmath35 and @xmath36 are 1-vectors .
( this also shows that the inner product on @xmath26 does not depend on the choice of orthonormal basis of @xmath37 . )
we then define the ( or ) @xmath38 , where the summands are declared orthogonal , making it into a hilbert space .
( throughout the paper , @xmath39 is used to indicate the sum of orthogonal summands , or , if there are an infinite number of orthogonal summands , the closure of their sum . )
vectors @xmath40 are linearly independent iff @xmath41 . for a @xmath18-element subset @xmath3 with ordering @xmath42 in @xmath17 , write @xmath43 .
we also write @xmath44 for any function @xmath45 .
although there is an isometric isomorphism @xmath46 for @xmath47 , this does not simplify matters in the discrete case
. it will be very useful in the continuous case later , however .
if @xmath48 is a closed linear subspace of @xmath37 , written @xmath49 , then we identify @xmath50 with its inclusion in @xmath51 .
that is , @xmath52 is the closure of the linear span of the @xmath18-vectors @xmath53 . in particular ,
if @xmath54 , then @xmath55 is a 1-dimensional subspace of @xmath51 ; denote by @xmath56 a unit multivector in this subspace .
note that @xmath56 is unique up to a scalar factor of modulus 1 ; which scalar is chosen will not affect the definitions below .
we denote by @xmath15 the orthogonal projection onto @xmath48 for any @xmath49 or , more generally , @xmath57 .
l.projection for every closed subspace @xmath49 , every @xmath16 , and every @xmath58 , we have @xmath59 write @xmath60 and expand the product .
all terms but @xmath61 have a factor of @xmath62 in them , making them orthogonal to @xmath50 by e.ipdet/. a multivector is called or if it is the wedge product of 1-vectors .
b.whitney:book/ , p. 49 , shows that e.whitney .
we shall use the defined by duality : @xmath63 in particular , if @xmath64 and @xmath65 is a multivector that does not contain any term with @xmath66 in it ( that is , @xmath67 ) , then @xmath68 and @xmath69 . more generally , if @xmath70 with @xmath71 and @xmath72 , then @xmath73 and @xmath74 .
note that the interior product is sesquilinear , not bilinear , over @xmath25 .
for @xmath70 , write @xmath75 $ ] for the subspace of scalar multiples of @xmath12 in @xmath37 . if @xmath48 is a finite - dimensional subspace of @xmath37 and @xmath76 , then e.hwedge _ h
e = p_h^e_h + [ e ] ( up to signum ) . to see this ,
let @xmath77 be an orthonormal basis of @xmath48 , where @xmath78 .
put @xmath79 .
then @xmath80 is an orthonormal basis of @xmath81 $ ] , whence @xmath82 } = u_1 \wedge u_2 \wedge \cdots \wedge u_r \wedge v = \mv_h \wedge v = \mv_h \wedge e/\|p_h^\perp e\|\ ] ] since @xmath83 .
this shows e.hwedge/. similarly , if @xmath84 , then e.hvee _ h e = p_h e_h e^ ( up to signum ) .
indeed , put @xmath85 .
let @xmath86 be an orthonormal basis of @xmath48 with @xmath87 .
then @xmath88 ( up to signum ) , as desired .
finally , we claim that e.reverse = .
indeed , @xmath89 , so this is equivalent to @xmath90 thus , it suffices to show that @xmath91 by sesquilinearity , it suffices to show this for @xmath92 members of an orthonormal basis of @xmath48 .
but then it is obvious . for a more detailed presentation of exterior algebra ,
see b.whitney : book/. let @xmath48 be a subspace of @xmath37 of dimension @xmath93 . define the probability measure @xmath94 on subsets @xmath95 by e.xihpr ^h(\{b } ) : = ||^2 .
note that this is non-0 only for @xmath96 .
also , by l.projection/ , @xmath97 for @xmath96 , which is non-0 iff @xmath98 are linearly independent .
that is , @xmath99 iff the projections of the elements of @xmath100 form a basis of @xmath48
. let @xmath101 be any basis of @xmath48 .
if we use e.ipdet/ and the fact that @xmath102 for some scalar @xmath103 , then we obtain another formula for @xmath94 : we use @xmath104 to denote a random subset of @xmath0 arising from a probability measure @xmath94 . to see that e.dpm/ holds for the matrix of @xmath15 , observe that for @xmath96 , @xmath105 = \bigip{p_{\ext(h ) } \theta_b , \theta_b } = \bigip{\bigwedge_{e \in b } p_h e , \bigwedge_{e \in b } e } = \det [ \ip{p_h e , f}]_{e , f \in b}\ ] ] by e.ipdet/.
this shows that e.dpm/ holds for @xmath106 since @xmath107 @xmath94-a.s .
the general case is a consequence of multilinearity , which gives the following extension of e.dpm/. we use the convention that @xmath108 and @xmath109 for any multivector @xmath65 .
t.genprs if @xmath110 and @xmath111 are ( possibly empty ) subsets of a finite set @xmath0 , then e.genprs ^h[a_1 , a_2 = ] = . in particular , for every @xmath3
, we have e.included ^h[a ] = p_(h ) _ a^2 .
c.dualrep if @xmath0 is finite , then for every subspace @xmath49 , we have e.dualrep ^h^(\{e b } ) = ^h(\{b } ) .
these extend to infinite @xmath0 . in order to define @xmath94
when @xmath48 is infinite dimensional , we proceed by finite approximation .
let @xmath112 be infinite .
consider first a finite - dimensional subspace @xmath48 of @xmath37 .
define @xmath113 as the image of the orthogonal projection of @xmath48 onto the span of @xmath114 . by considering a basis of @xmath48 , we see that @xmath115 in the weak operator topology ( wot ) , i.e. , matrix - entrywise , as @xmath116 .
it is also easy to see that if @xmath117 , then @xmath118 for all large @xmath18 and , in fact , @xmath119 in the usual norm topology .
it follows that e.genprs/ holds for this subspace @xmath48 and for every finite @xmath120 .
now let @xmath48 be an infinite - dimensional closed subspace of @xmath37 .
choose finite - dimensional subspaces @xmath121 .
it is well known that @xmath115 ( wot ) . then e.detgenprs a ( p_h_k a ) ( p_h a ) , whence @xmath122 has a weak@xmath123 limit that we denote @xmath94 and that satisfies e.genprs/. we also note that for _ any _ sequence of subspaces @xmath113 , if @xmath124 ( wot ) , then @xmath125 weak@xmath123 because e.detgenprs/ then holds .
we call @xmath1 a if @xmath1 is a self - adjoint operator on @xmath37 such that for all @xmath126 , we have @xmath127 . a of @xmath1 is an orthogonal projection @xmath15 onto a closed subspace @xmath128 for some @xmath129 such that for all @xmath126 , we have @xmath130 , where we regard @xmath131 as the orthogonal sum @xmath132 . in this case , @xmath1 is also called the of @xmath15 to @xmath37 .
choose such a dilation ( see e.vecdilate/ or e.dilate/ ) and define @xmath9 as the law of @xmath133 when @xmath104 has the law @xmath94 .
then e.dpm/ for @xmath1 is a special case of e.dpm/ for @xmath15 .
of course , when @xmath1 is the orthogonal projection onto a subspace @xmath48 , then @xmath134 .
basic properties of @xmath9 follow from those for orthogonal projections , such as : t.q if @xmath1 is a positive contraction , then for all finite @xmath135 , e.qgenprs ^q= .
if e.dpm/ is given , then e.qgenprs/ can be deduced from e.dpm/ without using our general theory and , in fact , without assuming that the matrix @xmath1 is self - adjoint . indeed , suppose that @xmath136 is any diagonal matrix .
denote its @xmath137-entry by @xmath138 . comparing coefficients of @xmath138
shows that e.dpm/ implies , for finite @xmath3 , e.xe = ( ( q + x ) a ) . replacing @xmath5 by @xmath139 and
choosing @xmath140 gives e.qgenprs/. on the other hand , if we substitute @xmath141 , then we may rewrite e.xe/ as e.ze = ( ( q z + i - q ) a ) , where @xmath142 is the diagonal matrix of the variables @xmath143 .
let @xmath0 be finite .
write @xmath144 for @xmath3
. then e.ze/ is equivalent to e.affine _ a e ^q[= a ] z^a = ( i - q+qz ) .
this is the same as the laplace transform of @xmath9 after a trivial change of variables .
when @xmath145 , we can write @xmath146 with @xmath147 . thus , for all @xmath3 , we have a probability measure @xmath8 on @xmath148 is called if its generating polynomial @xmath149 z^a$ ] satisfies the inequality for all @xmath150 and all real @xmath151 .
this property is satisfied by every determinantal probability measure , as was shown by b.bbl:rayleigh/ , who demonstrated its usefulness in showing other properties , such as negative associations and preservation under symmetric exclusion processes . for a set @xmath152 ,
denote by @xmath153 the @xmath154-field of events that are measurable with respect to the events @xmath155 for @xmath156 .
define the @xmath154-field to be the intersection of @xmath157 over all finite @xmath158 .
we say that a measure @xmath8 on @xmath148 has if every event in the tail @xmath154-field has measure either 0 or 1 .
t.tail b.lyons:det/ if @xmath1 is a positive contraction , then @xmath9 has trivial tail . for finite @xmath0 and a positive contraction @xmath1 ,
define the of @xmath9 to be @xmath159 numerical calculation supports the following conjecture b.lyons:det/ : g.concave for all positive contractions @xmath160 and
@xmath161 , we have e.concave ( ( q_1+q_2)/2 ) ( ( q_1 ) + ( q_2))/2 . let @xmath0 be denumerable .
a function @xmath162 is called if for all @xmath163 and all @xmath164 , we have @xmath165 . an event is called increasing or if its indicator is increasing . given two probability measures @xmath166 , @xmath167 on @xmath148 , we say that and write @xmath168 if for all increasing events @xmath169 , we have @xmath170 .
this is equivalent to @xmath171 for all bounded increasing @xmath172 .
a of two probability measures @xmath166 , @xmath167 on @xmath148 is a probability measure @xmath173 on @xmath174 whose coordinate projections are @xmath166 , @xmath167 ; it is if @xmath175 by strassen s theorem @xcite , stochastic domination @xmath176 is equivalent to the existence of a monotone coupling of @xmath166 and @xmath167 .
t.dominate-infinite b.lyons:det/ if @xmath177 , then @xmath178
. it would be very interesting to find a natural or explicit monotone coupling .
a coupling @xmath173 has @xmath8 if for all events @xmath179 , we have @xmath180 .
q.unioncoupling @xcite given @xmath181 , is there a coupling of @xmath182 and @xmath183 with union marginal @xmath94 ? a positive answer is supported by some numerical calculation .
it is easily seen to hold when @xmath184 by c.dualrep/. in the sequel , we write @xmath185 if @xmath186 for all @xmath126 .
t.dominate @xcite if @xmath187 , then @xmath188 . by t.dominate-infinite/ , it suffices that there exist orthogonal projections @xmath189 and @xmath190 that are dilations of @xmath160 and @xmath161 such that @xmath191 .
this follows from namark s dilation theorem @xcite , which says that any measure whose values are positive operators , whose total mass is @xmath192 , and which is countably additive in the weak operator topology dilates to a spectral measure .
the measure in our case is defined on a 3-point space , with masses @xmath160 , @xmath193 , and @xmath194 , respectively . if we denote the respective dilations by @xmath195 , @xmath196 , and @xmath197 , then we set @xmath198 and @xmath199
. a positive answer in general to q.unioncoupling/ would give the following more general result by compression : if @xmath160 , @xmath161 and @xmath200 are positive contractions on @xmath37 , then there is a coupling of @xmath201 and @xmath202 with union marginal @xmath203
. it would be very useful to have additional sufficient conditions for stochastic domination : see the end of s.orthogpoly/ and g.fkdom/. for examples where more is known , see t.gmdom/. we shall say that the events in @xmath153 are @xmath158 and likewise for functions that are measurable with respect to @xmath153 .
we say that @xmath8 has if for every pair @xmath204 , @xmath205 of increasing functions that are measurable with respect to complementary subsets of @xmath0 , e.negass .
@xcite if @xmath206 , then @xmath9 has negative associations .
the details for finite @xmath0 were given in b.lyons : det/. for infinite @xmath0 , let @xmath204 and @xmath205 be increasing bounded functions measurable with respect to @xmath207 and @xmath208 , respectively . choose finite @xmath209 .
the conditional expectations @xmath210 $ ] and @xmath211 $ ] are increasing functions to which e.negass/ applies ( because restriction to @xmath212 corresponds to a compression of @xmath1 , which is a positive contraction ) and which , being martingales , converge to @xmath204 and @xmath205 in @xmath213 .
write @xmath214 for the distribution of a bernoulli random variable with expectation @xmath215 . for @xmath216 $ ] ,
let @xmath217 be the distribution of a sum of independent @xmath218 random variables . recall that @xmath75 $ ] is the set of scalar multiples of @xmath12 .
t.eigmix @xcite ; lemma 3.4 of @xcite ; ( 2.38 ) of @xcite ; @xcite let @xmath1 be a positive contraction with spectral decomposition @xmath219}$ ] , where @xmath220 are orthonormal .
let @xmath221 be independent .
let @xmath222 $ ] ; thus , @xmath223 . then @xmath224 .
hence , if @xmath225 , then @xmath226 . by t.dominate/
, it suffices to prove it when only finitely many @xmath227 . then by t.q/
, we have @xmath228 = \bigip{\bigwedge_{e \in a } q e , \theta_a } $ ] for all @xmath3 . now
@xmath229 } e & = \sum_{j \colon a \to \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e \\ & = \sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e\end{aligned}\ ] ] because @xmath230 and @xmath231 } e$ ] is a multiple of @xmath12 , so none of the terms where @xmath232 is not injective contribute .
thus , @xmath233 } e = \ebig{\sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } \\ & = \ebig{\sum_{j \colon a \to \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } = \be\bigwedge_{e \in a } \sum_k i_k p_{[v_k ] } e = \be \bigwedge_{e \in a } p_{\rh } e \,.\end{aligned}\ ] ] we conclude that @xmath234 = \be \leftip{\bigwedge_{e \in a } p_{\rh } e , \theta_a } = \ebig { \bp^{\rh}\left [ a \subseteq \ba \right ] } $ ] by e.qgenprs/. we sketch another proof : let @xmath235 be disjoint from @xmath0 with the same cardinality .
choose an orthonormal sequence @xmath236 in @xmath131 .
define then @xmath1 is the compression of @xmath15 to @xmath37 . expanding @xmath237 in the obvious way into orthogonal pieces and restricting to @xmath0 , we obtain the desired equation from e.xihpr/. the first proof shows more generally the following : let @xmath238 be a positive contraction .
let @xmath220 be ( not necessarily orthogonal ) vectors such that @xmath239 } \lloew i$ ] .
let @xmath240 be independent bernoulli random variables with @xmath241 .
write @xmath242}$ ]
. then @xmath243 .
this was observed by ghosh and krishnapur ( personal communication , 2014 )
. note that in the mixture of t.eigmix/ , the distribution of @xmath244 is determinantal corresponding to the diagonal matrix with diagonal @xmath245 .
thus , it is natural to wonder whether @xmath246 can be taken to be a general determinantal measure .
if such a mixture is not necessarily determinantal , must it be strongly rayleigh or at least have negative correlations ? here
, we say that a probability measure @xmath8 on @xmath148 has if for every pair @xmath5 , @xmath100 of finite disjoint subsets of @xmath0 , we have @xmath247 \le \bp [ a \subseteq \qba ] \bp [ b \subseteq \qba ] $ ] .
note that negative associations is stronger than negative correlations .
the most well - known example of a ( nontrivial discrete ) determinantal probability measure is that where @xmath6 is a uniformly chosen random spanning tree of a finite connected graph @xmath248 with @xmath249 . here ,
we regard a spanning tree as a set of edges .
the fact that holds for the uniform spanning tree is due to b.burpem/ and is called the transfer current theorem .
the case with @xmath250 was shown much earlier by b.kirchhoff/ , while the case with @xmath251 was first shown by b.bsst/. write @xmath252 for the uniform spanning tree measure on @xmath253 . to see that @xmath252 is indeed determinantal , consider the vertex - edge incidence matrix @xmath254 of @xmath253 , where each edge is oriented ( arbitrarily ) and the @xmath255-entry of @xmath254 equals 1 if @xmath256 is the head of @xmath66 , @xmath257 if @xmath256 is the tail of @xmath66 , and 0 otherwise .
identifying an edge with its corresponding column of @xmath254 , we find that a spanning tree is the same as a basis of the column space of @xmath254 .
given @xmath258 , define the at @xmath256 to be the @xmath256-row of @xmath254 , regarded as a vector @xmath259 in the row space , @xmath260 .
it is easy that the row - rank of @xmath254 is @xmath261 .
let @xmath262 and let @xmath65 be the wedge product ( in some order ) of the stars at all the vertices other than @xmath263 .
thus , @xmath264 for some @xmath265 .
since spanning trees are bases of the column space of @xmath254 , we have @xmath266 iff @xmath5 is a spanning tree .
that is , the only non - zero coefficients of @xmath65 are those in which choosing one edge in each @xmath259 for @xmath267 yields a spanning tree ; moreover , each spanning tree occurs exactly once since there is exactly one way to choose an edge incident to each @xmath267 to get a given spanning tree .
this means that its coefficient is @xmath268 .
hence , @xmath269 is indeed uniform on spanning trees .
simultaneously , this proves the matrix tree theorem that the number of spanning trees equals @xmath270_{x , y \ne x_0}$ ] , since this determinant is @xmath271
. one can define analogues of @xmath252 on infinite connected graphs @xcite by weak limits . for brevity ,
we simply define them here as determinantal probability measures .
again , all edges of @xmath253 are oriented arbitrarily .
we define @xmath272 as the closure of the linear span of the stars .
an element of @xmath273 that is finitely supported and orthogonal to @xmath272 is called a ; the closed linear span of the cycles is @xmath274 .
the is @xmath275 , while the is @xmath276 .
our discussion of the continuous " case includes the discrete case , but the discrete case has the more elementary formulations given earlier .
let @xmath0 be a measurable space .
as before , @xmath0 will play the role of the underlying set on which a point process forms a counting measure . while before we implicitly used counting measure on @xmath0 itself , now we shall have an arbitrary measure @xmath173 ; it need not be a probability measure . the case of lebesgue measure on euclidean space is a common one .
the hilbert spaces of interest will be @xmath277
. there may be no natural order in @xmath0 , so to define , e.g. , a probability measure on @xmath278 points of @xmath0 , it is natural to use a probability measure on @xmath279 that is symmetric under coordinate changes and that vanishes on the diagonal @xmath280 .
likewise , for exterior algebra , it is more convenient to identify @xmath281 with @xmath282 for @xmath283 .
thus , @xmath284 is identified with the function @xmath285_{i , j \in \{1 , \ldots , n\}}/\sqrt{n ! } $ ] .
note that @xmath286 \det [ { v_i(x_j ) } ] = \det [ u_i(x_j ) ] \det [ { v_i(x_j)}]^t \nonumber \\ & = \det [ u_i(x_j)][{v_i(x_j)}]^t = \det [ k(x_i , x_j)]_{i , j \in \{1 , \ldots , n\}}\end{aligned}\ ] ] with @xmath287 .
here , @xmath288 denotes transpose .
suppose from now on that @xmath0 is a locally compact polish space ( equivalently , a locally compact second countable hausdorff space ) .
let @xmath173 be a radon measure on @xmath0 , i.e. , a borel measure that is finite on compact sets .
let @xmath289 be the set of radon measures on @xmath0 with values in @xmath290 .
we give @xmath289 the vague topology generated by the maps @xmath291 for continuous @xmath172 with compact support ; then @xmath289 is polish .
the corresponding borel @xmath154-field of @xmath289 is generated by the maps @xmath292 for borel @xmath3 .
let @xmath293 be a simple point process on @xmath0 , i.e. , a random variable with values in @xmath289 such that @xmath294 for all @xmath295 .
the power @xmath296 lies in @xmath297 .
thus , @xmath298 $ ] is a borel measure on @xmath299 ; the part of it that is concentrated on @xmath300 is called the of @xmath293 .
if the intensity measure is absolutely continuous with respect to @xmath301 , then its radon - nikodym derivative @xmath302 is called the or the : since the intensity measure vanishes on the diagonal @xmath303 , we take @xmath302 to vanish on @xmath303
. we also take @xmath302 to be symmetric under permutations of coordinates .
intensity functions are the continuous analogue of the elementary probabilities e.dpm/. since the sets @xmath304 generate the @xmath154-field on @xmath300 for pairwise disjoint borel @xmath305 , a measurable function @xmath306 is the " @xmath18-point intensity function iff since @xmath293 is simple , @xmath307 , where @xmath308 . since @xmath302 vanishes on the diagonal , it follows from e.rn/ that for disjoint @xmath309 and non - negative @xmath310 summing to @xmath18 , again , this characterizes @xmath302 , even if we use only @xmath311 . in the special case that @xmath312 a.s .
for some @xmath313 , then the definition e.rn/ shows that a random ordering of the @xmath278 points of @xmath293 has density @xmath314 .
more generally , e.rn/ shows that for all @xmath315 , whence in this case , we call @xmath293 if for some measurable @xmath316 and all @xmath16 , @xmath317 @xmath301-a.e . here
, @xmath318 is the matrix @xmath319_{i , j \le k}$ ] . in this case
, we denote the law of @xmath293 by @xmath320 .
we consider only @xmath158 that are locally square integrable ( i.e. , @xmath321 is radon ) , are hermitian ( i.e. , @xmath322 for all @xmath323 ) , and are positive semidefinite ( i.e. , @xmath324 is positive semidefinite for all finite @xmath325 , written @xmath326 ) . in this case , @xmath158 defines a positive semidefinite integral operator @xmath327 on functions @xmath328 with compact support . for every borel @xmath3 , we denote by @xmath329 the measure @xmath173 restricted to borel subsets of @xmath5 and by @xmath330 the compression of @xmath158 to @xmath5 , i.e. , @xmath331 for @xmath332 .
the operator @xmath158 is locally trace - class , i.e. , for every compact @xmath3 , the compression @xmath330 is trace class , having a spectral decomposition @xmath333 , where @xmath334 are orthonormal eigenfunctions of @xmath330 with positive summable eigenvalues @xmath335 . if @xmath110 is the set where @xmath336 , then @xmath337 and @xmath338 converges on @xmath339 , with sum @xmath340-a.e .
equal to @xmath158 .
we normally redefine @xmath158 on a set of measure 0 to equal this sum .
such a @xmath158 defines a determinantal point process iff the integral operator @xmath158 extends to all of @xmath341 as a positive contraction @xcite .
the joint intensities determine uniquely the law of the point process ( * ? ? ?
* lemma 4.2.6 ) .
poisson processes are not determinantal processes , but when @xmath173 is continuous , they are distributional limits of determinantal processes . to see that a positive contraction defines a determinantal point process , we first consider @xmath158 that defines an orthogonal projection onto a finite - dimensional subspace , @xmath48
. then @xmath342 for every orthonormal basis @xmath343 of @xmath48 and @xmath344 is a unit multivector in the notation of s.ext/. because of e.prodtensor/ , we have i.e. , @xmath345/n!$ ] is a density with respect to @xmath346 .
although in the discrete case , the absolute squared coefficients of @xmath347 give the elementary probabilities , now coefficients are replaced by a function whose absolute square gives a probability density .
as noted already , e.firstdensity/ means that @xmath348 is the @xmath278-point intensity function . in order to show that this density gives a determinantal process with kernel @xmath158
, we use the cauchy - binet formula , which may be stated as follows : for @xmath349 matrices @xmath350 $ ] and @xmath351 $ ] with @xmath352_{\substack{i \le k \\ j \in j}}$ ] , we have @xmath353 [ b_{i , j}]^t\big ) = \sum_{|j| = k } \det a^j \cdot \det b^j = \sum _ { \substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k a_{i , \sigma(i ) } b_{i , \tau(i ) } \,,\ ] ] where @xmath354 denotes the image of @xmath154 and the sums extend over all pairs of injections @xmath355 here , the sign @xmath356 of @xmath154 is defined in the usual way by the parity of the number of pairs @xmath357 for which @xmath358 .
we have @xmath359 \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) \nonumber \\ & = \frac{1}{(n - k ) ! }
\int_{e^{n - k } } \sum_{\sigma \in \sym(n ) } ( -1)^\sigma \prod_{i=1}^n \phi_{\sigma(i)}(x_i ) \cdot { } \nonumber \\
\noalign{$\displaystyle \hfill \cdot \sum_{\tau \in \sym(n ) } ( -1)^\tau \prod_{i=1}^n \overline{\phi_{\tau(i)}(x_i ) } \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) $ } & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k \phi_{\sigma(i)}(x_i ) \overline{\phi_{\tau(i)}(x_i ) } \nonumber \\ & = \det \big(k \restrict ( x_1 , \ldots , x_k)\big ) \,.\end{aligned}\ ] ] here , the first equality uses e.integrate/ , the second equality uses e.prodtensor/ , the third equality uses the fact that @xmath360 is 1 or 0 according as @xmath361 or not , and the fourth equality uses cauchy - binet .
note that a factor of @xmath362 arises because for every pair of injections @xmath363 with equal image , there are @xmath362 extensions of them to permutations @xmath364 with @xmath361 for all @xmath365 ; in this case , @xmath366 .
we write @xmath94 for the law of the associated point process on @xmath0 .
l.weaklimit let @xmath367 with @xmath368 for some @xmath369 .
then @xmath370 is tight and every weak limit point of @xmath371 is simple . by using the kernel @xmath372 with respect to the measure @xmath373 , we may assume that @xmath374 .
tightness follows from @xmath375 \le \be[\sx_n(a ) ] = \int_a k_n(x , x ) \,d\mu(x)\,.\ ] ] for the rest , we may assume that @xmath0 is compact and @xmath376 .
let @xmath293 be a limit point of @xmath371 .
let @xmath377 be the atomic part of @xmath173 and @xmath378 .
choose @xmath379 and partition @xmath0 into sets @xmath380 with @xmath381 .
let @xmath5 be such that @xmath382 and @xmath383 .
let @xmath384 be open such that @xmath385 and @xmath386 .
then @xmath387 & \le \limsup_n \big(\bp[\sx_n(u \setminus a ) \ge 1 ] + \bp[\texists i \sx_n(a_i ) \ge 2]\big ) \\ & \le \limsup_n \big(\be[\sx_n(u \setminus a ) ] + \sum_i \be[(\sx_n(a_i))_2]\big ) \\ & \le \muc(u ) + \sum_i \mu(a_i)^2 < 2/m \ , .
\tag*{\qedhere}\end{aligned}\ ] ] now , given any locally trace - class orthogonal projection @xmath158 onto @xmath48 , choose finite - dimensional subspaces @xmath388 with corresponding projections @xmath389 .
clearly @xmath390 @xmath391-a.e.and @xmath392 @xmath173-a.e .
thus , the joint intensity functions converge a.e . by dominated convergence , if @xmath393 is relatively compact and borel , then @xmath394 \to \int_a \det ( k \restrict f ) \,d\mu^k(f)$ ] . by uniform exponential moments of @xmath395 (
* ? ? ?
* proof of lemma 4.2.6 ) , it follows that all weak limit points of @xmath396 are equal , and hence , by l.weaklimit/ , define @xmath94 with kernel @xmath158 .
( in s.cinequalities/ , we shall see that @xmath397 is stochastically increasing . ) finally , let @xmath158 be any locally trace - class positive contraction . define the orthogonal projection on @xmath398 whose block matrix is take an isometric isomorphism of @xmath277 to @xmath131 for some denumerable set @xmath235 and interpret the above as an orthogonal projection @xmath399 on @xmath400 .
then @xmath399 is clearly locally trace - class and @xmath158 is the compression of @xmath399 to @xmath0 .
thus , we define @xmath320 by intersecting samples of @xmath401 with @xmath0 .
we remark that by writing @xmath399 as a limit of increasing finite - rank projections that we then compress , we see that @xmath320 may be defined as a limit of determinantal processes corresponding to increasing finite - rank positive contractions .
g.ctail if @xmath158 is a locally trace - class positive contraction , then @xmath320 has trivial tail in that every event in @xmath402 is trivial . rather than using compressions as in the last paragraph above , an alternative approach to defining @xmath320 uses mixtures and starts from finite - rank projections , as in s.mix/. this approach is due to b.hkpv : survey/. consider first a finite - rank @xmath403 .
let @xmath404 be independent .
let @xmath405 $ ] ; thus , @xmath406 . we claim that @xmath407 is determinantal with kernel @xmath158 .
indeed , it is clearly a simple point process .
write @xmath408 , @xmath409 , and @xmath410 .
let @xmath411 .
combining cauchy - binet with e.prodtensor/ yields @xmath412 .
similarly , the joint intensities of @xmath413 are the expectations of the joint intensities of @xmath414 , which equal @xmath415 essentially the same works for trace - class @xmath416 ; we need merely take , in the last step , a limit in the above equation as @xmath417 for @xmath418 , since all terms are non - negative and @xmath419 a.e .
given this construction of @xmath320 for trace - class @xmath158 , one can then construct @xmath320 for a general locally trace - class positive contraction by defining its restriction to each relatively compact set @xmath5 via the trace - class compression @xmath330 . as noted by b.hkpv:survey/ ,
a consequence of the mixture representation is a clt due originally to b.soshnikov:gauss/ : t.clt let @xmath389 be trace - class positive contractions on spaces @xmath420 .
let @xmath367 and write @xmath421 . if @xmath422 as @xmath417 , then @xmath423 obeys a clt .
in order to simulate @xmath320 when @xmath158 is a trace - class positive contraction , it suffices , by taking a mixture as above , to see how to simulate @xmath424 when @xmath425 . the following algorithm ( * ? ? ? * algo .
18 ) gives a uniform random ordering of @xmath293 as @xmath426 . since @xmath427 , the measure @xmath428/n = n^{-1 } k(x , x)\,d\mu(x)$ ] is a probability measure on @xmath0 .
select a point @xmath429 at random from that measure .
if @xmath430 , then we are done .
if not , then let @xmath431 be the orthogonal complement in @xmath48 of the function @xmath432 , where @xmath343 is an orthonormal basis for @xmath48 .
then @xmath433 and we may repeat the above for @xmath431 to get the next point , @xmath434 , then @xmath435 , etc .
the conditional density of @xmath436 given @xmath437 is @xmath438 by e.densityktuple/ , i.e. , @xmath439 times the squared distance from @xmath440 to the linear span of @xmath441 .
it can help for rejection sampling to note that this is at most @xmath442 .
one can also sample faster by noting that the conditional distribution of @xmath436 is the same as that of @xmath443 , where @xmath444 is a uniformly random vector on the unit sphere of @xmath113 .
note that if @xmath445 are bounded @xmath446-valued random variables , then the function @xmath447 determines the joint distribution of @xmath448 since it gives the derivatives at @xmath449 of the probability generating function @xmath450 .
let us re - examine e.falling/ in the context of a finite - rank @xmath451 .
given disjoint @xmath452 and non - negative @xmath310 summing to @xmath18 , it will be convenient to write @xmath453 for @xmath454 .
we have by cauchy - binet @xmath455 = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \rho_k \,d\mu^k = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \det ( k \restrict ( x_1 , \ldots , x_k ) ) \,\prod_{j=1}^k d\mu(x_j ) \\ & = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,\prod_{j=1}^k d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \int_{a_{\kappa(j ) } } \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\
\operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \lambda^{\operatorname{im}(\sigma ) } \prod_{j=1}^k \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_{\tau(j ) } } } \\ & = \sum_{\sigma \in \sym(k , n ) } ( -1)^\sigma \lambda^{\operatorname{im}(\sigma ) } \det \big [ \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_\ell } } \big]_{\substack{j \le k \hfill \\ \ell
\in \operatorname{im}(\sigma ) } } \,.\end{aligned}\ ] ] as an immediate consequence of this formula , we obtain the following important principle of goldman ( * ? ? ?
* proposition 12 ) that allows one to infer properties of continuous determinantal point processes from corresponding properties of discrete determinantal probability measures : t.transfer let @xmath456 and @xmath457 be two radon measure spaces on locally compact polish sets .
let @xmath458 be pairwise disjoint borel subsets of @xmath0 and @xmath459 be pairwise disjoint borel subsets of @xmath325 .
let @xmath460 $ ] with @xmath461 .
let @xmath462 be orthonormal in @xmath277 and @xmath463 be orthonormal in @xmath464 .
let @xmath465 and @xmath466 . if @xmath467 for all @xmath468 , then the @xmath320-distribution of @xmath469 equals the @xmath470-distribution of @xmath471 .
when only finitely many @xmath227 , this follows from our previous calculation .
the general case follows from weak convergence of the processes corresponding to the partial sums , as in the paragraph following l.weaklimit/. this permits us to compare to discrete measures via ( * ? ? ?
* lemma 16 ) : l.compare let @xmath173 be a radon measure on a locally compact polish space , @xmath0 .
let @xmath458 be pairwise disjoint borel subsets of @xmath0 .
let @xmath472 for @xmath16 .
then there exists a denumerable set @xmath325 , pairwise disjoint subsets @xmath459 of @xmath325 , and @xmath473 such that @xmath474 and @xmath475 for all @xmath468 . without loss of generality
, we may assume that @xmath476 .
for each @xmath477 , fix an orthonormal basis @xmath478 for the subspace of @xmath277 spanned by @xmath479 . here ,
@xmath480 .
define @xmath481 and @xmath482 .
let @xmath483 be the isometric isomorphism from the span of @xmath484 to @xmath485 that sends @xmath486 to @xmath487 .
defining @xmath488 yields the desired vectors .
we now show how the discrete models of s.transf/ allow us to obtain the analogues of the stochastic inequalities known to hold for discrete determinantal probability measures . for a borel set @xmath7 , let @xmath207 denote the @xmath154-field on @xmath289 generated by the functions @xmath489 for borel @xmath490 .
we say that a function that is measurable with respect to @xmath207 is , more simply , measurable with respect to @xmath5 .
the obvious partial order on @xmath289 allows us to define what it means for a function @xmath491 to be . as in the discrete case
, we say that @xmath8 has if @xmath492 \le \be[f_1 ] \be[f_2 ] $ ] for every pair @xmath204 , @xmath205 of bounded increasing functions that are measurable with respect to complementary subsets of @xmath0 .
an event is increasing if its indicator is increasing .
then @xmath8 has negative associations iff for every pair @xmath493 , @xmath494 of increasing events that are measurable with respect to complementary subsets of @xmath0 .
we also say that and write @xmath495 if @xmath496 for every increasing event @xmath169 .
call an event if it has the form @xmath497 , where @xmath100 is a relatively compact borel set and @xmath498 .
write @xmath499 for the closure under finite unions and intersections of the collection of elementary increasing events with @xmath500 ; the notation @xmath501 is chosen for upwardly closed " .
note that every event in @xmath499 is measurable with respect to some finite collection of functions @xmath502 for pairwise _ disjoint _ relatively compact borel @xmath503 .
write @xmath504 for the closure of @xmath499 under monotone limits , i.e. , under unions of increasing sequences and under intersections of decreasing sequences ; these events are also increasing .
this is the same as the closure of @xmath499 under countable unions and intersections .
l.approxincr let @xmath5 be a borel subset of a locally compact polish space , @xmath0 .
then @xmath504 is exactly the class of increasing borel sets in @xmath207 .
we give a proof at the end of this subsection .
first , we derive two consequences . a weaker version ( negative correlations of elementary increasing events ) of the initial one is due to b.ghosh/. t.cfm let @xmath173 be a radon measure on a locally compact polish space , @xmath0 .
let @xmath158 be a locally trace - class positive contraction on @xmath277 .
then @xmath320 has negative associations .
let @xmath505 be borel .
let @xmath506 and @xmath507 .
then @xmath508 for some compact @xmath100 by definition of @xmath509 .
we claim that e.cnegass/ holds for @xmath493 , @xmath494 , and @xmath510 , i.e. , for @xmath511 .
now @xmath493 is measurable with respect to a finite number of count functions @xmath502 for some disjoint @xmath512 ( @xmath513 ) and likewise @xmath494 is measurable with respect to a finite number of functions @xmath514 for some disjoint @xmath515 ( @xmath513 ) .
thus , there are functions @xmath516 and @xmath517 such that @xmath518 and @xmath519 . by t.transfer/ and l.compare/
, there is some discrete determinantal probability measure @xmath9 on some denumerable set @xmath325 and pairwise disjoint sets @xmath520 such that the joint @xmath521-distribution of all @xmath522 and @xmath523 is equal to the joint @xmath9-distribution of all @xmath524 and @xmath525 .
define the corresponding events @xmath526 by @xmath527 and @xmath528 .
since @xmath526 depend on disjoint subsets of @xmath325 , t.fm/ gives that @xmath529 .
this is the same as e.cnegass/ by t.transfer/. the same e.cnegass/ clearly then holds in the less restrictive setting @xmath530 by taking monotone limits .
l.approxincr/ completes the proof .
t.cdom theorem 3 of b.goldman/ suppose that @xmath531 and @xmath532 are two locally trace - class positive contractions such that @xmath533
. then @xmath534 .
it suffices to show that @xmath535 for every @xmath536 .
again , it suffices to assume that @xmath537 are trace class .
l.compare/ applied to all eigenfunctions of @xmath531 and @xmath532 yields a denumerable @xmath325 and two positive contractions @xmath538 on @xmath485 , together with an event @xmath539 , such that @xmath540 for @xmath541 .
furthermore , by construction , every function in @xmath485 is the image of a function in @xmath542 under the isometric isomorphism @xmath483 used to prove l.compare/ , whence @xmath543 .
therefore t.dominate/ yields @xmath544 , as desired .
again , it would be very interesting to have a natural monotone coupling of @xmath545 with @xmath546 .
for some examples where this would be desirable , see s.orthogpoly/. l.approxincr/ will follow from this folklore variant of a theorem of dyck b.dyck/ : t.dyck let @xmath136 be a polish space on which @xmath547 is a partial ordering that is closed in @xmath548 . let @xmath501 be a collection of open increasing sets that generates the borel subsets of @xmath136 .
let @xmath549 be the closure of @xmath501 under countable intersections and countable unions .
suppose that for all @xmath550 , either @xmath551 or there is @xmath552 and an open set @xmath553 such that @xmath554 , @xmath555 , and @xmath556 . then @xmath549 equals the class of increasing borel sets .
obviously every set in @xmath549 is borel and increasing . to show the converse
, we prove a variant of lusin s separation theorem .
namely , we show that if @xmath557 is increasing and analytic ( with respect to the paving of closed sets , as usual ) and if @xmath558 is analytic with @xmath559 , then there exists @xmath560 such that @xmath561 and @xmath562 . taking @xmath563 to be borel and @xmath564 forces @xmath565 and gives the desired conclusion . to prove this separation property , we first show a stronger conclusion in a special case
: suppose that @xmath566 are compact such that @xmath110 is contained in an increasing set @xmath563 that is disjoint from @xmath111 ; then there exists an open @xmath560 and an open @xmath567 such that @xmath568 , @xmath569 , and @xmath556 .
indeed , since @xmath563 is increasing , for every @xmath570 , we do _ not _ have that @xmath551 , whence by hypothesis , there exist @xmath571 and an open @xmath572 with @xmath573 , @xmath574 , and @xmath575 . because @xmath111 is compact , for each @xmath576 , we may choose @xmath577 such that @xmath578 .
define @xmath579 .
then @xmath580 is open , contains @xmath256 , and is disjoint from @xmath581 , whence compactness of @xmath110 ensures the existence of @xmath582 with @xmath583 .
then @xmath584 is open , contains @xmath111 , and is disjoint from @xmath384 , as desired . to prove the general case ,
let @xmath585 and @xmath586 be the two coordinate projections on @xmath587 .
define @xmath588 for @xmath589 to be 0 if there exists @xmath560 such that @xmath590 and @xmath591 ; and to be 1 otherwise .
we claim that @xmath192 is a capacity in the sense of ( * ? ? ?
* ( 30.1 ) ) .
it is obvious that @xmath592 if @xmath593 and it is simple to check that if @xmath594 , then @xmath595 .
suppose for the final property that @xmath5 is compact and @xmath596 ; we must find an open @xmath597 for which @xmath598 .
there exists some @xmath599 with @xmath600 and @xmath601 .
then the result of the second paragraph yields sets @xmath384 and @xmath567 that give @xmath602 as desired
. now let @xmath563 and @xmath603 be as in the first paragraph .
if @xmath604 is compact , then setting @xmath605 and applying the second paragraph shows that @xmath596 .
thus , by the choquet capacitability theorem ( * ? ? ?
* ( 30.13 ) ) , @xmath606 .
l.approxincr/ clearly every set in @xmath504 is increasing and in @xmath207 .
for the converse , endow @xmath5 with a metric so that it becomes locally compact polish while preserving its class of relatively compact sets and its borel @xmath154-field : choose a denumerable partition of @xmath5 into relatively compact sets @xmath607 and make each one compact and of diameter at most 1 ; make the distance between @xmath256 and @xmath608 be 1 if @xmath256 and @xmath608 belong to different @xmath607 .
let @xmath609 with the vague topology and let @xmath501 be the class of elementary increasing events defined with respect to ( relatively compact ) sets @xmath500 that are open for this new metric .
apply t.dyck/. since @xmath610 , the result follows .
natural examples of determinantal point processes arise from orthogonal polynomials with respect to a probability measure @xmath173 on @xmath25 .
assume that @xmath173 has infinite support and finite moments of all orders .
let @xmath389 denote the orthogonal projection of @xmath611 onto the linear span @xmath612 of the functions @xmath613 .
there exist unique ( up to signum ) polynomials @xmath614 of degree @xmath18 such that for every @xmath278 , @xmath615 is an orthonormal basis of @xmath612 . by elementary row operations
, we see that for variables @xmath616 , the map @xmath617_{i , j \le n}$ ] is a vandermonde polynomial up to a constant factor , whence @xmath618 [ \phi_i(z_j)]^ * = c_n \prod_{1 \le i < j \le n } |z_i - z_j|^2\ ] ] for some constant @xmath619 . therefore , the density of @xmath620 ( with points randomly ordered ) with respect to @xmath346 is given by @xmath621 times the square of a vandermonde determinant .
classical examples include the following : 1
. if @xmath173 is gaussian measure on @xmath24 , i.e. , @xmath622 , then @xmath614 are the hermite polynomials , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath624 , where @xmath625 is an @xmath626 matrix whose entries are independent standard complex gaussian .
( a standard complex gaussian random variable is the same as a standard gaussian vector in @xmath627 divided by @xmath628 in order that the complex variance equal 1 .
its density is @xmath629 with respect to lebesgue measure on @xmath25 . )
this is due to wigner ; see b.mehta/. 2 .
if @xmath173 is unit lebesgue measure on the unit circle @xmath630 , then @xmath631 , so @xmath632 , and @xmath620 is the law of the , which is the set of eigenvalues of a random matrix whose distribution is haar measure on the set of @xmath626 unitary matrices .
this ensemble was introduced by dyson , but the law of the eigenvalues is due to weyl ; see b.hkpv : book/. 3 .
if @xmath173 is standard gaussian measure on @xmath25 , then @xmath633 , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of an @xmath626 matrix whose entries are independent standard complex gaussian .
this is due to ginibre ; see b.hkpv : book/. 4 .
if @xmath173 is unit lebesgue measure on the unit disk @xmath634 , then @xmath635 , so @xmath636 , and the limit of @xmath620 is the law of the zero set of the random power series whose coefficients are independent standard complex gaussian , which converges in the unit disk a.s .
this is due to peres and virg b.peresvirag/. 5 .
if @xmath173 has density @xmath637 with respect to lebesgue measure on @xmath25 , then @xmath638 for @xmath315 , so @xmath639 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath640 when @xmath641 are independent @xmath626 matrices whose entries are independent standard complex gaussian .
( here , we are limited to @xmath612 since the larger spaces do not lie in @xmath341 . )
this is due to krishnapur b.krishnapur:thesis/ ; see b.hkpv : book/. the process was studied earlier by b.caillol/ and b.fjm/ , but without observing the connection to eigenvalues .
inverting stereographic projection , we identify this process with one whose density with respect to lebesgue measure on the unit sphere in @xmath642 is proportional to @xmath643 . for additional information on such processes ,
see @xcite . for an extension to complex manifolds ,
see @xcite . by t.cdom/ ,
the processes @xmath620 stochastically increase in @xmath278 for each of the examples above except the last
. it would be interesting to see natural monotone couplings .
perhaps the last example also increases stochastically in @xmath278 .
the is the limit of the @xmath278th ginibre processes as @xmath417 ; it has the kernel @xmath644 with respect to standard gaussian measure on @xmath25 .
this process is invariant under all isometries of @xmath25 . for each of the plane , sphere , and hyperbolic disk ,
there is only a 1-parameter family of determinantal point processes having a kernel @xmath645 that is holomorphic in @xmath646 and in @xmath647 and whose law is isometry invariant ( * ? ? ?
* theorem 3.0.5 ) . for the sphere , that family
has already been given above ; the parameter is a positive integer . for the other two families ,
the parameter is a positive real number , @xmath648 . in the case of the plane ,
the processes are related simply by homotheties , @xmath649 .
the push - forward of the ginibre process with respect to @xmath650 has kernel @xmath651 with respect to the measure @xmath652 , where @xmath173 is lebesgue measure on @xmath25 .
do these processes increase stochastically in @xmath648 , like poisson processes do ? in the hyperbolic disk , the processes have kernel @xmath653 with respect to the measure @xmath654 , where @xmath173 is lebesgue measure on @xmath655 .
( we fix a branch of @xmath656 for @xmath657 . )
these give orthogonal projections onto the generalized bergman spaces .
the case @xmath658 is that of the limiting ope4 above .
do these processes stochastically increase in @xmath648 ?
recall that when @xmath48 is a finite - dimensional subspace of @xmath37 , the measure @xmath94 is supported by those subsets @xmath95 that project to a basis of @xmath48 under @xmath15 .
similarly , when @xmath158 is the kernel of a finite - rank orthogonal projection onto @xmath659 , define the functions @xmath660 .
then the measure @xmath320 is supported by those @xmath661 such that @xmath662 is a basis of @xmath48 , since @xmath663 .
here , @xmath664 means that @xmath665 .
the question of extending this to infinite - dimensional @xmath48 turns out to be very interesting .
a basis of a finite - dimensional vector space is a minimal spanning set .
although @xmath666 is @xmath94-a.s .
linearly independent , minimality does not hold in general , even for the wired spanning forest of a tree , as shown by the examples in b.heicklenlyons/. see also c.ell2min/. however , the other half of being a basis does hold in the discrete case and is open in the continuous case .
let @xmath667 $ ] be the closed linear span of @xmath668 .
t.basis b.lyons:det/ for every @xmath49 , we have @xmath669 = h$ ] @xmath94-a.s .
we give an application of t.basis/ for @xmath670 , but it has an analogous statement for every countable abelian group . let @xmath671 be the unit circle equipped with unit lebesgue measure
. for a measurable function @xmath672 and @xmath673 , the of @xmath172 at @xmath278 is @xmath674 .
let @xmath675 denote the restriction of @xmath676 to @xmath677 .
if @xmath678 is measurable , we say @xmath679 is @xmath5 if the set @xmath680 is dense in @xmath681 , where we identify @xmath681 with the set of functions in @xmath682 that vanish outside @xmath5 .
the case where @xmath5 is an interval is quite classical ; see b.redheffer/ for a review . a crucial role in that case is played by the following notion of density of @xmath677 .
d.bm for an interval @xmath683 \subset \bbr \setminus \ { 0 \}$ ] , define its @xmath684\big ) : = \max \big\ { |a| , |b| \big\}/ \min \big\ { |a| , |b| \big\ } \,.\ ] ] for a discrete @xmath685 , the of @xmath677 , denoted @xmath686 , is the supremum of those @xmath687 for which there exist disjoint nonempty intervals @xmath688 with @xmath689 for all @xmath278 and @xmath690 ^ 2 = \infty$ ] .
a simpler form of the beurling - malliavin density was provided by b.red:two/ , who showed that e.bmred ( s ) = \ { c : s _ k s | - | < } .
c.seqdual b.lyons:det/ let @xmath691 be lebesgue measurable with measure @xmath692 .
then there is a set of beurling - malliavin density @xmath692 in @xmath693 that is complete for @xmath5 .
indeed , let @xmath694 be the determinantal probability measure on @xmath695 corresponding to the toeplitz matrix @xmath696
. then @xmath694-a.e .
@xmath697 is complete for @xmath5 and has @xmath698 .
when @xmath5 is an interval , the celebrated theorem of beurling and malliavin b.bm/ says that if @xmath677 is complete for @xmath5 , then @xmath699 , and that if @xmath700 , then @xmath677 is complete for @xmath5 .
( this holds for @xmath677 that are not necessarily sets of integers , but we are concerned in this subsection only with @xmath679 . )
c.seqdual/ can be compared ( take @xmath701 and @xmath702 ) to a theorem of bourgain and tzafriri b.btz/ , according to which there is a set @xmath697 of ( schnirelman ) density at least @xmath703 such that if @xmath704 and @xmath676 vanishes off @xmath677 , then @xmath705 it would be interesting to find a quantitative strengthening of c.seqdual/ that would encompass this theorem of @xcite .
the following theorem is equivalent to t.basis/ by duality : t.morris b.lyons:det/ for every @xmath49 , we have @xmath706 } h } = [ \ba]$ ] @xmath94-a.s . as an example , consider the wired spanning forest of a graph , @xmath253 . here , @xmath707 .
in this case , @xmath708 } \star(g ) } = \star(b)$ ] for @xmath709 .
thus , the conclusion of t.morris/ is that @xmath710 , which equals @xmath711 , is concentrated on the singleton @xmath712 for @xmath713-a.e .
@xmath714 .
this was a conjecture of , established by b.morris/. c.ell2min for every @xmath49 , @xmath94-a.s . the maps @xmath715 \to h$ ] and @xmath716 } \colon h \to [ \ba]$ ] are injective with dense image .
both statements are equivalent to @xmath717 \cap h^\perp = \{0\ } = h \cap \ba^\perp$ ] , and these are the contents of theorems [ thm : basis ] and [ thm : morris ] . proved that on any network @xmath718 ( where @xmath253 is the underlying graph and @xmath719 is the function assigning conductances , or weights , to the edges ) , for @xmath720-a.e .
forest @xmath714 and for every component tree @xmath483 of @xmath714 , the @xmath721 of @xmath722 equals @xmath483 a.s .
this suggested b.lyons:det/ the following extension .
given a subspace @xmath48 of @xmath37 and a set @xmath95 , the subspace of @xmath723 $ ] most like " or closest to " @xmath48 is the closure of the image of @xmath48 under the orthogonal projection @xmath724}$ ] ; we denote this subspace by @xmath725 .
for example , if @xmath726 , then @xmath727 since for each @xmath728 , we have @xmath724 } ( \star_x^g ) = \star_x^b$ ] .
to say that @xmath729 is concentrated on @xmath730 is the same as to say that @xmath731 $ ] .
this motivated the following theorem and shows how it is an extension of morris s theorem .
if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath732 , we have @xmath733 in other words , @xmath158 is a reproducing kernel for @xmath48 .
a subset @xmath677 of @xmath48 is called if the closed linear span of @xmath677 equals @xmath48 ; equivalently , the only element of @xmath48 that is orthogonal to @xmath677 is 0 .
an analogue of t.basis/ was conjectured by lyons and peres in 2010 : g.cbasis if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath320-a.e .
@xmath293 , @xmath734 = h$ ] , i.e. , if @xmath735 and @xmath736 , then @xmath737 .
just as in the discrete case , this appears to be on the critical border for many special instances , as we illustrate for several processes where @xmath738 : 1 .
let @xmath173 be lebesgue measure on @xmath24 and @xmath739 , the .
denote the fourier transform on @xmath24 by @xmath740 for @xmath741 , and , by isometric extension , for @xmath742 .
write @xmath743}$ ] . since @xmath744 , we have @xmath745 , where @xmath746 is the inverse fourier transform of @xmath172 .
therefore , the induced operator @xmath158 arises from the orthogonal projection onto the paley - wiener space @xmath747 .
the sine - kernel process arises frequently ; e.g. , it is various scaling limits of the @xmath278th gaussian unitary ensemble in the bulk " as @xmath417 .
( a related scaling limit of the gue is wigner s semicircle distribution . )
we may more easily interpret g.cbasis/ for fourier transforms of functions in @xmath748 $ ] : it says that for @xmath320-a.e .
@xmath293 , the only @xmath749 $ ] such that @xmath750 is @xmath751 .
although the beurling - malliavin theorem applies , no information can be deduced because @xmath752 a.s .
however , ghosh b.ghosh/ has proved this case .
2 . let @xmath173 be standard gaussian measure on @xmath25 and @xmath753 .
this is the ginibre process .
it corresponds to orthogonal projection onto the @xmath754 consisting of the entire functions that lie in @xmath611 ; this is the space of power series @xmath755 such that @xmath756 .
completeness of a set of elements @xmath757 in @xmath754 is equivalent to completeness in @xmath758 ( with lebesgue measure ) of the gabor system of windowed complex exponentials @xmath759 \st \lambda \in \sqrt{2}\lambda\big\ } \,,\ ] ] which is used in time - frequency analysis of non - band - limited signals
. the equivalence is proved using the bargmann transform @xmath760 \,dt \big ) \,,\ ] ] which is an isometry from @xmath758 to @xmath754 . that the critical density is 1 was shown in various senses going back to von neumann ; see b.clp/. this case has also been proved by ghosh b.ghosh/. 3 .
let @xmath173 be unit lebesgue measure on the unit disk @xmath761 and @xmath762 .
this process is the limiting ope4 in s.orthogpoly/. it corresponds to orthogonal projection onto the @xmath763 consisting of the analytic functions that lie in @xmath764 .
what is known about the zero sets of functions in the bergman space b.duren/ is insufficient to settle g.cbasis/ in this case and it remains open .
the two instances above that have been proved by ghosh b.ghosh/ follow from his more general result that g.cbasis/ holds whenever @xmath173 is continuous and @xmath320 is , which means that @xmath765 is measurable with respect to the @xmath320-completion of @xmath766 for every ball @xmath767 .
the limiting process ope4 is not rigid b.hs : tolerance/. ghosh and krishnapur ( personal communication , 2014 ) have shown that @xmath320 is rigid only if @xmath158 is an orthogonal projection .
it is not sufficient that @xmath158 be a projection , as the example of the bergman space shows .
a necessary and sufficient condition to be rigid is not known .
let @xmath158 be a locally trace - class orthogonal projection onto @xmath768 .
for a function @xmath172 , write @xmath769 for the function @xmath770
. let @xmath771 .
clearly @xmath772 for a.e .
@xmath293 . also , for @xmath773
, the function @xmath774 is bounded .
a conjecture analogous to c.ell2min/ is that @xmath293 is a sort of set of interpolation for @xmath48 in the sense that given any countable dense set @xmath775 , for a.e .
@xmath293 , the set @xmath776 is dense in @xmath777 .
one may also ask about completeness for appropriate poisson point processes .
suppose @xmath778 is a group that acts on @xmath0 and that @xmath158 is @xmath778-invariant , i.e. , @xmath779 for all @xmath780 , @xmath295 , and @xmath781 .
( this is equivalent to the operator @xmath158 being @xmath778-equivariant . )
then the probability measure @xmath320 is @xmath778-invariant .
this contact with ergodic theory and other areas of mathematics suggests many interesting questions .
lack of space prevents us from considering more than just a few aspects of the case where @xmath0 is discrete and from giving all definitions .
let @xmath782 . in this case
, @xmath158 is invariant iff @xmath783 for some @xmath784 $ ] , where @xmath785 .
we write @xmath786 in place of @xmath320 . some results and questions from b.ls:dyn/
follow .
t.bern for all @xmath172 , the process @xmath786 is isomorphic to a bernoulli process .
this was shown in dimension 1 by b.shitak:ii/ for those @xmath172 such that @xmath787 by showing that those @xmath786 are weak bernoulli ( wb ) , also called @xmath788-mixing " and absolutely regular " . despite its name
, it is known that wb is strictly stronger than bernoullicity . the precise class of @xmath172 for which @xmath786 is wb
is not known . as usual ,
the of a nonnegative function @xmath172 is @xmath789 .
t.gmdom for all @xmath172 , the process @xmath786 stochastically dominates product measure @xmath790 and is stochastically dominated by product measure @xmath791 .
these bounds are optimal .
we conjecture that ( kolmogorov - sinai ) entropy is concave , as would follow from g.concave/. g.invconcave for all @xmath172 and @xmath792 , we have @xmath793 .
q.block let @xmath794 $ ] be a trigonometric polynomial of degree @xmath795 .
then @xmath786 is @xmath795-dependent , as are all @xmath796-block factors of independent processes .
is @xmath797 an @xmath796-block factor of an i.i.d .
process ?
this is known when @xmath798 b.broman/. let @xmath778 be a sofic group , a class of groups that includes all finitely generated amenable groups and all finitely generated residually amenable groups .
no finitely generated group is known not to be sofic .
let @xmath0 be @xmath778 or , more generally , a set acted on by @xmath778 with finitely many orbits , such as the edges of a cayley graph of @xmath778 .
the following theorems are from b.lyonsthom/. t.sbern for every @xmath778-equivariant positive contraction @xmath1 on @xmath37 , the process @xmath9 is a @xmath799-limit of finitely dependent ( invariant ) processes . if @xmath778 is amenable and @xmath800 , then @xmath9 is isomorphic to a bernoulli process . even if @xmath166 and @xmath167 are @xmath778-invariant probability measures on @xmath801 with @xmath168 , there need not be a @xmath778-invariant monotone coupling of @xmath166 and @xmath167 b.mester : mono/. the proof of the preceding theorem depends on the next one : t.monojoin if @xmath160 and @xmath161 are two @xmath778-equivariant positive contractions on @xmath37 with @xmath185 , then there exists a @xmath778-invariant monotone coupling of @xmath201 and @xmath202 .
the proof of t.sbern/ also uses the inequality @xmath802 for equivariant positive contractions , @xmath1 and @xmath803 , where @xmath804 is the schatten 1-norm .
when @xmath1 and @xmath803 commute , one can improve this bound to @xmath805 we do not know whether this inequality always holds .
write @xmath806 for the fuglede - kadison determinant of @xmath1 when @xmath1 is a @xmath778-equivariant operator
. the following would extend t.gmdom/. it is open even for finite groups .
g.fkdom for all @xmath778-equivariant positive contractions @xmath1 on @xmath807 , the process @xmath9 stochastically dominates product measure @xmath808 and is stochastically dominated by product measure @xmath809 , and these bounds are optimal .
it turns out that the expected degree of a vertex in the free uniform spanning forest of a cayley graph depends only on the group , via its first @xmath810-betti number , @xmath811 , and not on the generating set used to define the cayley graph b.lyons:betti/ : t.betti in every cayley graph @xmath253 of a group @xmath778 , we have @xmath812 = 2 \beta_1(\gp ) + 2 \,.\ ] ] this is proved using the representation of @xmath813 as a determinantal probability measure .
it can be used to give a uniform bound on expansion constants b.lpv/ : t.lpv for every finite symmetric generating set @xmath677 of a group @xmath778 , we have @xmath814 for all finite non - empty @xmath815 .
there are extensions of these results to higher - dimensional cw - complexes and higher @xmath810-betti numbers b.lyons : betti/. in unpublished work with d. gaboriau @xcite , we have shown the following : t.damien let @xmath253 be a cayley graph of a finitely generated group @xmath778 and @xmath816 . then there exists a @xmath778-invariant finitely dependent determinantal probability measure @xmath9 on @xmath817 that stochastically dominates @xmath818 and such that @xmath819 \le \be_\fsf\big[\deg_\fo(\bp)\big ] + \epsilon \,.\ ] ] in addition , if @xmath778 is sofic , then @xmath820 .
if it could be shown that @xmath9 , or indeed every invariant finitely dependent probability measure that dominates @xmath813 , yields a connected subgraph a.s . , then it would follow that @xmath821 is equal to the cost of @xmath778 , a major open problem of b.gaboriau : invar/.
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in this appendix , we explain how the line width was extracted from the numerical data .
we begin by determining the spectral function , defined by @xmath115 this consists of a set of delta functions .
we then define the integrated spectral function @xmath116 .
this consists of a set of step functions ( see fig .
[ steps](a ) ) . for each step
, we identify the energy values corresponding to @xmath117 of the step , @xmath118 of the step , and @xmath119 of the step . the energy spacing between the @xmath117 and @xmath119 points
is taken to be the linewidth of this spectral line .
we track how this line width scales with @xmath0 .
we note that there is in general a wide distribution of line widths for any @xmath0 ( fig .
[ steps](b ) ) . as a result ,
the mean and the median linewidth scale very differently ( see fig.5 of the main text ) .
an understanding of the difference between the scaling of the mean and typical line width is an important challenge for future work .
( a ) the procedure for determining the linewidth .
the blue curve is an integrated spectral function .
the green squares divide each step into half , the red diamonds mark @xmath117 and the light blue circles mark @xmath120 of each step .
( b ) probability distribution of the linewidth @xmath109 for different values of coupling to the bath @xmath0 for a system with @xmath69 and @xmath121 averaged over 10 disorder configurations .
lines are a guide to the eye . ]
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we use exact diagonalization to study the breakdown of many - body localization in a strongly disordered and interacting system coupled to a thermalizing environment .
we show that the many - body level statistics cross over from poisson to goe , and the localized eigenstates thermalize , with the crossover coupling decreasing with the size of the bath in a manner consistent with the hypothesis that an infinitesimally small coupling to a thermodynamic bath should destroy localization of the eigenstates .
however , signatures of incomplete localization survive in spectral functions of local operators even when the coupling to the environment is sufficient to thermalize the eigenstates .
these include a discrete spectrum and a gap at zero frequency .
both features are washed out by line broadening as one increases the coupling to the bath .
we also determine how the line broadening scales with coupling to the bath .
isolated quantum systems with quenched disorder can enter a ` localized ' regime where they fail to ever reach thermodynamic equilibrium @xcite . while we have an essentially complete understanding of localization in non - interacting systems @xcite ,
the theory of many - body localization ( mbl ) is still under construction @xcite .
numerical investigations using exact diagonalization @xcite _ do _ indicate that all eigenstates of a strongly interacting disordered system can be localized .
most of the theoretical research so far has been in the limit of a perfectly isolated system .
however , experimental tests of mbl ( @xcite ) will always include some finite coupling to the environment .
what then can we expect to see in experiments designed to probe many body localization ?
a recent theory of mbl systems weakly coupled to heat baths proposed that while eigenstates are delocalized by an infinitesimally weak coupling to a heat bath , signatures of localization persist in spectral functions of local operators for weak coupling to a bath @xcite .
this theory has yet to face stringent numerical tests .
moreover , it did not discuss the spectral functions of the physical degrees of freedom , the quantities of direct relevance for experiments , focusing instead on the spectral functions of certain localized integrals of motion that are believed to exist @xcite , but which are related to the physical degrees of freedom by an unknown unitary transformation .
this work directly addresses these issues .
we use exact numerical diagonalization to establish the behavior of many body localized systems weakly coupled to heat baths .
we show that coupling @xmath0 to a bath results in a crossover from poisson to gaussian orthogonal ensemble ( goe ) eigenvalue statistics , which becomes exponentially steeper with increasing bath size . a similar rapid crossover to thermalization is seen in the eigenstates .
however , the prospect for seeing mbl in experiments is still realistic because signatures of incomplete localization remain in the spectral functions of local ( in real space ) operators .
indeed , we find that the spectral functions of the microscopic degrees of freedom look completely different in the localized and thermal phases ( see fig .
1 ) . the thermal phase has a continuous spectrum whereas the local spectral function in the localized phase is discrete , with a hierarchy of gaps , and a gap at zero frequency that survives even after spatial averaging .
increasing @xmath0 causes lines to broaden and fill in these gaps .
however , as long as the typical line broadening is less than the largest gaps , gap - like features remain .
our work also reveals how the line broadening scales with @xmath0 . _ the model _ : we choose the antiferromagnetic heisenberg spin-@xmath1 chain with random fields along @xmath2 : @xmath3 we set the interaction @xmath4 .
the on - site fields @xmath5 are independent random variables , uniformly distributed between @xmath6 and @xmath7 ; @xmath7 measures the disorder strength in the system .
this model with periodic boundary conditions has been shown to have a many - body localization transition at @xmath8 in the infinite temperature limit @xcite .
the hamiltonian in eq .
[ eq : h_pbit ] is written in terms of the physical degrees of freedom @xmath9 ( ` @xmath10-bits , ' in the language of @xcite , where @xmath10=physical ) .
in general , its eigenstates are quite complicated and non - trivial .
as shown @xcite , one can perform a unitary transformation to rewrite @xmath11 in terms of localized constants of motion @xmath12 .
the @xmath13 are dressed versions of the @xmath9 operators , which are localized in real space , with exponential tails , and are referred to in @xcite as ` @xmath14-bits ' ( @xmath14=localized )
. a unitary transformation to this ` @xmath14-bit ' basis can always be performed , if the system is in the regime where all the many body eigenstates are localized . in this @xmath14-bit basis
, the hamiltonian becomes @xmath15 the values of the coefficients @xmath16 and @xmath17 will depend upon the parent hamiltonian ( 1 ) , although these coefficients all fall off exponentially with distance .
the eigenstates of ( 2 ) are just products of @xmath18 .
motivated by the representation ( 2 ) of the hamiltonian ( 1 ) , it is instructive to consider the simpler hamiltonian @xmath19 where the @xmath20 and @xmath21 as independent random variables taken from a log - normal distribution with @xmath22 and @xmath23 , and similarly for @xmath24 .
we take @xmath25 and work with open - boundary conditions .
this hamiltonian also has the feature that eigenstates are product states of @xmath26 , and is simpler to work with numerically .
for the bath , we use a non - integrable hamiltonian that has been recently studied @xcite .
it consists of @xmath27 interacting spins with the hamiltonian : @xmath28 while using open boundary conditions , we add a boundary term @xmath29 to @xmath30 .
we use @xmath31 , @xmath32 and @xmath33 , values for which @xmath30 has been numerically shown by @xcite to have fast entanglement spreading .
( we use periodic boundary conditions only for @xmath10-bits with @xmath34 . ) the interaction between the system and bath should be local for both @xmath10- and @xmath14-bits .
we first study @xmath14-bit eigenstates , choosing the coupling : @xmath35 later we examine @xmath10-bit spectra , using the coupling @xmath36 the total hamiltonian is thus @xmath37 , where @xmath11 and @xmath38 are given by eq .
( 3 ) and ( 5 ) in the first part of this work , and by eq .
( 1 ) and ( 6 ) in the latter part of this work .
we will indicate the transition clearly in the text .
we use open boundary conditions except where periodic boundaries are explicitly mentioned .
we start by analyzing the breakdown of localization when the @xmath14-bit hamiltonian ( 3 ) is coupled to a bath according to ( 5 ) , by examining the many - body eigenvalue statistics as @xmath0 is increased from @xmath39 .
we perform exact diagonalization on a system with @xmath40 spins coupled to @xmath41 spins in the bath .
the many body level - spacing is @xmath42 , where @xmath43 is the energy of the @xmath44th eigenstate .
following @xcite , we define the ratio of adjacent gaps as @xmath45 .
we average this over eigenstates and several different realizations of the disorder to get a probability distribution @xmath46 at a particular value of @xmath0 . in fig .
[ fig : level_space ] , we show how @xmath46 evolves from poisson to goe like as @xmath0 is increased . in a localized system
we expect that @xmath47 , and for a thermalizing system , we expect that @xmath48 .
-bit hamiltonian as @xmath0 is increased .
results are for a system with @xmath40 spins and bath with @xmath41 spins averaged over @xmath49 eigenstates obtained from several disorder configurations .
the dark blue solid line is the poisson distribution expected for localized systems , and the light blue dashed line is the goe distribution expected for thermalizing systems . ]
the transition from poisson to goe statistics happens gradually for this finite size system . a simple analytical estimate of the characteristic value of @xmath0 at the crossover point proceeds as follows ( see also @xcite ) : if @xmath50 is the bandwidth of the bath and @xmath51 is the many body level spacing in the bath , then the system couples to @xmath52 states , with a typical matrix element to each state of order @xmath53 .
the coupling to the bath will be effective in thermalizing the system when this matrix element becomes of order the level spacing in the bath , i.e. when @xmath54 .
this indicates that the crossover coupling @xmath55 . since @xmath56
, the critical value of @xmath0 is expected to scale as @xmath57 . to quantitatively compare this crossover estimate to the data ,
we define @xmath58 . after averaging over disorder distributions
, @xmath59 should be @xmath60 in the goe regime and @xmath61 in the localized regime @xcite .
it is convenient to define the normalized quantity @xmath62=(@xmath63 , such that @xmath64 if the level statistics are goe and @xmath65 if they are poisson .
fig .
3(a ) shows how @xmath66 varies with @xmath0 for systems of size @xmath67 .
fig .
[ fig : r_trans](b ) shows that scaling of the form @xmath68 is successful in making the data for different @xmath27 in fig .
[ fig : r_trans](a ) collapse onto one curve .
data collapse occurs also for @xmath69 and @xmath70 , indicating clearly that it is @xmath27 which controls the finite size scaling .
we get the best collapse when the constant in the exponential is @xmath71 which is in good agreement with the analytical estimate @xmath72 .
this implies that the crossover to thermalization is at a coupling @xmath73 that is exponentially small in system size , so that level statistics become goe at infinitesimal @xmath0 in the thermodynamic limit .
( defined in the text ) in the @xmath14-bit hamiltonian as @xmath0 is increased for system sizes @xmath74 and @xmath75 .
data is averaged over @xmath49 eigenstates obtained from several disorder configurations .
( b ) collapse of data in ( a ) is in good agreement with analytic arguments for the finite size scaling presented in the main text , and depends only on @xmath27 . ]
another test of thermalization is checking whether the eigenstates obey the eigenstate thermalization hypothesis ( eth ) @xcite . the eth states that the expectation value of a local operator should be the same in every eigenstate within a small energy window . for a localized system
this will not be the case . in fig .
[ fig : eth ] , we show how eigenstate thermalization sets in as @xmath0 is increased .
we choose an energy window around the center of the band and calculate the standard deviation of the expectation value of @xmath76 for all eigenstates within the window .
explicitly , we define @xmath77,\ ] ] where the overline denotes averaging over an energy window of width @xmath78 in the middle of the band and @xmath79 is an eigenstate of the coupled system and bath .
we choose @xmath80 . after averaging over disorder distributions ,
we expect to find @xmath81 for a thermalized system .
fig .
[ fig : eth](a ) shows how @xmath82 approaches 0 as @xmath0 is increased for different system sizes . fig .
[ fig : eth](b ) shows that @xmath82 scales with @xmath0 similar to @xmath66 .
the exponent here is @xmath83 , also close to the estimated analytical value .
-bit hamiltonian as @xmath0 is increased for system sizes @xmath69 , @xmath84 , @xmath85 , @xmath86 , @xmath87 and @xmath75 .
@xmath82 as defined in the text is measured at the site of the central spin .
data is averaged over @xmath49 eigenstates obtained from several disorder configurations .
( b ) collapse of data in ( a ) agrees with analytical estimates of finite size scaling for @xmath88 . for a finite size system with @xmath27 spins in the bath ,
the eigenstates become effectively thermal for @xmath89 , implying that eigenstates in the thermodynamic limit become thermal for infinitesimal @xmath0 . ]
we now turn to an analysis of the spectral functions of local operators .
henceforth we are working with the physical degrees of freedom , eq .
( 1 ) and ( 6 ) .
we examine the spectral function from an exact eigenstate @xmath90 where @xmath91 is the @xmath92 eigenstate of the combined system and bath .
we note that since we are working with a finite size system with a discrete spectrum , the spectral function will always consist of a set of delta functions . at @xmath93 ,
the delta functions should have minimum spacing @xmath94 , equal to the many body level spacing in the system . at non - zero @xmath0 , each ` parent ' delta function will split into exponentially many descendants , with a typical spacing @xmath95 . a fine binning in energy with bin size greater than @xmath95
will then yield a smooth spectral function , with the ` parent ' delta functions of the system having been ` broadened ' by coupling to the bath . to investigate this broadening ,
it is convenient to take @xmath96 .
we therefore take @xmath97 and @xmath98 , and investigate how the ` line broadening ' evolves with @xmath0 for @xmath99 .
details of the procedure are outlined in the supplementary material , and the results are illustrated in fig .
[ fig : linewidth ] for @xmath100 .
the mean and median linewidth at a particular value of @xmath0 are significantly different .
this is a result of the long tails in the distribution of the linewidth ( see supplement ) .
fig .
[ fig : linewidth ] shows that at the larger values of @xmath0 we study , a log - log plot of the median vs @xmath0 appears to fit well to a straight ( dashed ) line . for
the system sizes that we are able to access , the straight line fit suggests @xmath101 , where @xmath102 increases as the size of the bath increases , reaching @xmath103 for @xmath104 .
we note that while a simple application of the golden rule predicts @xmath105 , a more careful analysis @xcite suggests that the true scaling should be @xmath106 .
the solid lines in fig .
[ fig : linewidth ] are a fit to this theoretical prediction , and are consistent with the data , except at smallest @xmath0 .
the discrepancy at smallest @xmath0 and the difference between median and mean are worthwhile topics for future work . for a system of @xmath10-bits with @xmath69 and
@xmath107 averaged over more than 38000 eigenstates obtained from several disorder configurations at @xmath100 .
@xmath108 for the sizes shown here .
the mean and the median of the probability distribution of the linewidth @xmath109 are extracted from the data as discussed in the appendix .
the dotted lines are linear fits to the data .
the solid lines are fits to the theoretical prediction .
[ fig : linewidth ] ] finally , we analyze the behavior of the spectral function averaged over all sites and eigenstates of the system , for @xmath110 .
we note that the hamiltonian ( 1 ) has a delocalization - localization phase transition at @xmath8 .
fig .
[ fig : pbits](a ) shows @xmath111 on the delocalized side of the transition for a small value of @xmath0 .
@xmath111 is smooth everywhere .
( the graininess is a result of the small system size . )
fig .
[ fig : pbits](b ) is on the localized side of the transition , with the system almost decoupled from the bath . here
, @xmath111 consists of clusters of narrow spectral lines , with a hierarchy of energy gaps , just as was shown to be the case for @xmath14-bit spectral functions in @xcite .
@xmath111 vanishes at @xmath112 .
thus , local spectral functions can distinguish between extended and localized phases . in fig .
[ fig : pbits](c - e ) we examine how the @xmath10-bit spectral functions evolve as @xmath0 increases .
we see that the line broadening increases and different lines start to overlap with each other , washing out the weaker spectral features , but larger gaps remain .
the zero - frequency gap also fills in with increasing @xmath0 .
the spectral functions retain signatures of localization even for @xmath113 when the eigenstates of the combined system and bath are effectively thermal , and get washed out when @xmath0 becomes comparable to the characteristic energy scales in the system ( i.e. @xmath114 ) . in conclusion
, we have investigated the signatures of localization in a disordered system weakly coupled to a heat bath using exact diagonalization .
the wave functions are found to exhibit a crossover to thermalization as a function of coupling to the bath .
the crossover coupling is proportional to the many body level spacing in the bath , and vanishes exponentially fast in the limit of a large bath size .
in contrast , the spectral functions of local operators are found to show more robust signatures of proximity to a localized phase .
while the spectral functions are smooth and continuous in the delocalized phase ( after coarse graining on the scale of the many body level spacing ) , the spectral functions in the localized phase consist of narrow spectral lines , and contain a hierarchy of gaps , as well as a gap at zero frequency that persists even after spatial averaging .
increasing the coupling to the bath increases the line broadening ( in a manner that we calculate ) and washes out these features .
however , signatures of localization survive in the spectral functions even at couplings to the bath where the exact eigenstates are effectively thermal ( fig .
1 ) . _
acknowledgments : _ rn would like to thank sarang gopalakrishnan and david huse for a collaboration on related ideas .
this work was supported by doe grant de - sc0002140 .
rnb . acknowledges the hospitality of the institute for advanced study , princeton while this work was being done .
rn was supported by a pcts fellowship .
sj was supported by the porter ogden jacobus fellowship of princeton university .
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in this supplementary material , we compare the value of the penetration depth obtained from experiments @xcite with the prediction from homes law ; for the latter , we use a combination of the experimental data obtained from optical - conductivity and dc transport . for each value of the doping ( @xmath8 )
, we estimate the ( approximate ) dc resistivity ( @xmath80 ) by extrapolating the curves to @xmath9 , from the transport data in fig.1(b ) of ref.@xcite .
we estimate the value of @xmath81 , where @xmath23 is the superconducting gap , from the data for optical conductivity in the superconducting state , as shown in fig .
3(b ) of ref . @xcite . since @xmath7 remains relatively unchanged as a function of @xmath8 in the vicinity of optimal doping , we assume @xmath82 to be independent of @xmath8 such that @xmath83@xmath84s@xmath85 .
then , in the dirty limit , _ s = _ . in order to obtain the penetration depth
, we need to restore various dimensionful constants such that , _
l^2(0)= , where @xmath86 m / s ) is the speed of light and @xmath87 f / m ; 1 f=1 @xmath88s ) is the permitivity of free space .
the values obtained are shown in the table below and have been presented in fig . 2 of the main text , along with a comparison to the experimental data @xcite .
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we present a theory for the large suppression of the superfluid - density , @xmath0 , in bafe@xmath1(as@xmath2p@xmath3)@xmath1 in the vicinity of a putative spin - density wave quantum critical point at a p - doping , @xmath4 .
we argue that the transition becomes weakly first - order in the vicinity of @xmath5 , and disorder induces puddles of superconducting and antiferromagnetic regions at short length - scales ; thus the system becomes an electronic micro - emulsion .
we propose that frustrated josephson couplings between the superconducting grains suppress @xmath0 .
in addition , the presence of ` normal ' quasiparticles at the interface of the frustrated josephson junctions will give rise to a highly non - trivial feature in the low - frequency response in a narrow vicinity around @xmath6 .
we propose a number of experiments to test our theory . _
introduction.- _ an important focus of the study of high temperature superconductivity ( sc ) has been on the role of antiferromagnetism ( afm ) and its relation to sc @xcite .
there is clear evidence across many different families of compounds that sc appears in close proximity to an afm phase @xcite ; these families include the iron - pnictides , the electron - doped cuprates and the heavy - fermion superconductors .
moreover , the optimal transition temperature ( @xmath7 ) of the sc is often situated where the normal state afm quantum critical point ( qcp ) would have been located , in the absence of superconductivity .
the experimental detection of the qcp is often challenging in the normal state , and more so in the superconducting state .
recently , a number of measurements were reported in a member of the pnictide family , bafe@xmath1(as@xmath2p@xmath3)@xmath1 , as a function of the isovalent p - doping , @xmath8 .
the experiments show a phase transition involving onset of spin - density wave ( sdw ) order in the normal state above @xmath7 , which extrapolates to a @xmath9 sdw qcp ( see @xcite and references therein ) .
these experiments include : ( _ i _ ) a sharp enhancement in the effective mass , @xmath10 , upon approaching a critical doping from the overdoped side , as obtained from de haas - van alphen oscillations @xcite and from the jump in the specific - heat at @xmath7 @xcite , and , ( _ ii _ ) a vanishing curie - weiss temperature ( @xmath11 ) , extracted from the @xmath12 measurements using nmr .
as we will review below , a number of puzzling results have appeared from experiments investigating whether the sdw qcp actually survives `` under the sc dome . '' here we propose a resolution of these puzzles by postulating a weakly first - order transition for the onset of sdw order in the presence of sc order ( see fig .
[ ph]a ) .
our results are independent of the specific microsopic mechanism responsible for rendering the transition weakly first - order @xcite .
it is well known that ` random bond ' disorder has a strong effect on symmetry - breaking first - order transitions @xcite , and ultimately replaces them with a disorder - induced second order transition in two dimensional systems .
our main claim is that the inhomogeneities associated with these highly relevant effects of disorder can resolve the experimental puzzles .
the possiblity of a qcp within the sc state was investigated by measurements @xcite of the zero temperature london penetration depth , @xmath13 ( @xmath14 superfluid - density ) , as a function of @xmath8 .
a sharp peak in @xmath15 was observed at @xmath16 and interpreted as evidence for a qcp @xcite .
however , this interpretation is at odds with general theoretical considerations @xcite concerning a qcp associated with the onset of sdw order in the presence of a superconductor with gapped quasiparticle excitations @xcite .
these considerations suggest that such systems will display a _ monotonic _ variation in @xmath15 across the qcp , rather than a sharp peak ( see dashed - blue / solid - red curves in fig .
[ ph]b ) @xcite . as a first step toward resolving this discrepancy ,
it is useful to place measurements of @xmath0 in the context of what is known about the normal state conductivity of the bafe@xmath1(as@xmath2p@xmath3)@xmath1 system , as these quantities are intimately related through a sum rule .
the low temperature superfluid density of a spatially homogeneous superconductor can be estimated from the missing area " relation , _ s_0 ^ 2/ ( z)dz , [ homese ] where @xmath17 is the elastic scattering rate and @xmath18 . in the dirty limit where @xmath19 , the above relation yields homes law @xcite , @xmath20 , whereas in the clean limit @xmath21 where @xmath22 is the conductivity spectral weight in the normal state
. eqn .
[ homese ] is particularly useful when the normal state resistivity data can reasonably be extrapolated to @xmath9 . by combining dc transport data as a function of @xmath8 @xcite and a measurement of 2@xmath23 from optical conductivity @xcite , eq .
[ homese ] provides a lower bound on @xmath15 ( with the assumption that @xmath23 is independent of @xmath8 ) .
fig.[homes ] shows @xmath15 as a function of @xmath8 obtained under this assumption ( details of the procedure are presented as supplementary information ) .
the decrease of superfluid density on the underdoped side reflects the growth in residual resistivity that begins as @xmath8 drops below about 0.33 .
the values of @xmath15 estimated from eq .
[ homese ] form a baseline for comparison with the experimental results presented in ref .
@xcite . on the same graph in fig .
[ homes ] , we show the experimentally measured @xmath15 @xcite . the data generally reflect the trend expected from the variation in the residual resistivity , with the exception of the sample with @xmath24 , in which the condensate spectral weight is suppressed by about 40% from the homes law estimate . given the constraints imposed by the sum rule ,
there are two possible sources of this discrepancy : ( _ i _ ) the quasiparticle mass could be renormalized at this value of @xmath8 , corresponding to an intrinsic decrease in @xmath22 , or , ( _ ii _ ) a considerable fraction of the ( unrenormalized ) @xmath22 could fail to contribute to the low temperature superfluid density .
the latter possibility is suggested within the scenario that we develop here .
we analyze the above experiments by assuming a weakly first - order transition @xcite , and argue that the presence of quenched disorder leads to formation of a _ micro - emulsion _ at small scales @xcite .
the system consists of sc puddles , where some of the puddles additionally have sdw order ( see fig .
[ ph]a inset ) .
the sdw(+sc ) regions , which have a locally well - developed antiferromagnetic moment but no long - range orientational order , act as barriers between the different sc grains . upon moving deeper into the ordered side of the transition , the sdw(+sc )
regions start to percolate and crossover to a state with long - range sdw order ; this is the regime with a microscopically coexistent sc+sdw . as a function of decreasing @xmath8 ,
the micro - emulsion is therefore a transitional state ( shown as grey region in fig . [ ph]a ) between a pure sc and a coexistent sc+sdw .
recent experiments in the vicinity of optimal doping using neutron - scattering and nmr have found results broadly consistent with our proposed phase diagram @xcite .
we note that the granular nature of superconductivity should have no effect on the bulk @xmath7 in the presence of percolating sc channels . _
model.- _ when the system is well described in the vicinity of @xmath5 by a micro - emulsion as explained above , the phase fluctuations associated with the sc grains ( shown as purple regions in fig .
[ ph]a inset ) , can be modeled by the following effective theory , h_= - _ a , bj_ab(_a-_b ) , where @xmath25 represent the josephson junction ( jj ) couplings between grains ` @xmath26 ' and ` @xmath27 ' .
we have ignored the capacitive contributions .
the josephson current across the junction will be given by @xmath28 , and @xmath25 may therefore be interpreted as the lattice version of the local superfluid density , @xmath29 , i.e. @xmath30 , with @xmath31 representing the superfluid - current and velocity respectively . having a frustrated jj ( also known as a @xmath32junction ) with a negative value of @xmath25 leads to a local suppression in @xmath0 .
similar ideas have been discussed in the past in a variety of contexts ( see refs .
@xcite for a specific example ) , though the mechanism considered here will be different .
we shall now propose an explicit scenario under which a suppression in @xmath0 arises in the vicinity of putative magnetic qcps , utilizing the sc gap structure in the material under question .
the basic idea is as follows : suppose that the tunneling of electrons between the two grains is mediated by the sdw moment in the intervening region @xcite , and is accompanied by a transfer of finite momentum that scatters them from a hole - like to an electron - like pocket . because the sc gaps on the two pockets have a relative phase - difference of @xmath33 , the jj coupling will be frustrated @xcite .
let us first focus on a single grain .
in order to capture the multi - band nature of the scs , we introduce two superconducting order parameters , @xmath34 with @xmath35 to model the @xmath36 state on the two pockets .
microscopically , these belong to regions in the grain having different momenta , @xmath37 , parallel to the junction .
the gaps are related to the microscopic degrees of freedom @xcite via the following relation , _
i(z)=___i v__,____- _ , where @xmath38 creates an electron at position @xmath39 with momentum @xmath37 parallel to the junction and spin @xmath40 .
@xmath41 is the pairing interaction in the cooper channel and @xmath39 is the coordinate perpendicular to the junction with area @xmath42 .
the regions @xmath43 are defined as , @xmath44 and @xmath45 , where @xmath46 is an arbitrary momentum scale chosen such that @xmath47 ( see fig . [ jj ] for an illustration ) .
we ll assume that such a prescription is valid for each grain , with possibly different values of @xmath46 .
let us then write down a model for the two coupled sc grains with an intervening proximity coupled sdw that has a well developed moment , @xmath48 .
our notation is as follows : we use @xmath49 to denote the grain index and @xmath35 to denote the band index within each grain . from now on , we relabel @xmath37 as @xmath50 .
we introduce the nambu spinor , @xmath51 , where now @xmath52 creates an electron with momentum @xmath50 parallel to the junction and at a position @xmath39 ( label suppressed ) , which belongs to a region of band @xmath53 " within grain @xmath54 " .
the effective hamiltonian is given by , [ heff ] h_&=&h_+h_t , + h_&=&_,i , _
i,,^_i,,^ , + h_t&=&g_k ( ^a_+,,[_^0]_-,,^b + & & + ^a_-,,[_^0]_+,,^b ) + , where @xmath55 is the tunneling matrix element , @xmath56 @xmath57 act in nambu space and @xmath58 @xmath57 act in spin space . in the above
, @xmath59 corresponds to the bare pairing hamiltonian written for the @xmath60 bands within each of the two grains .
@xmath61 represents the sdw moment mediated hopping of electrons from one grain to the other ( represented by the @xmath62 superscripts ) and simultaneously scattering from one band to the other ( represented by the @xmath60 subscripts ) .
therefore , @xmath48 imparts a finite momentum ( along the interface ) to the electrons when it scatters them from the electron ( hole ) pocket on one grain to the hole ( electron ) pocket on the other grain ( shown as the black arrows in fig .
[ jj ] ) . _
results.- _ using the ambegaokar - baratoff relation @xcite , we can write the josephson coupling ( at @xmath9 ) between the two grains as , j_ab= where @xmath63 and @xmath64 represent the band indices on the different grains . since @xmath65 , the coupling @xmath66 .
note that the specific nature of the frustrated tunneling arises from the same spin - fluctuation mediated mechanism that is predominantly responsible for the @xmath67 pairing symmetry @xcite .
however , there will also be a direct tunneling term ( not included in eqn .
[ heff ] ) in the hamiltonian , which does not scatter the electrons from one pocket to the other , as they hop across the junction .
the contribution to the jj coupling from this term will be unfrustrated ( i.e. @xmath68 ) .
the ratio of the tunneling amplitudes in the two different channels is non - universal and depends on various microscopic details .
in particular , the emulsion is associated with a distribution of josephson - couplings , @xmath69 , with a mean coupling strength , @xmath70 .
if a substantial fraction of the jj couplings become negative due to the mechanism proposed above , @xmath71 will be small , and the superfluid density will be suppressed ( see green curve in fig.[ph]b ) .
we now propose a resolution as to the fate of the uncondensed spectral weight ( highlighted in fig .
[ homes ] ) , which can potentially be tested by measurements of the low frequency optical conductivity .
frustrated @xmath32junctions host gapless states at the interface between the two grains @xcite , giving rise to a finite density of states around zero energy ( see fig [ sigw ] inset ) . as a result of the gapless ` normal'-fluid component at the interface , a fraction @xmath72 of the spectral weight will be displaced from the superfluid - density to non - zero frequencies ( shaded region in fig . [ sigw ] ) .
given that the weight of the condensate is proportional to @xmath73 , the 40% suppression in @xmath0 for bafe@xmath1(as@xmath2p@xmath3)@xmath1 in the vicinity of the putative qcp corresponds to @xmath74 .
our proposed optical conductivity , @xmath75 , in the vicinity of penetration depth anomaly is shown in in fig [ sigw ] .
the spectrum shows clearly that the connection between normal state conductivity and superfluid density implied by eq .
[ homes ] will break down .
in particular , @xmath76 ( which is a property of the normal state ) , could vary monotonically with isovalent - doping across @xmath6 , while the abundance of low - energy excitations in the immediate vicinity of @xmath6 would give rise to a non - monotonic variation in the superfluid density .
this allows for an unusual way of rearranging spectral weight in the _ superconducting _ state below the gap , without violating optical sum - rules
. the above scenario will give rise to a number of interesting low temperature thermodynamic and transport properties , as we now discuss .
first of all , there should be a striking enhancement in the low - temperature thermal conductivity and specific - heat , as a function of @xmath8 in the narrow vicinity of @xmath6 , due to the ` normal'-component .
it is important to recall that this material has loop - like nodes on the electron - pockets @xcite .
however , the geometry of the electron - pockets and the magnitude of the gap do not change substantially in the vicinity of @xmath6 , and therefore it is unlikely that the contribution to the above quantities from the nodal - quasiparticles will have a drastic modificiation .
it should therefore be relatively straightforward to disentangle the contribution arising from the nodal versus the ` normal ' quasiparticles .
studying the nmr - spectra as a function of decreasing temperature ( across @xmath7 ) and down to sufficiently low temperatures in the vicinity of @xmath6 should also reveal the spatial inhomogeneity associated with the sdw regions .
a large residual density of states in the superconducting state has been detected at a particular p - doping via the power - law temperature dependence of @xmath77 @xcite . within our scenario
, there should be a striking enhancement in this quantity as a function of doping around @xmath6 .
finally , we note that a promising direction for future studies would be to measure the magnetic - field distribution due to the propagating currents in the emulsion using nv - based magnetometers @xcite . _
discussion.- _ the theoretical study in this paper was motivated by a number of remarkable experiments carried out in bafe@xmath1(as@xmath2p@xmath3)@xmath1 , as a function of @xmath8 in the normal and superconducting phases .
our primary objective was to provide an explanation for the striking enhancement of the london penetration depth in the vicinity of a putative sdw qcp in the sc state .
we developed a scenario based on the idea that true sdw criticality is masked by a weak first - order phase transition in the superconducting state at @xmath9 . in this picture ,
quenched disorder naturally gives rise to an _ emulsion _ at small length scales with puddles of sc and sdw(+sc ) .
it is then , in principle , possible for sdw moments at the interface of the sc grains to generate frustrated josephson couplings , which deplete the local superfluid - density .
our proposed scenario naturally calls for a number of experimental tests that should be carried out in the near future , which should directly look for both the spatial inhomogeneities associated with the emulsion @xcite , and probe the gapless excitations using thermodynamic probes , as explained above .
in addition to experiments on bafe@xmath1(as@xmath2p@xmath3)@xmath1 , it should be important to further investigate the contrasting behavior of the electron - doped system , ba(fe@xmath2co@xmath3)@xmath1as@xmath1 , where @xmath78 behaves monotonically as a function of @xmath8 across the putative qcp @xcite .
electron - doping leads to significantly higher amounts of disorder compared to the isovalently - doped case , and would therefore lead to puddles with typically much smaller size @xcite .
our proposed mechanism for the strong suppression of the superfluid - density in the isovalently - doped material relies on the existence of an emulsion with puddles of appreciable size , in the presence of an optimal amount of disorder .
a comparison of the nmr spectra in the narrow vicinity of the putative qcp in the electron and isovalently doped materials would shed light on these microscopic differences between the two families . finally , though we have hypothesized that the sdw onset transition _ inside _ the sc is , in the absence of disorder , a weak first order transition , we emphasize that the normal state properties are consistent with the presence of a hidden " qcp around optimal doping @xcite .
it is plausible that in the normal state , different experimental techniques are probing the critical fluctuations associated with not one , but distinct qcps as a function of @xmath8 . for instance
, @xmath10 extracted from high - field quantum oscillations is dominated by the vicinity of ` hot - spots ' , where quasiparticles are strongly damped due to coupling to the sdw fluctuations @xcite . on the other hand , strong critical fluctuations associated with the nematic order - parameter @xcite , that couple to the entire fermi - surface ,
would dominate @xmath10 extracted at zero - field from the jump in the specific heat at @xmath7 . _
acknowledgements.- _ we thank a. carrington , a. chubukov , n. curro , j.c .
davis , r. fernandes , k. ishida , m .- h .
julien , s. kivelson , y. matsuda , a. millis and a. vishwanath for useful discussions .
we thank k. hashimoto and y. matsuda for providing us with the data shown in fig.[homes ] .
dc is supported by the harvard - gsas merit fellowship and acknowledges the boulder summer school for condensed matter physics - modern aspects of superconductivity " , where some preliminary ideas for this work were formulated .
dc and ss were supported by nsf under grant dmr-1360789 , the templeton foundation , and muri grant w911nf-14 - 1 - 0003 from aro .
ts was supported by department of energy desc-8739- er46872 , and partially by a simons investigator award from the simons foundation .
jo acknowledges the office of basic energy sciences , materials sciences and engineering division , of the u.s .
department of energy under contract no .
de - ac02 - 05ch11231 for support .
part of this work was completed when jo was visiting mit as a moore visitor supported by grant gbmf4303 .
research at perimeter institute is supported by the government of canada through industry canada and by the province of ontario through the ministry of research and innovation .
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we study the coupling of magneto - acoustic waves to alvn waves using 2.5d numerical simulations . in our experiment
, a fast magnetoacoustic wave of a given frequency and wavenumber is generated below the surface .
the magnetic field in the domain is assumed homogeneous and inclined .
the efficiency of the conversion to alfvn waves near the layer of equal acoustic and alfven speeds is measured calculating their energy flux .
the particular amplitude and phase relations between the oscillations of magnetic field and velocity help us to demonstrate that the waves produced after the transformation and reaching upper atmosphere are indeed alfvn waves .
we find that the conversion from fast magneto - acoustic waves to alfvn waves is particularly important for the inclination @xmath0 and azimuth @xmath1 angles of the magnetic field between 55 and 65 degrees , with the maximum shifted to larger inclinations for lower frequency waves .
the maximum alfvn flux transmitted to the upper atmosphere is about 23 times lower than the corresponding acoustic flux .
conversion from fast - mode high-@xmath2 magneto - acoustic waves ( analog of @xmath3 modes ) to slow - mode waves in solar active regions is relatively well studied both from analytical theories and numerical simulations ( e.g. , @xcite ) , see @xcite for a review . in a two - dimensional situation ,
the transformation from fast to slow magnetoacoustic modes is demonstrated to be particularly strong for a narrow range of the magnetic field inclinations around 2030 degrees to the vertical .
however , no generalized picture exists so far for conversion from magneto - acoustic to alfvn waves in a three - dimensional situation .
studies of this conversion were initiated by cally & goossens @xcite , who found that the conversion is most efficient for preferred magnetic field inclinations between 30 and 40 degrees , and azimuth angles between 60 and 80 degrees , and that alfvnic fluxes transmitted to the upper atmosphere can exceed acoustic fluxes in some cases .
newington & cally @xcite studied the conversion properties of low - frequency gravity waves , showing that large magnetic field inclinations can help transmitting an important amount of the alfvnic energy flux to the upper atmosphere .
time - height variations of the three projected velocity components corresponding to @xmath4 ( alfven wave , left ) , @xmath5 ( fast wave , middle ) and @xmath6 ( slow wave , right ) for @xmath7 mhz in a simulation with @xmath8 inclined by @xmath9 and @xmath10 .
the solid line marks the position @xmath11 , and the dashed line marks the cut - off layer @xmath12 .
the colour scaling is the same in all panels .
the amplitudes are scaled with @xmath13 ( first two panels ) @xmath14 ( last panel).,width=566 ] motivated by these recent studies , here we attack the problem by means of 2.5d numerical simulations .
the purpose of our study is to calculate the efficiency of the conversion from fast - mode high-@xmath2 magneto - acoustic waves to alfvn and slow waves in the upper atmosphere for various frequencies and wavenumbers as a function of the field orientation .
we limit our study to a plane parallel atmosphere permeated by a constant inclined magnetic field , to perform a meaningful comparison with the work of cally & goossens @xcite
. numerical simulation will allow generalization to more realistic models in our future work .
we numerically solve the non - linear equations of ideal mhd assuming all vectors in three spatial directions and all derivatives in two directions ( i.e. 2.5d approximation , see @xcite ) , though perturbations are kept small to approximate the linear regime .
an acoustic wave of a given frequency and wave number is generated at @xmath15 mm below the solar surface in a standard model atmosphere permeated by a uniform inclined magnetic field .
the top boundary of the simulation box is 1 mm above the surface , and 0.8 mm above the layer where the acoustic speed , @xmath16 , and the alfvn speed , @xmath17 , are equal .
we consider frequencies @xmath18 and 5 mhz and wave numbers @xmath19 mm@xmath20 and @xmath21 .
the simulation grid covers field inclinations @xmath0 from 0@xmath22 to 80@xmath22 and field azimuths @xmath1 from 0@xmath22 to 160@xmath22 .
the field strength is kept at @xmath23 g. to separate the alfvn mode from the fast and slow magneto - acoustic modes in the magnetically dominated atmosphere we use velocity projections onto three characteristic directions : @xmath24 ; \nonumber\\ { \hat\mathbf{e}}_{\rm perp } & = & [ - \cos\phi \sin^2\theta \sin\phi , \ , 1-\sin^2\theta \sin^2\phi , \ , - \cos\theta \sin\theta \sin\phi ] ; \\
\nonumber { \hat\mathbf{e}}_{\rm trans } & = & [ -\cos\theta , \ , 0 , \ , \cos\phi \sin\theta].\end{aligned}\ ] ] to measure the efficiency of conversion to alfvn waves near and above the @xmath11 equipartition layer , we calculate acoustic and magnetic energy fluxes , averaged over time : @xmath25 figure [ fig : modes ] shows an example of the projected velocities in our calculations as a function of space and time . in this representation
the larger inclination of the ridges mean lower propagation speeds and vice versa . note , that by projecting the velocities , we are able to separate the modes only in the magnetically dominated atmosphere , i.e. above the solid line in fig . [
fig : modes ] .
the figure shows how the incident fast mode wave propagates to the equipartition layer and then splits into several components .
the alfvn wave is produced by mode conversion above 0.2 mm ( left panel ) and propagates upwards with the ( rapid ) alfvn speed , confirmed by almost vertical inclination of the ridges .
conversely , the essentially magnetic fast - mode low-@xmath2 wave produced in the upper atmosphere ( middle panel ) is reflected , and its velocity variations in the upper layers vanish with height . the ( acoustic ) slow - mode low-@xmath2 wave escapes to the upper atmosphere tunnelling over the cut - off layer due to the field inclination of @xmath9 .
the amplitudes of the velocity variations of the alfvn wave are comparable to those of the slow wave .
left panel : log@xmath26 of the ratio @xmath27 to @xmath28 for projected velocities and magnetic field variations , averaged over all @xmath1 , as a function of @xmath0 .
black line : fast mode ( @xmath5 projection ) ; red line : alfvn mode ( @xmath29 ) ; blue line : slow mode ( @xmath6 ) .
right panel : phase shift between the projected variations of @xmath30 and @xmath27 , as a function of @xmath0 for selected @xmath1 .
red lines : alfvn mode ; black lines : fast mode.,title="fig : " ] left panel : log@xmath26 of the ratio @xmath27 to @xmath28 for projected velocities and magnetic field variations , averaged over all @xmath1 , as a function of @xmath0 .
black line : fast mode ( @xmath5 projection ) ; red line : alfvn mode ( @xmath29 ) ; blue line : slow mode ( @xmath6 ) .
right panel : phase shift between the projected variations of @xmath30 and @xmath27 , as a function of @xmath0 for selected @xmath1 .
red lines : alfvn mode ; black lines : fast mode.,title="fig : " ] to confirm the alfvn nature of the transformed waves , as revealed by the projection calculations , we checked the amplitude and phase relations for all three modes reaching the upper atmosphere . for the alfvn mode the magnetic field @xmath27 and velocity variations
@xmath30 should be in equipartition ( i.e. @xmath31 ) , and both magnitudes should oscillate in phase ( see priest @xcite ) .
figure [ fig : phases ] presents the calculations of the amplitude ratio @xmath32 and temporal phase shift between @xmath27 and @xmath30 , where both velocity and magnetic field variations are projected in the corresponding characteristic direction for each mode ( eq . [ eq : directions ] ) .
this calculation confirms that , indeed , for all magnetic field orientations @xmath0 and @xmath1 , the amplitude ratio for the alfvn mode ( @xmath29 projection ) is around one ( left panel ) .
this is clearly not the case for the slow and fast modes . for the fast mode ,
the amplitude ratio is two orders of magnitude larger , and for the slow mode , it is two orders of magnitude lower than one . for the alfvn mode the phase shifts group around zero for all @xmath1 , unlike the case of the fast mode ( right panel ) .
we did not calculate the phase shifts for the slow mode as the variations of the magnetic field are negligible .
thus , we conclude that the properties of the simulated alfvn mode separated by the projection correspond to those expected for a classical alfvn mode . examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq .
[ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 .
solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 .
, title="fig : " ] examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 .
solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 .
, title="fig : " ] examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 .
solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 .
, title="fig : " ] an example of the height variations of the acoustic and magnetic fluxes is given in figure [ fig : fluxes2 ] .
the total vertical flux ( dotted line ) is conserved in the simulations except for the limitations caused by the finite grid resolution not resolving slow small - wavelength waves in the deep layers ( see fig . [
fig : modes ] ) .
both acoustic and magnetic fluxes show strongest variations near the conversion layer and become constant above it between 0.5 and 1 mm height .
the fluxes reaching the upper atmosphere depend crucially on the orientation of the field . in this example
, the acoustic flux decreases with @xmath0 whilst the magnetic flux increases with @xmath0 and becomes larger than the acoustic fluxes for @xmath33 . as the fast wave
is already reflected in the upper atmosphere ( see fig . [
fig : modes ] ) , the magnetic flux at these heights is due to the propagating alfvn wave .
vertical fluxes measured at the top of the atmosphere at 1 mm for waves with @xmath7 mhz ( left panels ) and 3 mhz ( right panels ) .
upper panels give magnetic fluxes and lower panels give acoustic fluxes .
, title="fig:",width=226 ] vertical fluxes measured at the top of the atmosphere at 1 mm for waves with @xmath7 mhz ( left panels ) and 3 mhz ( right panels ) .
upper panels give magnetic fluxes and lower panels give acoustic fluxes .
, title="fig:",width=226 ] finally , figure [ fig : fluxes ] gives the time averages of the vertical magnetic and acoustic fluxes at the top of the atmosphere as a function of the field orientation .
as proven above , the magnetic flux at 1 mm corresponds to the alfvn mode . at @xmath7 mhz
, the maximum of the magnetic flux corresponds to @xmath34 and @xmath35 .
this maximum is shifted to larger inclinations @xmath36 for waves with @xmath18 mhz .
the presence of the sharp maximum of the alfvnic flux transmission agrees well with the conclusions made previously by cally & goossens @xcite , though the exact position of the maximum is shifted to somewhat larger inclinations .
the maximum of the transmitted acoustic flux corresponds to inclinations @xmath37 for @xmath7 mhz waves , and to @xmath38 for @xmath18 mhz waves , again , in agreement with previous calculations @xcite .
the absolute value of the fluxes is about 30 times lower for 3 mhz compared to 5 mhz . at some angles the afvn magnetic flux transmitted to the upper atmosphere is larger than the acoustic flux .
however , at angles corresponding to the maximum of the transmission , the alfvn flux is 2 - 3 times lower than the corresponding acoustic flux .
it is important to realize that quantitatively simulating mode transformation numerically is a challenge , as any numerical inaccuracies are amplified in such second - order quantities as wave energy fluxes .
the tests presented in this paper prove the robustness of our numerical procedure and offer an effective way to separate the alfvn from magneto - acoustic modes in numerical simulations .
this will allow us in future to study the coupling between magneto - acoustic and alfvn waves in more realistic situations resembling complex solar magnetic structures .
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we would like to thank f.s .
navarra for fruitiful conversations .
this work has been partly supported by fapesp and cnpq - brazil . for a review and references to original works ,
see e.g. , s. narison , _ qcd as a theory of hadrons , cambridge monogr . part .
* 17 * , 1 ( 2002 ) [ hep - h/0205006 ] ; _ qcd spectral sum rules , world sci .
notes phys . _
* 26 * , 1 ( 1989 ) ; acta phys . pol . *
b26 * , 687 ( 1995 ) ; riv .
* 10n2 * , 1 ( 1987 ) ; phys . rept . *
84 * , 263 ( 1982 ) .
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we use the qcd sum rules to evaluate the mass of a possible scalar mesonic state that couples to a molecular @xmath0 current .
we find a mass @xmath1 gev , which is in a excellent agreement with the recently observed @xmath2 charmonium state .
we consider the contributions of condensates up to dimension eight , we work at leading order in @xmath3 and we keep terms which are linear in the strange quark mass @xmath4 . we also consider a molecular @xmath5 current and we obtain @xmath6 , around 200 mev above the mass of the @xmath7 charmonium state .
we conclude that it is possible to describe the @xmath2 structure as a @xmath8 molecular state .
there is growing evidence that at least some of the new charmonium states recently discovery in the b - factories are non conventional @xmath9 states . some possible interpretations for these states
are mesonic molecules , tetraquarks , or / and hybrid mesons .
some of these new mesons have their masses very close to the meson - meson threshold like the @xmath10 @xcite and the @xmath11 @xcite .
therefore , a molecular interpretation for these states seems natural .
the most recent aquisiton for this list of peculiar states is the narrow structure observed by the cdf collaboration in the decay @xmath12 .
the mass and width of this structure is @xmath13 , @xmath14 @xcite .
since the @xmath2 decays into two @xmath15 vector mesons , it has positive @xmath16 and @xmath17 parities .
there are already some theoretical interpretations for this structure .
its interpretation as a conventional @xmath9 state is complicated because , as pointed out by the cdf collaboration @xcite , it lies well above the threshold for open charm decays and , therefore , a @xmath9 state with this mass would decay predominantly into an open charm pair with a large total width . in ref .
@xcite , the authors interpreted the @xmath2 as the molecular partner of the charmonium - like state @xmath7 , which was observed by belle and babar collaborations near the @xmath18 threshold @xcite .
they concluded that the @xmath2 is probably a @xmath19 molecular state with @xmath20 or @xmath21 . in ref .
@xcite they have interpreted the @xmath2 as an exotic hybrid charmonium with @xmath22 . in this work ,
we use the qcd sum rules ( qcdsr ) @xcite , to study the two - point function based on a @xmath8 current with @xmath20 , to see if the new observed resonance structure , @xmath2 , can be interpreted as such molecular state . in previous calculations ,
the hidden charm mesons @xmath23 and @xmath24 have been studied using the qcdsr approach as tetraquark or molecular states @xcite . in some cases a very good agreement with the experimental mass was obtained . the starting point for constructing a qcd sum rule to evaluate the mass of a hadronic state , @xmath25 ,
is the correlator function ( q)=id^4x e^iq.x0 |t[j_h(x)j_h^(0)]|0 , where the current @xmath26 creates the states with the quantum numbers of the hadron @xmath25 .
a possible current describing a @xmath27 molecular state with @xmath28 is j=(|s_a_c_a)(|c_b^s_b ) , [ field ] where @xmath29 and @xmath30 are color indices .
the qcd sum rule is obtained by evaluating the correlation function in eq .
( [ 2po ] ) in two ways : in the ope side , we calculate the correlation function at the quark level in terms of quark and gluon fields .
we work at leading order in @xmath3 in the operators , we consider the contributions from condensates up to dimension eight and we keep terms which are linear in the strange quark mass @xmath4 . in the phenomenological side , the correlation function is calculated by inserting intermediate states for the @xmath27 molecular scalar state . parametrizing the coupling of the scalar state , @xmath31 , to the current , @xmath32 , in eq .
( [ field ] ) in terms of the parameter @xmath33 : [ eq : decay ] 0 | j|h=. [ lam ] the phenomenological side of eq .
( [ 2po ] ) can be written as ^phen(q^2)=^2m_h^2-q^2+_0^ds ^cont(s)s - q^2 , where the second term in the rhs of eq.([phe ] ) denotes higher scalar resonance contributions .
it is important to notice that there is no one to one correspondence between the current and the state , since the current in eq .
( [ field ] ) can be rewritten in terms of sum a over tetraquark type currents , by the use of the fierz transformation .
however , the parameter @xmath33 , appearing in eq . ( [ lam ] ) , gives a measure of the strength of the coupling between the current and the state .
the correlation function in the ope side can be written as a dispersion relation : ^ope(q^2)=_4m_c^2^ds
^ope(s)s - q^2 , where @xmath34 is given by the imaginary part of the correlation function : @xmath35 $ ] . as usual in the qcd sum rules method
, it is assumed that the continuum contribution to the spectral density , @xmath36 in eq .
( [ phe ] ) , vanishes bellow a certain continuum threshold @xmath37 . above this threshold , it is given by the result obtained with the ope .
therefore , one uses the ansatz @xcite ^cont(s)=^ope(s)(s - s_0 ) , to improve the matching between the two sides of the sum rule , we perfom a borel transform . after transferring the continuum contribution to the ope side , the sum rules for the scalar meson , considered as a scalar @xmath38 molecule , up to dimension - eight condensates , using factorization hypothesis , can be written as : ^2e^-^2/m^2=_4m_c^2^s_0ds e^-s / m^2 ^ope(s ) , [ sr1 ] where ^ope(s)=^pert(s)+^(s ) + ^g^2(s)+^mix(s)+^^2(s)+^mix(s ) , with [ eq : pert ] & & ^pert(s)=32 ^ 9 ^6_^ d^3 _ ^1-d^2(1 - - ) ^3(-4m_cm_s ) , + & & ^(s)=32 ^ 5 ^ 4_^ d\{m_s(m_c^2-(1-)s)^21 - - m_c_^1-d .
+ & & . }
, + & & ^g^2(s)=m_c^22 ^ 8 ^ 6_^ d^3_^1-d(1 - - ) , + & & ^mix(s)=-m_0 ^ 22 ^ 6 ^ 4\ { 3m_c_^d [ m_c^2-(1-)s ] -m_s(8m_c^2-s ) } , + & & ^^2(s)=m_c^28 ^ 2\ { ( 2m_c - m_s)-m_sm_c^2_0 ^ 1d ( s - m_c^2(1- ) ) } , [ dim6 ] where the integration limits are given by @xmath39 , @xmath40 , @xmath41 , and we have used @xmath42 .
we have neglected the contribution of the dimension - six condensate @xmath43 , since it is assumed to be suppressed by the loop factor @xmath44 .
we also include a part of the dimension-8 condensate contributions , related with the mixed condensate - quark condensate contribution : ^mix(s)&=&-m_cm_0 ^ 2 ^ 216 ^ 2_0 ^ 1 d ( s - m_c^2(1- ) ) .
[ dim8 ] it is important to point out that a complete evaluation of the dimension-8 condensate , and higher dimension condensates contributions , require more involved analysis @xcite , which is beyond the scope of this calculation . to extract the mass @xmath45 we take the derivative of eq .
( [ sr ] ) with respect to @xmath46 , and divide the result by eq .
( [ sr ] ) . for a consistent comparison with the results obtained for the other molecular states using the qcdsr approach , we have considered here the same values used for the quark masses and condensates as in refs .
@xcite : @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 with @xmath52 , @xmath53 .
the borel window is determined by analysing the ope convergence and the pole contribution . to determine the minimum value of the borel mass we impose that the contribution of the dimension-8 condensate should be smaller than 20% of the total contribution . in fig .
[ figconv ] we show the contribution of all the terms in the ope side of the sum rule . from this figure
we see that for @xmath54 gev@xmath55 the contribution of the dimension-8 condensate is less than 20% of the total contribution .
therefore , we fix the lower value of @xmath56 in the sum rule window as @xmath57 gev@xmath55 . the maximum value of the borel mass is determined by imposing that the pole contribution must be bigger than the continuum contribution . in table
i we show the values of @xmath58 . in our numerical analysis , we will consider the range of @xmath56 values from 2.3 @xmath59 until the one allowed by the pole dominance criterion given in table i. + [ cols="^,^",options="header " , ] taking into account the incertainties given above we finally arrive at = ( 4.140.09 ) , [ ymass ] in an excellent agreement with the mass of the narrow structure @xmath2 observed by cdf .
one can also deduce , from eq .
( [ sr1 ] ) , the parameter @xmath33 defined in eq .
( [ lam ] ) .
we get : = ( 4.220.83 ) 10 ^ -2 ^5 , [ la1 ] from the above study it is very easy to get results for the @xmath60 molecular state with @xmath20 . for this
we only have to take @xmath61 and @xmath62 in eqs .
( [ dim6 ] ) , ( [ dim8 ] ) .
this study was already done in ref .
@xcite considering @xmath63 .
although in the case of the @xmath64 scalar molecule we get a worse borel convergence than for the @xmath38 scalar molecule , as can be seen by fig .
[ opedd ] , there is still a good ope convergence for @xmath65 . if we allow also for the @xmath60 molecule values of the continuum threshold in the range @xmath66 we get @xmath67 .
therefore , from a qcd sum rule study , the difference between the masses of the states that couple with scalar @xmath8 and @xmath60 currents , is consistent with zero . the mass obtained with the @xmath60 scalar current is about 100 mev above the @xmath68 threshold .
this could be an indication that there is a repulsive interaction between the two @xmath69 mesons .
strong interactions effects might lead to repulsive interactions that could result in a virtual state above the threshold .
therefore , this structure may or may not indicate a resonance .
however , considering the errors , it is not compatible with the observed @xmath70 charmonium - like state . in fig .
[ dif ] we show the relative ratio @xmath71 as a function of the borel mass for @xmath72 .
from this figure we can see that the ratio is very stable as a function of @xmath56 and the difference between the masses is smaller than 0.5% .
although the ratio is shown for @xmath72 , the result is indiscernible from the one shown in fig .
[ dif ] for other values of the continuum threshold in the range @xmath73 .
this result for the mass difference is completely unexpected since , in general , each strange quark adds approximately 100 mev to the mass of the particle .
therefore , one would naively expect that the mass of the @xmath38 state should be around 200 mev heavier than the mass of the @xmath64 state .
this was , for instance , the result obtained in ref .
@xcite for the vector molecular states @xmath74 and @xmath75 , where the masses obtained were : @xmath76 and @xmath77 . for the value of the parameter @xmath33
we get : _ d^*d^ * = ( 4.200.96)10 ^ -2 ^5 . [ la2 ] therefore , comparing the results in eqs .
( [ la1 ] ) and ( [ la2 ] ) we conclude that the currents couple with similar strength to the corresponding states , and that both , @xmath8 and @xmath60 scalar molecular states have masses compatible with the recently observed @xmath2 narrow structure .
however , since the @xmath2 was observed in the decay @xmath78 , the @xmath8 assignment is more compatible with its quark content . in conclusion , we have presented a qcdsr analysis of the two - point function for possible @xmath8 and @xmath60 molecular states with @xmath20 .
our findings indicate that the @xmath2 narrow structure observed by the cdf collaboration in the decay @xmath12 can be very well described by using a scalar @xmath8 current . although the authors of ref .
@xcite interpreted the @xmath2 as a @xmath27 molecular scalar state and the @xmath7 as a @xmath5 molecular scalar state , we have obtained similar masses for the states that couple with the scalars @xmath27 and @xmath5 currents .
therefore , from a qcd sum rule point of view , the charmonium - like state @xmath7 , observed by belle and babar collaborations , has a mass around 200 mev smaller than the state that couples with a @xmath5 scalar current and , therefore , can not be well described by such a current .
while this work has been finalized , a similar calculation was presented in ref .
@xcite .
however , the author of ref .
@xcite arrived to a different conclusion .
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we thank g. kohring , d. stauffer and c. tsallis for interesting discussions .
this work was performed within the sfb 341 kln
aachen jlich supported by the dfg .
[ fig2 ] time - dependence of the activity for @xmath42 and various values of @xmath13 .
the system size is @xmath72 . concerning the statistical error we observe that all runs using different random numbers yield curves that are indistinguishable on this scale .
[ fig3 ] determination of @xmath43 : the system size is @xmath73 , @xmath42 and @xmath68 ( lower curve ) , @xmath57 ( middle curve ) and @xmath74 ( upper curve ) .
the straight line in the middle is the function @xmath75 .
we conclude that @xmath76 .
[ fig5 ] time - dependence of the distance for @xmath77 and @xmath78 ( lower curve ) , @xmath69 ( upper curve ) .
the system size is @xmath79 .
the curve for @xmath69 shows that @xmath80 is well inside the chaotic region , whereas @xmath81 lies within the frozen phase .
this fact stronly supports reentrance .
the emerging phase diagram in the vicinity of the tricritical point is depticted in the insert the two black dots represent the parameter values for the two curves shown .
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we present numerical and analytical results for a special kind of one - dimensional probabilistic cellular automaton , the so called domany - kinzel automaton .
it is shown that the phase boundary separating the active and the recently found chaotic phase exhibits reentrant behavior .
furthermore exact results for the @xmath0=0-line are discussed .
pacs numbers : 87.10.+e , 02.50.+s , 89.80.+h cellular automata have been an intensive research field in recent years @xcite due to their computational simplicity and the wide range of applications in various areas . even in one dimension
a particular probabilistic variant ( domany - kinzel automaton ) of the originally deterministic cellular automata shows a rich phase diagram including directed percolation and other critical phenomena @xcite . only recently
a new phase in this model has been explored numerically exhibiting chaotic behavior @xcite .
this region of the diagram , up to a deterministic corner - point , is not accessible to exact treatments up to now .
nevertheless sophisticated approximation - methods , which systematically go beyond mean - field theory , have been applied successfully @xcite . in the so called tree - aproximation @xcite one
finds reentrant behavior in two directions , which is not fully understood yet .
this phenomenon has never been observed in numerical simulations up to now @xcite .
therefore one might ask , whether this reentrant behavior is a real feature of the model or just an artifact of the tree - approximation .
this issue is the main topic of the present paper , where we try to clearify this point with an alternative approximation method ( the cluster - approximation ) as well as with large scale monte - carlo simulations ( up to @xmath1 sites ) . to state the final results already at this place : the cluster - approximation again yields reentrant behavior in two directions and the simulations show clear evidence for reentrance near the tricritical point .
the model we consider is defined as follows : the domany - kinzel pca consists of a one - dimensional chain of @xmath2 binary variables , @xmath3 , @xmath4 taking on the values @xmath5 ( empty , occupied ) .
all sites are updated simultaneously ( i.e.parallel ) at discrete time steps and the state of each site at time @xmath6 depends only upon the state of the two nearest neighbors at time @xmath7 according to the following rule : @xmath8 \right\ } \end{aligned}\ ] ] where @xmath9 is the ( time - independent ) conditional probability that site @xmath10 takes on the value @xmath4 given that its neighbors have the values @xmath11 and @xmath12 at the previous time step .
@xmath13 ( @xmath0 ) is the probability that site @xmath10 is occupied if exactly one ( both ) of its neighbors is ( are ) occupied .
if neither neighbor is occupied , the site @xmath10 will also become empty , therefore the state with all sites empty is the absorbing state of the pca . the @xmath14-phase diagram , as it is known up to now , consists of three different phases .
most of it ( small enough @xmath13 ) is dominated by the _ frozen _
phase , where all initial conditions eventually lead into the absorbing state .
with other words , the activity @xmath15 tends to zero for @xmath16 within the frozen phase . for large enough @xmath13
one enters the @xmath17 phase , where , starting from a random initial condition , the system ends up in a state with a finite density of active sites . within this active phase
one can distinguish between a chaotic and a non - chaotic part .
this difference can be seen by starting with two slighly different ( random ) initial conditions @xmath18 and
@xmath19 subjected to the same external noise ( local updating rules ) . calculating the normalized distance @xmath20 of these two systems @xmath21 during the update of the replicated systems according to the rule displayed in equation ( 2 ) of reference @xcite one
observes a sharp transition from the chaotic phase , characterized by @xmath22 , to the active phase with @xmath23 ( in the following we call the active / non - chaotic phase simply the active phase ) .
the underlying picture is that in the latter case the system is characterized by only one attractor , which nevertheless depends strongly on the external noise . with other words , in this phase the noise ( and not the initial condition ) dominates the dynamics completely .
this is not true for the chaotic phase , where the system memorizes the initial state even after infinite time .
first we present analytical results obtained by the application of the so - called cluster - approximation already known in different contexts @xcite as probability path method @xcite or local structure theory @xcite . in this way we check earlier results @xcite derived with a different approximation scheme ( the tree - approximation ,
see @xcite ) .
the problem with the dynamical rules defined above is that one can not write down the probability distribution of the stationary state since no simple detailed balance condition can be derived .
therefore , in principle , it is necessary to solve the dynamics completely in order to obtain the equilibrium properties .
this is not possible in general .
one way out of this dilemma is to take into account systematically all possible correlations between @xmath24 neighboring sites ( @xmath24-cluster approximation ) and to treat interactions over longer distances by conditional probabilities .
more formally , given the probability @xmath25 for the configuration @xmath26 in an @xmath24-cluster - approximation the probabilitiy for configuration @xmath27 with @xmath28 is approximated to be : @xmath29 here @xmath30 denotes the conditional probability to find site @xmath31 in state @xmath32 given that the @xmath33 sites to the left are in the state @xmath34 .
a factorisation of this kind can describe the stationary state exactly only if the interactions extend over not more than @xmath24 sites .
a natural choice for the conditional probaility @xmath35 is @xmath36 with @xmath37 simple examples of one - dimensional systems which can be described exactly by a finite value of @xmath24 are the @xmath38-spin - ising - model where one needs @xmath39 for the exact equilibrium distribution ( @xmath40 being the standard one - dimensional ising model with next - neighbour interactions only ) @xcite .
another example is the the parallel asymmetric exclusion process where again @xmath41 leads to the exact result for the stationary state @xcite the phase diagram resulting from a calculation based on the cluster approximation with @xmath41 is shown in figure 1 .
since during one update step according to the rules equation 1 the even ( odd ) sites only depend on the odd ( even ) sites at the timestep before we performed two timesteps at once to deal with sites of only one fixed parity .
one firstly observes that @xmath41 is still far from the exact solution for the stationary state .
unfortunately higher approximations are very hard to obtain due to the exponentially growing number of equations to be analysed simultaneously ( especially for the distance @xmath20 with two replicated systems ) .
furthermore even for @xmath41 the resulting equations can not be solved analytically with final closed expressions but have to be iterated until one finds a fixed point of the system of equations . in order to obtain a better localisation of the phase boundaries we applied the same method described below to analyse the numerical data from the monte - carlo - simulations . as can be seen from the figure we find reentrant behaviour both in @xmath13- and @xmath0-direction comparable to the result from the tree approximation @xcite .
it seems that the tricritical point has moved upwards , but a detailed analysis of the results suggests that it remains on the @xmath42-line . for the frozen / active - phase boundary one can go to larger clusters with higher values of @xmath24 . in tabular 1 the critical values @xmath43 ( @xmath42 ) for of @xmath44
are given : @xmath45 a simple least square fit leads to a limiting value for @xmath43 of about @xmath46 which is significantly larger than the known values from the simulations @xcite . in order to test the predictions of both approximation schemes mentioned above we performed large - scale monte - carlo simulations of the domany - kinzel cellular automaton with probabilities @xmath13 and @xmath0 in the vicinity of the two end - points of the phase boundary of the chaotic phase ( i.e. : @xmath47 and @xmath48 ) , where reentrance could occur according to the above calculations .
the system - sizes were up to @xmath49 sites with periodic boundary conditions , and the number of iterations @xmath50 were maximally @xmath51 . in this way
one avoids self - correlations ( finite size effects ) , since after @xmath7 updates those sites separated by a distance smaller than @xmath7 are correlated .
therefore @xmath50 has to be smaller than @xmath2 . by choosing @xmath2 much larger than @xmath50 one improves the statistics significantly ( for obvious reasons , since one can devide the system into many statistically independent subsystems ) .
therefore no finite - size effects are present in our data ( which was checked by comparing results for different system sizes ) and we need not to perform a ( non - trivial ) extrapolation the infinite system @xmath52 .
furthermore the probability that the system gets trapped by the absorbing state ( @xmath53 ) after time @xmath7 increases with decreasing system size .
this renders the simultaneous limit @xmath52 and @xmath54 to a delicate point , which we also avoid by our approach .
looking at the data obtained from the simulations it turned out to be rather unreliable to try to discriminate between two phases by looking at the long - time limit of the order parameter ( activity @xmath55 or distance @xmath20 ) . apart from the two phase - boundaries we expect exponential decay of @xmath55 and @xmath20 to their asymptotic values .
exactly on the phase - boundary we expect the spectrum of relaxation times to extend to infinity and thus the decay to become algebraic .
this behavior is illustrated in figure 2 : the activity as a function of time is depicted in a log - log plot for increasing values of @xmath13 ( @xmath42 ) .
we see that below a certain value the curves are bended downwards , whereas above this value the curves are bended upwards reflecting exactly the behaviour explained above .
the curve just in the middle corresponding to @xmath56 is closest ( as defined quantitatively by a least square fit ) to a straight line . to determine @xmath43 more accurately we performed longer runs with larger system sizes and depict the result in figure 3 . the middle curve , corresponding to @xmath57 , is nicely approximated by an algebraic decay with an exponent @xmath58 .
this exponent agrees well with the universal order parameter exponent @xmath59 determined in reference @xcite . from figure 3
we determined @xmath43 to be @xmath60 .
this is the most accurate estimate of @xmath43 so far .
surprinsingly it is significantly larger than the value @xmath61 obtained with a different method but with system sizes of around @xmath62 @xcite .
it seems that in the latter reference the long transient times ( @xmath63 ) together with the small system sizes lower the critical value due to larger correlations in the system as known from similar systems @xcite . as we have mentioned above the finite size scaling analysis of small systems
is by no means straightforward and can not be done without further ad hoc assumptions , from which our method is free .
hence , from our point of view , the results we quote seem to be more reliable .
note that there is no overlap even of the error bars of the two critical values . in figure 4
we show the same scenario for @xmath64 . by the same arguments as above we now locate the critical value of @xmath13 ( i.e.the value at which the transition from vanishing to finite activity takes place ) to be @xmath65 , which is significantly lower than @xmath43 . for larger increasing values of @xmath0
the phase boundary between frozen and active phase bends down monotonically to smaller values of @xmath13 terminating at the point ( @xmath66 , @xmath67 ) which is exactly known since the whole @xmath67-line is exactly solvable . in figure 5
a comparison of the two curves for @xmath20 at @xmath68 , which is below @xmath43 , for @xmath42 and @xmath69 is shown .
note that ( 0.8090,0 ) lies within the frozen phase .
the upper curve bends upwards , which means that ( 0.8090,0.03 ) lies within the chaotic phase .
this is indicated by the schematic phase diagram depicted in the insert of figure 5 and which has been supported by simulations of various parameters @xmath14 in this region .
the two black dots represent the two curves shown and along the arrow connecting them one finds clear evidence for reentrant behavior .
the phase boundary of the chaotic phase therefore bends to the left up to values around @xmath69 and for larger values of @xmath0 it bends monotonically to the right until it terminates at the point @xmath48 .
one reason for the fact that this phenomenon was not seen in earlier simulations is that it is in fact a marginal effect observable only in high - precision simulation - data .
we also performed large scale simulations around the other endpoint of the chaotic / active phase boundary . here
it is quite evident that the reentrant behaviour parallel to the @xmath13-axis at @xmath48 is in fact an artifact of the approximation schemes and not existent in the actual system . concerning the question of the conjugate field for the order - parameter of the chaotic phase posed in reference @xcite one can make on the @xmath0=0-line exact statements .
since both the activity and the chaos order parameter obey exactly the same evolution equations @xcite it is easy to conclude that the conjugated fields also should be equivalent . for the activity one chooses independent random numbers at each site and each timestep ( on the @xmath0=0-line
this is just the role of @xmath13 ) .
accordingly one chooses for the chaotic order parameter independant random numbers at each site and each timestep ( rule @xmath70 in ref.@xcite ) .
the absorbing state now corresponds to identical variables states in the two replicas yielding the same update since the same noise has to be applied for equal configurations in the two systems .
this picture remains valid for @xmath71 although the evolution equations are no longer identical , but the absorbing state has the same properties .
note that on the line @xmath42 the critical exponents of the order - paramater are also the same .
if universality holds away from this line this statement should be true also for @xmath71 ( for @xmath67 it is known , that the critical exponents are different @xcite ) . in summary
we have shown in this letter that , in contradiction to previous findings , the chaotic phase in fact shows reentrant behaviour in the vicinity of the tricritical point as predicted by approximative analytical methods .
the effect was not seen before since it is relatively small and large scale simulations have to be made to detect it . on the other hand in the near of the @xmath13=1-line the predicted reentrant behaviour
is absent .
furthermore one can see that simulations of small systems with long transient times can lead to erroneous conclusions about the locations of the critical point as well as the shape of the phase boundary since it is difficult to estimate the error due to self correlations .
therefore the error - bars in reference @xcite seem to neglect these systematic errors and should be larger ( which could lead to an agreement with our results ) .
finally we have seen that the conjugate field to the chaotic order parameter can directly be identified from the equivalence to the activity order parameter on the @xmath0=0-line .
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we would like to thank bruce reed for introducing us to the classical version of the locker puzzle and richard cleve for pointing out the perfect quantum search of @xcite .
this work was partially supported by an an nserc discovery grant and an nserc postdoctoral fellowship .
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the _ locker puzzle _ is a game played by multiple players against a referee .
it has been previously shown that the best strategy that exists can not succeed with probability greater than , no matter how many players are involved .
our contribution is to show that quantum players can do much better they can succeed with . by making the rules of the game significantly stricter ,
we show a scenario where the quantum players still succeed perfectly , while the classical players win with vanishing probability .
other variants of the locker puzzle are considered , as well as a cheating referee .
* keywords : quantum complexity , grover search , locker puzzle * 10000 10000 grover s quantum algorithm @xcite provides a quadratic speedup over the best possible classical algorithm for the problem of unsorted searching in the query model . while grover s search method has been shown to be optimal @xcite , our results reveal that in the context of multi - player query games , applying grover s algorithm yields success probabilities that are much better than the success probabilities of classical optimal protocols .
specifically , we show that in the case of the _ locker puzzle _ , quantum players succeed with probability 1 while the known optimal classical success probability is bounded above by . in order to amplify this separation , we prove that a significantly stricter version of the locker puzzle has vanishing classical success probability , while still admitting a perfect quantum strategy .
we also consider the empty locker and the coloured slips versions of the locker puzzle , and the possibility of a cheating referee .
[ sec : locker puzzle ] the _ _ locker puzzle _ _ is a cooperative game between a team of @xmath0 players numbered @xmath1 and a referee . in the initial phase of the game , the referee chooses a random permutation @xmath2 of @xmath3 , and for each player @xmath4 she places number @xmath4 in locker @xmath5 . in the following phase , each player
is individually admitted into the locker room .
once in the room , each player is allowed to open @xmath6 lockers , one at a time , and look at their contents ( for simplicity , we ll take @xmath0 to be even ) .
after the player leaves the room , all lockers are closed .
the players are initially allowed to discuss strategy , but once the game starts , they are separated and can not communicate .
an individual player @xmath4 _ wins _ if he opens a locker containing number @xmath4 , while the team of @xmath0 players _ wins _ if all individual players win .
we would like to know what is the best strategy for the team of @xmath0 players .
a nave approach is for each player to independently choose @xmath6 lockers to open .
each players wins independently with probability @xmath7 , hence the team wins with probability @xmath8 .
surprisingly , it is known that the players can do much , much better !
we will review in section [ section : optimal - classical - locker ] an optimum strategy by which , for any @xmath0 , the players can win with probability at least @xmath9 .
the locker puzzle was originally considered by peter bro miltersen , and was first published in @xcite ; a journal version appears in @xcite .
sven skylum is credited for the pointer - following strategy that we will give in the next section .
a proof of optimality for this strategy is given by eugene curtin and max warshauer @xcite .
our presentation of the classical puzzle and its solution follows along the lines of their article .
many variations have been proposed @xcite .
we will consider the variations of _ empty lockers _ in section [ section : empty - lockers ] , _ coloured slips _ in section [ section : coloured - slips ] ( to be accurate , the locker and the coloured slips puzzles are variants of the empty locker puzzle ) , and a _ cheating referee _ in section [ sec : cheating ] .
[ section : optimal - classical - locker ] we saw that a nave solution allows the players to win with an exponentially small probability .
how can we devise a strategy that does better ? the reader avid to search for a solution on his or her own is encouraged to do so now .
the key is to find a solution where the individual success probabilities are not independent .
consider the following strategy : when first entering the locker room , player @xmath4 opens locker number @xmath4 .
a number is revealed ; this is used to indicate which locker to open next ( i.e. if number @xmath10 is revealed , the next locker opened is locker @xmath10 ) .
each player executes this pointer - following strategy until @xmath6 lockers are opened . to analyze the success probability , note that the team will win provided that the _ permutation _ that corresponds to the placement of numbers in lockers by the referee does not contain a cycle of length longer than @xmath6 .
the probability of such a long cycle occurring is : @xmath11 it can be shown that as @xmath12 , @xmath13 and that the sum increases with @xmath0 .
hence the probability that the team wins is decreasing to @xmath14 . using a reduction to another game
, this strategy can be shown to be optimal @xcite .
[ sec : quantum solution ] we now present our first contribution : a quantum solution to the locker puzzle , which performs better than the classical solution . as before the referee chooses a random permutation @xmath2 and she places numbers in the lockers according to this permutation . in the quantum solution , we allow the players to open locker doors in _ superposition _ , each player working with his own quantum register .
this is analogous to the quantum query model . for the quantum case
, we need to modify the goal of the game which , for player @xmath4 , becomes to _ correctly guess _ the locker containing number @xmath4 after @xmath6 queries , and _ not _ to open locker containing number @xmath4 , because this would be too easy to do in superposition ! we show that quantum players can always win at the locker game .
in fact , our results are stronger : we give a stricter version of the locker puzzle for which the optimal classical solution succeeds with vanishing probability , while a quantum strategy always succeeds ! [ imp ]
the main idea is to apply grover s quantum search algorithm to the locker puzzle . for player @xmath4
, we consider the action of opening a locker as a query to the oracle which when input locker number @xmath15 , @xmath16 , outputs the following : @xmath17 note that this oracle is weaker than the oracle in the original puzzle which would output @xmath18 .
we discuss this further in section [ subsection : optimality ] and in the conclusion .
grover s search algorithm @xcite was thoroughly analyzed in @xcite , where it was shown that in a black - box search scenario where it is known that a single solution exists , @xmath19 queries yield a failure probability no greater than @xmath20 , where @xmath0 is the number of elements in the search space ( here , @xmath0 is assumed to be large ) .
this was further improved in @xcite , where is was shown that the same amount of queries is sufficient to find a solution with _
certainty_.
applying this directly to the quantum players of the locker puzzle yields the following : 1 .
[ step : groverquery]each player performs @xmath19 queries ( this is less than the @xmath21 queries in the classical solution ) .
2 . each player wins independently with certainty , implying that the team wins with certainty .
[ sec : reducing - number - queries ] we ve seen that quantum players of the locker game can succeed with probability 1 .
our solution only requires @xmath19 oracle queries per player .
hence , we now consider the asymptotically stricter version of the locker puzzle , where players are allowed to open at most @xmath22 lockers .
the next theorem state that the success probability for classical players goes quickly to 0 .
[ thm : classic ] in the locker puzzle with @xmath22 queries , classical players win with probability at most @xmath23 .
let @xmath24 .
we upper bound the success probability of the first @xmath25 players , when each player is allowed to open @xmath26 lockers . since @xmath27 ,
this upper bounds the success probability of all @xmath0 players .
consider a new game where the first player opens exactly @xmath26 lockers and publicly reveals all of their contents . if the first player s number is not revealed the players lose and the game is over .
otherwise the @xmath26 revealed players have successfully located their lockers .
these @xmath26 lockers and players are now removed from the game .
the first player has success probability at most @xmath28 . in successive rounds ,
a player is chosen from amongst those not yet removed from the game .
he continues in the same way by choosing @xmath26 of the remaining lockers and revealing their contents .
if he finds his label , again @xmath26 lockers and players are removed from the game .
the game stops whenever a chosen player does not find his label .
otherwise it continues for @xmath26 rounds and terminates with a win for the players .
the success probability of the new game is at most @xmath29 the original game with no revealing of numbers can not do better . [
subsection : optimality ] in the quantum query model with oracle ( [ eq : oracle ] ) the total number of queries required to obtain a success probability of one for the players is in @xmath30 .
first consider a variation of the quantum game where the players act sequentially in the order @xmath31 and are allowed to announce their results to the other players .
the number of queries performed by player 1 must be in @xmath32 or he will not succeed with probability one .
this follows from the analysis of grover s algorithm , see @xcite . the only information given by the oracle @xmath33 is the location of the locker containing label @xmath34 .
suppose player 2 is allowed to receive this information and remove that locker from consideration .
the permutation @xmath2 induces a random permutation on the remaining @xmath35 lockers .
player 2 s success probability is then one only if his number of queries is in @xmath36 .
continuing , the @xmath4-th player must ask a number of queries in @xmath37 .
the total number of queries is therefore in @xmath30 . in the modified game we share all information available to all players that have not already played .
so this shows a lower bound of the same order for the original version of the quantum game where no information is shared .
let us now compare the strength of oracle ( [ eq : oracle ] ) with the stronger oracle where @xmath38 . in the classical setup
, the weaker oracle ( [ eq : oracle ] ) merely tells a given player whether or not his label is in a requested locker .
there are an even number @xmath0 of lockers and he can ask @xmath39 queries . again
we consider a sequential version of the game as described above , where each player reveals his results .
if he succeeds , he reveals the locker with his number and that locker is removed . for the other lockers he queried
, the only information he has is that they did not contain his label .
therefore after his locker is removed , the other players have no further information .
the success probability of this variation of the locker game is : @xmath40 where we have used stirling s formula twice .
this is exponentially small and provides an upper bound on the success probability of the classical locker game with the weak oracle ( [ eq : oracle ] ) . by comparison , as we saw in section [ section : optimal - classical - locker ] the players can win with constant probability using the stronger oracle .
an open question is whether the quantum algorithm can be improved by using this stronger oracle .
the original motivation for the locker puzzle came from the study of time - space tradeoffs for the substring search problem in the context of _ bit probe complexity _
@xcite . there , a version with both _
empty lockers _ and _ coloured slips _ was presented .
we now examine these two variations separately and consider the quantum case .
[ section : empty - lockers ] suppose there are a total of @xmath41 lockers .
the referee selects an unordered subset @xmath2 of @xmath42 with cardinality @xmath0 and she puts label @xmath4 into locker @xmath43 for @xmath44 .
the remaining @xmath45 lockers are empty .
assume @xmath46 is even , and we allow the players to open up to @xmath47 lockers .
an optimum winning strategy for this more general situation is unknown : the pointer algorithm fails if an empty locker is opened .
even for the case @xmath48 , where half of the lockers are empty , it is still unknown if there is a classical strategy with success probability bounded away from zero @xcite .
however , the quantum strategy given in section [ sec : quantum solution ] still succeeds with probability one with a number of queries in @xmath49 per player , for a total of @xmath50 queries .
it suffices to modify the oracle ( [ eq : oracle ] ) so that @xmath15 runs over the range @xmath51 , and query it @xmath52 times .
if it turns out that for these same parameters , the classical success probability vanishes , then the power of the quantum world would be once more confirmed , as in section [ sec : reducing - number - queries ] . and section [ subsection : optimality ] .
[ section : coloured - slips ] consider the empty lockers game with @xmath41 lockers , again with @xmath0 players and @xmath0 slips of paper , each labelled @xmath53 .
this time the referee colours each slip either red or blue as she chooses , and places them in a randomly selected subset of @xmath0 lockers . as before ,
each player @xmath4 may open @xmath47 lockers using any adaptive strategy , and based on this , must make a guess about the colour of the slip labelled @xmath4 .
the players win if every player correctly announces the colour of his slip . with @xmath54 ,
this can be solved with the pointer - following algorithm and the players have success probability about 0.31 . in the quantum setting
, the players can win with probability one at the colour guessing game also , by changing the oracle ( [ eq : oracle ] ) .
let @xmath55 be the colour of the slip for player @xmath4 .
define for @xmath51 and @xmath56 : @xmath57 now we use the protocol described in section [ imp ] with each player querying this new oracle @xmath52 times .
if for player @xmath4 @xmath58 , then there is exactly one @xmath15 for which @xmath59 and grover s algorithm returns @xmath60 with probability one .
otherwise , if @xmath61 then @xmath62 is identically zero and grover s algorithm may return any value @xmath15 .
the player now makes one further call to oracle ( [ eq : oracle1 ] ) with the returned value @xmath15 and guesses red if the oracle returns one and blue otherwise .
[ sec : cheating ] a cheating referee can obviously beat the players in the locker game .
she simply has to omit the label of one of the players .
this could be easily exposed by requiring that all the lockers be opened and checked at the end of the game .
a more subtle way of cheating is if the referee can somehow choose the permutation @xmath2 . in the original locker game ,
let @xmath63 , and let @xmath64 be a random unordered subset of @xmath65 players .
she may set @xmath66 , @xmath67 , and fill out the rest of @xmath2 at random from the remaining players .
it is easy to verify that , using the pointer algorithm , player @xmath68 opens @xmath6 lockers @xmath69 and does not find his label .
he has to guess and loses with probability about @xmath70 .
the same thing happens for each of the players @xmath71 .
( incidentally , the reason for not choosing @xmath72 is that the players not finding their label may guess the locker number they see in the last locker they open , winning the game with probability one ! ) .
using variants of this idea the referee may cheat successfully for some time before the players catch on .
if the players have access to shared randomness ( which is unknown to the referee ) , they can circumvent this problem by first applying their own permutation on the lockers before opening any of them .
interestingly , our quantum protocol is impervious to a referee who maliciously chooses the permutation , and does not require shared randomness .
[ sec : conclusion ] it was previously known that the locker puzzle has an intriguing classical optimal solution .
now we know that the locker puzzle and its variants also have interesting quantum solutions which perform significantly better than the classical ones . we have given a quantum solution in the black - box query complexity model that _ does not use the pointer - following technique that is crucial to the classical optimal solution_. it would be interesting to see if using the stronger classical oracle could lead to a quantum solution that works with a reasonable probability of success using @xmath73 total queries . with this stronger oracle ,
perhaps shared entanglement could help the players ?
it would also be interesting to see if , analogous to the classical case , our results have any consequences for time - space tradeoffs for data structures @xcite .
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we note that linear stability analysis can not be used to probe the stability of the synchronization , as we can not perform a valid taylor expansion when @xmath81 , where @xmath82 . for equal driving force profiles that we linearize , @xmath182 , @xmath183 ,
if we were to taylor expand , then the linearized expression for @xmath184 would be @xmath185 which has a singularity at @xmath81 .
the apparent singularity actually occurs at @xmath186 and at @xmath187 in the full expression , but the choice of constraining force ensures this zero in the denominator is canceled by the numerator . however , when we expand in @xmath37 and shift the singularity so that it occurs at @xmath81 , then the numerator is no longer zero at this point .
the reason we have this zero in the denominator is the following : the torque free condition ( [ eq : forcetorquefree ] ) is @xmath188 along with equations ( [ eq : rl]-[eq : rb],[eq : rld]-[eq : rbd ] ) , we use ( [ eq : torquefree ] ) to solve for the constraining forces @xmath189 , @xmath24 .
however , at @xmath190 , @xmath189 is multiplied by a term which vanishes , so the torque free condition can be satisfied without specifying @xmath189 .
we over - constrain the system when we divide by zero and specify @xmath189 at @xmath190 .
geometrically , @xmath191 corresponds to the phase where @xmath192 is parallel to @xmath193 .
our numerical analysis of the full expression avoids this singularity .
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the green alga _ chlamydomonas _ swims with synchronized beating of its two flagella , and is experimentally observed to exhibit run - and - tumble behaviour similar to bacteria .
recently we studied a simple hydrodynamic three - sphere model of _ chlamydomonas _ with a phase dependent driving force which can produce run - and - tumble behaviour when intrinsic noise is added , due to the non - linear mechanics of the system . here
, we consider the noiseless case and explore numerically the parameter space in the driving force profiles , which determine whether or not the synchronized state evolves from a given initial condition , as well as the stability of the synchronized state .
we find that phase dependent forcing , or a beat pattern , is necessary for stable synchronization in the geometry we work with .
introduction microorganisms swim in the low reynolds number regime where viscous forces dominate , inertia is negligible and the familiar propulsion methods of larger organisms become ineffective @xcite .
fluid flow is governed by the stokes equation , which is time reversible . a necessary condition on a periodic swimming stroke in order to achieve net propulsion is that it is non - time reversible @xcite .
inspired by sperm cells , which achieve propulsion by propagation of bending waves through their flagellum , taylor demonstrated that propulsion is possible in a viscous environment by studying the propagation of waves on an infinite sheet @xcite .
purcell showed that a swimmer needs at least two compact degrees of freedom to break the time reversal symmetry and achieve net propulsion @xcite .
many microorganisms swim using flagella @xcite ; there are two fundamentally different types of flagella : bacterial flagella and eukaryotic flagella ( or cilia ) .
eukaryotic flagella form bends when microtubules on one side of the flagella ` walk ' or ` slide ' along the microtubules on the other side @xcite .
the propagation of bends allows the flagella to form beat patterns that can break the time reversal symmetry .
for example , the first half of an individual cilium s beat cycle , called the _ power stroke _
, has the cilium sticking out and pushing the fluid , while the second half , called the _ recovery stroke _ , has the cilium bent as it returns to its original position @xcite .
our understanding of propulsion at low reynolds number has been developed by theoretical model microswimmers .
lighthill demonstrated a model that can achieve net propulsion by studying periodic shape deformations of a nearly spherical swimmer , showing that the swimming velocity is at most of the order of the square of the amplitude of the deformations @xcite .
purcell s three - link swimmer was studied by becker _
et al . _ , who determined the swimming direction and velocity for different angle amplitudes and relative link lengths @xcite .
a useful one - dimensional model is the linear three - sphere swimmer , where three beads are connected by two rods that change length with a non - reciprocal pattern @xcite .
dreyfus _
et al . _
studied a rotational analogue of the three - sphere swimmer @xcite .
avron _
et al . _
presented a more efficient swimmer consisting of a pair of bladders which exchange their volume and vary the distance between them @xcite . there have been several experimental realisations of artificial low reynolds number swimmers @xcite .
when two sperms swim close to each other , their tails beat in synchrony @xcite and taylor studied this using his waving sheet model with hydrodynamic interactions @xcite .
coordinated beating of flagella or cilia is important for a range of processes including motility , efficient pumping of fluid and symmetry breaking in developing embryos @xcite .
theoretical and experimental models have been studied to show that synchronization can occur through hydrodynamic interaction and that it is relevant to bacterial swimming and pumping by arrays @xcite .
flagellar synchronization is observed in _ chlamydomonas _
, a unicellular green alga that swims using two flagella that beat with a breaststroke pattern @xcite .
the cell has diameter @xmath0 and swims with velocity @xmath1 so the reynolds number is @xmath2 and inertia is negligible . during normal swimming ,
the flagella beat in synchrony .
these periods of synchrony are interrupted by periods of asynchronous beating and during these asynchronies , there is a large change in the cell s orientation @xcite .
this is analogous to run - and - tumble behaviour observed in bacteria .
simple models have helped us understand better the intricacies of low reynolds number swimming and hydrodynamic synchronization , and a recent development has been to combine these two effects in the context of a simple three - sphere model for the swimming of _ chlamydomonas _ @xcite .
this simple model captures some of the important features of _ chlamydomonas _ , namely , the ability to swim , the exact role of hydrodynamic interactions @xcite , the existence of stable synchronized states and an emergent run - and - tumble behaviour which is observed when we add white noise to the driving force @xcite . here , we consider the model without added noise and explore its phase diagram and full parameter space to see when the model evolves into the synchronized state .
we also investigate the stability of the synchronized state under various conditions , and the different types of behaviour that can be obtained form the model .
these studies have revealed a number of intriguing features .
the model we arrange three beads in the @xmath3 plane , each of radius @xmath4 , on a frictionless scaffold , as shown in figure [ fig : model ] . in a lab frame , but it does not interact with the fluid . the green underlay is a schematic of a _ chlamydomonas _ cell . ]
we refer to the left , right and back beads with the subscripts ` @xmath5 ' , ` @xmath6 ' and ` @xmath7 ' , respectively .
let @xmath8 be the origin of the cell reference frame with respect to a lab frame .
the cell axes @xmath9 make an angle @xmath10 with the lab axes @xmath11 .
the left and right beads model the flagella and move on circular trajectories in the cell frame of radius @xmath7 in opposite directions and with phases @xmath12 and @xmath13 ; the back bead models the cell body and is fixed with respect to the cell frame .
the positions and velocities of the beads are @xmath14 where the unit vectors @xmath15 and @xmath16 are in the normal and tangent directions of the circular trajectory of @xmath17 .
the left and right beads are driven by tangential forces @xmath18 and @xmath19 respectively .
normal forces @xmath20 and @xmath21 are exerted by the beads in order to be constrained to the circular trajectories .
the force on the back bead is such that the swimmer is force free and torque free : @xmath22 where @xmath23 for @xmath24 and @xmath25 for @xmath26 .
the forces and velocities are related through hydrodynamic interactions between the beads : @xmath27 where @xmath28 is the friction coefficient of each bead ( @xmath29 is viscosity of the ambient fluid ) . in the limit when @xmath4 is small
compared with all other length scales , the hydrodynamic interaction is described by the oseen tensor @xmath30 with @xmath31 @xcite .
the phase difference @xmath32 evolves according to @xmath33 , \label{eq : deltadot}\end{aligned}\ ] ] where @xmath34 and @xmath35 .
\label{eq : thetadot}\end{aligned}\ ] ] we solve equation [ eq : deltadot ] numerically for a choice of stroke pattern ( driving forces ) @xmath36 , @xmath24 .
the long term behaviour of @xmath37 depends on the stroke pattern and in many cases the initial condition .
we compute @xmath38 and @xmath39 to leading order in @xmath40 , since we do not require hydrodynamic interactions for synchronization @xcite , but we include the next order hydrodynamic term when computing the velocities , since we need this second order affect to achieve a net swimming velocity . swimming velocity in the synchronized state
first we consider the synchronized state where @xmath41 and @xmath42 , so that @xmath43 .
we do not worry about the stability of the synchronized state , which we consider in the next section , and assume that the swimmer stays in this state .
since the reynolds number is low , we need to ask ` does the model achieve net propulsion ? '
if hydrodynamic interactions are not included , then the cell just moves forwards and backwards and there is no net motion .
however , if we include hydrodynamic interactions , which vary in strength around the cycle , then the symmetry in the swimming stroke is broken and net propulsion is achieved .
the magnitude and direction of the net swimming velocity depends on the ratios @xmath44 and @xmath45 .
figure [ fig : hlvel ] shows the net swimming velocity in the @xmath46 direction for a range of @xmath47 and constant driving force @xmath48 . for @xmath49 , swimming is in the positive direction , otherwise the cell swims in the negative direction .
polotzek and friedrich in reference @xcite give the following explanation of why the cell may swim in either direction .
the instantaneous velocity @xmath50 is the ratio of the force @xmath51 , which has to be applied to the back bead to prevent it from moving , and the friction coefficient @xmath52 associated with towing the swimmer in the @xmath46 direction . the force @xmath53 oscillates during a stroke cycle and the hydrodynamic interactions , which reduce the magnitude of @xmath53 , are strongest when the beads are closest together . on the other hand ,
the friction coefficient is largest when the beads are furthest apart and smallest when the beads are close together .
the geometry of the swimmer determines which effect dominates and therefore whether the net swimming is in the positive or negative direction . .
the zero contour lies approximately on the line @xmath54 for sufficiently large @xmath44 . ]
( 10,10)(0,0 ) ( 327,13 ) ( 88,246 ) ( 142,180 ) ( 312,180 ) ( 218,174 ) ( 218,96 ) ( 117,115 ) ( 314,115 ) ( 117,85 ) ( 314,85 ) ( 117,62 ) ( 314,62 ) herein we fix the values @xmath55 and @xmath56 , which results in forwards swimming for @xmath57 . the the net velocity is also affected by the force profile and we consider driving forces @xmath58 such that @xmath59 , where @xmath60 is a fixed average force . the net velocity @xmath61 can be written as @xmath62 , where @xmath63 , @xmath64 is the period , and we can write @xmath65 . in the synchronized state the force dependence cancels in the ratio @xmath66 , so the force only enters the net velocity expression through the @xmath67 term . in order to maximise the net velocity , we must minimise the period @xmath68 , where we can write @xmath69 .
minimising @xmath70 with the constraint @xmath59 tells us that a constant force profile @xmath71 maximises the net velocity .
clearly , increasing @xmath60 increases the net velocity .
however as we shall see in the next section , the synchronized state is not stable when we choose a constant force profile .
friedrich _
et al .
_ showed that a constant driving force can give a stable synchronized state if we change the direction of rotation of the beads , so @xmath72 @xcite , which is equivalent to changing the sign of @xmath73 , but here we choose to work with @xmath57 and @xmath74 .
synchronization and stability we consider force profiles of the forms @xmath75 and @xmath76 , where @xmath77 and @xmath24 .
the definition of the synchronized state that we use here is zero ( or integer multiple of @xmath78 ) phase difference between the two flagella , i.e. when @xmath79 , @xmath80 .
initially we tried to analyst the the synchronization stability by linear stability analysis , but we are unable to do this because we can not perform a valid taylor expansion when @xmath81 , where @xmath82 .
we work numerically to avoid this taylor expansion ; for further details see the appendix .
we identify five main types of stability of the synchronized state by looking at the evolution of @xmath37 from different initial conditions @xmath83 for a number of @xmath84 equally spaced in the range @xmath85 : ( i ) all the initial conditions @xmath83 evolve into the synchronized state for all @xmath84 ( the synchronized state is stable ) .
( ii ) some choices of @xmath84 evolve into the synchronized state and others choices evolve into an oscillating state , but there is a larger number of @xmath84 that lead to synchronization than the number of @xmath84 that leads to oscillations .
( iii ) some choices of @xmath84 evolve into the synchronized state and others choices evolve into an oscillating state ; the numbers of @xmath84 that lead to each type of behaviour are similar .
( iv ) some choices of @xmath84 evolve into the synchronized state and others choices evolve into an oscillating state , but there is a larger number of @xmath84 that lead to oscillations than the number of @xmath84 that leads to synchronization .
( v ) all choices of @xmath84 evolve into the oscillating state , ( the synchronized state is unstable ) .
although the choice of initial condition @xmath86 is arbitrary , we want to know how likely it is that a small perturbation from the synchronized state will decay back to synchronization , or whether it is likely to evolve into an oscillating state .
this choice of @xmath87 is suitable for this purpose . for type ( v )
stability , if we start in the synchronized state , then it is likely that some numerical noise will kick @xmath37 into an oscillating state .
it is possible for a small amount of noise to kick the synchronized state into an oscillating state for types ( ii ) , ( iii ) , ( iv ) , with low probability for type ( ii ) , then increasing probability for type ( iii ) and then type ( iv ) .
first we consider the case where @xmath88 . for each choice of coefficient and initial condition
, @xmath37 evolves either to an integer multiple of @xmath78 and remains at this value ( synchronization ) ; or it reaches a state where it oscillates about @xmath89 sinusoidally ; or it reaches a periodic state near zero , but never reaching zero .
figure [ fig : devolutions1 ] shows examples of these three cases and the corresponding orientation @xmath90 . for driving force @xmath91 and initial condition @xmath92 with ( a ) @xmath93 , ( b ) @xmath94 , ( c ) @xmath95 .
the insets show a small part of the plot in more detail .
the bottom axis @xmath96 shows the number of cycles which increases monotonically with time .
bottom row : corresponding @xmath97.,title="fig : " ] for driving force @xmath91 and initial condition @xmath92 with ( a ) @xmath93 , ( b ) @xmath94 , ( c ) @xmath95 .
the insets show a small part of the plot in more detail .
the bottom axis @xmath96 shows the number of cycles which increases monotonically with time .
bottom row : corresponding @xmath97.,title="fig : " ] for driving force @xmath91 and initial condition @xmath92 with ( a ) @xmath93 , ( b ) @xmath94 , ( c ) @xmath95 .
the insets show a small part of the plot in more detail .
the bottom axis @xmath96 shows the number of cycles which increases monotonically with time .
bottom row : corresponding @xmath97.,title="fig : " ] + for driving force @xmath91 and initial condition @xmath92 with ( a ) @xmath93 , ( b ) @xmath94 , ( c ) @xmath95 .
the insets show a small part of the plot in more detail .
the bottom axis @xmath96 shows the number of cycles which increases monotonically with time .
bottom row : corresponding @xmath97.,title="fig : " ] for driving force @xmath91 and initial condition @xmath92 with ( a ) @xmath93 , ( b ) @xmath94 , ( c ) @xmath95 .
the insets show a small part of the plot in more detail .
the bottom axis @xmath96 shows the number of cycles which increases monotonically with time .
bottom row : corresponding @xmath97.,title="fig : " ] for driving force @xmath91 and initial condition @xmath92 with ( a ) @xmath93 , ( b ) @xmath94 , ( c ) @xmath95 .
the insets show a small part of the plot in more detail .
the bottom axis @xmath96 shows the number of cycles which increases monotonically with time .
bottom row : corresponding @xmath97.,title="fig : " ] ( 10,10)(0,0 ) ( 145,127 ) ( 290,126 ) ( 434,126 ) ( 8,192 ) ( 154,192 ) ( 300,192 ) ( 14,88 ) ( 147,93 ) ( 159,79 ) ( 289,82 ) ( 306,99 ) ( 435,59 ) ( 80,110 ) ( 80,10 ) ( 225,110 ) ( 225,10 ) ( 370,110 ) ( 370,10 ) when @xmath37 oscillates about @xmath89 , then the orientation oscillates about some fixed value .
the cycle averaged motion is in a straight line , but the cell jiggles from side to side as well as backwards and forwards as it moves along .
when @xmath94 and the oscillations are near zero , there is a net drift in the orientation so the net motion of the cell is along a curved trajectory . for many choices of @xmath58
, the initial condition determines whether @xmath37 evolves into the synchronized state or the oscillating state .
figure [ fig : icphasediagram ] shows the the dependence of @xmath37 evolution on initial condition for @xmath98 .
a 63 @xmath99 69 grid is shown where each square represents an initial condition @xmath100 .
a black square represents an initial condition for which @xmath101 after a sufficiently long time ; a white square represents an initial condition for which @xmath37 continues to oscillate periodically as @xmath102
. evolves for different initial conditions for @xmath103 with ( a ) @xmath104 , ( b ) @xmath105 , ( c ) @xmath106 .
black squares represent initial conditions which lead to the synchronized state and white squares represent initial conditions which lead to a periodic oscillating state .
the stability of the synchronized state is : ( a ) type ( v ) unstable ; ( b ) type ( i ) stable ; ( c ) type ( ii).,title="fig : " ] evolves for different initial conditions for @xmath103 with ( a ) @xmath104 , ( b ) @xmath105 , ( c ) @xmath106 .
black squares represent initial conditions which lead to the synchronized state and white squares represent initial conditions which lead to a periodic oscillating state .
the stability of the synchronized state is : ( a ) type ( v ) unstable ; ( b ) type ( i ) stable ; ( c ) type ( ii).,title="fig : " ] evolves for different initial conditions for @xmath103 with ( a ) @xmath104 , ( b ) @xmath105 , ( c ) @xmath106 .
black squares represent initial conditions which lead to the synchronized state and white squares represent initial conditions which lead to a periodic oscillating state .
the stability of the synchronized state is : ( a ) type ( v ) unstable ; ( b ) type ( i ) stable ; ( c ) type ( ii).,title="fig : " ] ( 10,10)(0,0 ) ( 120,10 ) ( 0,130 ) ( 268,10 ) ( 147,130 ) ( 418,10 ) ( 298,130 ) ( 70,4 ) ( 218,4 ) ( 369,4 ) for @xmath104 , all initial conditions lead to an oscillating state and the synchronized state is unstable ( type ( v ) ) .
the black squares in figure [ fig : icphasediagram ] are initially in the synchronized state .
many squares along the line @xmath42 are white because a small amount of numerical noise drives the system away from the synchronized state . for @xmath105 ,
initial conditions close to @xmath42 lead to the synchronized state , but initial conditions far from @xmath42 lead to an oscillating state .
the synchronized state is stable ( type ( i ) ) . for @xmath106 , most initial conditions close to @xmath42 lead to the synchronized state , but a few initial conditions close to @xmath42 lead to an oscillating state and we have type ( ii ) stability ;
if @xmath37 starts close to the synchronized state , it is likely to evolve into the synchronized state and it will stay in the synchronized state if there is no noise , but it is also possible for the cell to start close to the synchronized state and move away into an oscillating state . in an oscillating state
the cell can still swim , but there will be more side to side movement .
there appear to be a few white squares on the line @xmath107 , however this is because the grid does not lie exactly on the @xmath78 line , the grid contains points @xmath108 and @xmath109 and this small deviation from the synchronized state @xmath110 is enough for evolution into an oscillating state for a few choices of @xmath84 .
similar behaviour is observed for other harmonics , but with different ranges of coefficients giving the different stability types for the synchronized state .
for example , figure [ fig : harm4 ] shows the 4th harmonic for three different coefficients of cosine and initial condition @xmath111 .
figure [ fig : harm4](a ) shows that @xmath37 oscillate about @xmath89 for @xmath112 , oscillations are close to zero for @xmath113 ( it is interesting to note the 4 peaks in every cycle ) , and @xmath37 evolves into the synchronized state for @xmath114 . for the 4th harmonic with initial condition @xmath111 and coefficient ( a ) @xmath112 , ( b ) @xmath113 , ( c ) @xmath114 . (
a ) , ( b ) @xmath37 evolves into different oscillating states .
( c ) @xmath37 evolves into the synchronized state .
the insets show a small part of the plot in more detail.,title="fig : " ] for the 4th harmonic with initial condition @xmath111 and coefficient ( a ) @xmath112 , ( b ) @xmath113 , ( c ) @xmath114 .
( a ) , ( b ) @xmath37 evolves into different oscillating states .
( c ) @xmath37 evolves into the synchronized state .
the insets show a small part of the plot in more detail.,title="fig : " ] for the 4th harmonic with initial condition @xmath111 and coefficient ( a ) @xmath112 , ( b ) @xmath113 , ( c ) @xmath114 .
( a ) , ( b ) @xmath37 evolves into different oscillating states .
( c ) @xmath37 evolves into the synchronized state .
the insets show a small part of the plot in more detail.,title="fig : " ] ( 10,10)(0,0 ) ( 81,7 ) ( 222,7 ) ( 370,7 ) ( 10,88 ) ( 144,26 ) ( 151,88 ) ( 286,26 ) ( 291,86 ) ( 424,28 ) figure [ fig : harmstab ] shows the stability of the synchronized state for the first 10 harmonics for a discrete range of coefficients when we choose the cosine term , i.e. @xmath115 . .
orange blocks represent stability ( i ) ; dark blue blocks represent stability ( ii ) ; pale blue blocks represent stability ( iii ) ; pale green blocks represent stability ( iv ) ; and bright green blocks represent stability ( v ) . ]
( 10,10)(0,0 ) ( 96,147 ) ( 225,12 ) we see that there are more type ( i ) and ( ii ) force profiles for negative coefficients than for positive , showing that we are more likely to end up in the synchronized state when the coefficient is negative than when the coefficient is positive , for an arbitrary choice of harmonic .
we see that for @xmath116 there is no type ( i ) behaviour , so we can not guarantee that we will reach the synchronized states for these higher harmonics .
the unstable type ( v ) band around @xmath117 gets narrower as @xmath118 increases , so we only need weak phase dependence to reach a synchronized state for higher harmonics , but the region of initial conditions which does lead to the synchronized state is very small . even after reaching the synchronized state ,
it is likely that noise will drive @xmath37 away from the synchronized state .
usually the stability moves towards the lower types of stability ( more stable types ) as @xmath119 increases for each harmonic , although there are some exceptions . for example , when @xmath120 we can start in a type ( i ) region , then as @xmath121 increases we move into a type ( ii ) region .
we also see a type ( iii ) region surrounded by type ( iv ) regions on both sides for @xmath122 and @xmath123 . for each type of stability shown in figure
[ fig : harmstab ] , we select an arbitrary force profile and show the full initial condition phase diagram in figure [ fig : icselection ] . in the type ( i )
stable case , we see that oscillating states can still evolve ( see also figure [ fig : icphasediagram ] ) , but the initial conditions for an oscillating state are not close to the synchronized case . in a few cases when we replace the cosine with sine , all initial conditions lead to the synchronized state , for example , the force profile @xmath124 for @xmath125 , except for a very thin dotted white curve through the middle of the phase diagram indicating an unstable oscillating state .
however , if we look at this oscillating state for long enough , after some time the numerical noise causes it to move into the synchronized state . .
( a ) type ( i ) stable diagrams for @xmath126 and @xmath127 .
( b ) type ( ii ) stability for @xmath128 and @xmath129 .
( c ) type ( iii ) stability for @xmath128 and @xmath130 .
( d ) type ( iv ) stability for @xmath131 and @xmath132 .
( e ) type ( v ) stability for @xmath133 and @xmath134.,title="fig : " ] .
( a ) type ( i ) stable diagrams for @xmath126 and @xmath127 .
( b ) type ( ii ) stability for @xmath128 and @xmath129 .
( c ) type ( iii ) stability for @xmath128 and @xmath130 .
( d ) type ( iv ) stability for @xmath131 and @xmath132 .
( e ) type ( v ) stability for @xmath133 and @xmath134.,title="fig : " ] .
( a ) type ( i ) stable diagrams for @xmath126 and @xmath127 .
( b ) type ( ii ) stability for @xmath128 and @xmath129 .
( c ) type ( iii ) stability for @xmath128 and @xmath130 .
( d ) type ( iv ) stability for @xmath131 and @xmath132 .
( e ) type ( v ) stability for @xmath133 and @xmath134.,title="fig : " ] + .
( a ) type ( i ) stable diagrams for @xmath126 and @xmath127 .
( b ) type ( ii ) stability for @xmath128 and @xmath129 .
( c ) type ( iii ) stability for @xmath128 and @xmath130 .
( d ) type ( iv ) stability for @xmath131 and @xmath132 .
( e ) type ( v ) stability for @xmath133 and @xmath134.,title="fig : " ] .
( a ) type ( i ) stable diagrams for @xmath126 and @xmath127 .
( b ) type ( ii ) stability for @xmath128 and @xmath129 .
( c ) type ( iii ) stability for @xmath128 and @xmath130 .
( d ) type ( iv ) stability for @xmath131 and @xmath132 .
( e ) type ( v ) stability for @xmath133 and @xmath134.,title="fig : " ] ( 10,10)(0,0 ) ( 73,161 ) ( 219,161 ) ( 367,161 ) ( 143,2 ) ( 297,2 ) ( 0,288 ) ( 122,169 ) ( 148,288 ) ( 269,169 ) ( 295,288 ) ( 416,169 ) ( 71,134 ) ( 192,12 ) ( 225,134 ) ( 345,12 ) we see from the bright green band down the center of figure [ fig : harmstab ] that we need some form of phase dependence in order to achieve synchronization . if we choose a constant driving force then @xmath37 evolves into an oscillating state for all initial conditions .
the synchronized state is unstable , so even if we start in the synchronized state , a small amount of numerical noise can drive the system into the oscillating state .
friedrich _
et al .
_ showed that synchronization can occur with constant forcing when the direction of rotation of the beads is reversed ( equivalent to @xmath135 and reversing swimming direction ) but here we focus on the case @xmath136 and @xmath74 .
the inspiration for our run - and - tumble model in reference @xcite came from the initial condition phase diagrams . to see run - and - tumble we want to start in a stable synchronized state , then allow noise to move us temporarily into a white region of the phase diagram , before moving back into a black region . in reference @xcite , we allowed noise to vary the coefficients @xmath137 , so that a black square in the noiseless phase diagram can change to a white square when the instantaneous effect of the noise changes the value of @xmath137 , then changes back to a black square when the instantaneous effect of the noise is smaller .
this allows us to start in the synchronized state ( with fluctuations due to the noise ) , then move away from the synchronized state when the instantaneous noise is large and we are at a suitable point in the phase diagram , then move back into the fluctuating synchronized state when the instantaneous noise is small . in reference
@xcite , we chose to work with the first harmonic and use @xmath138 , where @xmath139 is the noise term .
we chose the value 0.7 because it is in the stability type ( i ) region , but lies close to the type ( ii ) region .
in the fluctuating synchronized state , when we move through the values of @xmath140 which are surrounded by white squares in the type ( ii ) phase diagram , there is the possibility to move into an oscillating state , but there are still plenty of black squares surrounding the line @xmath42 , so we can have long periods in the _ run _ phase .
the noise causes fluctuations of the position in the phase diagram , and this could also be a cause of run - and - tumble behaviour .
for example , consider the phase diagram in figure [ fig : icselection](a ) .
if we are in the synchronized state and noise is small enough such that fluctuations in @xmath37 are within the black region , then tumbles will not occur .
if the noise is larger , so @xmath37 fluctuations move into the white region , then the cell could begin to move into the oscillating state , and after a few oscillations noise could kick the oscillations into the black region and the cell would move back towards a synchronized state .
elsewhere , we will consider the effects of adding noise to @xmath39 and @xmath141 , without any noise in the coefficient , to see if we obtain run - and - tumble behaviour this way and compare the statistics to the run - and - tumble obtained when noise is added to the coefficient .
if we start in the synchronized state , then @xmath142 which is non - zero for @xmath143 , so the system does not stay in the synchronized state .
we focus on the first harmonic and consider the case @xmath144 and @xmath145 where @xmath146 ( and we have dropped the upper index on the coefficient ) .
when @xmath147 and @xmath148 , synchronization is frustrated and @xmath37 oscillates about @xmath149 , @xmath80 , shown in figure [ fig : mc](a ) for @xmath150 .
the orientation of the cell drifts , shown in figure [ fig : mc](b ) so the cell swims along a curved trajectory . for @xmath151 ,
the centre of @xmath37 oscillations drifts away from the synchronized state , but remains close to @xmath152 .
swapping the signs of the coefficients swaps the direction of the orientation drift .
when the coefficients have the same sign but different magnitudes , @xmath37 oscillates about @xmath153 and the orientation oscillates about some fixed value .
figure [ fig : mc](c ) shows the evolution of @xmath37 for @xmath154 .
this type of behaviour is also seen for some choices of equal coefficients , for example , in figure [ fig : devolutions1](e ) , ( f ) where @xmath155 .
choices of coefficients which give this type of behaviour can be used to model a mutant of _ chlamydomonas _ which swims with antiphase synchrony @xcite . when the model swims with antiphase beating , the beat frequency is higher than when it swims with in phase beating , which has been observed in real _ chlamydomonas _ cells @xcite . and ( b ) , ( d ) , ( f ) corresponding orientation @xmath90 for force profiles @xmath156 and initial conditions @xmath157 , @xmath158 and with ( a ) , ( b ) @xmath150 ; ( c ) , ( d ) @xmath154 ; ( e ) , ( f ) @xmath159 . each inset shows a small part of the main plot in more detail.,title="fig : " ] and ( b ) , ( d ) , ( f ) corresponding orientation @xmath90 for force profiles @xmath156 and initial conditions @xmath157 , @xmath158 and with ( a ) , ( b ) @xmath150 ; ( c ) , ( d ) @xmath154 ; ( e ) , ( f ) @xmath159 .
each inset shows a small part of the main plot in more detail.,title="fig : " ] and ( b ) , ( d ) , ( f ) corresponding orientation @xmath90 for force profiles @xmath156 and initial conditions @xmath157 , @xmath158 and with ( a ) , ( b ) @xmath150 ; ( c ) , ( d ) @xmath154 ; ( e ) , ( f ) @xmath159 .
each inset shows a small part of the main plot in more detail.,title="fig : " ] + and ( b ) , ( d ) , ( f ) corresponding orientation @xmath90 for force profiles @xmath156 and initial conditions @xmath157 , @xmath158 and with ( a ) , ( b ) @xmath150 ; ( c ) , ( d ) @xmath154 ; ( e ) , ( f ) @xmath159 .
each inset shows a small part of the main plot in more detail.,title="fig : " ] and ( b ) , ( d ) , ( f ) corresponding orientation @xmath90 for force profiles @xmath156 and initial conditions @xmath157 , @xmath158 and with ( a ) , ( b ) @xmath150 ; ( c ) , ( d ) @xmath154 ; ( e ) , ( f ) @xmath159 .
each inset shows a small part of the main plot in more detail.,title="fig : " ] and ( b ) , ( d ) , ( f ) corresponding orientation @xmath90 for force profiles @xmath156 and initial conditions @xmath157 , @xmath158 and with ( a ) , ( b ) @xmath150 ; ( c ) , ( d ) @xmath154 ; ( e ) , ( f ) @xmath159 .
each inset shows a small part of the main plot in more detail.,title="fig : " ] ( 10,10)(0,0 ) ( 83,107 ) ( 83,9 ) ( 226,107 ) ( 226,9 ) ( 368,107 ) ( 368,9 ) ( 12,188 ) ( 148,125 ) ( 16,84 ) ( 149,80 ) ( 155,188 ) ( 288,125 ) ( 165,94 ) ( 290,41 ) ( 295,182 ) ( 430,159 ) ( 304,85 ) ( 431,78 ) when the coefficients have opposite signs and different magnitudes , there are two main types of behaviour that occur . for @xmath160 and @xmath161 or @xmath162 and @xmath163 ,
then @xmath37 oscillates periodically and the corresponding orientation drifts in the negative direction in the former case and in the positive direction in the latter case .
this is shown in figure [ fig : mc](e ) , ( f ) for @xmath159 . for @xmath160 and @xmath164 or @xmath162 and @xmath165 ,
then oscillations in phase difference , @xmath37 , drift in the negative ( positive ) direction and the orientation drifts in the positive ( negative ) direction in the former ( latter ) case . figure [ fig : deldrift ] shows examples of this this case where one bead completes more cycles than the other bead .
there is a difference in the mean angular velocity of the two beads without choosing a different @xmath60 for each bead . and corresponding orientation for ( a ) , ( b ) @xmath166 and @xmath167 ; ( c ) , ( d ) @xmath168 and @xmath169 .
the insets show a small part of the plot in more detail.,title="fig : " ] and corresponding orientation for ( a ) , ( b ) @xmath166 and @xmath167 ; ( c ) , ( d ) @xmath168 and @xmath169 .
the insets show a small part of the plot in more detail.,title="fig : " ] + and corresponding orientation for ( a ) , ( b ) @xmath166 and @xmath167 ; ( c ) , ( d ) @xmath168 and @xmath169 .
the insets show a small part of the plot in more detail.,title="fig : " ] and corresponding orientation for ( a ) , ( b ) @xmath166 and @xmath167 ; ( c ) , ( d ) @xmath168 and @xmath169 .
the insets show a small part of the plot in more detail.,title="fig : " ] ( 10,10)(0,0 ) ( 128,136 ) ( 128,6 ) ( 325,136 ) ( 325,6 ) ( 40,234 ) ( 207,236 ) ( 32,108 ) ( 209,65 ) ( 239,239 ) ( 406,154 ) ( 228,109 ) ( 408,80 ) so far we have considered force profiles with only one harmonic term .
now we consider driving forces with contributions from two harmonics . for simplicity
we choose equal profiles for the left and right beads , @xmath170 , and of the form @xmath171 , @xmath172 and with the @xmath173 s chosen such that @xmath174 for all real @xmath140 .
figure [ fig : harm12 ] shows the stability of the synchronized state for @xmath175 and @xmath176 .
each grid square represents a choice of driving force with coefficients @xmath177 , @xmath178 and the colour represents the stability of the synchronized state for that particular driving force . ,
( b ) @xmath179 .
the colour of each square represents the following stability : orange represents type ( i ) ; dark blue represents type ( ii ) ; pale blue represents type ( iii ) ; pale green represents type ( iv ) ; bright green represents type ( v).,title="fig : " ] , ( b ) @xmath179 .
the colour of each square represents the following stability : orange represents type ( i ) ; dark blue represents type ( ii ) ; pale blue represents type ( iii ) ; pale green represents type ( iv ) ; bright green represents type ( v).,title="fig : " ] ( 10,10)(0,0 ) ( 125,4 ) ( 322,4 ) ( 193,16 ) ( 392,16 ) ( 34,170 ) ( 232,170 ) we see that there are large regions for which the synchronized state is type ( i ) stable .
conclusion this simple mechanical model is able to evolve into a stable synchronized state for certain choices of parameters in the driving force when the initial condition is within some region of the synchronized state .
we do not need hydrodynamic interactions to achieve stable synchronization ; we include hydrodynamic friction on each bead , with force free and torque free conditions and a phase dependent driving force , which can be constructed with a suitable combination of harmonic terms . for many choices of force profile ,
some initial conditions allow the model to evolve into the synchronized state , while other initial conditions that are very close to the synchronized state lead to an oscillating state .
there are some force profiles , including constant forcing , where there are no initial conditions that evolve into the synchronized state , and if the system starts in the synchronized state when the driving forces are equal , even a small amount of numerical noise can drive the system into an oscillating state .
there are different types of periodic behaviour for different choices of parameters in the driving force ; often the phase difference oscillates about @xmath89 , but sometimes the oscillations can occur close to zero with multiple peaks per cycle . when the parameter in the driving force is different for the left and right beads , we can get periodic oscillating states about a range of values , or we can get a drifting oscillating state , where one bead has a higher average angular velocity than the other . when the coefficients have equal magnitude and opposite sign , this can lead to oscillations about the synchronized state .
this frustrated synchronization is interesting when we add intrinsic noise to the driving force , because then the behaviour of @xmath37 is very similar for both opposite coefficients and equal coefficients , although the behaviour in the orientation is different in the two cases .
the nonlinear mechanics of the system make it difficult to study analytically and it is not easy to predict the parameter ranges which give stable synchronization .
our numerical results have highlighted some of the main types of behaviour of the model .
an important feature is that it is necessary to have some sort of phase dependent driving force in order to have a stable type ( i ) or type ( ii ) synchronized state .
when the phase dependence is only weak , then the synchronized state is unstable , which we see from figure [ fig : harmstab ] when the coefficient is small .
the value of the coefficient at which the phase dependence becomes strong enough to give synchronization depends on the harmonic , whether we choose a positive or negative coefficient , and whether we choose sine or cosine .
these latter choices are equivalent to adding a constant phase @xmath180 , @xmath89 or @xmath181 in the harmonic term .
this simple mechanical model shows a wide range of behaviour when we vary the parameter in the driving forces .
the variety of stabilities suggests possibilities for developing run - and - tumble models , where noise can be used to to jump between regions of a phase diagram that lead to synchronization or oscillations , or jump between phase diagrams as we did in reference @xcite .
we would like to thank nariya uchida for fruitful discussions and the epsrc for financial support .
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during the expansion of the outcoupled atoms , optical levitation is performed with a blue - detuned @xmath67 nm laser beam to compensate for gravity and a radio frequency dressing @xcite is used to keep the out - coupled fraction confined and clearly detectable after the @xmath9 ms expansion . in particular , the rf field is such to produce a mexican - hat potential which limits the radial expansion to about @xmath68 @xmath25 m , whereas the slower axial expansion is barely perturbed .
@xmath25 m .
the in - situ value can be obtained considering a scale factor of @xmath69 , given by the ratio between the in - situ and expanded tf radius at @xmath46 ms ; this because the assumption of a constant @xmath70 during the expansion .
this gives a mean @xmath71 of @xmath72 with a standard deviation of @xmath73 .
there is no statistical difference between the single - vortex distribution and the double - vortex one . ]
a precise statistical analysis is not possible here because information on the phase shift can be extracted only in the data subset where the crossing point occurs at about half of the inspected time evolution ( @xmath74 of the cases ) .
clear phase shifts are present in about half of this subset .
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we study the real - time dynamics of vortices in a large elongated bose - einstein condensate ( bec ) of sodium atoms using a stroboscopic technique .
vortices are produced via the kibble - zurek mechanism in a quench across the bec transition and they slowly precess keeping their orientation perpendicular to the long axis of the trap as expected for solitonic vortices in a highly anisotropic condensate .
good agreement with theoretical predictions is found for the precession period as a function of the orbit amplitude and the number of condensed atoms . in configurations with two or more vortices ,
we see signatures of vortex - vortex interaction in the shape and visibility of the orbits .
in addition , when more than two vortices are present , their decay is faster than the thermal decay observed for one or two vortices .
the possible role of vortex reconnection processes is discussed .
vortex dynamics is an essential feature of quantum fluids @xcite and plays a key role in superfluid helium @xcite , superconductors @xcite , neutron stars @xcite and magnetohydrodynamics @xcite .
the interaction between vortices is crucial for understanding the formation of vortex lattices in rotating superfluids and is the basic mechanism leading to quantum turbulence _ via _ vortex reconnection @xcite .
vortices have been extensively investigated in atomic gases @xcite , where a variety of techniques permits the observation of single ones up to a few hundreds , interacting in a clean environment and on a spatial scale ranging from the healing length ( core size ) @xmath0 to a few tens of @xmath0 .
the fact that atoms are confined by external fields of tunable geometry makes them suitable to explore the physics of reconnection and dissipation in inhomogeneous systems and in the presence of boundaries .
seminal experiments were performed in rotating bose - einstein condensates ( becs ) , where the effect of rotation and long - range interaction favors vortex alignment and the formation of vortex lattices @xcite and hence crossing and reconnection mechanisms are inhibited .
interacting vortices have been observed in nonrotating oblate becs , where vortex lines are short and either parallel or antiparallel , thus behaving as pointlike particles dominated by their long - range interaction in a quasi-2d background @xcite .
in our experiment we use a cigar - shaped bec which is particularly suitable for studying the dynamics of vortex lines in 3d . because of the boundary conditions imposed by the tight radial confinement each vortex line lies in a plane perpendicular to the long axis @xmath1 of the trap , such to minimize its length and therefore its energy , as in the solitonic vortex configuration predicted in refs . @xcite and recently observed both in a bec @xcite and in a superfluid fermi gas @xcite .
the line is randomly oriented in the plane and away from it , at distances of the order of the system transverse size , the superfluid flow quickly vanishes and the long - range part of the vortex - vortex interaction is suppressed .
hence , vortices can move almost independently along elliptic orbits except when they approach each other and may collide with a random relative angle . at the scale of the healing length , where reconnection can take place , the system is still equivalent to a uniform superfluid , like liquid he , but with the advantage that vortex filaments collide at measurable relative velocities .
the experimental apparatus is described in ref .
@xcite .
we evaporate sodium atoms in a magnetic harmonic trap with frequencies @xmath2 hz .
vortices with random position and velocity spontaneously originate _ via _ the kibble - zurek mechanism @xcite from phase defects in the condensate when crossing the bec transition and their average number scales as a power law with the evaporation rate . at the end of the evaporation we have an almost pure prolate bec with about @xmath3 atoms at @xmath4 nk in the state @xmath5 . in refs .
@xcite we counted and characterized defects using destructive absorption imaging . here
we apply a stroboscopic technique , similar to that in refs .
@xcite , which allows us to observe the real - time dynamics .
starting from an initial number of atoms @xmath6 , we remove a small fraction @xmath7 by outcoupling them to the antitrapped state @xmath8 _ via _ a microwave pulse , short enough to provide a resonance condition throughout the whole sample .
outcoupled atoms are imaged along a radial direction after a @xmath9 ms expansion @xcite without affecting the trapped ones .
the extraction mechanism is repeated @xmath10 times with time steps @xmath11 , keeping @xmath12 fixed .
raw images are fitted to a thomas - fermi ( tf ) profile @xcite and the residuals are calculated .
because of the peculiar structure of the superfluid flow of solitonic vortices @xcite , after expansion the whole radial plane containing a vortex exhibits a density depletion and vortices are seen as dark stripes independently of their in - plane orientation . during the extraction sequence
the remaining condensate evolves in trap , only weakly affected by atom number change , provided @xmath13 is sufficiently small .
we can then identify the axial position of the vortex in each image of the outcoupled atoms and analyze its oscillation as a faithful representation of the in - trap dynamics .
typical examples are shown in figs .
[ figure1](a)-[figure1](i ) .
alternatively we image the full bec along the axial direction after a long expansion with a destructive technique as in @xcite and directly see the shape and orientation of the vortex lines as in figs .
[ figure1](j)-[figure1](m ) .
images of the density distribution of the atoms extracted from three becs ; frames are taken every @xmath14 ms , each after a @xmath9 ms expansion .
( a ) static vortex .
( b)-(c ) vortices precessing with different amplitudes .
each vortex is randomly oriented in the @xmath15 plane and , after expansion , it forms a planar density depletion @xcite which is visible as a stripe .
( d)-(i ) sequences with two and three vortices , with @xmath16 ms ; here frames are not to scale and vertically squeezed to enhance visibility .
( j)-(m ) destructive absorption images of the whole bec taken along the axial direction @xmath1 after @xmath17 ms of expansion , showing ( j ) a single vortex filament crossing the condensate from side to side and ( k)-(m ) two vortices with different relative orientation and shape .
all images show the residuals after subtracting the fitting tf profile . ]
we first choose an evaporation rate of @xmath18 khz / s , yielding one vortex in each bec on average .
from the sequence of radial images we extract the axial position of each vortex @xmath19 .
frames are recorded every @xmath14 ms . figures [ figure2](a ) and [ figure2](b ) show two examples corresponding to the raw images of figs .
[ figure1](b ) and [ figure1](c ) , respectively .
the observations are consistent with a vortex precession around the trap center , as the one observed in oblate becs @xcite . in a nonrotating elongated condensate , a straight vortex line , oriented in a radial plane ,
is expected to follow an elliptic orbit in a plane orthogonal to the vortex line , corresponding to a trajectory at constant density @xcite .
the observed motion of each dark stripe in figs .
[ figure1](a)-[figure1](c ) is the axial projection of such a precession .
given @xmath20 the in - trap amplitude of the orbit normalized to the tf radii @xmath21 and @xmath22 @xcite , the precession period is predicted to be @xmath23 where @xmath24 is the axial trapping period and @xmath0 is related to the chemical potential @xmath25 by @xmath26 .
this result , which is valid to logarithmic accuracy , has been derived for a disk - shaped nonaxisymmetric condensate in refs .
@xcite within the gross - pitaevskii theory at @xmath27 and in the tf approximation , corresponding to @xmath28 ( in our case , @xmath29 ranges from @xmath30 to @xmath10 ) .
it can also be obtained by means of the superfluid hydrodynamic approach introduced in ref .
@xcite to describe the motion of vortex rings in elongated condensates , appropriately generalized to the case of solitonic vortices as in ref .
@xcite .
the quantity @xmath31 is the local chemical potential along the vortex trajectory and we assume @xmath32 to be constant during expansion , as distances are expected to scale in the same way in the slow axial expansion . in comparing the observed period with eq .
( [ eqn : period ] ) we must consider that the number of atoms is decreasing from shot to shot . since extraction is spatially homogeneous , the gradients of the density , and hence the equipotential lines for the vortex precession and the orbit amplitude remain almost unchanged .
however , @xmath33 ( hence @xmath34 ) decreases in time and so does the vortex orbital period @xmath35 , as is clearly visible in figs .
[ figure2](a ) and [ figure2](b ) .
we define an instantaneous period at time @xmath36 as the period obtained from a sinusoidal fit to the measured position in a time interval centered at @xmath36 and containing about one oscillation .
such @xmath37 is plotted in fig .
[ figure2](c ) and [ figure2](d ) and compared to eq .
( [ eqn : period ] ) , where we include the effect of the observed @xmath36 dependence on @xmath38 , shown in fig .
[ figure2](e ) , both in @xmath25 and @xmath0 .
the agreement is good , the major limitation being the experimental uncertainty in @xmath38 .
we also show the period expected for the oscillation of a dark or grey soliton , which is @xmath39 independently of @xmath38 @xcite . in fig .
[ figure2](f ) we plot the period of vortices orbiting with different amplitude @xmath32 .
the agreement with theory is again good and can be further appreciated by considering the ratio between each value of @xmath35 measured at a given @xmath32 and the theoretical value in eq .
( [ eqn : period ] ) obtained for the same @xmath32 and @xmath38 .
figure [ figure2](g ) shows the histogram of all values obtained by extracting @xmath35 and @xmath32 from a fit to the first oscillation , using @xmath40 in eq .
( [ eqn : period ] ) .
the histogram gives @xmath41 .
this remarkable agreement with theory is nontrivial since eq .
( [ eqn : period ] ) assumes @xmath42 and a rigid straight vortex line , while off - centered vortices actually bend toward the curved bec surface . for rotating condensates the bending mechanism has been discussed in refs . @xcite and observed in ref .
@xcite .
examples of straight and bent vortices in our condensate are given in figs .
[ figure1](j)-[figure1](m ) . in our elongated bec , with strong radial inhomogeneity
, this bending mechanism is expected to be more effective than in oblate becs .
our observations seem to indicate that its effect on the period is small , possibly of the same order of the logarithmic corrections to eq .
( [ eqn : period ] ) predicted for a straight vortex in a 2d geometry @xcite . this may be due to the fact that the difference in length between a bent and a straight vortex , at a comparable @xmath32 , is relatively small and the overall structure of the vortical flow is also quite similar , so that the key quantities entering the hydrodynamic description ( i.e , the force acting on a unit of length of the vortex and the momentum of the vortex , in the language of ref .
@xcite ) are almost the same in the two cases . ,
remaining in a condensate at time @xmath36 starting from configurations with @xmath43 ( circles ) , @xmath44 ( triangles ) and @xmath45 ( diamonds ) at @xmath46 .
solid lines are exponential fits . ]
vortex lifetime in nonrotating becs is limited by scattering of thermal excitations , which causes the dissipation of the vortex energy into the thermal cloud . since a vortex behaves as a particle of negative mass , dissipation causes an antidamping of the orbital motion and vortices decay at the edge of the condensate @xcite .
we can measure the lifetime @xmath47 by counting the average number of vortices @xmath48 remaining in the condensate at time @xmath36 , starting with @xmath49 .
if @xmath50 we find a clear exponential decay with @xmath51 ms ( fig .
[ figure3 ] ) , close to that measured in refs . @xcite and of the same order of the one observed in a fermionic superfluid @xcite .
( d)-[figure1](g ) , respectively .
solid and empty symbols are used to distinguish high and low density contrast , respectively . ] using a faster evaporation ramp ( @xmath52 khz / s ) , we produce more vortices and search for signatures of mutual interaction .
examples are shown in fig .
[ figure1](d)-[figure1](i ) and typical trajectories are also reported in fig .
[ figure4 ] . in some cases
, vortices perform unperturbed oscillations [ fig .
[ figure4](a ) ] ; in others , we clearly see a shift in their trajectories at the crossing point [ fig .
[ figure4](b ) ] .
the average relative velocity at the crossing in the latter case is systematically smaller ( @xmath53 mm / s ) than in the former ( @xmath54 mm / s ) @xcite .
the shift has a consequence also in the determination of the orbital period as it causes a broadening of the probability distribution of the ratio @xmath55 which now gives @xmath56 , with a standard deviation three times larger than for the single vortex [ fig .
[ figure2](g ) ] . in addition
, crossings are frequently associated with a sudden change of visibility of one or both vortices [ figs .
[ figure1](e)-[figure1](h ) ) .
finally , by analyzing the lifetime of vortices for the initial condition @xmath57 and @xmath58 we observe a lifetime @xmath59 ms for the two - vortex configuration , consistent with the one - vortex configuration .
the situation instead changes in the three - vortex configuration , where a faster decay is observed , @xmath60 ms ( fig .
[ figure3 ] ) .
the frequent observation of unperturbed orbits for multiple vortices is intriguing .
two vortex lines moving back and forth in the condensate with random radial orientations should have large probability to cross each other at some point .
if crossings occur , reconnections are expected to take place @xcite with possible drastic ( and almost temperature independent @xcite ) effects on the vortical dynamics .
the actual dynamics can strongly depend on the relative angle @xmath61 between vortex lines as well as the relative velocity @xmath62 between the planes where they lie .
when @xmath61 is close to @xmath63 ( @xmath64 ) , the vortex lines tend to align ( antialign ) , thus reducing the chance of reconnection for vortices on different orbits .
but when vortices approach with @xmath65 reconnection can be hardly avoided .
the fact that we observe the same vortex lifetime for @xmath50 and @xmath44 implies that such reconnections are either suppressed or they induce a negligible dissipation .
a possible explanation is the occurrence of double reconnection processes @xcite .
vortex reconnection corresponds to the switching of a pair of locally coplanar vortex lines , accompanied by a change of topology . in our geometry
a finite @xmath62 implies that the newly formed filaments must stretch in the condensate while the two planes separate again after reconnection .
the consequent energy cost is instead avoided if vortices perform a consecutive second reconnection when they are still at close distance .
this would preserve the vortex number , consistent with our observation of an equal vortex lifetime for @xmath50 and @xmath44 .
it is worth mentioning that a similar scenario has also been recently suggested for the collision of cosmic strings @xcite .
the occurrence of a shift in the trajectories , that apparently depends on @xmath62 , could be associated with the role of the collision time : faster vortices have less time to interact and their trajectories are marginally affected , and this scenario may be applicable both to fly - by vortices and double reconnections . also kelvin modes can be excited in the collision @xcite but , if present , they seem not to affect the lifetime , while they are likely responsible for the change of visibility of the vortices , as they can produce out - of - plane distortions and hence a change of contrast in the density distribution .
finally , the observation of a shorter lifetime in configurations with @xmath66 can be understood by considering the role of a third vortex in the collision of two other vortices , whose tendency to rotate in the radial plane is frustrated by three - body interaction , thus enhancing the probability of collisions and reconnections .
a similar role of three - body interactions in the dynamics of vortices was recently investigated in the context of 2d classical turbulence @xcite .
our experimental results demand new theoretical models .
so far , numerical simulations of vortex reconnection are usually performed with vortex lines initially at rest , at small distance , which then evolve in time @xcite , while in our case the role of the relative velocity seems to be crucial .
shedding light on this , and generally on the dynamics of few vortices in such a relatively simple configuration , can help to understand the physics of vorticity in more complex settings , like those of refs .
@xcite , in the search of a satisfactory comprehension of quantum turbulence in superfluids with boundaries .
we thank l.p .
pitaevskii , n.p .
proukakis , i - kang liu , n.g .
parker and c.f .
barenghi for insightful discussions .
we acknowledge provincia autonoma di trento for funding .
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our task is to compute the maximal quantum violation @xmath80 of a two - party bell inequality defined by the vector of coefficients @xmath81 , given a fixed set of measurements operators @xmath82 for alice .
let s write @xmath83 for the assemblage created on alice s side by bob s measurements on the state @xmath0 . with this
, we have the conditional probabilities @xmath84 and we have to maximize @xmath85 for fixed @xmath86 and @xmath1 .
the following sdp program is a relaxation of the above problem : @xmath87 it is well known that one can always find a quantum state @xmath0 and quantum measurements @xmath88 for bob which attain the maximum @xmath89 .
hence , @xmath90 , and the above sdp provides the exact quantum bound of @xmath80 for a fixed set of alice s measurements @xmath1 on the bell inequality defined by coefficients @xmath81 .
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we investigate the relation between the incompatibility of quantum measurements and quantum nonlocality .
we show that a set of measurements is not jointly measurable ( _ i.e._incompatible ) if and only if it can be used for demonstrating einstein - podolsky - rosen steering , a form of quantum nonlocality .
moreover , we discuss the connection between bell nonlocality and joint measurability , and give evidence that both notions are inequivalent .
specifically , we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of bell inequalities .
this suggest the existence of incompatible quantum measurements which are bell local , similarly to certain entangled states which admit a local hidden variable model . the correlations resulting from local measurements on an entangled quantum state can not be explained by a local theory .
this aspect of entanglement , termed quantum nonlocality , is captured by two inequivalent notions , namely bell nonlocality @xcite and epr steering @xcite .
the strongest form of this phenomenon is bell nonlocality , witnessed via the violation of bell inequalities .
steering represents a strictly weaker form of quantum nonlocality @xcite , witnessed via violation of steering inequalities @xcite .
both aspects have been extensively investigated in recent years , as they play a central role in the foundations of quantum theory and in quantum information processing .
interestingly quantum nonlocality is based on two central features of quantum theory , namely entanglement and incompatible measurements . specifically , performing ( i ) arbitrary local measurements on a separable state , or ( ii ) compatible measurements on an ( arbitrary ) quantum state
can never lead to any form of quantum nonlocality .
hence the observation of quantum nonlocality implies the presence of both entanglement and incompatible measurements .
it is interesting to explore the converse problem .
two types of questions can be asked here ( see fig .
[ fig1 ] ) : ( a ) do all entangled states lead to quantum nonlocality ?
( b ) do all sets of incompatible measurements lead to quantum nonlocality ?
an intense research effort has been devoted to question ( a ) .
first , it was shown that all pure entangled states violate a bell inequality @xcite , hence also demonstrating epr steering .
for mixed states , the situation is much more complicated .
there exist entangled states which are local , in the sense that no form of quantum nonlocality can be demonstrated with such states when using non - sequential measurements @xcite .
these issues become even more subtle when more sophisticated measurement scenarios are considered @xcite .
question ( b ) has received much less attention so far . in the case of projective measurements
, it was shown that incompatible measurements can always lead to bell nonlocality @xcite .
note that in this case , compatibility is uniquely captured by the notion of commutativity @xcite .
however , for general measurements , _ i.e._positive - operator - valued - measures ( povms ) , no general result is known . in this case
, there are several inequivalent notions of compatibility . here
we focus on the notion of joint measurability , see e.g.@xcite , as this represents a natural choice in the context of quantum nonlocality .
several works discussed question ( b ) for povms @xcite .
the strongest result is due to wolf et al .
@xcite , who showed that any set of two incompatible povms with binary outcomes can always lead to violation of the clauser - horne - shimony - holt bell inequality .
however , this result may not be extended to the general case ( of an arbitrary number of povms with arbitrarily many outcomes ) , since pairwise joint measurability does not imply full joint measurability in general @xcite . here
we explore the relation between compatibility of general quantum measurements and quantum nonlocality .
we start by demonstrating a direct link between joint measurability and epr steering .
specifically , we show that for any set of povms that is incompatible ( _ i.e._not jointly measurable ) , one can find an entangled state , such that the resulting statistics violates a steering inequality .
hence the use of incompatible is a necessary and sufficient ingredient for demonstrating epr steering .
this raises the question of how joint measurability relates to bell nonlocality . specifically , the question is whether , for any set of incompatible povms ( for alice ) , one can find an entangled state and a set of local measurements ( for bob ) , such that the resulting statistics violates a bell inequality .
here we give evidence that the answer is negative .
in particular , we exhibit sets of incompatible measurements which can provably not violate a large class of bell inequalities ( including all full correlation bell inequalities , also known as xor games , see @xcite ) .
we therefore conjecture that non joint measurability and bell nonlocality are inequivalent . hence , similarly to local entangled states
, there may exist incompatible quantum measurements which are bell local . _ steering vs joint measurability .
_ we start by defining the relevant scenario and notations .
we consider two separated observers , alice and bob , performing local measurements on a shared quantum state @xmath0 .
alice s measurements are represented by operators @xmath1 such that @xmath2 , where @xmath3 denotes the choice of measurement and @xmath4 its outcome . upon performing measurement @xmath3 , and obtaining outcome @xmath4 , the ( unnormalized ) state held by bob
is given by @xmath5 the set of unnormalised states @xmath6 , referred to as an _ assemblage _ , completely characterizes the experiment , since @xmath7 is the probability of alice getting the output @xmath4 ( for measurement @xmath3 ) and given that information and bob s state is described by @xmath8 .
importantly , one has that @xmath9 for all measurements @xmath10 and @xmath11 , ensuring that alice can not signal to bob . in a steering test @xcite ,
alice want to convince bob that the state @xmath0 is entangled , and that she can steer his state .
bob does not trust alice , and thus wants to verify alice s claim . asking alice to perform a given measurement @xmath3 , and to announce the outcome @xmath4
, bob can determine the assemblage @xmath12 via local quantum tomography . to ensure that steering did indeed occur ,
bob should verify that the assemblage does not admit a decomposition of the form @xmath13 where @xmath14 . clearly ,
if a decomposition of the above form exists , then alice could have cheated by sending the ( unentangled ) state @xmath15 to bob and announce outcome @xmath4 to bob according to the distribution @xmath16 .
note that here @xmath17 represents a local variable of alice , representing her choice of strategy .
assemblages of the form are termed unsteerable and form a convex set @xcite .
hence any steerable assemblage can be detected via a set of linear witnesses called steering inequalities @xcite . by observing violation of a steering inequality
, bob will therefore be convinced that alice can steer his state . for a demonstration of steering , it is necessary for the state @xmath0 to be entangled .
however , not all entangled states can be used to demonstrate steering @xcite ; at least not when non - sequential measurements are performed on a single copy of @xmath0 .
moreover , steering also requires that the measurements performed by alice are incompatible . to capture the compatibility of a set of quantum measurements we use here the notion of joint measurability , see e.g. @xcite
a set of @xmath18 povms @xmath1 is called jointly measurable if there exists a measurement @xmath19 with outcome @xmath20 $ ] where @xmath21 gives the outcome of measurement @xmath3 , that is @xmath22 where @xmath23 stands for the elements of @xmath24 except for @xmath25 .
hence , all povm elements @xmath1 are recovered as marginals of the _ mother observable _
@xmath19 .
importantly , the joint measurability of a set of povms does not imply that they commute @xcite .
hence joint measurability is a strictly weaker notion of compatibility for povms . moreover , joint measurability is not transitive .
for instance ,
pairwise joint measurability does not imply full joint measurability in general @xcite ( see below ) .
our main result is to establish a direct link between joint measurability and steering .
specifically , we show that a set of povms can be used to demonstrate steering if and only if it is not jointly measurable .
more formally we prove the following result .
the assemblage @xmath26 , with @xmath27 , is unsteerable for any state @xmath0 acting in @xmath28 if and only if the set of povms @xmath29 acting on @xmath30 is jointly measurable .
the if part is straightforward .
our goal is to show that @xmath26 admits a decomposition of the form when @xmath29 is jointly measurable , for any state @xmath0 .
consider @xmath19 , the mother observable for @xmath29 , and define alice s local variable to be @xmath31 , distributed according to @xmath32 , where @xmath33 .
next alice sends the local state @xmath34 . when asked by bob to perform measurement @xmath3 , alice announces an outcome @xmath4 according to @xmath35 .
we now move to the only if part .
consider an arbitrary pure state @xmath36 with schmidt number @xmath37 .
notice that we can always write @xmath38 , where @xmath39 is an ( unormalized ) maximally entangled state in @xmath40 , and @xmath41 is diagonal matrix that contains only strictly positive numbers .
the assemblage resulting from a set of povms @xmath29 on @xmath0 is given by @xmath42 where @xmath43 is the transpose of @xmath44 .
our goal is now to show that if @xmath45 is unsteerable then @xmath29 is jointly measurable .
as @xmath45 is unsteerable , we have that @xmath46 which allows us to define the positive definite operator @xmath47 form which we can recover the assemblage @xmath48 as marginals , _
i.e._@xmath49 .
since the diagonal matrix @xmath41 is invertible , we can define @xmath50 .
it is straightforward to check that @xmath19 is a mother observable for @xmath51 : ( i ) it is positive , ( ii ) sums to identity , and ( iii ) has povm elements @xmath44 as marginals .
hence @xmath51 is jointly measurable , which concludes the proof .
note finally an interesting point that follows from the above . considering a set of incompatible measurements acting on @xmath30 , any pure entangled state of the schmidt number @xmath37
can be used to demonstrate epr steering .
_ bell nonlocality vs joint measurability .
_ it is natural to ask whether the above connection , between joint measurability and steering , can be extended to bell nonlocality .
recall that in a bell test , both observers alice and bob are on the same footing , and test the strength of the shared correlations . specifically
, alice chooses a measurement @xmath3 ( bob chooses @xmath52 ) and gets outcome @xmath4 ( bob gets @xmath53 ) .
the correlation is thus described by a joint probability distribution @xmath54 .
the latter can be reproduced by a pre - determined classical strategy if it admits a decomposition of the form @xmath55 where @xmath17 represents the shared local ( hidden ) variable , and @xmath56 .
any distribution that does not admit a decomposition of the above form is said to be bell nonlocal .
the set of local distributions , _
i.e._of the form is convex , and can thus be characterized by a set of linear inequalities called bell inequalities @xcite .
hence violation of a bell inequality implies bell nonlocality . in quantum theory , bell
nonlocal distributions can be obtained by performing suitably chosen local measurements , @xmath1 and @xmath57 , on an entangled state @xmath0 . in this case
, the resulting distribution @xmath58 does not admit a decomposition of the form .
bell nonlocality is however not a generic feature of entangled quantum states .
that is , there exist mixed entangled states which are local , in the sense that the statistics resulting from arbitrary non - sequential local measurements can be reproduced by a local model @xcite .
given the above , we investigate now how joint measurability relates to bell nonlocality .
first the above theorem implies that , if the set of povms @xmath59 used by alice is jointly measurable , then the statistics @xmath54 can always be reproduced by a local model , for any state @xmath0 and measurements of bob @xmath60 .
the converse problem is much more interesting .
the question is whether for any set of povms @xmath59 that is not jointly measurable , there exists a state @xmath0 and a set of measurements @xmath60 such that the resulting statistics @xmath54 violates a bell inequality .
this was shown to hold true for the case of sets of two povms with binary outcomes @xcite . in this case ,
joint measurability is equivalent to violation of the chsh bell inequality .
here we give evidence that this connection does not hold in general .
specifically , we exhibit a set of povms which is not jointly measurable but nevertheless can not violate a large class of bell inequalities .
consider the set of three dichotomic povms ( acting on @xmath61 ) given by the following positive operators @xmath62 for @xmath63 , where @xmath64 are the pauli matrices , and @xmath65 .
indeed , one has that @xmath66 .
this set of povms should be understood as noisy pauli measurements .
the set is jointly measurable if and only if @xmath67 , although any pair of povms is jointly measurable for @xmath68 @xcite ( see also @xcite ) .
hence in the range @xmath69 , the set @xmath70 forms a _ hollow triangle _ : it is pairwise jointly measurable but not fully jointly measurable .
we now investigate whether the above hollow triangle can lead to bell inequality violation .
the most general class of bell inequalities to be considered here are of the form : @xmath71 where @xmath72 all ( tight ) bell inequalities of the above form for @xmath73 are known ( see appendix ) . using a numerical method based on semi - definite - programming ( sdp ) @xcite ( see appendix )
we could find the smallest value of the parameter @xmath74 for which a given inequality can be violated using the set of povms .
the results are summarized in table i. notably , we could not find a violation in the range @xmath69 where the set @xmath70 is a hollow triangle .
in fact , no violation was found for @xmath75 , whereas pairwise joint measurability is achieved for @xmath76 , thus leaving a large gap .
note also that pairwise joint measurability implies violation of the chsh inequality here , since we have povms with binary outcomes @xcite .
we thus conjecture that there is a threshold value @xmath77 , such that all hollow triangles with @xmath78 do not violate any bell inequality .
[ table ] .
bell inequality violation with incompatible povms .
specifically , we consider the sets given in equations and .
for each set , we determine the smallest value of the parameter @xmath74 , such that the set becomes jointly measurable ( jm ) , and achieve bell inequality violation .
we consider tight bell inequalities with up to @xmath79 measurements for bob ( see appendix ) .
note that pairwise joint measurability is equivalent to violation of the chsh bell inequality . [ cols="^,^,^",options="header " , ]
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i am very grateful to pauchy hwang and the organizers for their invitation and warm hospitality .
i also thank pauchy hwang and bingkan xue for the collaborated results in this talk .
this work is partially supported by national natural science foundation of china ( nos . 10721063 , 10575003 , 10528510 ) , by the key grant project of chinese ministry of education ( no .
305001 ) , and by the research fund for the doctoral program of higher education ( china ) .
j. r. horandel , j. phys .
g * 29 * , 2439 ( 2003 ) . c. a. ayre _ et al .
_ , j. phys .
a * 5 * , l102 ( 1972 ) .
t. hebbeker and c. timmermans , astropart .
phys . * 18 * , 107 ( 2002 ) .
s. ostapchenko , nucl .
suppl . * 151 * , 147 ( 2006 ) .
see , e.g. , w. d. apel _ et al .
_ [ kascade collaboration ] , arxiv : astro - ph/0510810 .
p. hansen , t. k. gaisser , t. stanev and s. j. sciutto , phys .
d * 71 * , 083012 ( 2005 ) ; a. cillis and s. j. sciutto , arxiv : astro - ph/9908002 .
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the muon charge ratio of ultrahigh energy cosmic rays may provide information to detect the composition of the primary cosmic rays .
we propose to extract the charge information of high energy muons in very inclined extensive air showers by analyzing their relative lateral positions in the shower transverse plane .
the most high energy particles can be observed by human being are from cosmic rays .
the study of them belongs to frontiers of human knowledge in combination of cosmology , astrophysics , and particle physics , and can provide better understanding of the universe from most small to most big , i.e. , connecting quarks to the cosmos .
the universe is not empty , but full of background relic particles from the big bang .
it has long been anticipated that the highest energy cosmic rays would be protons from outside the galaxy , and there is an upper limit of the highest energy in the observed proton spectrum , commonly referred to as the gzk cutoff @xcite , as the protons traveling from intergalactic distances should experience energy losses owing to pion productions by the photons in the cosmic background radiation .
although there have been attentions for the cosmic ray events above the gzk cutoff , it is natural to expect that these ultrahigh energy cosmic rays come from sources within the gzk zone @xcite , i.e. , not far from us in more than tens of mpc .
recently there are also reports on the observation of the gzk cut - off by new experiments @xcite .
however , questions about the composition of such ultrahigh energy cosmic ray particles , e.g. , whether they are protons , neutrons , or anti - nucleons @xcite , are still open to investigations . muons in the air showers are mainly from decays of pions and kaons produced in the interactions of the primary cosmic rays with the atmosphere .
the very high energy secondary pion and kaon cosmic rays can be considered as from the current fragmentation of partons in deep inelastic scattering of the primary cosmic rays with the nucleon targets of the atmosphere in a first approximation @xcite .
we also consider only the favored fragmentation processes , i.e. , the @xmath0 , which is composed of valence @xmath1 and @xmath2 quarks , is from the fragmentation of @xmath1 and @xmath2 quarks in the nucleon beam , and the @xmath3 , which is composed of valence @xmath4 and @xmath5 quarks , is from the fragmentation of @xmath4 and @xmath5 quarks @xcite .
similarly , the @xmath6 , which is composed of valence @xmath1 and @xmath7 , is from the fragmentation of @xmath1 and @xmath7 quarks , and the @xmath8 , which is composed of valence @xmath4 and @xmath9 , is from the fragmentation of @xmath4 and @xmath9 quarks .
the @xmath10 is from the decay of a @xmath0 or a @xmath6 and the @xmath11 is from the decay of a @xmath3 or a @xmath8 .
we can roughly estimate the muon charge ratio by @xmath12+\kappa \left[u(x)+\bar{s}(x)\right]\right\}}{\int_0 ^ 1 { \mathrm d } x \left\{\left[d(x)+\bar{u}(x)\right]+\kappa \left[\bar{u}(x)+s(x)\right]\right\ } } , \label{qme}\ ] ] where @xmath13 is the quark distribution with flavor @xmath14 for the incident hadron beam and @xmath15 is a factor reflecting the relative muon flux and fragmentation behavior of @xmath16 .
secondary collisions do not influence the above estimation , since the current parton beams still keep their flavor content and act as the current partons after the strong interactions with the partons in the atmosphere targets . adopting a simple model estimation of the parton flavor content in the nucleon without any parameter @xcite , we find that @xmath17 for proton and @xmath18 for neutron .
this simple evaluation is in agreement with the empirical expectation of @xmath19 for proton and @xmath20 for neutron @xcite as well as that in an extensive monte carlo calculation @xcite , thus it provides a clear picture to understand the dominant features for the muon charge ratio by the primary hadronic cosmic rays . for the @xmath21 ratio for antiproton , it is equivalent to the @xmath22 ratio for proton by using eq .
( [ qme ] ) , thus we find @xmath23 for antiproton , which is close to that for neutron .
the @xmath21 ratio for antineutron is also equivalent to the @xmath22 ratio for neutron , and it is @xmath24 , which is close to that for proton .
it is hard to distinguish between the primary neutrons and antiprotons ( or protons and antineutrons ) by the @xmath21 ratio of the air shower , unless very high precision measurement is performed and also our knowledge of the muon charge ratio for each nucleon species is well established .
the study of cosmic rays with primary energies above @xmath25gev are typically based on the measurements of extensive air showers ( eas ) that they initiate in the atmosphere .
the ground detector array records the secondary particles produced in shower cascades , including photons , electrons ( positrons ) , muons , and some hadrons .
then their arrival times and density profiles are used to infer the primary energy and composition of the incident cosmic ray particle , usually through comparison with simulated results .
photons , electrons and positrons are the most numerous secondary particles in an eas event .
however , for very inclined showers , these electromagnetic components would travel a long slant distance and are almost completely absorbed before they reach the ground . on the other hand ,
muons are decay products of charged mesons in shower hadronic cascades .
most high energy muons survive their propagation through the slant atmospheric depth , during which they lose typically a few tens of gev s energy .
these high energy muons carry important information about the nature of the primary cosmic ray hadron , which will be extracted from their energy spectrum and lateral distribution .
as discussed by hwang and i @xcite , the ratio of positive versus negative muons @xmath26 is a significant quantity which can help to discern the primary composition , and at high energies this charge ratio also reflects important features of hadronic meson production in cosmic ray collisions . in order to obtain such muon charge information
, we would need a way to distinguish between positive and negative high energy muons .
unfortunately , existing muon detectors available at shower arrays , usually scintillators and water erenkov detectors , are not commonly equipped with magnetized steel to differentiate the muon charges .
even if they were , the limited region of the magnetic field prevents definite determination of high energy muons track curvature .
this invites us to think of the geomagnetic field as a huge natural detector for muon charge information . apparently , after being produced high in the atmosphere , a positively charged muon would bend east on its way down while a negatively charged muon would bend west , introducing an asymmetry into the density profile of the shower front . if their separation is large enough as compared with other circularly symmetric `` background '' deviations , it will be possible to distinguish the positive muons from the negative ones . to see such an effect , xue and i @xcite analyzed the possibility of obtaining the charge information of high energy muons in very inclined extensive air showers .
we have demonstrated that positive and negative high energy muons in sufficiently inclined air showers can be distinguished from each other through their opposite geomagnetic deviations in the transverse plane .
we developed a revised heitler model to calculate this distinct double - lobed distribution , and studied the condition for the two lobes of either positive or negative muons to be separable with confidence . from our criterion of resolvability
, we concluded that a zenith angle @xmath27 will be most suitable for our approach .
there are already some results from full air shower simulations that take into account the geomagnetic effect on muon propagation @xcite .
they illustrated remarkable double - lobed muon lateral density profile in very inclined air showers , which is in agreement with our expectation qualitatively .
however , no present study has fully considered the high energy part of muon content , which can be used to compare with our results .
thus we would like to propose future simulations of very inclined extensive air showers that focus on the behavior of high energy muons .
they also have to keep track of the muon charges and the relation to their lateral positions . for more detailed analysis and discussion
, please refer to ref.@xcite . in summary
, we propose to extract the charge information of high energy muons in very inclined extensive air showers by analyzing their relative lateral positions in the shower transverse plane .
this muon charge information is helpful to detect the composition of cosmic rays , e.g. , the neutron or antiproton content of the ultrahigh energy cosmic rays .
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* figure 1 : * reduction factor @xmath54 for the excitation probability as a function of the characteristic parameter @xmath26 defined in eq .
[ zdef ] for the two model wave functions ( i : , ii : - - - ) . + * figure 2 : * reduction @xmath102 of the total cross section for the excitation of the @xmath11 first excited state from the @xmath103 ground state of @xmath0be as a function of the projectile velocity @xmath17 for the two model wave functions ( i : , ii : - - - ) .
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we study the corrections of first order electromagnetic excitation due to higher order electromagnetic interactions .
an effective operator is introduced which takes these effects into account in the sudden approximation .
evaluating the matrix - elements of this operator between the relevant states corrections to the first order result are obtained in a simple way . as an example we discuss the excitation of the first excited state in @xmath0be .
it tends to improve the agreement between experiment and theory .
425.70 in 0.80 in 0.20 in electromagnetic excitation in the energy domain of several tens of mev / u up to relativistic energies is a growing field of study .
the cross - section can become large and irreducible nuclear effects can be kept under control . with increasing beam energy
the equivalent photon spectrum becomes harder , and also particle - unstable states can be reached .
the coulomb dissociation @xmath0li @xmath1 @xmath2li + 2n and @xmath3o @xmath1 @xmath4n + p , which is also astrophysically relevant , are examples @xcite . recently ,
bound states were also excited and their ( doppler shifted ) de - excitation @xmath5-rays were measured .
a large deformation of the neutron - rich nucleus @xmath6 mg was recently deduced from a measurement of the @xmath7 885 kev transition to the ground state after medium energy electromagnetic excitation @xcite .
the 320 kev @xmath8 @xmath5-transition in @xmath0be was recently observed .
the measured cross - section for the @xmath0be ( @xmath9 ) coulomb excitation was found to be noticeably less than expected from the known lifetime and @xmath10 order pure coulomb excitation @xcite .
apart from possible nuclear and coulomb - nuclear interference effects , a possible reason for this discrepancy is the influence of higher order electromagnetic interaction .
it is the purpose of this letter to describe a framework suitable for fast projectiles .
e.g. the rather loosely bound @xmath0be in its @xmath10 excited @xmath11 state could easily be excited electromagnetically into the continuum in a second step @xcite .
electromagnetic excitation is mainly characterized by two parameters , the adiabaticity parameter @xmath12 and the strength parameter @xmath13 the excitation energy is given by @xmath14 , the impact parameter in a straight - line approximation is denoted by @xmath15 , and @xmath16 , where @xmath17 is the projectile velocity .
the target charge number is denoted by @xmath18 , and @xmath19 the electric multipole operator . in coulomb excitation below the barrier ,
multiple electromagnetic excitation is usually treated in a coupled channels approach using the relevant states from appropriate nuclear models , like the harmonic vibrator or rigid rotor . for a review see ref .
@xcite .
the situation for coulomb excitation above the barrier becomes simpler , because the excitations tend to be sudden .
while @xmath14 is restricted to a few mev , the adiabaticity parameter @xmath20 is typically less than 1 , for the important range of impact parameters @xmath21 , where @xmath22 and @xmath23 are the nuclear radii of projectile and target .
thus fast collisions become the domain of the sudden approximation @xcite , or of a recently developed low-@xmath20 approximation @xcite . in this case
it can be advantageous to construct operators which take into account the influence of intermediate states . for simplicity ,
let us use the straight - line approximation and the dipole approximation .
the first order excitation amplitude is given by @xcite @xmath24 with @xmath25 where the projectile moves in the @xmath26-direction and the impact parameter points to the @xmath27-direction .
the dipole effective charge is given by @xmath28 using a model of two pointlike inert clusters @xmath15 and @xmath29 with charge numbers @xmath30 , @xmath31 and masses @xmath32 and @xmath33 .
@xmath34 and @xmath35 are the modified bessel functions .
for @xmath36 we get the classical coulomb push @xmath37 in the limit @xmath38 , the excitation amplitude can be evaluated easily to all orders in the sudden approximation .
it is given by @xmath39 this could be generalized to higher multipolarities and trajectories corrected for coulomb deflection . by comparing @xmath40 and
@xmath41 the influence of multiple electromagnetic excitation can be assessed .
this is a remarkably simple procedure , all intermediate states are included .
the reduction of the excitation probability due to higher order effects is given by @xmath42 for loosely bound states , e.g. in @xmath0be = @xmath43be + n , we use simple model wave functions to reveal the characteristic parameters .
we choose two models , with the correct asymptotic behaviour .
their differences give a feeling about the model dependence .
for the wave functions we make the ansatz for the radial part in the initial state ( with orbital angular momentum @xmath44 ) @xmath45 which corresponds to the solution of the schrdinger equation with a @xmath46-like potential .
for the final state ( @xmath47 ) we choose @xmath48 and the more extended wave function @xmath49 respectively .
the constants @xmath50 ( @xmath51 ) are calculated from the binding energies @xmath52 . with these wave functions
the calculated mean lifetimes of the @xmath11 state are 110 fs and 97 fs , respectively .
they are smaller than the experimental value of ( @xmath53 ) fs @xcite . despite this difference ,
the ratio of higher order to first order effects can be given with some confidence , where , e.g. , common spectroscopic factors cancel out .
the absolute value depends on more sophisticated details of the nuclear model ( see , e.g. , ref .
@xcite ) .
the reduction factor @xmath54 can be calculated analytically .
we get @xmath55 and @xmath56 resp . , where we have introduced the parameter @xmath57 the parameter @xmath26 is directly related to @xmath58 ( eq . [ chidef ] ) .
it describes the ratio of the strength of the coulomb push @xmath59 and the `` looseliness '' of the system , and is a measure of the importance of higher order effects .
the quantity @xmath54 is plotted in fig . 1 for the two model wave - functions described above . for large @xmath15 the sudden approximation fails ( @xmath60 ) , on the other hand , higher order effects diminish due to the decrease of the strength parameter @xmath58 .
the product @xmath61 is a very small number for low excitation energies @xmath14 and high projectile velocities @xmath17 . the ranges of validity for the first order calculation ( @xmath62 small , @xmath20 arbitrary ) and the sudden approximation ( @xmath20 small , @xmath62 arbitrary ) overlap . in a convenient and accurate interpolation procedure
we calculate the total cross section in the following way @xmath63 which should be an accurate expression for all values of b in the integrand . for small impact parameters we have @xmath64 and the first order approximation cancels out in the calculation of @xmath65 .
we compare it with the total cross section in the first order calculation @xmath66 both cross sections depend on the value of the minimum impact parameter @xmath67 .
the change in the two cross sections will be similar so that their ratio is less affected by a change in @xmath67 .
if the sudden approximation is not well enough fulfilled , one could use the low-@xmath20 approximation @xcite which takes second order electromagnetic effects into account . for the @xmath0be coulomb excitation a reduction of the cross section from @xmath68 mb ( expected from the first order calculation ) to the measured @xmath69 mb
was recently found in an experiment at ganil @xcite .
indeed , the parameter @xmath26 can become substantial in this case , and higher order effects are not negligible . for a collision with @xmath70 fm and an energy of @xmath71 mev ( @xmath72 )
we have @xmath73 and from fig .
1 we see a substantial reduction of the excitation probability .
in a recent experiment at riken @xcite , which is currently evaluated , an energy of about @xmath74 mev ( @xmath75 ) for the @xmath0be was used .
this leads to a value of @xmath76 for the same impact parameter corresponding to a smaller reduction .
the product @xmath77 takes on the small values @xmath78 and @xmath79 , respectively , for the two models and the ganil conditions . at @xmath80 fm
we get an excitation probability of 3.4% and 3.9% in the first order calculation , decreasing with @xmath81 . for impact parameters larger than @xmath82 fm
the excitation becomes adiabatic and the excitation probability drops off exponentially .
the excitation probability is small compared to 1 and the non - conservation of unitarity in the first order approximation will not affect the calculation of the cross section .
the apparent reduction of the b(e1)-value ( due to higher order effects ) is given by @xmath83 which is plotted in fig . 2 as a function of the projectile velocity @xmath17 for the two model wave functions described above .
we assume a minimum impact parameter @xmath84 fm corresponding to a grazing collision of the projectile and target .
this will give an estimate of the largest possible effect to be expected from the higher order contributions .
we obtain a reduction , depending on the particular model chosen , of @xmath85 or @xmath86 for the ganil energy and @xmath87 or @xmath88 for the riken energy .
the range of the reduction r in the two models gives a feeling of the reliabiliy of the results .
the use of more realistic wavefunctions is outside the scope of the present work .
the reduction of the excitation probability for the @xmath11 state by higher order effects will be accompanied by an increase of the cross section in the breakup channel . in the dipole approximation ,
the lowest order correction was of @xmath89 order .
e1-e2 excitation contributes already in second order .
nevertheless , it is smaller than the third order e1-e1-e1 correction .
for low @xmath20 we can estimate the ratio of the two amplitudes as @xmath90 for @xmath91 , @xmath92 , and @xmath93 at the ganil energy .
thus the dipole approximation is reasonable , at least for a first exploration .
possible higher order effects in the @xmath6 mg intermediate energy coulomb excitation @xcite can also be estimated .
assuming a rigid rotor model , high energy coulomb excitation was calculated in the sudden approximation in ref .
@xcite .
the characteristic strength parameter is @xmath94 using the b(e2)-value found in ref .
@xcite we have ( see eq . 6 of ref .
@xcite , where a factor @xmath95 is missing on the rhs . ) @xmath96 and @xmath97 with @xmath98 fm for an energy of @xmath99 mev . from fig .
1a of ref .
@xcite it can be seen that higher order effects are negligible for the value of the strength parameter @xmath100 .
this is in agreement with the result found in the coupled channel calculation of ref .
@xcite . in conclusion
, we provide a framework to apply corrections to 1@xmath101 order electromagnetic excitation .
it is appropriate for fast collisions .
we constructed operators which take the influence of intermediate states into account .
this can lead to a great simplification as compared , e.g. , to the coupled channels approach . in this approach ,
a set of states , considered to be relevant , has to be chosen with known electromagnetic matrix - elements . in the present approach ,
of course , the model dependence can not be altogether avoided ; it enters when the corresponding matrix - elements of the operator has to be calculated , or at least estimated . as an example we studied the excitation of the @xmath11 state in @xmath0be . the importance of higher order effects for this case of an extremely loosely bound nucleus
was established .
the estimate for the reduction of the cross section can only partly explain the observed reduction of the b(e1)-value in the ganil experiment .
however , for a final analysis , more accurate calculations with improved wave functions , including e2 and nuclear effects in the excitation , should be performed .
we wish to thank p. g. hansen and m. g. saint - laurent for stimulating discussions .
note added in revision : in the meantime we got to know about a coupled channel study of the @xmath0be coulomb excitation by c. a. bertulani , l. f. canto , and m. s. hussein .
they get very similar conclusions as compared to our findings .
99 see e.g. g. baur and s. typel , in : proceedings of the 6th international conference on clusters in nuclear structure and dynamics , 6 9 sept .
1994 , ed . by f.
haas , ( strasbourg , 1995 ) and further references given there . t. motobayashi et al . , phys .
lett .
b 346 ( 1995 ) 9 .
r. anne et al . , ganil preprint p-94 - 35 ( 1994 ) , z. phys .
a in print .
p. g. hansen , private communication .
k. alder and a. winther , electromagnetic excitation ( north - holland , amsterdam , 1975 ) . s. typel and g. baur , nucl .
phys . a 573 ( 1994 ) 486 .
s. typel and g. baur , phys .
rev .
c 50 ( 1994 ) 2104 .
d. j. millener , j. w. olness , and e. k. warburton , phys .
rev .
c 28 ( 1983 ) 497 .
t. motobayashi , private communication .
g. baur , z. phys .
a 332 ( 1989 ) 203 .
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1 . temperature dependence of the magnetization , measured with @xmath2 = 1 t with zero - field - cooling ( zfc ) , field - cooled - cooling ( fcc ) , and field - cooled - warming ( fcw ) procedures .
the inset shows the time evolution of the normalized magnetization after zfc to @xmath15 = 8 , 40 , 60 and 100k .
field dependence of the sample s temperature showing an abrupt warming from 2.5 to @xmath20 30 k at @xmath21 @xmath19 2.2 t. the inset shows a spontaneous magnetization jump , measured with a fixed magnetic field .
time dependence of the magnetization during a magnetic field sweep , for different waiting times between consecutive field increments : @xmath62 = 7.5 ( squares ) , 15 ( circles ) , 30 ( up triangles ) and 60 min .
( down triangles ) , at @xmath15 = 2.5 k. the inset shows an enlarged portion of the region just before the magnetization jump .
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the occurrence at low temperatures of an ultrasharp field - induced transition in phase separated manganites is analyzed .
experimental results show that magnetization and specific heat step - like transitions below 5 k are correlated with an abrupt change of the sample temperature , which happens at a certain critical field .
this temperature rise , a magnetocaloric effect , is interpreted as produced by the released energy at the transition point , and is the key to understand the existence of the abrupt field - induced transition .
a qualitative analysis of the results suggests the existence of a critical growing rate of the ferromagnetic phase , beyond which an avalanche effect is triggered .
mixed valent manganites show a great deal of fascinating properties , arising from the strong interplay between spin , charge , orbital , and lattice degrees of freedom @xcite .
the most intriguing one is the existence of a phase separated state , the simultaneous coexistence of submicrometer ferromagnetic ( fm ) metallic and charge ordered ( co ) insulating regions dagotto .
the phase separation scenario has its origin in the unusual proximity of the free energies of these very distinct fm and co states , and in the fact that the competition between both phases is resolved in mesoscopic length scales , giving rise to real space inhomogeneities in the material .
yet another surprising result more recently found in manganites is the appearance of ultrasharp magnetization steps at low temperatures ( below @xmath0 5 k ) in the isothermal magnetization @xmath1(@xmath2 ) curves @xcite .
this effect , the field induced transition of the entire compound from one phase to the other of the coexisting states , is included in the category of metamagnetic transitions @xcite .
however , unlike the broad continuous transitions expected for inhomogeneous granular systems , in this case it occurs in an extremely narrow window of magnetic fields .
these ultrasharp steps were observed in both single crystals and polycrystalline samples , indicating that it is not related to a particular micro - structure of the material . the actual existence of a phase separated state was recognized as a key parameter for the observation of these magnetization jumps @xcite .
the effect was first reported in manganites doped at the mn site , and the disorder in the spin lattice was thought to play a relevant role hebert1 .
however , a similar behavior was also found in pr@xmath3ca@xmath4mno@xmath5 , and the qualitative interpretation of the phenomenon shifted to the martensitic character of the phase separated state @xcite .
accommodation strains were shown to be relevant in the stabilization of phase separation @xcite , but their role in the magnetization steps is not clear , since it is expected that grain boundaries would act as a sort of firewall for the movement of the domain walls , stopping the avalanche process . additionally , despite its intrinsic first - order character , the martensitic transformation is spread over a large range of the external parameter driving the transition , the magnetic field in the present case , in strong disagreement with the abrupt character of the transition . the aim of this investigations is to address a basic question concerning this abrupt field - induced transition : why is this metamagnetic transition so sharp , and what is actually causing it ?
we report the occurrence of ultrasharp magnetization steps at low temperatures in a prototype phase separated manganite , which are accompanied by discontinuities in the magnetic field dependence of the specific heat . concomitantly with these facts
, we found that the field - induced transition is accompanied by a large increase in the temperature of the sample , by dozens of degrees .
this feature suggests a mechanism in which the abrupt first order transition in the whole sample is triggered by the released heat in a microscopic phase transformation .
a low temperature heat controlled magnetization avalanche was previously found in bulk disordered magnets @xcite due to the heat released by the fm domain wall motion during the reversal of the remnant magnetization .
also , local heating induced by non - uniform current flow was proposed as the origin of the mesoscopic fluctuations between coexisting phases observed in la@xmath6pr@xmath7ca@xmath8mno@xmath5@xcite .
we propose that in phase separated manganites the interplay between the growth of the fm phase induced by the magnetic field and the heat generated by this growth is the key to explain the avalanche process leading to an ultrasharp field - induced transition in these inhomogeneous strongly correlated systems .
the particular compound under study is a high quality polycrystalline sample of la@xmath6pr@xmath7ca@xmath8mno@xmath5 , synthesized by the sol - gel technique .
it belongs to the well known family of compounds la@xmath9pr@xmath10ca@xmath11mno@xmath12 whose tendency to form inhomogeneous structures in the range [email protected] is extensively documented .
uehara , uehararate , kim , balagurov , kiryukhin , nonvolatile scanning electron micrographs revealed a homogeneous distribution of grain sizes , of the order of 2 @xmath14 m .
an identification of the magnetic phases of the material can be made through the temperature dependence of the magnetization , @xmath1(@xmath15 ) .
the results were obtained on an extraction magnetometer with a field @xmath2 = 1 t , and are shown in fig .
1 . as the temperature
is lowered the sample changes from a paramagnetic to a charge - ordered antiferromagnetic state at @xmath16 = 220 k. just below , a small kink at 190 k is a signature of the onset of the formation of ferromagnetic clusters @xcite .
a more robust ferromagnetic phase appears at @xmath17 = 70 k ( 90 k on warming ) , which coexists with the majority co state in an inhomogeneous phase separated state @xcite . in a temperature window extending from @xmath17 down to a temperature @xmath18 @xmath19 20 k the magnetization shows considerable relaxation effects , as shown in the inset of fig .
1 , signaling the growth of the fm phase against the co background .
the temperature @xmath18 ( which depends on the applied field ) can be identified as a blocking temperature ; relaxation below @xmath18 is strongly reduced . additionally ,
the magnetic state below @xmath18 is highly dependent on the sample magnetic field and cooling history .
if the sample is cooled without an applied field ( zfc ) the magnetization at 2 k shows a significant low value , that remains unchanged while warming until @xmath18 , above with it shows a continuous increase and merges with the field cooling warming curve .
= 1 t with zero - field - cooling ( zfc ) , field - cooled - cooling ( fcc ) , and field - cooled - warming ( fcw ) procedures .
the inset shows the time evolution of the normalized magnetization after zfc to @xmath15 = 8 , 40 , 60 and 100k.,width=264 ] with the application of a large enough magnetic field the low temperature ( below @xmath20 5 k ) zero - field - cooled state is transformed into a fm phase in an abrupt step - like metamagnetic transition .
figure 2(a ) shows magnetization measurements as a function of applied field , @xmath1(@xmath2 ) , measured at @xmath15 = 2.5 k. at a certain critical field @xmath21 the entire system changes to a nearly homogenous fm state , which remains stable even after the field is removed .
the width of the transition , determined by repeating the measurements with lower field increments , is below 10 oe .
figure 2(b ) shows specific heat data as a function of applied field , @xmath22(@xmath2 ) , measured by the relaxation method at the same base temperature , @xmath15 = 2.5 k. as can be readily noticed , a discontinuous transition occurs at approximately the same magnetic field , indicating that a true thermodynamic transition is taking place .
= 2.5 k. both measurements show an abrupt change at the same critical field @xmath21 @xmath19 2.2 t.,width=226 ] since the observed transition is first order , it is expected that the latent heat involved should affect the thermodynamic state of the sample , for instance , its temperature . in order to gain some insights on the magnitude of the effect
the following experiment was performed : with the sample placed in a vacuum calorimeter with a weak thermal link to the temperature controlled surroundings ( kept at 2.5 k ) , the sample s temperature was measured while the magnetic field was increased , with field increments identical to the data of fig.2 . the obtained result , plotted in fig .
3 , shows the occurrence of a sudden and huge increase of the sample s temperature , greater than 25 k , at the same critical field of the magnetization jump .
since the relaxation time for temperature stabilization between the sample and the temperature controlled surroundings ( of the order of several seconds ) is much larger than the internal time constant between the sample and the sample holder ( of the order of milliseconds ) , the temperature rise measured is intrinsic to the sample .
the abrupt increase of the sample s temperature can be then doubtless ascribed to the heat generated when the non - fm fraction of the material is converted to the fm phase .
the same experiment was repeated with samples of la@xmath23pr@xmath24ca@xmath11mno@xmath5 with different pr content , as well as in samples of the series landcamno . whenever a magnetization jump occurs a sizable increase of the sample s temperature
was observed .
it is also worth mentioned that in the @xmath1(@xmath2 ) and @xmath22(@xmath2 ) data of figure 2 the sample s temperature is in fact not strictly constant ; there is also a sudden temperature rise at the field of the step transition , followed by a quick relaxation to the base temperature of system . the process which starts with nearly the whole material in the non - fm phase at @xmath25 = 2.5 k and ends with a nearly homogeneous fm state at @xmath26 @xmath27 30 k , is conceptually related with the magnetocaloric ( mc ) effect @xcite .
30 k at @xmath21 @xmath19 2.2 t. the inset shows a spontaneous magnetization jump , measured with a fixed magnetic field.,width=226 ] the mc effect consists basically in a temperature change @xmath28 induced by the application of a magnetic field , which , within the approximation of reversible process , is related with the entropy change @xmath29 generated by ordering the spin lattice . in our case
, however , the approximation of reversible adiabatic process is not valid , due to the strong irreversible character of the field - induced transformation .
also , as the phase transition from de co to the fm phases involves large changes in the magnetic , structural and electronic properties , which are strongly correlated , the magnetic field affects all the degrees of freedom of the system .
this fact makes inapplicable some of the usual basic equations employed in the description of the mc effect .
a more realistic approach is to use the conservation of the internal energy during the fast conversion process ( hypothesis of adiabaticity ) . neglecting small changes of the sample volume
, we can make the identity @xmath30 , where @xmath31 is the internal energy per volume unit . replacing the whole ( irreversible ) process by an isothermal plus an isobaric one
, we can write : @xmath32 where @xmath33 is the specific heat of the fm phase at constant pressure .
this yields an estimate for the released heat at the field induced transition given by @xmath34 .
this estimated value was obtained from specific heat measurements as a function of temperature performed at zero magnetic field after the sample was transformed to the fm phase by application of a field of 9 @xmath15 .
the magnetization and specific heat results , shown in fig . 2 ,
are macroscopic signatures of a phenomenon which must be understood at a microscopic level .
below the temperature @xmath18 the sample gets into a strongly blocked regime , in which the fm clusters can not grow against the co background ( see inset of fig 1 ) .
after zero- field - cooling , the sample reaches the blocked state with a small , time independent , fraction @xmath35 of fm phase , which can be thought as distributed in isolated regions or clusters surrounded by a co matrix .
the application of an external magnetic field @xmath2 weakens this frozen - in state , inducing the increase of each cluster of volume @xmath36 in an amount @xmath37 , which can depend on @xmath38 , @xmath15 , @xmath2 and time .
the released heat yielded by this particular process is @xmath39 .
part of this energy is used to locally increase the temperature of the fm volume @xmath36 , a process that can be considered as instantaneous , taking into account that the thermal conductivity of the fm phase is much greater than that of the co phase .
the remaining energy @xmath40 is evacuated through the surrounding co region .
this balance yields @xmath41 once a process involving a change of the local fm fraction happens , the further evolution of the system is determined by the interplay between the rates at which the system is generating heat , and the rate at which the co phase is releasing it . when the former is greater than the latter , a local temperature rise within the fm region is obtained .
if this temperature reaches values beyond the blocking temperature corresponding to the applied field @xmath2 the system becomes critical , in the sense that the adjacent co regions , which in turn will increase their local temperature too , become highly unstable .
these unstable co regions are now easily transformed to the fm state , releasing heat , and so on , inducing an avalanche - like chain reaction . at the end ,
all regions which have ferromagnetism as its equilibrium state at @xmath15=@xmath26 and field @xmath2 had been converted from co to fm .
the equilibrium fractions of the coexisting phases at that @xmath15 and @xmath2 values then determine the size of the avalanche process .
following equation ( 2 ) , we can make an estimation of the critical value of the volume change @xmath42 which is needed to _ turn - on _ the avalanche process .
the first condition to be accomplished is that the temperature of the local fm region increases beyond the blocking temperature @xmath18(@xmath2 ) . assuming that it occurs in a time scale @xmath43 , and that in this scale the heat transferred to the co region is negligible , we obtain @xmath44 where @xmath45 8 k at @xmath46 = 2.2 t was estimated from zfc magnetization measurements .
this calculation yields @xmath47 , i.e. , almost one per cent increase of the local fm volume is needed to initiate the abrupt transition .
this condition must be accompanied by another one , related with the time @xmath43 in which the volume enlargement occurs . as mentioned above
, this time has to be short enough to avoid the heat be released through the surrounding co region , i.e. , the condition @xmath48 must hold , where @xmath49 is the area of the cluster surface , @xmath50 is the thermal conductivity of the co phase and @xmath51 is a local spatial coordinate .
a crude estimation of @xmath43 can be done assuming that , within the adjacent co region , the temperature decays from @xmath52 to @xmath25 in a length @xmath53 , and taking into account that typical low temperature values for @xmath50 could range between 0.1 - 1 w/(mk ) .
these assumptions give , for instance , @xmath5410@xmath55 - 10@xmath56 s for clusters of volume @xmath36 = ( 50 nm)@xmath57 , and predicts a critical rate @xmath58 .
therefore , we estimate that thermal processes that happen within a narrow time window , involving a one percent increase of the local fm regions are needed to initiate the abrupt field - induced transition , the critical rate scaling as the linear size of the fm cluster .
the occurrence of the step transition is then governed by the probability of such an event , once the magnetic field has yielded the crossover between the free energies of the coexisting phases . one way to modify
the avalanche probability is allowing the system to relax before reaching the critical state . allowing relaxation an increase of the fm fraction as a function of the elapsed time occurs , and consequently the value of the @xmath59 needed to _ turn - on _ the process should also increases
this would be reflected on the dependence with the elapsed time of the critical field @xmath21 at which the jump occurs . to verify this hypothesis
we have measured the time dependence of the magnetization during a field sweep , @xmath1(@xmath2,@xmath60 ) at 2.5 k , starting with @xmath2 = 1.9 t , and waiting a time @xmath61 at fixed @xmath2 before changing to the next field value . in fig .
4 we show the @xmath1 vs @xmath60 curves obtained for different values of @xmath62 , confirming the above presumption : for larger @xmath62 the magnetization jump occurs at higher critical fields . as a remarkable result , we observed that in most cases the step transition occurs spontaneously within the time interval where the field was unchanged .
this fact signs unambiguously that the width of the step transition , beyond any experimental resolution , is strictly zero .
the inset of fig .
3 shows a spontaneous ( as opposed to field - induced ) magnetization jump , which happens at a fixed field and temperature values .
the occurrence of spontaneous magnetization jumps in phase separated manganites was also reported by another group.spontaneous = 7.5 ( squares ) , 15 ( circles ) , 30 ( up triangles ) and 60 min .
( down triangles ) , at @xmath15 = 2.5 k. the inset shows an enlarged portion of the region just before the magnetization jump.,width=321 ] the fact that the step transition can be reached spontaneously while the external parameters ( @xmath2 and @xmath15 ) are kept constant indicates that the abrupt transition is truly connected with the probability of occurrence of certain microscopic process , which within the above described scenario is a particular enlargement of the fm phase .
however , this process will initiate the avalanche only when the local increment of the fm phase is large and fast enough to yield the appropriate increase of the local temperature through a magnetocaloric effect .
figure 4 and its inset clearly show that not any enlargement process is able to trigger the step transition .
the relaxation effects displayed by the system before the occurrence of the magnetization jump indicate that the system can in fact increase its fm phase fraction without initiating the avalanche , i.e. , there are fm regions that starts to become unblocked for field values just below the critical field , increasing their local volume by overcoming energy barriers .
for instance , the curve for @xmath62 = 60 min .
( with @xmath21 = 2.56 t ) shows a sizable increase of the fm fraction before the occurrence of the magnetization jump . from inspection of fig .
4 , it is likely that for larger values of @xmath62 larger values of the fm fraction before the jump would be obtained .
the waiting time @xmath62 is a key parameter to determine the energy barriers values for which the system is blocked .
eventually , for an extremely large value of @xmath62 the whole system would behave as unblocked and the @xmath1(@xmath2 ) curve obtained in this hypothetical situation would display a continuous metamagnetic transition behavior , without jumps .
therefore , once the value of the minimal @xmath21 corresponding to the fastest experiment is established , the limit temperature above which the step transition no longer occurs is determined by the blocking temperature corresponding to this field , @xmath18(@xmath21 ) .
this suggests why the magnetization jump occurs only below a very specific temperature , mahendiran above which the system overcomes the energy barriers without turning on the avalanche process . in conclusion ,
we have presented evidence that the low temperature abrupt field - induced transition occurring in phase separated manganites is intimately related with the sudden increase of the sample temperature at the first order transition point , a feature which is crucial for the understanding of the phenomenon .
we proposed a simple model in which the close interplay between the local increase of the fm phase and the heat released in this microscopic transformation can turn - on the avalanche leading to the observed step - like transition . within this framework ,
the entity which is propagated is heat , not magnetic domain walls , so the roles of grain boundaries in ceramic samples or strains which exist between the coexisting phases are less relevant .
the observation of spontaneous transitions gives additional support to that view , demonstrating that the step transition is not only the result of a crossover between macroscopic free energies induced by the magnetic field , but must be triggered by a microscopic mechanism which initiates the avalanche process .
additionally , we have established that a critical relative increment of a fm region or cluster is needed for the system to reach the `` chain reaction '' state , i.e. , larger initial fm factions require larger critical fields to _ turn - on _ the process , a feature previously observed .
@xcite finally , it must be emphasized that the basic condition for the occurrence of the abrupt transition is that the system must reach the low temperature regime in a strongly blocked state . at temperatures
just a few degrees higher the abrupt step - like transition no longer occurs , and it is replaced by a standard continuous metamagnetic transition @xcite . in summary , we propose that some microscopic mechanism promotes locally a fm volume increase , which yield a local temperature rise , and triggers the observed avalanche process .
this work was partially supported by capes , cnpq , fapesp , conicet , and fundacin antorchas .
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this work was supported by the canada excellence research chairs ( cerc ) program , the natural sciences and engineering research council of canada ( nserc ) and the uk epsrc . o.s.m.l .
acknowledges support from conacyt and the mexican secretaria de educacion publica ( sep ) .
dklo , jj , and vp acknowledge quisco .
v.p . , f.m.m . , d.k.l.o . and j.j .
developed the theory . f.m.m .
, a.c.l . and r.w.b .
conceived the experiment . m.m . o.s.m.l . and a.c.l .
carried out the experiment .
f.m.m . performed the data analysis .
all authors contributed to writing the paper .
the authors declare no competing financial interests .
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in 1924 david hilbert conceived a paradoxical tale involving a hotel with an infinite number of rooms to illustrate some aspects of the mathematical notion of `` infinity '' . in continuous - variable quantum mechanics we routinely make use of infinite state spaces : here we show that such a theoretical apparatus can accommodate an analog of hilbert s hotel paradox .
we devise a protocol that , mimicking what happens to the guests of the hotel , maps the amplitudes of an infinite eigenbasis to twice their original quantum number in a coherent and deterministic manner , producing infinitely many unoccupied levels in the process .
we demonstrate the feasibility of the protocol by experimentally realising it on the orbital angular momentum of a paraxial field .
this new non - gaussian operation may be exploited for example for enhancing the sensitivity of n00n states , for increasing the capacity of a channel or for multiplexing multiple channels into a single one .
the `` hilbert hotel paradox '' demonstrates the counterintuitive nature of infinity @xcite .
the hilbert hotel has infinitely many rooms numbered @xmath0 , all of which are currently occupied .
each new visitor that arrives can be accommodated if every current guest in the hotel is asked to move up one room ( @xmath1 ) .
even if a countably infinite number of new guests arrives at once , they can still be accommodated if each of the existing occupants moves to twice their current room number ( @xmath2 ) leaving the odd - numbered rooms free .
we may ask whether such phenomena can exist physically .
one possibility is in continuous - variables systems where in principle we have infinite ladders of energy eigenstates .
previously @xcite , the first of the hilbert hotel paradoxes ( with a single new guest ) was proposed in cavity qed using the sudarshan - glogower bare raising operator @xmath3 that shifts all the amplitudes up one level leaving the vacuum state unoccupied . here
, we show how we can implement the extended case where every second level of an infinite set of states is vacated .
this can be performed coherently and deterministically , preserving all the initial state amplitudes by remapping them to twice their original levels using a short and simple sequence of instantaneous , dynamic , and adiabatic processes .
we first show how to map the eigenstate amplitudes of a infinite square potential well to twice their original level , and then we report results of a physical implementation of an analogous protocol on the orbital angular momentum ( oam ) eigenstates of light , where we coherently multiply any linear superposition by a fixed integer ( in our case , by three ) . in the supplementary material
we describe further details of the experiment and we show that the square well protocol can be generalised to implement a multiplication of the eigenstate numbers by any positive integer , not only by two .
consider a quantum system with an infinite ladder of energy eigenstates bounded from below , @xmath4 .
an arbitrary state can be then represented as @xmath5 .
our earlier work @xcite has introduced the hilbert hotel operator @xmath6 , transforming @xmath7 to @xmath8 our new aim is to extend the toolbox by an operator @xmath9 , @xmath10 representing the second hilbert hotel paradox by leaving every second energy level vacant .
both operators are non - unitary isometries , as @xmath11 .
we show that we can deterministically implement @xmath9 on a infinite square potential well with initial width @xmath12 with the following operations ( fig .
[ fig : protocol ] ) : _ i _ ) we instantaneously expand the well from @xmath12 to @xmath13 , _ ii _ )
we let it evolve for the original fundamental period , _ iii _ ) we divide the well into two sub - wells of width @xmath12 with a barrier , _ iv _ ) we let each half - well evolve with a relative potential offset , to correct the relative phase , _ v _ ) we merge the half - wells together into one well of width @xmath13 , _ vi _ )
we adiabatically shrink the well back to width @xmath12 .
in general , the amplitudes of an initial state can be mapped to any integer multiple ( @xmath14 ) using a slightly modified procedure ( see supplementary material for details ) . within an infinite square potential
well . * b * , we instantaneously expand the well to twice its original width .
the original wavefunction is not immediately changed but the eigenbasis is different . *
c * , we allow free evolution for a period corresponding to the original fundamental period .
the wavefunction is reflected around the centre of the expanded well , with an undesired phase shift . *
d * , we insert an infinite barrier in the centre ( where the wavefunction is zero ) to split it into two independent wells that evolve separately , an energy shift on one well corrects the relative phase . *
e * , after the phase correction we align the potentials and merge the two halves back together .
* f * , an adiabatic compression of the well maps the eigenstates of the expanded well to those of the original well . the original wavefunction has now been halved and reflected , corresponding to the hilbert hotel operation
@xmath9 being applied to the eigenstates @xmath15.[fig : protocol],width=288 ] ideally steps _ i _ ,
_ iii _ and _ v _ should be instantaneous while step _ vi _ should be adiabatic .
the fidelity of a physical implementation will depend on the accuracy of the timing and the quality of the approximations , especially the maximum effective excitation number @xmath16 of the initial state in comparison to the validity regime of the schrdinger equation approximation in any realistic system under consideration .
the hilbert space of a particle in a well of width @xmath12 consists of the set of square - integrable functions , @xmath17 and the free particle hamiltonian is @xmath18 with boundary conditions @xmath19 .
this describes a one - dimensional particle in an infinite square potential well , but it can also describe other situations , e.g. an ideal two - dimensional optical waveguide within the paraxial wave approximation .
the hamiltonian yields an infinite ladder of nondegenerate energy eigenfunctions of the form @xmath20 with eigenvalues @xmath21 where @xmath22 . the desired operation @xmath9 transforms an initial state @xmath23 into @xmath24 interleaving the amplitudes of the initial state in the energy eigenbasis with zeros .
the first step of the hilbert hotel protocol is to double the width of the well so the original wave function @xmath25 extends from @xmath26 to @xmath27 , filling the new interval by constant zero .
we denote this extended wave function by @xmath28 and the free hamiltonian with the new boundary conditions @xmath29 by @xmath30 .
this hamiltonian has a new set of eigenfunctions @xmath31 which we use to express @xmath32 .
we allow @xmath28 to evolve over a time @xmath33 into @xmath34 where @xmath35 is @xmath36 for even @xmath16 and @xmath37 for odd @xmath16 , thus @xmath38 where @xmath39 is the identity operator and @xmath40 the mirror reflection ( or parity ) operator . therefore , after step
_ ii _ we have ( up to a global phase factor ) the state @xmath41 this resembles the point symmetry extension of @xmath25 to @xmath27 but the phase factor in @xmath42 needs to be corrected . steps _ iii _ , _ iv _ and _ v _ remove the undesired @xmath43 factor while preventing cross - talk between the two sub - wells . after splitting the interval @xmath27 in two ,
each part will evolve separately under the hamiltonian @xmath44 with appropriate boundary conditions .
the two halves can be phase - matched by applying potentials @xmath45 in @xmath26 and @xmath46 in @xmath42 for a time @xmath47 .
after removing the barrier ( step _ v _ ) , the wave function of the system becomes @xmath48 substituting for @xmath49 from , we find that both branches allow for a common analytic expression , as the domain of @xmath31 is twice that of @xmath49 : @xmath50 the final step is an adiabatic compression of the well back to its original width @xmath12 . up to a relative phase due to free evolution , which can be corrected by matching the total time of the evolution to an integer number of full revolutions of the running eigenbasis , this adiabatically
transforms the basis states @xmath31 into @xmath49 of the same @xmath16 , keeping coherent superpositions intact .
this shows the resulting state is indeed .
the crucial step in the hilbert hotel operation is the coherent mapping @xmath51 ( for @xmath52 ) on a countably infinite set of basis states @xmath53 , as described above . instead of a particle in an infinite square potential well , we can use systems that share important characteristics in order to perform analogous operations . in our experimental realisation ( fig .
[ hhsetup ] ) we choose the set of oam eigenstates of a beam of light , denoted by @xmath54 , and the coherent multiplication makes use of two well - known optical devices in a novel configuration : an oam sorter and a `` fan - out '' refractive coherent beam copier @xcite .
the oam multiplier has four steps : _ i _ ) unwrapping the initial azimuthal phase ring into a linear phase ramp with a polar - to - cartesian mapping , _ ii _ ) branching out new copies of the linearised field and correcting their relative phase with a suitable grating , _ iii _ ) demagnifying the juxtaposed copies with a cylindrical lens , and _ iv _ ) wrapping the resulting field back to polar coordinates .
the combination of these four steps amounts to the transformation : @xmath55 where @xmath56 is the number of copies produced in step _
ii_. the first step is achieved by way of an oam sorter @xcite , which unwraps any oam mode into a linear gradient ( and therefore it turns a combination of oam modes into a combination of linear gradients ) by way of an extremely astigmatic lens @xmath57 followed by a phase - correcting element @xmath58 , which effectively stops the unwrapping after the transformation is complete .
these two elements can be described by the phase delay that they impose on the incoming field as a function of position : @xmath59 where @xmath60 is the focal length of the fourier lens connecting near - field and far - field , @xmath61 is the wavelength of the light beam , and the free parameters @xmath62 and @xmath63 determine the scaling and position of the transformation in the fourier plane of coordinates @xmath64 and @xmath65 . at this point
we produce equal - weighted copies of the unwrapped phase ramp using a fan - out element by way of a suitable 1d phase grating on the far field .
it is crucial that the copies have the same intensity in order to obtain the desired oam modes at the end of the process . in our experiment ,
the fan - out grating produces three copies and the equation describing the phase delay of the grating as a function of position in the far field is @xmath66,\end{aligned}\ ] ] where @xmath67 .
such a phase mask does not depend on the @xmath68 coordinate , as we are copying a linear field .
this grating is displayed on a spatial light modulator ( slm ) , so the output of the sorter needs to be fourier transformed onto the fan - out slm with a @xmath69 system , followed by another @xmath69 system which images it through a second sorter operated _ in reverse_. in order to wrap the field back correctly without leaving wide gaps or without wrapping more than necessary , we use a cylindrical lens to demagnify the horizontal cartesian coordinate before the beam enters the reverse - sorter . exploiting the flexibility of slms
, we achieve this by adding the phase of a cylindrical lens directly on top of the fan - out grating . in the first part of our experiment we test the coherence of the protocol , i.e. its ability to preserve superpositions .
to do this , we generate balanced superpositions of @xmath70 and @xmath71 , with @xmath72 ranging from 1 to 3 .
such initial modes display @xmath73 maxima , or `` petals '' .
we feed them to the multiplier ( here set to multiply by @xmath74 ) and a successful protocol results in @xmath75 petals with high visibility at the output , as can be seen in fig .
[ petals ] . * coherent oam multiplication . *
top row : near field of input coherent superpositions .
bottom row : tripled output states .
the number of petals is @xmath75 , as expected from a coherent operation.,width=288 ] * oam multiplication performance .
* for each input eigenmode we measure the composition of the multiplied output .
circle size is linearly proportional to the overlap with the output modes .
as can be seen , the small leakage onto the neighbouring output modes is contained within a few adjacent modes .
a sufficiently distant input superposition such as @xmath76 would maintain an effective orthogonality.,width=288 ] in the second part of our experiment we assess the accuracy of the protocol by measuring the leakage onto neighbouring oam eigenmodes . to do this
, we multiply single oam eigenmodes by @xmath74 and projectively measure the oam spectrum of the output .
the results show that the overlap decays quickly enough for suitably distant superpositions to maintain their orthogonality ( fig .
[ dataanalysis ] ) .
for instance , the superposition @xmath76 which ideally maps to @xmath77 , was mapped to a superposition of modes , peaked on @xmath78 , but nevertheless with negligible cross - talk ( details in supplementary material ) . in summary
, we showed how to implement the hilbert hotel `` paradox '' , where the rooms of the hotel are the excitation modes of an infinite square potential well .
we then reported the successful implementation of the core step of the operation ( the coherent multiplication of the basis states of a countably infinite basis ) on the oam eigenmodes of a paraxial beam of light .
we show that the operation is coherent and that even in our proof - of - principle experiment , the multiplication of sufficiently distant modes can be performed with negligible overlap .
mode multiplication could be implemented also in other quantum systems , such as becs in a box potential with predicted talbot carpet features , though nonlinear interactions may spoil the ideal free particle expansion required for perfect wavefunction mirroring @xcite .
nonetheless , we note that this idea could be used to enhance several state production schemes without the need to modify the existing apparatuses , because it can act as an extension .
for instance , it could prove useful in quantum and classical information processing as a means of multiplexing an arbitrary number of input channels into a single output channel , or to enhance the sensitivity of systems that use n00n states , or to distribute ordered gaps in the spectral profile of a state .
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j. a. noble is a royal commission for the exhibition of 1851 research fellow
* corresponding author : s. coussan , [email protected]
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in the quest to understand the formation of the building blocks of life , amorphous solid water ( asw ) is one of the most widely studied molecular systems . indeed , asw is ubiquitous in the cold interstellar medium ( ism ) , where asw - coated dust grains provide a catalytic surface for solid phase chemistry , and is believed to be present in the earth s atmosphere at high altitudes .
it has been shown that the ice surface adsorbs small molecules such as co , n@xmath0 , or ch@xmath1 , most likely at oh groups dangling from the surface .
our study presents completely new insights concerning the behaviour of asw upon selective infrared ( ir ) irradiation of its dangling modes .
when irradiated , these surface h@xmath0o molecules reorganise , predominantly forming a stabilised monomer - like water mode on the ice surface .
we show that we systematically provoke `` hole - burning '' effects ( or net loss of oscillators ) at the wavelength of irradiation and reproduce the same absorbed water monomer on the asw surface .
our study suggests that all dangling modes share one common channel of vibrational relaxation ; the ice remains amorphous but with a reduced range of binding sites , and thus an altered catalytic capacity .
+ asw is a molecular system which has long provoked interest due , in part , to its role in the formation of molecules key to the origins of life@xcite .
asw has long been known to accrete small molecules such as co , h@xmath0o , n@xmath0 , or ch@xmath1@xcite , initiating chemical and photochemical surface reactivity@xcite . in the ism ,
water in the form of asw is the most abundant solid phase molecular species@xcite .
the production of molecules , from the most simple , h@xmath0@xcite , to the more complex ch@xmath2oh@xcite , and even precursors to the simplest amino acid , glycine@xcite , is catalysed by the asw surface@xcite ; both the outer surface and surfaces within its porous structure are involved .
the selective ir irradiation of crystalline ice@xcite and water clusters@xcite has already been studied . in the former case , the desorption of h@xmath0o molecules , and in the latter , the dissociation of clusters , was stimulated .
we are interested in the behaviour of asw upon selective ir irradiation and have studied the irradiation of the four surface modes of this ice , assigned in the literature@xcite and illustrated in figure [ fig : cartoon ] .
theoretical calculations , supported by experimental studies , suggest that water molecules in the dh mode are bi- or tri - coordinated , presenting one free oh bond dangling at the surface ; do molecules present a free oxygen electronic doublet ; and s4 molecules have a tetrahedral structure at the surface , which is distorted compared to the tetrahedra of bulk asw . ) , do ( 3549 @xmath3 ) , and s4 ( 3503 @xmath3 ) are illustrated on a representative asw sample.,scaledwidth=50.0% ] in these experiments we prepared a pure asw sample ( figure [ fig : cartoon ] ) as follows : deionised water was subjected to multiple freeze - pump - thaw cycles under vacuum to remove dissolved gases .
mixtures of purified h@xmath0o and helium ( air liquide , @xmath4 99.9999 % ) gas were prepared in a stainless steel dosing line with base pressure 10@xmath5 mbar in ratios of @xmath6 .
ices were produced by depositing the gas mixture directly onto a gold - plated copper surface held at 50 k ( to avoid trapping of the vector gas or nitrogen ) then cooled to 3.7 k ( the cooling typically takes around five minutes due to the high cryogenic power of 0.5 w at 4 k ) .
the cooled surface is located in a high vacuum chamber with a base pressure of 10@xmath7 mbar at 3.7 k. ir spectra were recorded in reflection mode using a bruker 66/s ftir spectrometer equipped with a mct detector ( 4000 800 @xmath3 ) .
full details of the experimental setup are given in coussan _
_ et al.__@xcite .
the ices grown in our study were characterised as purely amorphous in nature due to the position of the bulk oh stretch and the characteristic dangling modes at 3720 and 3698 @xmath3 ( see figure [ fig : cartoon])@xcite .
some previous studies using a helium carrier gas have reported the formation of ice nanocrystals or clusters@xcite , but this can be ruled out due to the absence of an absorption feature centred at 3692 @xmath3 . this band was never observed in any deposition of a h@xmath0o : he mixture during the work detailed here .
after deposition , ices were selectively irradiated using a tunable ir opo laserspec ( 1.5 4 @xmath8 m ) , pumped at 10 hz by a pulsed nd : yag quantel brilliant b laser ( 1064 nm , pulse duration 6 ns ) .
the average laser power is @xmath9 35 mw in the @xmath10 domain , except in the range 3520 3500 @xmath3 where it is @xmath9 10 mw , with a fwhm @xmath4 1.5 @xmath3 .
each irradiation was performed for one hour to ensure saturation of the effects . , ( b ) 3698 @xmath3 , ( c ) 3549 @xmath3 , and ( d ) 3503 @xmath3 .
the spectra are coloured as follows : before irradiation , blue ; after irradiation , green ; difference spectrum , red .
the insets magnify the irradiated range.,scaledwidth=50.0% ] figure [ fig : irr ] shows the results of irradiations carried out on the four dangling mode bands .
the irradiations provoke permanent `` hole - burnings '' in the irradiated band which are slightly shifted ( by up to 3 @xmath3 ) compared to the irradiation frequency .
the `` hole - burning '' is clearest for the irradiation of dh at 3698 @xmath3 but is also easily visible for the irradiations at 3720 and 3549 @xmath3 .
interestingly , upon each irradiation , we observe the growth of a new band centred at @xmath11 3725 @xmath3 , with fwhm @xmath9 5 @xmath3 .
both the shifts in frequency upon irradiation , and the narrowness of the `` hole - burning '' effects and the newly created bands at 3725 @xmath3 illustrate the inhomogeneity of the bands ; each band contains a distribution of oscillators , but only one class of oscillator isomerises upon irradiation at a given frequency , producing one new oscillator class .
after irradiation at 3698 @xmath3 , a second , smaller peak at 3638 @xmath3 is also observed ( see figure [ fig : depot]a ) . unirradiated asw samples and their ir spectra were stable and remained unchanged over the timescale of an irradiation study .
the common feature of all the irradiations in figure [ fig : irr ] is the growth of the 3725 @xmath3 band .
what is the source of this new band , previously unidentified in asw spectra ?
considering energetics only , irradiating between 3720 and 3503 @xmath3 could potentially break one or two hydrogen bonds , as the average weak h - bond strength is around 1800 @xmath3 .
is the band , therefore , due to h@xmath0o surface molecules in a different conformation than those in dh , do and s4 modes , or is it due to water molecules which have desorbed then re - adsorbed at the surface ?
the latter response can be immediately discarded based on the results of molecular adsorption studies which have shown a red - shift of the oh dangling frequency upon adsorption of multiple molecular species at the dangling modes@xcite .
for example , in the case of nitrogen adsorption onto asw , manca _ _ et al.__@xcite observed a red shift of 22 @xmath3 of the dh mode@xcite . in our experiments , the new band at 3725
@xmath3 is blue - shifted with respect to the dh modes .
moreover , the dynamic vacuum of 10@xmath7 mbar rapidly evacuates any desorbing molecules , such that the residual pressure is too low to allow redeposition .
thermal effects are discounted based upon annealing of asw samples , which provoked a global decrease of the dangling bonds and no production of narrow peaks at 3725 @xmath3 , in agreement with previous studies@xcite . upon irradiation ,
the newly produced band is narrow , indicating a single , homogeneous vibrational mode .
thus , the 3725 @xmath3 band is clearly not due to a perturbed dh mode , but it rather has dangling oh character , and we propose that it is samples a bi - coordinated h@xmath0o molecule , with two dangling oh and two co - ordinated electron pairs ( d2h ) . this structure explains the blue shift of the peak with respect to the dh mode at 3720 @xmath3 as , because neither of the two oh oscillators is directly hydrogen bonded , the free oh oscillators are less perturbed than the surface dh modes . ,
reproduced from figure [ fig : irr ] ; ( b ) an asw sample plus background - deposited water ; ( c ) spectrum b plus background - deposited nitrogen ; ( d ) an asw sample plus background - deposited nitrogen ; ( e ) spectrum d plus background - deposited h@xmath0o : n@xmath0 mixture.,scaledwidth=50.0% ] we performed further experiments to verify the source of the two new peaks at 3725 and 3638 @xmath3 , as illustrated in figure [ fig : depot ] .
the background deposition of water on a pure asw sample does not result in a new band , but rather the global growth of the dh and bulk oh bands ( figure [ fig : depot]b ) .
however , subsequent background deposition of pure nitrogen , a molecule present as a low - level source of pollution in the chamber , provokes the appearance of a band at 3726 @xmath3 ( figure [ fig : depot]c ) , revealed thanks to the magnifying effect of nitrogen@xcite .
the background deposition of n@xmath0 upon a new asw sample does not produce a clear peak ( figure [ fig : depot]d ) , but the background deposition of a h@xmath0o : n@xmath0 ( 1:10 ) mixture provokes the appearance of two narrow bands at 3725 @xmath3 and 3638 @xmath3 ( figure [ fig : depot ] ) , with a peak area ratio of approximately 10:1 , compared to 6:1 seen after irradiation at 3698 @xmath3 . when this deposition spectrum is compared to the spectrum of a gas - phase water monomer ( see table [ table : frequencies ] ) , the two peaks show red shifts of nearly 200 @xmath3 with respect to the @xmath12 modes of the monomer ( one always compares gas - phase experimental results with those of a `` classical '' theoretical calculation , as classical calculations are carried out for isolated species .
it also allows us to approximate the perturbation induced by the medium ) .
these red shifts are almost precisely those observed in the solid phase for water in a nitrogen matrix , where the @xmath13 ratio is 8:1 , and a h@xmath0o - n@xmath0 complex.@xcite considering our experimental results and the literature , the band at 3725 @xmath3 is positively attributed to a water monomer interacting with the surface via its two electronic doublets ; its large intensity compared to the other dh bands is explained by the magnifying effect of nitrogen , as extensively investigated by hujo _ _ et al.__@xcite .
we suggest that nitrogen molecules , present as a low - level pollutant in the chamber , serendipitously complex the water molecule , as illustrated in figure [ fig : depot ] , stabilising the molecule , preventing any further adsorption , and magnifying the oh stretching bands
. however , the nitrogen must only be present as a trace pollutant on the surface , as no shifting of the dh peak positions of the asw is seen , either during cooling of the water ice sample from 50 k to 3.7 k , or during the irradiation period .
if nitrogen were present at multilayer concentrations , we would expect to see a red shift@xcite of up to 22 @xmath3 .
the @xmath14 peak at 3638 @xmath3 is not observed after irradiation of the other three surface modes , but this is likely due to its low intensity compared to that of @xmath15 .
.comparison between calculated and observed @xmath15 and @xmath14 water monomer frequencies .
frequencies are given in @xmath3 . [ cols="^,^,^,^,^,^ " , ] one explanation for the observed `` hole - burning '' is that amorphous ice is unable to relax all of the vibrational energy injected at the surface through bulk relaxation channels .
as a result , some fraction of this energy is accumulated at the surface and in the immediate sub - layers , where it induces reconstruction of the surface .
we observe saturation of the `` hole - burning '' events within the timescale of the irradiations performed , suggesting that the rearrangement of surface molecules is not an efficient relaxation channel , but is a minority effect compared to the main relaxation channels via the bulk ice .
thus , the production of the 3725 @xmath3 band is not the major result of irradiation , but is due to the inability of the ice to fully dissipate the injected energy .
it has previously been suggested that asw is a disordered material which has no long - range organisation@xcite , which could help to explain the lack of efficiency in the bulk relaxation channels . although it is possible that some h@xmath0o molecules desorb upon irradiation , as in the studies of focsa _ _ et al.__@xcite , we consider this effect to be minor because no increase in pressure is observed in the chamber and the energy injected is not enough to break more than two h - bonds .
it is curious that the absorption band at 3725 @xmath3 is produced upon irradiation of all dangling bonds .
the irradiation effects at 3720 and 3698 @xmath3 , in particular , are very similar , except that the `` hole - burning '' of the doubly - coordinated dh is less pronounced than that of the triply - coordinated dh .
such molecular rearrangement requires only a reorientation of the water molecule and thus is not `` energy consuming '' compared to the energy injected into the system .
these results also provide evidence of a local ordering to the asw structure . upon irradiation of the dh modes ,
both the `` hole - burning '' in absorption bands and the newly produced monomer band at 3725 @xmath3 are relatively narrow ( fwhm @xmath9 5 @xmath3 ) .
this suggests that each surface molecule is surrounded by a locally ordered oscillator network , resulting reproducibly in the production of one oscillator class upon selective irradiation .
if we consider irradiation of the s4 band , centred at 3503 @xmath3 , we were unable to observe a definitive `` hole - burning '' event , likely due to the width of the band .
however , we observed an increase at 3725 @xmath3 , as for the dh modes .
it is unlikely that the tetra - coordinated s4 molecules are themselves ejected from within the surface layer , as this would be highly energetically unfavourable .
however , as we see an increase at 3725 @xmath3 , it is likely that the relaxation channel involves the breaking of h - bonds , with two breaks being enough to `` transform '' a s4 molecule into a monomer - like water molecule .
the case of do is likely intermediate between those of dh and s4 . during each of these irradiations
, we see no interconversion between the modes , suggesting that there is only a single `` surface '' channel for the release of excess vibrational energy . in this work we have provided new insights into the bonding and structure of the surface molecules in amorphous solid water .
the ensemble of our results show that surface modes , in particular the dh dangling bonds , are sensitive to photo - induced rearrangement due to competition between surface reorganisation and the main relaxation channels in the bulk water ice . rather than desorbing from the surface , molecules embedded in the surface layer
become loosely associated with the surface in the form of monomer - like structures interacting through their two free electron pairs .
the fortuitous presence of nitrogen in the chamber both promoted the magnification of the oh stretching mode of the monomer - like molecule and stabilised it on the ice surface .
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lm , sl and ac acknowledge the italian miur for financial support .
pu wishes to thank a.j . dean and j.b .
stephen for useful scientific discussions .
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using recent polarimetric observations of the crab nebula in the hard x - ray band by integral , we show that the absence of vacuum birefringence effects constrains @xmath0 lorentz violation in qed to the level @xmath1 at @xmath2 cl , tightening by more than three orders of magnitude previous constraints .
we show that planned x - ray polarimeters have the potential to probe @xmath3 by detecting polarization in active galaxies at red - shift @xmath4 .
experimental constraints on the parameters quantifying lorentz invariance violation ( lv ) are of fundamental importance .
because the lowest order corrections predicted in the photon dispersion relation imply the vacuum is birefringent , observations of polarized photons from distant astronomical sources provide very promising tests . in this letter
we exploit the recently discovered linear polarization of hard x - rays from the crab nebula ( cn ) @xcite .
these observations show a remarkably high degree of linear polarization ( @xmath5 ) and very close alignment of the polarization vector with both the optical polarization vector and the projection on the sky of the spin axis of the central neutron star . the high degree of polarization together with the lack of detectable rotation of the polarization vector of these @xmath6kev photons whilst propagating over the intervening @xmath7 cm enables us to tighten existing constraints by three orders of magnitude .
recent years have witnessed a growing interest in the possible high energy violations of local lorentz invariance as well as a flourishing of observational tests .
indeed , specific hints of lv arose from various approaches to quantum gravity @xcite .
however , most tests require a well established theoretical framework to calculate reaction rates and describe the particle dynamics . here , we work within the framework of effective field theory with non - renormalizable , mass dimension 5 lv operators ( see @xcite and references therein ) restricted to qed , for which the most general dispersion relations for photons and electrons are @xmath8 where ( [ eq : disp_rel_phot ] ) refers to photons and ( [ eq : disp_rel_ferm ] ) to fermions . @xcite . ]
we assume @xmath9 to be comparable to the planck mass @xmath10 .
the constants @xmath11 and @xmath12 indicate the strength of the lv .
the @xmath13 signs denote right and left circular polarization in ( [ eq : disp_rel_phot ] ) , and positive and negative helicity states of the fermion in ( [ eq : disp_rel_ferm ] ) .
equation ( [ eq : disp_rel_phot ] ) implies that the direction of polarization rotates during propagation due to the different velocities of the right- and left - handed circular polarizations , @xmath14 .
this effect is known as vacuum birefringence ( vb ) .
although it may seem hopeless to search directly for effects suppressed by the planck energy scale , even tiny corrections can be magnified to measurable ones when dealing with high energies , long distances of signal propagation or peculiar reactions ( see , e.g. , @xcite ) .
recently @xmath15 have been constrained to have a magnitude less than @xmath16 at 95% confidence level ( cl ) by a detailed analysis of the synchrotron component of the cn broadband spectrum @xcite , while the constraint @xmath17 has been obtained by @xcite considering the absence of vb effects during the propagation of optical / uv polarized light from gamma - ray bursts ( grb ) , were claimed in @xcite from grb 021206 observations @xcite ; however the result was later contested @xcite . ] .
there are also preliminary indications , based on an analysis of the photon fraction in ultra - high - energy cosmic rays , that these coefficients might be less than @xmath18 , though nothing conclusive can be claimed yet @xcite . in this work
we tighten the current constraints on @xmath19 suppressed lv by about three orders of magnitude for photons , by considering the limits on vb effects implied by the recently detected @xcite polarized hard x - rays from the cn .
firstly , we set such constraints following the arguments by @xcite , an approach robust against systematic uncertainties related to astrophysical modeling .
we then infer tighter limits that exploit and rely on modeling of the crab nebula and pulsar . finally , we consider the constraints which future x - ray polarization measurements of extragalactic objects , e.g. active galactic nuclei ( agn ) will allow .
this is of particular interest in the light of current experimental efforts to build x - ray polarimeters @xcite . during propagation over a distance @xmath20 ,
the ( cosmological ) distance is given by @xmath21 , which includes a @xmath22 factor in the integrand to take into account the red - shift acting on the photon energies .
as usual , @xmath23 is the present value of the hubble parameter , while @xmath24 and @xmath25 represent the density fractions of cosmological constant and matter in the universe , respectively . ] , the polarization vector of a linearly polarized plane wave with momentum @xmath26 rotates through an angle @xcite , @xmath27 observations of polarized light from a distant source can constrain @xmath28 in two ways , depending on the amount of available information on both the observational and the theoretical ( i.e. source modeling ) side : 1 . since detectors have a finite energy bandwidth , eq .
( [ eq : theta ] ) is never probed in real situations .
rather , if some net amount of polarization is measured in the band @xmath29 , an order - of - magnitude constraint arises from the fact that if the angle of polarization rotation ( [ eq : theta ] ) were to differ by more than @xmath30 over this band , the detected polarization would fluctuate sufficiently for the net signal polarization to be suppressed @xcite . from ( [ eq : theta ] ) , this constraint is @xmath31 this just requires that any intrinsic polarization ( at source ) is not completely washed out during signal propagation .
it thus relies on the mere detection of a polarized signal , without considering the observed polarization degree .
a more refined limit can be obtained by calculating the maximum observable polarization degree , given the maximum intrinsic value @xcite : @xmath32 where @xmath33 is the maximum intrinsic degree of polarization , @xmath34 is defined in eq .
( [ eq : theta ] ) and the average is weighted over the source spectrum and instrumental efficiency , represented by the normalized weight function @xmath35 @xcite .
conservatively , one can set @xmath36 , but a lower value can sometimes be justified on the basis of source modeling .
using , one can then cast a constraint by requiring @xmath37 to exceed the observed value .
2 . suppose that polarized light measured in a certain energy band has a position angle @xmath38 with respect to a fixed direction .
at fixed energy , the polarization vector rotates by the angle ( [ eq : theta ] ) ; if the position angle is measured by averaging over a certain energy range , the final net rotation @xmath39 is given by the superposition of the polarization vectors of all the photons in that range : @xmath40 where @xmath34 is given by ( [ eq : theta ] ) . if the position angle at emission @xmath41 in the same energy band is known from a model of the emitting source , a constraint can be set by imposing @xmath42 although this limit is tighter than that obtained from the previous methods , it clearly hinges on assumptions about the nature of the source , which may introduce significant uncertainties . in the case of the crab nebula ,
a @xmath43% degree of linear polarization in the @xmath44 band has recently been measured by the integral mission @xcite .
this measurement uses all photons within the spi instrument energy band .
however the convolution of the instrumental sensitivity to polarization with the detected number counts as a function of energy , @xmath35 , is maximized and approximately constant within a narrower energy band ( 150 to 300 kev ) and falls steeply outside this range @xcite .
for this reason we shall , conservatively , assume that most polarized photons are concentrated in this band .
given @xmath45 , @xmath46 and @xmath47 , eq .
( [ eq : decrease_pol ] ) leads to the order - of - magnitude estimate @xmath48 .
a more accurate limit follows from ( [ eq : pol ] ) . in the case of the cn
there is a robust understanding that photons in the range of interest are produced via the synchrotron proces , for which the maximum degree of intrinsic linear polarization is about @xmath49 ( see e.g. @xcite ) .
figure [ fig : casea ] illustrates the dependence of @xmath50 on @xmath11 for the distance of the cn and for @xmath51 . the requirement @xmath52 ( taking account of a @xmath2 offset from the best fit value @xmath53 ) leads to the constraint ( at 99% cl ) @xmath54 on @xmath11 for the distance of the cn and photons in the 150300 kev range , for a constant @xmath35 . ]
it is interesting to notice that x - ray polarization measurements of the cn already available in 1978 @xcite , set a constraint @xmath55 , only one order of magnitude less stringent than that reported in @xcite .
constraint ( [ eq : constraint - degree ] ) can be tightened by exploiting the current astrophysical understanding of the source .
the cn is a cloud of relativistic particles and fields powered by a rapidly rotating , strongly magnetized neutron star . both the _ hubble space telescope _ and the _ chandra _ x - ray satellite have imaged the system , revealing a jet and torus that clearly identify the neutron star rotation axis @xcite . the projection of this axis on the sky lies at a position angle of @xmath56 ( measured from north in anti - clockwise ) .
the neutron star itself emits pulsed radiation at its rotation frequency of 30 hz . in the optical band
these pulses are superimposed on a fainter steady component with a linear polarization degree of 30% and direction precisely aligned with that of the rotation axis @xcite . the direction of polarization measured by integral - spi in the @xmath57-rays is @xmath58 ( @xmath59 error ) from the north , thus also closely aligned with the jet direction and remarkably consistent with the optical observations .
this compelling ( theoretical and observational ) evidence allows us to use eq .
( [ eq : constraint - caseb ] ) . conservatively assuming @xmath60 ( i.e. @xmath2 from @xmath41 , 99% cl ) , this translates into the limit @xmath61 and @xmath62 for a @xmath63 deviation ( 95% cl ) .
figure [ fig : caseb ] shows @xmath64 as function of @xmath11 .
the left hand panel reports the global dependence ( the spikes correspond to rotations by @xmath65 ) , while the right hand panel focuses on the interesting range of values . ) rules out the possibility that the polarization angle is close to the expected one after rotating by some multiple of @xmath66 ( the polarization angle is defined on the interval @xmath67 $ ] ) . ]
the constraints presented in ( [ eq : constraint - degree ] ) and ( [ eq : constraint - serious - crab ] ) are remarkably strong .
although based on a cumulative effect , they are achieved using a local ( galactic ) object .
the reason lies , on the one hand , in the quadratic dependence of @xmath34 on the photon energy , in constrast with the linear gain given by distance ( see e.g. eq .
( [ eq : theta ] ) ) . on the other hand ,
the robust theoretical understanding of the cn has enabled us to strengthen the constraints significantly .
further improvements on lv constraints via birefringenge are expected thanks to the forthcoming high - energy polarimeters , such as xeus @xcite , pogolite @xcite , polar - x @xcite and gamma ray imager @xcite which will provide an unprecedented sensitivity , sufficient to detect polarized light at a few % levels also in extragalactic sources .
the lv limits will be optimized by balancing between source distance and observational energy range depending on the detector sensitivity .
this is illustrated in fig .
[ fig : plot ] , where the strength of the possible constraints ( cast with the first , most general method described above ) is plotted versus the distance of sources ( in red - shift @xmath68 ) and for different energy bands ( medium x- and @xmath57-rays ) .
remarkably , constraints of order @xmath69 could be placed if some polarized distant sources ( @xmath70 ) will be observed by such instruments at 1 mev .
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luc moreau is a postdoctoral fellow of the fund for scientific research - flanders . work done while a recipient of an honorary fellowship of the belgian american educational foundation , while visiting the princeton university mechanical and aerospace engineering department .
this paper presents research results of the belgian programme on inter - university poles of attraction , initiated by the belgian state , prime minister s office for science , technology and culture .
the scientific responsibility rests with its authors .
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some biological systems operate at the critical point between stability and instability and this requires a fine - tuning of parameters .
we bring together two examples from the literature that illustrate this : neural integration in the nervous system and hair cell oscillations in the auditory system . in both examples
the question arises as to how the required fine - tuning may be achieved and maintained in a robust and reliable way .
we study this question using tools from nonlinear and adaptive control theory .
we illustrate our approach on a simple model which captures some of the essential features of neural integration . as a result
, we propose a large class of feedback adaptation rules that may be responsible for the experimentally observed robustness of neural integration .
we mention extensions of our approach to the case of hair cell oscillations in the ear .
persistent neural activity is prevalent throughout the nervous system .
numerous experiments have demonstrated that persistent neural activity is correlated with short - term memory .
a prominent example concerns the oculomotor system
see @xcite for a review and experimental facts .
the brain moves the eyes with quick saccadic movements . between saccades , it keeps the eyes still by generating a continuous and constant contraction of the eye muscles ; thus requiring a constant level of neural activity in the motor neurons controlling the eye muscles .
this constant neural activity level serves as a short - term memory for the desired eye position . during a saccade
, a brief burst of neural activity in premotor command neurons induces a persistent change in the neural activity of the motor neurons , via a mechanism equivalent to integration in the sense of calculus .
neural activity of an individual neuron , however , has a natural tendency to decay with a relaxation time of the order of milliseconds .
therefore the question arises as to how a transient stimulus can cause persistent changes in neural activity . according to a long - standing hypothesis , persistent neural activity
is maintained by synaptic feedback loops .
positive feedback can oppose the tendency of a pattern of neural activity to decay .
if the feedback is weak , then the natural tendency to decay dominates and neural activity decays . as the feedback strength is increased , the neural dynamics undergo a bifurcation and become unstable .
when the feedback is tuned to exactly balance the decay , then neural activity neither increases nor decreases but persists without change .
this , however , requires a fine - tuning of the synaptic feedback strength and the question arises as to how a biological system can achieve and maintain this fine - tuning @xcite . some gradient descent and function approximation algorithms performing this fine - tuning
have been proposed @xcite and a feedback learning mechanism based on differential anti - hebbian synaptic plasticity has been studied in @xcite .
nevertheless , it is still unclear how the required fine - tuning is physiologically feasible . for this reason , a different model for neural integration based upon bistability
has recently been proposed in @xcite . in the present paper
, we do not follow the line of research based upon bistability .
instead , we pursue the hypothesis of precisely tuned synaptic feedback .
the present paper proposes an adaptation mechanism that may be responsible for the fine - tuning of neural integrators and that may explain the experimentally observed robustness of neural integrators with respect to perturbations .
before we present this adaptation mechanism in detail , we first discuss a similar phenomenon in the auditory system . in order to detect the sounds of the outside world , hair cells in the cochlea operate as nanosensors which transform acoustic stimuli into electric signals .
in @xcite these hair cells are described as active systems capable of generating spontaneous oscillations .
ions such as @xmath0 are believed to contribute to the hair cell s tendency to self - oscillate . for low concentrations of the ions ,
damping forces dominate and the hair cell oscillations are damped .
as the concentration increases the system undergoes a hopf bifurcation , the dynamics become unstable , and the hair cells exhibit spontaneous oscillations . in @xcite
the hair cells are postulated to operate near the critical point , where the activity of the ions exactly compensates for the damping effects . as before , this requires a fine - tuning of parameters ( the ion concentrations ) and again the question arises as to how this fine - tuning can be achieved and maintained . in @xcite
a feedback mechanism has been proposed which could be responsible for maintaining this fine - tuning .
it thus seems that operating in the vicinity of a bifurcation is a recurrent theme in biology . and
the question as to how proximity to the bifurcation point may be achieved and maintained in a noisy environment may be of considerable , general interest .
we view the two presented examples as special instances of the following general problem .
consider a forced dynamical system , described by a differential equation @xmath1 .
the right - hand side of this equation depends on a parameter @xmath2 and the unforced dynamics @xmath3 are assumed to exhibit a bifurcation when @xmath2 equals a critical value @xmath4 .
the problem consists of finding a feedback adaptation rule for the parameter @xmath2 which guarantees proximity to the bifurcation point ; that is , which steers @xmath2 toward its critical value @xmath4 .
this adaptation law may depend on @xmath2 and @xmath5 but should be independent of @xmath4 , since this critical value is not known precisely .
this abstract formulation captures common features of both biological examples and suggests some unexpected links with the literature .
questions very similar to the present one have been studied extensively in the literature on adaptive control @xcite and stabilization @xcite ; and the general problem is closely related to extremum seeking @xcite , and to instability detection @xcite , where an operating parameter is adapted on - line in order to experimentally locate bifurcations .
although the above general formulation is convenient , there is little hope that a complete and satisfactory theory can be developed that applies to all possible instances of the problem . simplifying assumptions
make it more tractable . in this letter
, we study in detail what is probably the most simple but nontrivial instance of the general problem .
we consider the one - dimensional system @xmath6 which captures some of the essential features of neural integration and is in fact closely related to the autapse model from @xcite . with this interpretation
, @xmath5 is a strictly positive variable representing neural activity in the integrator network and @xmath7 represents the signal generated by the premotor command neurons .
the term @xmath8 corresponds to the natural decay of neural activity and @xmath9 represents a positive , synaptic feedback loop .
of course , when studying neural integration , questions can be investigated at varying levels of detail .
it is clear that a simple model as ( [ e : ni ] ) has several limitations .
because of its one - dimensional nature , the present model is , for example , unable of reproducing the distributed nature of persistent activity patterns observed in the brain .
nevertheless , eq . ( [ e : ni ] ) captures a key feature of neural integration : when the feedback is tuned to exactly balance the decay , eq .
( [ e : ni ] ) behaves as an integrator and produces persistent neural activity .
eq .
( [ e : ni ] ) is therefore a valuable model when studying fine - tuning of neural integrator networks @xcite .
we are interested in the fine - tuning of eq .
( [ e : ni ] ) and study this question using tools from nonlinear and adaptive control theory .
first , we ignore the presence of the input @xmath7 and consider the simpler equation @xmath10 we present a large class of feedback adaptation laws for ( [ e : pa ] ) which steer @xmath2 to its critical value @xmath4 ; thus enabling the automatic self - tuning of parameters and the spontaneous generation of persistent neural activity .
we consider adaptation laws could come from synaptic plasticity . in particular , the term @xmath11 might be related to types of synaptic plasticity that depend on the temporal ordering of presynaptic and postsynaptic spiking , as in @xcite .
] of the form @xmath12 we show that , under three very mild conditions , this adaptation rule guarantees convergence to the bifurcation point for ( [ e : pa ] ) .
the first condition requires that @xmath13 is a strictly increasing function .
this means that the term @xmath14 in ( [ e : adaptation ] ) acts as a negative feedback . as a consequence , if the neural activity @xmath5 were constant in ( [ e : adaptation ] ) , then the synaptic feedback gain @xmath2 would naturally relax to a rest value depending on @xmath5 via the equation @xmath15 .
the second condition states that there exists @xmath16 such that @xmath17 .
this condition implies that , if the neural activity would be constant and equal to @xmath16 in ( [ e : adaptation ] ) , then the synaptic feedback gain @xmath2 would naturally relax to its critical , desired value @xmath4 .
of course there is no guarantee that the neural activity would be equal to , or even converge to , this special value @xmath16 . instead
, the level of neural activity is governed by eq .
( [ e : pa ] ) . therefore , in order for the adaptation law ( [ e : adaptation ] ) to work , we need to impose a last condition , that @xmath18 is a decreasing function .
this means that the level of neural activity negatively regulates the synaptic feedback strength .
we now show that , under these three conditions , the feedback adaptation law ( [ e : adaptation ] ) indeed tunes the synaptic feedback gain @xmath2 to exactly balance the natural decay rate @xmath4 .
we begin with noticing that the combined system of equations ( [ e : pa])([e : adaptation ] ) has a unique rest point .
this equilibrium is determined by setting the right - hand sides of ( [ e : pa])([e : adaptation ] ) equal to zero , yielding @xmath19 and @xmath20 .
although the precise value of @xmath4 is unknown , if we are able to prove that all trajectories of ( [ e : pa])([e : adaptation ] ) converge to this ( unknown ) fixed point , then it follows that @xmath2 indeed converges to its desired , critical value @xmath4 . in order to prove this
, we introduce a coordinate transformation @xmath21 and @xmath22 .
this transforms ( [ e : pa])([e : adaptation ] ) into @xmath23 and @xmath24 . in these new coordinates ,
the dynamics take the form of a nonlinear mass - spring - damper system ( with unit mass , nonlinear spring characteristic @xmath25 and nonlinear damping function @xmath26 ) .
it follows from physical energy considerations that this system exhibits damped oscillations @xcite .
this shows that all trajectories of ( [ e : pa])([e : adaptation ] ) indeed converge to the unique fixed point , where @xmath20 .
the above coordinate transformation reveals a subtle relationship between self - tuning of bifurcations and the internal model principle ( `` imp '' ) from robust control theory ( see @xcite for a discussion of the imp from a systems biology perspective ) .
this relation is made explicit by the equation @xmath23 , which represents an integrator and corresponds to integral action studied in robust control theory .
one regards the constant @xmath4 as an unknown perturbation acting on the system .
the imp implies that , in order to track this constant perturbation , the system dynamics should contain integral action .
the integral action is generated by the biological system itself , and not by the feedback adaptation law .
we have so far ignored the presence of the signal @xmath7 .
we showed that the adaptation law ( [ e : adaptation ] ) tunes the synaptic feedback gain to exactly compensate for the natural decay rate , resulting in the spontaneous generation of persistent neural activity . at these equilibrium conditions ,
the action potential firing rate equals @xmath16 , which is related to @xmath4 by @xmath17 . in the next paragraphs
, we take into account the effect of the input @xmath7 . in this case
, the value @xmath16 will play the role of a parameter that influences the accuracy with which the feedback adaptation law guarantees proximity to the bifurcation point .
the signal @xmath7 will in general result in a time - varying action potential firing rate @xmath27 .
the mechanism with which this happens , is determined by the neural integrator equation ( [ e : ni ] ) and the adaptation law ( [ e : adaptation ] ) . for the purpose of analysis ,
we make two simplifying assumptions , both of which seem to be natural and physically relevant for neural integration .
first , we assume that , over any sufficiently large time interval @xmath28 $ ] , the time spent by @xmath27 in any interval @xmath29 $ ] is approximately independent of @xmath30 . in more mathematical terms , we assume the existence of a function @xmath31 such that for every test function @xmath32 , the time average @xmath33 converges to @xmath34 as @xmath35 , uniformly with respect to @xmath30 .
secondly , we assume that the adaptation law acts on a much slower time scale than the time variations in @xmath27 . under these assumptions ,
the effect of the action potential firing rate @xmath27 on the adaptation law ( [ e : adaptation ] ) may be approximated by the average effect @xmath36 .
it is now clear when the adaptation law guarantees proximity to the bifurcation point : if the compatibility condition @xmath37 is satisfied , then time scale separation arguments suggest that @xmath2 will converge approximately to @xmath4 and the neural integrator will approximately behave as a perfect integrator .
the compatibility condition may by interpreted as follows @xcite . when the premotor command signal @xmath7 has zero time - average and the adaptation law acts on a slow time scale , then eq .
( [ e : ni ] ) behaves as a good integrator and the firing rate @xmath27 equals the time - integral of @xmath7 plus an integration constant .
the compatibility condition ensures that this integration constant is compatible with the desired range for the firing rate @xmath27 .
we illustrate this result on a particular example representative for saccadic eye movements .
we consider the case of periodic saccadic eye movements asking for an action potential firing rate in the motor neurons alternating between @xmath38 and @xmath39 every second . at each saccade
, a brief burst of neural activity in premotor command neurons changes the actual firing rate .
we assume that this change is such that immediately after each saccade , the actual firing rate equals the desired firing rate . between saccades
, we assume that no input is applied . at its desired level between saccades ,
which is consistent with experimental observations . ]
if the neural integrator is perfectly tuned , then the actual firing rate will remain constant between saccades and equal to the desired firing rate ( eyes are fixed ) .
if the neural integrator is not perfectly tuned , then the actual firing rate will deviate from the desired firing rate ( eyes drift ) until a new saccade occurs which brings the actual firing rate to its new desired value .
fig .
[ f2 ] shows the results of a simulation where the adaptation law satisfies the compatibility condition of the previous paragraph . in the beginning of the simulation
, we have mis - tuned the neural integrator .
clearly , after a short transient , the adaptation law achieves excellent tuning and the drift between two successive saccades becomes negligible .
we have thus shown that an adaptation law can tune a neural integrator with great accuracy to its bifurcation point . in order to achieve perfect tuning
, however , the adaptation law itself needs to satisfy a compatibility condition .
it seems that we have merely moved the problem of fine - tuning from the neural integrator to the adaptation law . the crucial observation and one of the main contributions of the present paper , however , is that this results in a significant decrease in sensitivity . _
the adaptation law is robust with respect to perturbations in its parameters . _ in order to illustrate this significant increase in robustness , let us first summarize the well - known @xcite sensitivity properties of neural integration .
experiments suggest that the actual time constant obtained in a tuned neural integrator circuit is typically greater than @xmath40 ; that is , @xmath41 .
this requires for the fine - tuning of @xmath2 a relative precision @xmath42 ranging from @xmath43 to @xmath44 , depending on whether the intrinsic time constant @xmath45 equals @xmath46 or @xmath47 ( typical values suggested in the literature ) . the required precision for @xmath2
should be contrasted with the required precision for the parameters of the adaptation law proposed in the present paper .
the simulations of fig .
[ f3 ] show that , in order to have @xmath41 as observed in experiments , the parameters of the adaptation law need to be tuned with a precision of @xmath48 , independent of the intrinsic time constant @xmath45 . comparing this with the originally required precision for the synaptic feedback strength @xmath2
, we conclude that _ the proposed adaptation mechanism could improve the robustness of neural integration with a factor ranging from @xmath49 to @xmath50_. we have studied a simple model for neural integration and proposed a class of feedback adaptation rules that could explain the experimentally observed robustness of neural integration with respect to perturbations . the analysis tools that we have introduced extend to the study of fine - tuning involved in other systems such as hair cell oscillations in the ear @xcite .
consider the nonlinear oscillator equation @xmath51 , which captures some of the essential features of hair cell oscillations @xcite .
inspired by our previous analysis , we consider a feedback adaptation law for the parameter @xmath2 of the form @xmath52 , with @xmath53 a positive variable characterizing the magnitude of oscillations and related to @xmath5 and @xmath54 via the expression @xmath55 .
fig .
[ f4 ] shows that , in the absence of the stimulus @xmath7 , this type of adaptation law is indeed able to bring and keep the bifurcation parameter close to its critical value , resulting in the spontaneous generation of oscillations .
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the b - v colour difference between young and evolved bars is 0.4 mag , which can be translated to an age difference of 10 gyr .
this means that bars can be robust structures , in agreement with recent @xmath0-body simulations and observations of barred galaxies at higher redshifts .
the young bars in our sample have an average length of [email protected] kpc , while the evolved bars have an average length of [email protected] kpc , consistent with recent theoretical expectations that bars grow longer while aging .
young bars are preferentially found in late - type spirals , indicating that bar recurrence may be more frequent in gas - rich , disk - dominated galaxies .
we also found that agn are preferentially hosted by galaxies with young bars , suggesting that the fueling of agn by bars happens in short timescales and that a clearer bar - agn connection would be found in a sample of galaxies with young bars .
we have also found that bar colours might be used as a proxy for bar ages .
enlarging the sample of bars with measured ages is paramount to calibrate this relation , confirm these results , compare in more detail observations and models , and better understand secular evolution .
see ( * ? ? ?
* ; * ? ? ?
* gadotti & de souza ( 2005 , 2006 ) ) for further details .
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in an effort to obtain further observational evidences for secular evolution processes in galaxies , as well as observational constraints to current theoretical models of secular evolution , we have used bvri and ks images of a sample of 18 barred galaxies to measure the lengths and colours of bars , create colour maps and estimate global colour gradients .
in addition , applying a method we developed in a previous article , we could distinguish for 7 galaxies in our sample those whose bars have been recently formed from the ones with already evolved bars .
we estimated an average difference in the optical colours between young and evolved bars that may be translated to an age difference of the order of 10 gyr , meaning that bars may be long standing structures .
moreover , our results show that , on average , evolved bars are longer than young bars .
this seems to indicate that , during its evolution , a bar grows longer by capturing stars from the disk , in agreement with recent numerical and analytical results .
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the author thanks prof .
j. katriel for helpful comments on the manuscript .
j. katriel , through correspondence , showed that eqns [ eq : evenzeros ] and [ eq : oddzeros ] agree with the first asymptotic term from dominici for @xmath51 for low @xmath40 ( not for maximal @xmath40 ) .
the latter result makes sense from the point of view that the maximal @xmath8 is always close to the edge of the hilbert space where the wavefunction goes to zero for any finite @xmath12 whereas that of the harmonic oscillator decays forever .
[ eq : evenzeros ] and [ eq : oddzeros ] do not agree with higher order terms ( w.r.t .
@xmath52 ) in dominici s asymptotic expansion .
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the holstein - primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of hermite polynomials @xmath0 as @xmath1 .
this establishes a correspondence between the zeros of the hermite polynomials and the boundaries of the position basis of finite - dimensional hilbert spaces .
the hermite polynomials are prevalent in many fields .
they can be defined as @xmath2 in the physics community , they are perhaps best recognized as the gaussian - weighted eigenfunctions ( in position representation ) of the quantum harmonic oscillator ( with @xmath3 , a convention that will be used for the rest of the paper ) : @xmath4 as such , they are orthogonal over the gaussian - weighted whole domain , @xmath5 .
this last property allows their use in gaussian quadrature , a useful and popular numerical integration technique where @xmath6 is approximated as @xmath7 where @xmath8 are the zeros of @xmath0 and @xmath9 is a well - behaved function . for this and many other reasons ,
an analytic formula for the asymptotic zeros of hermite and other orthogonal polynomials has been a subject of much interest@xcite , especially in the applied mathematics community and the field of approximation theory . in this paper ,
i examine the position state representation of the eigenstates of finite dimensional @xmath10-spin systems , as expressed in the holstein - primakoff transformation . as @xmath11 ,
the system becomes the infinite dimensional harmonic oscillator .
this association allows me to derive the simple main results presented in eqs [ eq : evenzeros ] and [ eq : oddzeros ] , with solutions that i conjecture become the asymptotic zeros of the hermite polynomials ( as @xmath1 ) .
furthermore , i numerically show that this convergence is rather quick and so the expressions can frequently be used , in many instances of finite - precision application , as the effective zeros of @xmath0 with finite @xmath12 , such as in applications of gaussian quadrature . in a more aesthetic sense
, these results also establish a beautiful correspondence between the boundaries of equal area partitions of circles with radii that are increasing in a certain manner and the hermite polynomial zeros .
spin systems are defined by the fundamental commutation relations between operators @xmath13 , @xmath14 and @xmath15 : @xmath16 = \hat s^+ , \hskip 5pt \left [ \hat s^z , \hat s^- \right ] = \hat s^- , \hskip 5pt \left [ \hat s^+ , \hat s^- \right ] = 2 \hat s^z.\ ] ] associating a spin with a boson @xmath17 , holstein and primakoff showed that to satisfy these commutation relations , the operators can be expressed as@xcite @xmath18 @xmath19 this is a very useful association and has found many applications in the condensed matter field s study of many - body spin systems .
each boson excitation represents the `` ladder up '' finitesimal excitation away from the spin s extremal @xmath10 state .
the hilbert space is finite - dimensional and possesses @xmath20 states @xmath21 .
in fact , considering eq .
[ eq : hkladder ] it is clear that the hilbert space outside this defined space is not even hermitian . transforming from the holstein - primakoff bosonic representation to position ( and its conjugate momentum ) space ( using the relations @xmath22 and @xmath23 ) reveals that the trivial hamiltonian is the harmonic oscillator : @xmath24 .
moreover , transformation of the @xmath14 and @xmath15 in eq .
[ eq : hkladder ] reveals that the hilbert space spans the domain @xmath25 .
just as in the @xmath26 representation , @xmath27 states all with the same area must exist within this domain .
fig .
[ fig : qbasis ] sketches out what they look like for the @xmath28-spin systems .
-basis representation of a ) @xmath29 , b ) @xmath30 and c ) @xmath31 systems is shown .
the radius of the hilbert space s domain is equal to @xmath32 and so grows along with the number of allowed basis elements . ] for a particular @xmath10-spin system , the lowest eigenstate must have the same sign at all @xmath33-basis elements since it must be nodeless . on the other hand
, the highest eigenstate must have @xmath34 nodes and so the @xmath33-basis elements must alternate in sign such that the eigenfunction passes through zero between them .
this latter behavior is sketched in fig .
[ fig : qbasis ] in red by the hermite polynomial @xmath0 denoting the value of the overlying @xmath33-basis element for the highest eigenstate . for @xmath11 ,
the hilbert space becomes infinite - dimensional and the hamiltonian becomes that of the harmonic oscillator defined over @xmath35 with the associated eigenfunctions proportional to @xmath36 .
it therefore follows that as @xmath11 , the boundaries of the @xmath33-basis elements become the zeros of the hermite polynomial @xmath0 where @xmath37 since the highest eigenstate must still have alternating sign with each @xmath33-basis element .
hermite polynomial zeros @xmath8 are real and symmetric around @xmath38 . to determine these boundary points , the @xmath39-dimensional hilbert space s circular shape in position space can be exploited . for even @xmath39 , the area of the all the @xmath33-basis elements up until the @xmath40th boundary ( measuring from the origin ) is @xmath41 . for odd @xmath39 ,
the area is @xmath42 .
this is illustrated in fig .
[ fig : areas ] . or b ) @xmath43 @xmath33-basis elements that approximately determine the @xmath40th zero of the hermite polynomial @xmath0 for @xmath12 even and odd respectively is shaded in blue .
the approximate @xmath40th zero is at the right boundary of these regions . ] using simple relations for the area of circle sectors and rectangles , it is possible to relate these @xmath33-basis element areas to @xmath8 ; the equation involving the approximate zeros of hermite polynomials @xmath44 with @xmath12 even is : @xmath45 while for odd @xmath12 it is : @xmath46 where @xmath47 and @xmath48 . solving these equations for @xmath8 yields the approximate @xmath40th zero for the @xmath12th hermite polynomial .
the results for the zeros of the first @xmath49 hermite polynomials are compared to the exact zeros in fig .
[ fig : zeros ] . in both cases ,
eqs .
[ eq : evenzeros ] and [ eq : oddzeros ] converge to the zeros of the hermite functions quite quickly and [ eq : oddzeros ] agree with the first asymptotic term from dominici@xcite for @xmath1 for low @xmath40 ( not for maximal @xmath40 ) .
the latter result makes sense from the point of view that the maximal @xmath8 is always close to the edge of the hilbert space where the wavefunction goes to zero for any finite @xmath12 whereas that of the harmonic oscillator decays forever .
eqs .
[ eq : evenzeros ] and [ eq : oddzeros ] do not agree with higher order terms ( w.r.t .
@xmath50 ) in dominici s asymptotic expansion . ] . th zeros of the hermite polynomials @xmath0 for @xmath12 a ) even and b ) odd compared to those obtained from solving eqs .
[ eq : evenzeros ] and [ eq : oddzeros].,title="fig : " ] th zeros of the hermite polynomials @xmath0 for @xmath12 a ) even and b ) odd compared to those obtained from solving eqs .
[ eq : evenzeros ] and [ eq : oddzeros].,title="fig : " ] the finding that the boundaries of equal area partitions of growing circles correspond to the asymptotic zeros of the hermite functions appears to be a novel one from a search of the literature .
it is all the more surprising that the origin of this one - to - one correspondance stems from the holstein - primakoff representations for finite - dimensional spin systems .
furthermore , on a practical level , the apparently rapid convergence of these solutions suggests that they may be useful for more efficient determination of hermite polynomial zeros for large - dimensional implementations of gaussian quadrature .
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99 bohigas o 1991 _ random matrix theories and chaotic dynamics _ , in giannoni m j , voros a and zinn - justin j ( eds ) , _ proceedings of the 1989 les houches summer school on chaos and quantum physics _ , pages 88 - 199 .
( amsterdam : elsevier ) .
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the spectral statistics of the circular billiard with a point - scatterer is investigated . in the semiclassical limit
, the spectrum is demonstrated to be composed of two uncorrelated level sequences .
the first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point - scatterer .
the nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found .
classical dynamics may be illuminating for the understanding of the corresponding quantum mechanical systems .
one of the most studied aspects of the relation between classical and quantum mechanics is the connection between the spectral statistics of the quantum system and the dynamical properties of its classical counterpart .
classically integrable systems typically exhibit poisson - like spectral statistics @xcite while classically chaotic systems exhibit spectral statistics of random matrix ensembles @xcite .
the spectral statistics of integrable and chaotic systems are universal , that is , they do not depend on specific details of the system but rather on the type of motion and its symmetries .
there are systems which are intermediate between integrable and chaotic ones and their spectral properties are not known to be universal .
such systems are of experimental relevance .
the spectral statistics of mixed systems , for which the phase space is composed of both integrable and chaotic regions , were studied by berry and robnik @xcite .
the spectrum can be viewed as a superposition of uncorrelated level sequences , corresponding to the various regions , which are either chaotic or integrable .
the nearest neighbor spacing distribution ( nnsd ) of such a superposition of sequences was calculated in @xcite .
the resulting statistics are , in some sense , intermediate between those of integrable and chaotic systems .
other types of systems with intermediate statistics include pseudointegrable systems and integrable systems with flux lines or point - scatterers @xcite .
the spectral statistics of billiards with flux lines @xcite and of some pseudointegrable billiards @xcite were recently studied .
a possible route towards an understanding of the spectral statistics of these systems is based on their classical periodic orbits .
it is possible to compute the correlation function of the energy levels from these orbits by using trace formulae @xcite . for the billiards with flux lines or for pseudointegrable billiards ,
the orbits include not only the periodic orbits but also diffracting orbits which are built from segments that start and end at some singularity of the system .
while the contributions from periodic orbits were easily calculated , those from diffracting orbits turn out to be much more involved .
the spectral statistics of these systems , that can be obtained numerically , appear to be intermediate between those of integrable and chaotic systems .
in particular , the nnsd show level repulsion at small spacings and an exponential falloff at large spacings . since the contributions of diffracting orbits to the spectral statistics of pseudointegrable systems and of billiards with flux lines is far from being understood , it is of interest to study simpler systems which exhibit intermediate statistics . a class of such systems is given by integrable systems with a point - scatterers .
a point - scatterer is the self adjoint extension of a `` @xmath0-function potential '' in two or three dimensions @xcite .
the spectral statistics of an integrable system with such a point - scatterer was first studied by eba @xcite .
it is a rectangular billiard with the perturbation at its center .
this `` eba billiard '' also exhibits intermediate statistics which differs from that of psedointegrable systems .
integrable systems with point - scatterers are much easier to study analytically compared to integrable systems with flux lines or to pseudointegrable systems .
the contributions of diffracting orbits to the correlation function of the energy levels were recently calculated for the rectangular billiard with a point - scatterer @xcite .
exact results for the nnsd were also obtained @xcite .
one of the intriguing features of the spectral statistics of the rectangular billiard with a point - scatterer ( and dirichlet boundary conditions ) is its dependence on the location of the scatterer . if the coordinates of the scatterer ( divided by the sides of the rectangle ) are rational numbers @xmath1 ( where @xmath2 ) then the spectral statistics depend in a non trivial way on @xmath3 @xcite .
in contrast , for typical locations , the spectral statistics seem to be location independent .
the cause for this dependence on location is that many wavefunctions vanish at rational values of the coordinates . at these locations
there are many degeneracies in the lengths of the diffracting orbits ( including repetitions ) .
since such dependence on location is atypical , it is of interest to study the dependence of the spectral statistics on the location of the perturbation in other systems .
for instance , it may be possible that for typical systems the location of the point scatterer affects the spectral statistics only smoothly ( and does not depend on the rationality of the coordinates of the scatterer ) .
the system that is studied in this work is the circle billiard perturbed by a point - scatterer and the dependence of the spectral statistics on its location is studied .
the ( two dimensional ) circle billiard with radius @xmath4 is described by the schrdinger equation @xmath5 with dirichlet boundary conditions @xmath6 .
( the units where @xmath7 are used in most of this work . )
the hamiltonian with the point - scatterer is the self - adjoint extension of a hamiltonian where one point , say @xmath8 , is removed from its domain .
it can be considered as the self - adjoint extension of a hamiltonian with a @xmath0-function potential at @xmath8 . given the eigenvalues ( and eigenfunctions ) of the unperturbed system @xmath9 ( and @xmath10 ) ,
namely the system in absence of the @xmath0-scatterer , the eigenvalues of the system with the point - scatterer are given by the roots of @xcite @xmath11 where @xmath12 and @xmath13 are two parameters .
for a more complete discussion regarding this equation , the roles of the parameters as well as the method of its numerical solution see , for example , @xcite .
this equation turns out to be very convenient for numerical solution since every root is located between two eigenvalues of the unperturbed system .
the ( unnormalized ) eigenfunctions of the circle billiard are @xmath14 where @xmath15 are bessel functions of the first kind and the angular momentum @xmath16 is any nonnegative integer .
the energy levels @xmath17 are determined by the boundary condition @xmath18 .
it is obvious that all the energy levels with @xmath19 are doubly degenerate . as a result
there is a linear combination of the two degenerate wavefunctions which vanishes at @xmath8 and an orthogonal linear combination which does not vanish at @xmath8 .
the perturbation breaks this degeneracy . the linear combination which vanishes is also an eigenfunction of the hamiltonian _ with _ the point - scatterer and
thus @xmath20 is an eigenvalue of the perturbed problem .
therefore , half of the spectrum is unchanged by the perturbation . to avoid this trivial part of the spectrum we choose to work with the non vanishing
linear combinations ( which are eigenfunctions of the unperturbed hamiltonian ) and only the half of the spectrum which is affected by the perturbation will be considered in this work . for convenience ,
the location of the perturbation is chosen at @xmath21 and therefore the eigenfunctions of the unperturbed hamiltonian which do not vanish there are @xmath22 the spectrum is determined by substituting these eigenfunctions and the corresponding energies @xmath17 in equation ( [ solvept ] ) .
we are interested in the dependence of the spectral statistics on the location of the scatterer .
this dependence can be easily understood in terms of the properties of the wavefunctions of the unperturbed system .
the quantum numbers @xmath23 correspond to a state with an angular momentum @xmath24 and an energy @xmath25 . in the semiclassical limit where @xmath26 , but @xmath27 and @xmath28 are kept fixed ,
the values of the wavefunctions are small in the classically forbidden region @xmath29 .
therefore , one can divide the states into two groups .
the first consists of states for which the point - scatterer is located in the classically forbidden region .
as will be demonstrated , the eigenvalues of these states change only slightly due to the perturbation ( and do not change at all in the semiclassical limit ) .
the second group includes the states for which the perturbation is located in the classically allowed region .
these states will be ( strongly ) affected by the perturbation .
this separation of the spectrum into a superposition of two sequences is the cause for the dependence of the spectral statistics on the location of the scatterer .
this separation into ( semiclassically ) affected and unaffected states is justified in the following . to demonstrate that some of the eigenvalues are almost unchanged by the perturbation one should solve equation ( [ solvept ] ) and show that for ( exponentially ) small eigenfunctions the corresponding eigenvalues are almost unaffected . for simplicity , instead of equation ( [ solvept ] ) it is sufficient to consider the finite sum @xmath30 where @xmath31 is assumed to be slowly varying function of @xmath32 .
we denote @xmath33 . to further simplify the argument
let us assume that @xmath34 and @xmath35 are of order unity while for @xmath36 the wavefunctions on the scatterer @xmath37 are all small .
the solutions of ( [ finitept ] ) are to a good approximation given by the solutions of @xmath38 for @xmath39 and by @xmath40 for @xmath36 .
this is true since substituting a solution of the form @xmath41 in ( [ finitept ] ) leads to @xmath42 expanding with respect to @xmath43 and then solving to the leading order in @xmath44 results in @xmath45 when @xmath46 , as is the case in the semiclassical limit , @xmath43 also approaches @xmath47 . for the other eigenenergies one can substitute @xmath48 ( with @xmath36 ) and find that in the leading order @xmath49 which also vanish when @xmath46 .
note that we have just found @xmath50 ( approximate ) solutions which are all the solutions between @xmath51 to @xmath52 .
it is not hard to generalize this calculation for more wavefunctions which are of order unity . in the semiclassical limit
, the resulting spectrum always consist of such affected and unaffected components . in this limit
the values of the wavefunctions at the scatterer are exponentially small if it is located in the classically forbidden region and the corresponding eigenvalues can be treated as unchanged by the perturbation for any semiclassical consideration . consider a circle billiard with the point - scatterer at @xmath53 .
the spectral statistics of its energy levels in an energy window of width @xmath54 around @xmath55 are studied in what follows .
assume that @xmath55 is very large compared to @xmath54 , i.e. that all the levels in the window have similar energies which are high enough to be considered semiclassical . a natural question to ask is how many of these levels are affected by the point - scatterer and how many are unaffected by it .
as was argued , the semiclassically unaffected levels are those for which the classical turning point , @xmath56 , satisfies @xmath57 .
equivalently , for a given energy @xmath55 , for a state to be unaffected , the angular momentum @xmath28 should satisfy @xmath58 ( for the estimate of @xmath59 , the values of the energies are approximated by @xmath55 ) .
the fraction of such states was calculated in @xcite where it was used to determine how many levels are poorly approximated in the wkb method .
this fraction is @xmath60.\ ] ] it is clear that when @xmath61 then @xmath62 , as expected , while for @xmath63 , @xmath64 .
as one approaches the semiclassical limit these states are less and less affected .
the predictions of equation ( [ xunaff ] ) can be checked numerically .
the number of unaffected levels was calculated for two energy windows as a function of the location of the perturbation .
the results are presented in figure [ unfig ] . , width=453,height=377 ] the radius of the billiard was chosen so that the mean level spacing is @xmath65 .
therefore both energy windows contain @xmath66 levels .
a level was counted as unaffected if its difference from an energy level of the unperturbed system was less than @xmath67 of the mean level spacing .
this criterion is somewhat arbitrary , since in the semiclassical limit the difference can be taken to be arbitrarily small .
figure [ unfig ] indicates that equation ( [ xunaff ] ) correctly describes the number of unaffected levels .
there is a slight deviation which is smaller for the levels from the higher energy window .
this deviation is caused by the fact that for any finite energy the wavefunctions do not vanish at the turning point @xmath56 but rather exhibit an airy - like structure ( in the radial direction ) near the turning point .
this means that for states with @xmath57 , for which @xmath56 is close to @xmath68 , @xmath69 might not be small at a finite ( but large ) energy .
the deviations are expected to vanish in the semiclassical limit as indicated by figure [ unfig ] .
the spectrum of the circle billiard with a point - scatterer can therefore be viewed as composed of two uncorrelated components .
one is unaffected by the point - scatterer and its relative fraction is @xmath70 while the other is affected and its relative fraction is @xmath71 .
the nnsd of a spectrum which is composed of several uncorrelated level sequences was computed by berry and robnik @xcite and is applied to the circle billiard with the point - scatterer in what follows .
the unaffected spectrum consists of many levels with different angular momentum quantum numbers and thus its statistics are poissonian @xcite . since the density of levels in this sequence is @xmath70 , and the radius was chosen so that the total level density is unity , its nnsd is @xmath72 the second level sequence contains the levels which are influenced by the point - scatterer and their density is @xmath71 .
the exact form of its nnsd is unknown and an exact computation of this nnsd is complicated and beyond of the scope of this letter . instead , following experience with other systems @xcite
, we will _ assume _ that the nnsd can be _ approximated _ by a semi - poisson distribution , that is , @xmath73 this distribution exhibits level repulsion at small spacings and exponentially small probability to find large spacings . in these works
it was found numerically to describe the distribution of spacings reasonably well but there is no analytical justification for its use .
note that even if the semi - poisson distribution is an approximation for the nnsd it may not approximate other spectral measures well . for instance , the form factor , which is the fourier transform of the energy - energy correlation function , satisfies @xmath74 for the semi - poisson distribution @xcite while for a billiard with point - scatterer one expects to find @xmath75 @xcite . in particular , for the rectangular billiard with a point - scatterer , the nnsd was computed analytically under some assumptions in @xcite and was found
_ not _ to be given by the semi - poisson or the poisson distributions .
the nth neighbor spacing distributions were also calculated there and found to be those of the poisson distribution at large spacings .
following @xcite the nnsd of the circle billiard with the point - scatterer , obtained by superposing the two sequences , is given by @xmath76 e^{-(2-x_{un})s}.\ ] ] when the perturbation is at the center , @xmath62 , and @xmath77 approaches the poisson distribution .
alternatively , when the perturbation is near the boundary , and @xmath64 , the nnsd approaches the semi - poisson distribution .
note that the distribution ( [ totalps ] ) does not exhibit complete level repulsion since its value at @xmath78 , @xmath79 , does not vanish and it is a manifestation of the existence of an infinite class of states that are unaffected by the perturbation . the nnsd of ( [ totalps ] ) is compared to numerical results in figure [ pofs ] . , ( c ) @xmath80 and ( d ) @xmath81 .
[ pofs],width=453,height=377 ] the nnsd was computed using the levels @xmath82-@xmath83 for three locations of the point - scatterer as well as for the circle billiard without the perturbation .
it is clear that the agreement is very good .
the main features of the distribution are captured by the simple argument leading to ( [ totalps ] ) .
there are slight deviations from the predictions of equation ( [ totalps ] ) , mainly at large @xmath84 .
these can be attributed to the fact that the nnsd of the affected spectrum differs from the semi - poisson distribution .
the results presented in figures [ unfig ] and [ pofs ] suggest that the spectrum of the circle with a point - scatterer consists of a superposition of two uncorrelated level sequences .
the relative densities of these sequences are determined by the way the classical tori of the integrable system are projected into coordinate space .
states for which the perturbation is in the classically allowed region are affected while states for which the perturbation is in the classically forbidden region are nearly unaffected .
we expect this behavior to be typical of systems where the scatterer affects only a fraction of the tori of the otherwise classically integrable systems .
this differs from the rectangular billiard where all tori are affected by the scatterer .
another important difference compared to the rectangle billiard results of the different nature of the wavefunctions . for the rectangular billiard
there are infinite classes of wavefunctions that have common zeros at rational points . consequently ,
if the scatterer is placed at such a location the wavefunctions are not affected , and the distribution depends strongly on the rationality of the location of the scatterer . for the circle billiard studied in the present work , on the other hand , there is one class of eigenfunctions that vanish on the scatterer since they are antisymmetric in @xmath85 .
these are not considered in the present work
. the symmetric eigenfunctions always satisfy @xmath86 . to obtain many functions , symmetric in @xmath85 , that vanish at the same location
is equivalent to finding many bessel functions which satisfy @xmath87 and @xmath88 for integer @xmath16 , @xmath89 and for @xmath90
. finding an infinite number of such solutions , for the same @xmath91 ( corresponding to the same location of the perturbation ) , is unlikely .
however , since any bessel function of large argument is asymptotically given by a cosine one can find states with close zeros , that is where @xmath92 and @xmath93 are zeros of @xmath15 , while @xmath94 is a zero of @xmath95 and @xmath96 is close to a zero of @xmath95 . in this case when the scatterer is at a zero of one of these states , the square of the wave function of the other state is much smaller there than its average value .
many such close zeros should exist to affect the spectral statistics .
this question is beyond the scope of the present letter and is left for future research .
our numerical results are not sensitive enough to resolve this issue .
the numerical results are used here just to verify that the mean dependence on the location of the scatterer is given by equation ( [ totalps ] ) .
we believe that the behavior of the circle billiard rather than that of the rectangular billiard is typical of integrable systems perturbed by a localized potential . in summary ,
the spectral statistics of the circle billiard , perturbed by a point - scatterer are intermediate between those of the poisson distribution , characteristic of integrable systems , and of the semi - poisson distribution .
the spectrum was shown to be composed of two uncorrelated components .
the first contains energy levels which are nearly unaffected by the perturbation , since the point - scatterer is in a classically forbidden region where the wavefunctions are exponentially small .
the relative fraction of such states was computed analytically and found to depend smoothly on the location of the point - scatterer .
the second contribution is from states which are affected by the perturbation .
the exact statistics of this level sequence are complicated but can be approximated by the semi - poisson statistics .
the nearest neighbor spacing distribution results of combination of the two and is a manifestation of the berry - robnik statistics .
other integrable systems should also exhibit qualitatively similar behavior when a localized perturbation is added to them .
this research was supported in part by the us - israel binational science foundation ( bsf ) and by the minerva center of nonlinear physics of complex systems .
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20 p. blaha , k. schwarz , g. k. h. madsen , d. kvasnicka and j. luitz , _ an augmented plane wave plus local orbitals program for calculating crystal properties _ , vienna univ . of technology , austria ( 2001 )
isbn 3 - 950131 - 1 - 2
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we have undertaken a study of diluted magnetic semiconductors @xmath0 and @xmath1 with @xmath2 , using the all electron linearized augmented plane wave method ( lapw ) for different configurations of mn as well as cr .
we study four possible configurations of the impurity in the wurtzite gan structure to predict energetically most favorable structure within the 32 atom supercell and conclude that the near - neighbor configuration has the lowest energy .
we have also analyzed the ferro - magnetic as well as anti - ferromagnetic configurations of the impurity atoms .
the density of states as well as bandstructure indicate half metallic state for all the systems . @xmath3 has also been estimated for the above systems .
introduction gallium nitride is one of the most promising materials among the diluted magnetic semiconductor ( dms ) material for application in spintronics . by doping transition metal ( tm ) atoms , mn or cr , local magnetic moment
are introduced in semiconductor which mediate ferromagnetically .
( ga , cr)n based dms was predicted to show high @xmath3 @xcite for high enough concentration of cr and further hashimoto _
et al . _
@xcite observed that ( ga , cr)n based dms grown by ecr molecular beam epitaxy showed @xmath3 above @xmath4k . cr@xmath5-implanted gan , studied by photoluminescence and superconducting quantum interference device ( squid ) reveal that the implanted cr@xmath5 incorporates substitutionally at ga site and the ferromagnetic order is retained upto @xmath6k @xcite .
takeuch _
et al . _
@xcite . have reported a systematic study of changes in the occupied and unoccupied n - partial density of states ( dos ) and confirm the wurtzite n @xmath7 dos and substitutional doping of cr into ga sites using sxes and xas .
recently , ferromagnetism above @xmath8k was reported in cr - gan thin films @xcite .
theoretically it was predicted that the ferromagnetic ( fm ) interaction in ( ga , mn)n may be retained upto room temperature @xcite .
the initial reports of high @xmath3 in ( ga , mn)n were followed by controversial results where the reported @xmath3 varied between @xmath9k - @xmath10k @xcite .
zajac and coworkers observed mn ions in ga@xmath11mn@xmath12n ( @xmath13 ) crystals coupled anti - ferromagnetically ( afm ) @xcite .
electronic structure and magnetic properties of zinc blende ga@xmath11mn@xmath12n for several values of @xmath14 with varied spatial distribution of dopant atoms to understand the magnetic interaction for explanation of fm - afm competition is discussed by uspenskii _
et al . _
@xcite where the calculations were done using the tight binding lmto method in the local spin density approximation .
sanyal and mirbt @xcite have studied mn doped gaas and gan dms using the _ ab - initio _ plane wave code ( vasp ) within density functional theory ( dft ) .
they have determined the interatomic exchange interactions by substituting mn in various positions in the unit cell and have attributed the origin of ferromagnetism in ( ga , mn)n to double - exchange mechanism involving the hopping of mn@xmath15 electrons .
raebiger _
et al . _
@xcite used the full potential linearized augmented plane wave ( fp - lapw ) method to investigate the interplay between clustering and exchange coupling in magnetic semiconductor ga@xmath11mn@xmath12as .
they have studied all possible arrangements of the two mn atoms on ga sublattice for @xmath16 and found that clustering of mn atoms at near neighbour ga sites is energetically preferred .
our analysis of the wurtzite gan doped with mn or cr is motivated by the latter study .
method and computational details we have employed the spin - polarized linearized augmented plane wave method ( fp - lapw ) as implemented in the wien2k package @xcite with the generalized gradient approximation ( gga ) for the exchange - correlation potential proposed by perdew , burke and ernzerof ( pbe96 ) @xcite .
this is state - of - the - art electronic structure method , which does not use any shape approximation for the potential , to solve the kohn - sham type of equations self - consistently .
gan normally occurs in the wurtzite structure with lattice constants @xmath17 and @xmath18 , giving @xmath19 ratio of 1.62 .
each ga is tetrahedrally bonded to n atoms at an average distance of @xmath20 and each n in turn is surrounded by four ga neighbors .
the calculations for dms were performed within a 32 atom supercell , constructed from @xmath21 standard unit cell of wurtzite structure wherein the dopant is substituted at various cation sites , since it has been shown that the formation energy for interstitial mn doping is higher than substitutional doping @xcite .
the supercell approach is used to restrict the dopant concentration to a small value , which is of interest for studying magnetic properties of the system , without altering the original underlying lattice structure .
our interest was in observing the changes in the electronic structure of the dms with respect to the possible different geometries of the dopants within the host semiconductor .
self - consistent electronic structure calculations were performed using the apw + local orbitals ( lo ) basis set for the valence and semi - core electrons with @xmath22 , @xmath23 and total energy convergence of @xmath24 the muffin - tin radii for ga , mn and cr were kept at @xmath25 and that for n at @xmath26 .
spin - polarized calculations were carried out to observe the effect of spin - splitting and to calculate the on - site magnetic moment at tm site .
we have studied wurtzite gan doped with one tm atom impurity , which is @xmath27 doping and two identical tm atoms in the @xmath28 atom unit cell amounting to @xmath29 doping . to simulate different surroundings for the transition metal ( tm ) atoms we have spanned certain geometries within the @xmath28 atom unit cell wherein the distance between the dopants is varied . in the case of the single impurity substitution ,
the nearest distance between two tm atoms is @xmath30 in plane and @xmath31 along the @xmath32-axes .
we have studied four different geometries of two tm atom substitutions at @xmath33 , near neighbor ( nn ) , @xmath34 , @xmath35 and @xmath36 separations .
when two near neighbor ( nn ) ga atoms are substituted by tm atoms the in - plane tm - tm atoms distance is @xmath33 and along the @xmath32-axes it is @xmath31 . for the second case
the out - of - plane distance is @xmath34 and the in - plane distance between the dopants is @xmath37 . in the third case ,
two ga atoms lying one above the other , along the @xmath38-axes , separated by a distance of @xmath35 are substituted by tm atoms and the in - plane tm atoms are at @xmath30 .
the last case is such that the in - plane separation ( @xmath30 ) and out of plane separation ( @xmath39 ) between the tm atoms is comparable . for estimating the magnetically favorable system , the spins of the dopants are aligned along the same direction , corresponding to the fm configuration , and aligned in the opposite directions corresponding to afm configuration .
the self consistency was achieved on a mesh of @xmath40 k - points .
structural relaxation for the tm site and the nn n sites was carried out to observe changes in the bond lengths between tm and the first shell of n atoms .
very small change ( @xmath41 ) was observed in the bond lengths and no significant changes were seen in the band structure in agreement with the earlier reported results @xcite .
thus the calculations reported here are for systems without allowing any relaxation .
results in the wurtzite gan semiconductor , each ga ( n ) is tetrahedrally bonded to 4 n ( ga ) atoms .
pure gan is a direct band gap semiconductor with top of the valence band consisting of n _ p_-states and the bottom of the conduction band having ga _
sp_-character .
the ga _
d _ levels are deep and do not take part in the bonding .
thus they are treated as core states .
the band gap of gan , which is underestimated by density functional theory within the approximation used for the exchange correlation energy functional , is @xmath42ev .
the experimentally determined band gap of undoped
gan is @xmath43ev .
das _
et al . _
@xcite have shown that , for mn atoms to couple ferromagnetically , they need to be kept apart by more than the critical distance of @xmath44 . similar calculations on clusters of ( gan)cr indicate that the critical cr - cr distance is @xmath45 @xcite . in all our calculations the distance between the dopants was greater than the corresponding critical distances .
mn doped systems localized magnetic moments are introduced within the gan system by substituting the cations with tm impurity atom(s ) .
mn atom with @xmath46 and @xmath47 electrons in the valence region replaces ga atom with valency @xmath48 . on substitution mn atoms contributing five @xmath49 levels per atom
are thus expected to contribute to the observed magnetic moment . 1.0 in 2.0 in 0.2 in 2.0 in + 0.02
cm + 2.1 in ( a ) 0.2 in 1.5 in since three of the valence electrons from mn go into compensating the three electron states of substituted ga , one hole per mn is introduced into the system .
figure [ mnet2 ] shows the mn - projected @xmath50 and @xmath51 majority spin electronic structure for @xmath52 .
the mn-@xmath49 states lie at the top of the valence band and cross the @xmath53 in some places .
these are split into @xmath51 and @xmath50 states , the @xmath50 level is two thirds filled and the @xmath51 is almost occupied .
the minority spin levels are empty and lie above the @xmath53 indicating @xmath54 spin - polarized states .
the mn induced states lie in the gap region of gan .
the top of the valence band in gan is composed of the n-@xmath55 levels and the unique properties , particularly the half metallic state of dms , thus arise from the tm _
d _ and host _ p _ interactions that couple the two subsystems .
mn - projected @xmath49-dos in @xmath56 ( a ) with mn - mn distance equal to @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37 .
the upper and lower panels represent the majority and minority spin dos respectively.,title="fig:",height=321 ] + ( a ) + mn - projected @xmath49-dos in @xmath56 ( a ) with mn - mn distance equal to @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37 .
the upper and lower panels represent the majority and minority spin dos respectively.,title="fig:",height=321 ] + ( b ) in order to understand the variation of exchange interaction among the tm impurity with the distance between the tm atoms , the concentration of mn atoms was increased to @xmath58 , equivalent to introducing @xmath59 mn atoms in the supercell .
self consistent calculations were carried out for two different magnetic configurations of mn electrons in which the electrons are parallel or antiparallel corresponding to fm or afm configuration .
for all the geometries of the dopants , systems , as described in section ii , with fm configuration of the mn atoms were found to have lower energy .
since the mn-@xmath49 levels are responsible for observed half metallic behavior , a comparison of mn-@xmath49 dos in various geometries is shown in figure [ mndos ] , ( a ) for separations @xmath33 and @xmath60 and ( b ) for @xmath57 and @xmath37 .
here the half metallic state is evident in all the cases .
the tm - tm distance of @xmath37 corresponds to single tm doping ( @xmath61 ) .
the mn @xmath62dos is broad for substitution at nn distance . in all the other cases
the band is split indicating that the majority spin @xmath49-bands of the two mn atoms at nn overlap . on increasing mn - mn distance , @xmath49-band splitting takes place implying a reduction in the interaction between the tm atoms .
it may be noted that the minority spin conduction band overlaps with the majority spin band for nn substitution .
a gap of @xmath63 is present between the minority spin conduction band and majority spin band for mn - mn distance greater than nn . the minority spin valence band
as well as conduction band is far apart from the @xmath53 thus retaining the highly spin polarized state also seen in the single mn doped @xmath64 system .
the down spin gap is @xmath65 for nn configuration and increases to @xmath66 at larger separations .
on increasing the distance between the mn atoms , splitting of d - level increases .
this is consistent with the observation that in single impurity doping , tm - tm atom distance is @xmath57 and splitting of the mn-@xmath49 band is larger as seen in figure [ mndos](b ) .
the magnetic moment at mn - site does not depend on the distance between the dopant atoms and has a value @xmath67 for all the geometries as indicated in table [ table1 ] . @llll system & total@xmath68 & dopant@xmath68 & n@xmath68 + @xmath52 & 4.00 & 3.33 & 0.001 + @xmath69 & 8.00 & 3.34 & 0.001 + mn - mn = @xmath33 & & & + @xmath69 & 8.00 & 3.34 & 0.005 + mn - mn = @xmath34 & & & + @xmath69 & 8.00 & 3.33 & 0.006 + mn - mn
= @xmath35 & & & + @xmath69 & 8.00 & 3.34 & 0.005 + mn - mn = @xmath57 & & & + the presence of localized moment influences the near neighbor n atoms within the gan system , such that dos of the nn n - atoms around the impurity atom becomes as shown in figure [ ndos](a ) . due to the @xmath70 interaction , induced states are seen on n atoms .
the magnitude of the induced states is maximum at nn n atoms and decreases as the distance from the tm atom increases .
this again indicates a localized nature of the tm states .
figure [ ndos](b ) shows the spin charge density ( scd ) in a plane containing three nn n atoms for single mn doping .
the plot shows that the scd on the n atoms lying above the mn atom is negative whereas it is positive for the n atom lying below the mn atom .
average magnetic moment on the nn n atoms is positive as shown in table [ table1 ] for all the different geometries .
( a ) variation in partial - dos at 3 n - sites in @xmath52 .
the solid lines : nn - n along @xmath71axis .
dotted lines :
nn - n in plane .
dashed - dotted lines : next nn - n ( b ) scd plot in a plane of three nn n atoms.,title="fig:",height=245 ] + ( a ) + 0.2 cm ( a ) variation in partial - dos at 3 n - sites in @xmath52 . the solid lines : nn - n along @xmath71axis .
dotted lines : nn - n in plane .
dashed - dotted lines : next nn - n ( b ) scd plot in a plane of three nn n atoms.,title="fig:",width=188,height=188 ] ( a ) variation in partial - dos at 3 n - sites in @xmath52 . the solid lines : nn - n along @xmath71axis .
dotted lines :
nn - n in plane .
dashed - dotted lines : next nn - n ( b ) scd plot in a plane of three nn n atoms.,title="fig:",width=75,height=188 ] + ( b ) cr doped systems electronic structure calculation for substitutional doping of cr in the gan system was also done and is analyzed in a similar fashion . for each cr
doped in the @xmath28 atom supercell , equivalent to @xmath72 doping , there are five spin up @xmath49-states which are introduced in the gan band gap .
since the cr atom has @xmath73 valence electrons , only three of the electron states out of the five @xmath74 levels are occupied , thus creating two hole states per cr substitution .
single cr doping into the @xmath28 atom supercell at cation site results in the cr-@xmath49 levels appearing in the band gap of the semiconductor host as seen in figure [ cret2 ] . 1.0 in 2.0 in 0.2 in 2.0 in 0.02 cm + 2.1 in ( a ) 0.2 in 1.5 in the cr d@xmath75 levels split ( figure [ cret2](a ) ) , out of which two energy levels lie below the fermi level ( @xmath76 ) and are occupied .
the third level which is @xmath77ev above is unoccupied .
one of the d@xmath78 level is occupied and the other lies just above the @xmath76 as seen in figure [ cret2](b ) .
however , the hybridization of @xmath50 and @xmath51 majority spin states is negligible and these levels are well separated as opposed to mn doped case . as in the case of mn doping ,
the cr minority spin @xmath49-states are above the @xmath76 and so the impurity states at the @xmath76 are @xmath54 spin polarized .
cr - projected @xmath49-dos in @xmath79 for cr - cr separation of ( a ) @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37.,title="fig:",height=321 ] + ( a ) + cr - projected @xmath49-dos in @xmath79 for cr - cr separation of ( a ) @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37.,title="fig:",height=321 ] + ( b ) in the @xmath79 system , figure [ cr2 ] shows that even though the cr atoms are substituted at nn sites there is a gap seen between the split cr-@xmath49 levels , unlike in the nn mn - doped system . the minority spin conduction band overlaps with the majority spin band for nn cr case , as in nn mn case . for all the geometries of the dopant atoms the system is half metallic .
scd plot in a plane of three nn n atoms around the tm atom in @xmath80.,title="fig:",width=188,height=188 ] scd plot in a plane of three nn n atoms around the tm atom in @xmath80.,title="fig:",width=75,height=188 ] the band gap for the minority spin , in case of two cr substitution at @xmath33 is @xmath81ev which is larger than the corresponding mn case which has a gap of @xmath82ev . when the tm - tm distance is @xmath35 , in the cr case the gap is @xmath83ev whereas for mn substitution it is @xmath84ev .
the lowest energy configuration for cr substitution occurs for cr - cr distance of @xmath33 and for all the geometries studied the fm configuration of tm atoms has lower energy compared to afm configuration .
the magnetic moments at the cr site in various geometries is as shown in table [ tab2 ] and it is seen that the magnetic moment does not show much variation depending on the distance , indicating that the direct interaction between the cr atoms is minimal . @llll system & total@xmath68 & dopant@xmath68 & n@xmath68 + @xmath85 & 3.00 & 2.47 & -0.025 + @xmath86 & 6.00 & 2.48 & -0.031 + cr - cr = @xmath33 & & & + @xmath86 & 6.00 & 2.47 & -0.022 + cr - cr = @xmath34 & & & + @xmath86 & 6.00 & 2.48 & -0.025 + cr - cr = @xmath35 & & & + @xmath86 & 6.00 & 2.48 & -0.026 + cr - cr = @xmath57 & & & + magnetic moments observed at the various dopant sites from our calculation are shown in the table [ tab2 ] .
the total magnetic moment per unit cell per mn atom is @xmath87 and the average magnetic moment on the nn n atoms in case of mn - doping is parallel to the mn - moment .
this can be understood as penetration of the spin - polarized mn states to the neighboring host which does not have any of its own states in the gap region .
the magnetic moment per unit cell per cr atom is @xmath88 .
the average magnetic moment on the nn n atom is anti - parallel to cr - moment .
the difference in the orientation of the average magnetic moment on nn n atoms of mn and cr is due to the difference in the @xmath55 dos of the nn n along the z axis ( figure not shown here ) compared to the nn n atoms lying in the xy plane above the tm atoms .
there is not much variation of magnetic moment with increase in cr - cr distance and shows a similar trend as mn doped systems .
the magnitude of average nn - n magnetic moment is greater in case of cr substitution , which contribute one less electron to the hybridized valcen band .
scd in figure [ crscd ] on all of the nn - n atoms of the single cr ( only 3 nn n atoms shown in figure [ crscd ] ) doped system is negative thus showing that the tm atom and the nn - n are anti - ferromagnetically coupled .
estimation of @xmath3 we have predicted the @xmath3 for the dms based on ga@xmath89mn@xmath90n@xmath89 and ga@xmath89cr@xmath90n@xmath89 considering the mean field approximation .
@xmath91 for mn and cr doping .
inset shows the mean field @xmath3 variation with distance between dopants.,height=321 ] figure [ evsd ] shows the @xmath91 for the mn / cr doped systems , where @xmath92 is the total energy for the antiferromagnetic ( afm ) configuration and @xmath93 is the total energy for the ferromagnetic ( fm ) configuration
. observed variation @xmath94 vs distance for mn substituted dms agrees with the one reported by sanyal @xcite .
@xmath94 is a measure of the exchange interaction in the system .
highest @xmath94 is seen for the case where the tm atoms are substituted as near neighbors , signifying larger overlap of the magnetic impurity orbitals . for mn doping at nn @xmath95ev ,
this compares well with the value calculated for dimer substitution by uspenskii _ et . al _
@xcite which compared qualitatively with the high @xmath96k measured @xcite . as for the identical cr case @xmath97 also compares well with the observed @xmath3 but is lower than the @xmath8k observed by liu and co - workers @xcite . from figure [ evsd ]
it also emerges that the exchange interaction decreases sharply as the distance between the tm atoms increases .
thus the exchange interaction is short range and could be interpreted as the double exchange mechanism .
summary and conclusions we have analyzed the electronic structure of gan doped with tm mn and cr with @xmath72 and @xmath58 doping for various possible geometries to replicate the situation where the tm atoms would appear either to cluster or be separated .
the self consistent fp - lapw calculations predict half metallic state for @xmath72 as well as @xmath58 doping . comparing the total energies of the fm and afm configurations for @xmath58 doping ,
the fm state is found to be lower in energy and is predicted to be the preferred state .
on - site magnetic moment at the tm site shows insignificant variation with distance between the dopants .
the near neighbor n atoms contribute to the states in energy gap of the semiconductor due to the influence of the tm atoms .
average magnetic moment at nn n site is parallel to the mn magnetic moment where as it is anti - parallel to the cr atoms .
we observe that both the systems with nn substitution of mn / cr atom would show high @xmath3 .
the energy gap between the minority spin band in mn is @xmath98 lower than in cr doped system and we think this could be an important factor in determining a suitable system .
but since the magnetic moment at mn site is higher than cr , it would be of interest to study mixed systems of mn and cr to incorporate the salient features of both tm atoms .
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this work was supported by the national basic research program of china under grants nos .
2012cb821305 , 2010cb923200 and 2013cb922403 , the national natural science foundation of china under grants nos . 11374375 , 11204043 , 11274399 and 61078027 , and the ph.d .
programs foundation of ministry of education of china under grant nos .
b.a.m . appreciates hospitality of the sun yat - sen university ( guangzhou , china ) .
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we demonstrate that in - bulk vortex localized modes , and their surface half - vortex ( horseshoe " ) counterparts self - trap in two - dimensional ( 2d ) nonlinear optical systems with @xmath0-symmetric photonic lattices ( pls ) .
the respective stability regions are identified in the underlying parameter space .
the in - bulk states are related to truncated nonlinear bloch waves in gaps of the pl - induced spectrum .
the basic vortex and horseshoe modes are built , severally , of four and three beams with appropriate phase shifts between them .
their stable complex counterparts , built of up to 12 beams , are reported too .
nonlinear spatially periodic systems support diverse types of self - trapped in - gap states .
in particular , spatial gap solitons @xcite originate from the interplay between the periodicity and nonlinearity .
further , surface gap solitons @xcite appear at the interface between a uniform medium and a photonic lattice ( pl ) built into a nonlinear material .
extended self - trapped waves with steep edges also exist in these settings , being related to truncated nonlinear bloch waves @xcite .
modes of the latter type provide a link between extended nonlinear bloch waves @xcite and tightly localized gap solitons @xcite .
recently , a great deal of interest has been drawn to the realization of parity - time ( @xmath0 ) symmetry in optics .
originally , this concept was developed in quantum mechanics , where it was demonstrated that , beyond the conventional hermitian hamiltonians , their @xmath0-symmetric non - hermitian counterparts may also give rise to purely real ( hence physically relevant ) spectra @xcite .
following the similarity between quantum mechanics and paraxial optics @xcite , @xmath0-symmetric optical systems with complex refractive indices @xcite have been extensively studied theoretically @xcite and experimentally @xcite . in this context ,
@xmath0-symmetric pls play an important role .
taking into account the non - orthogonality of the respective eigenmodes , their coupled - mode description had to be reformulated via the variational principle @xcite .
the light propagation in @xmath0-symmetric pls embedded into linear media were analyzed preliminarily @xcite .
further , it has been found that 1d and 2d spatial gap solitons exist in pls built into a nonlinear material @xcite .
although bloch waves @xcite and gap solitons @xcite were studied before in the context of some @xmath0-symmetric pls , the comprehensive study of self - trapped states in 2d @xmath0-symmetric systems combining lattices and nonlinearity was not reported yet .
in particular , such self - trapped nonlinear states may serve as a necessary link between spatially localized gap solitons and extended nonlinear bloch waves under the @xmath0 symmetry .
self - trapped vortices and surface modes are of great interest in the context of the @xmath0-symmetric settings .
indeed , the study of nonlinear surface modes pinned on the interface of a @xmath0-symmetric system opens a way to explore the interplay between surface effects , the nonlinearity , and the @xmath0-symmetry . on the other hand ,
the analysis of localized vortices supported by @xmath0-symmetric pls should shed light on the cooperation and competition of the @xmath0-symmetry with the azimuthal instability and spatial periodicity . in this work ,
we show the existence of in - bulk and surface self - trapped states in 2d nonlinear systems with @xmath0-symmetric pls .
in particular , we report in - bulk solitary vortices and novel half - vortex surface modes .
stable half - vortex surface modes appear as horseshoes `` pinned on the interface between a uniform linear medium and a nonlinear medium with built - in @xmath0-symmetric pl .
the in - bulk vortices and surface horseshoes '' have a common linear stability region at intermediate values of propagation constants .
we consider the light propagations in two nonlinear systems with @xmath0-symmetric pls : a uniform setting of the nonlinear @xmath0-symmetric pl , and a composite setting of the nonlinear @xmath0-symmetric pl at the left side ( @xmath1 ) and a uniform linear medium at the right side ( @xmath2 ) . assuming that the light propagates along the @xmath3-axis , the amplitude of the electromagnetic field is written as @xmath4 , with carrier wavenumber @xmath5 and frequency @xmath6 . with the effective refractive index including contributions from the complex pl and the kerr effect , @xmath7
, the amplitude obeys the nonlinear schrdinger equation with the complex potential , @xmath8 e \notag \\ & & + 2k_{0}^{2}\left [ n_{0}^{\mathrm{pl}}\left ( n^{\mathrm{r}}+in^{\mathrm{i}}\right ) + n_{0}^{\mathrm{pl}}n^{\mathrm{nl}}|e|^{2}\right ] e=0 .
\label{eq : one}\end{aligned}\]]here , @xmath9 is a constant , @xmath10 is the background refractive index , @xmath11 and @xmath12 are real and imaginary ( gain / loss ) parts of the spatial modulation of the local index , and @xmath13 is the kerr coefficient .
similarly , the light propagation in the linear uniform medium obeys the paraxial equation @xmath14 e=0 , \label{eq : two}\]]with the respective real refractive index , @xmath15 .
we normalize the equations by defining @xmath16 , @xmath17 , @xmath18 , and @xmath19 z}$ ] with an arbitrary scaling factor @xmath20 .
the accordingly rescaled form of eqs .
( [ eq : one ] ) and ( [ eq : two ] ) is@xmath21with @xmath22 , and @xmath23 $ ] .
the complex @xmath0-symmetric potential , @xmath24 , is chosen as@xmath25 , \\
w(\xi , \eta ) & = & \theta \{\sin [ \sqrt{2}(\eta -\xi ) ] + \sin [ \sqrt{2}(\eta+\xi ) ] \},\end{aligned}\]]with amplitudes @xmath26 and @xmath27 of the modulation of the real and imaginary parts of the refractive index .
this pl is a @xmath28 counterclockwise rotation of the one considered in refs .
@xcite .
the configurations of pls are shown by the white - blue circles in the @xmath29-plane .
its band - gap structure can be derived by using the plane - wave expansion method based on the floquet - bloch theorem , see fig .
[ fig : one ] ( a ) .
-symmetric photonic lattice : @xmath30 is the propagation constant , and @xmath31 , @xmath32 are bloch wavenumbers in @xmath33 and @xmath34 directions .
( b , c ) intensity profiles of self - trapped modes for propagation constant @xmath35 in the uniform and truncated systems , respectively .
the white dot - dash line depicts the interface in the truncated system .
( d ) power @xmath36 and ( e ) the real part of the instability growth rate , @xmath37 , of the self - trapped modes versus the propagation constant , @xmath38 .
green triangles and red circles in ( d ) correspond to the modes shown in ( b ) and ( c ) , respectively .
the green dashed and red solid lines in ( e ) represent , severally , in - bulk and surface self - trapped modes .
( f ) the density profile at @xmath39 , evolved from the initial self - trapped mode ( c ) with @xmath40 noise .
parameters are @xmath41 , @xmath42 and @xmath43 . ] to combine eqs .
( [ eq : seven ] ) and ( [ eq : eight ] ) into a single equation , we define a step function , @xmath44 at @xmath45 and @xmath46 at @xmath47 : @xmath48 gq+u(\xi ) |q|^{2}q=0 .
\label{eq : jia_1}\end{aligned}\]]the stationary solution with real propagation constant @xmath38 is looked for as @xmath49 , where complex function @xmath50 obeys equation @xmath51 gu \nonumber\\ & + & u(\xi)|u|^{2}u - bu=0.\label{eq : five}\end{aligned}\ ] ] to find the stationary self - trapping solutions , we used numerical simulations with the modified squared - operator method @xcite . while the existence and stability of the simplest single - beam solitons in the present setting is quite evident , as the first step of the analysis we produce double - beam self - trapped states . for @xmath35 and @xmath43 , the in - bulk and surface double modes
are displayed in fig .
[ fig : one](b , c ) . due to the presence of the interface
, the intensity of the surface self - trapped states is larger than the in - bulk ones , at the same propagation constant .
the dependence of the total power , @xmath52 , on the propagation constant @xmath38 demonstrates that the power of the surface modes is also larger than that of the in - bulk ones , see fig .
[ fig : one](d ) .
different from the single - beam solitons @xcite , both surface and in - bulk self - trapped states in the semi - infinite gap do not exist near the first bloch band in fig .
[ fig : one](d ) .
stability of the self - trapped modes was investigated by means of the linearization for small perturbations . to a given stationary state ,
@xmath53 , the perturbation is added as @xmath54e^{ib\zeta } $ ] with infinitesimal @xmath55 @xcite , where @xmath56 and @xmath57 are two perturbation eigenfunctions , @xmath58 is the corresponding growth rate , and the star ( @xmath59 ) stands for the complex conjugate . from eq .
( [ eq : jia_1 ] ) , the following linearized equations are derived : @xmath60f \nonumber\\ & & + u(\xi ) u^{2}g , \\ + i\delta g&=&\left [ \nabla _ { \perp } ^{2}+u(\xi ) r^{\ast } ( \xi,\eta)u+\psi+2u(\xi ) |u|^{2}\right]g \nonumber\\ & & + u(\xi)\left(u^{\ast}\right)^{2}f,\end{aligned}\ ] ] with @xmath61g - b$ ] . as usual , the self - trapped mode is linearly unstable if there is an eigenvalue with @xmath62 . as seen in fig .
[ fig : one](e ) , the two - beam in - bulk and surface self - trapped modes have a common stable region , with @xmath63 , at intermediate values of propagation constants @xmath38 .
the predicted stability of the modes has been verified in direct simulations of eq .
( [ eq : jia_1 ] ) with @xmath40 random noise added as an initial perturbation , see an example for @xmath35 in fig .
[ fig : one](f ) . .
( c , d ) the intensity profile and phase distribution for the three - beam surface self - trapped state at @xmath64 .
( e ) power @xmath36 of the self - trapped states versus @xmath38 .
( f ) the real part of the instability growth rate , @xmath65 , versus @xmath38 .
green triangles and the red circles correspond to the modes shown in ( a ) and ( c ) , respectively .
parameters are @xmath41 , @xmath42 , and @xmath43 . ]
adding more beams with phase shifts between them , one can construct composite vortices .
for an example , a composite vortex with the total phase circulation of @xmath66 may appear as a four - beam complex with the off - site vortex core in the center and the phase shifts @xmath67 between adjacent beams @xcite .
we have found that the composite vortices can exist in the system of the uniform setting . the intensity profile and phase distribution of a typical stable four - beam vortex in the uniform setting system
are shown in fig .
[ fig : two](a ) and ( b ) for @xmath64 . near the interface in of the composite setting system , there are no complete vortex modes , while there appear essentially new surface modes , in the form of _ half - vortices _ ( horseshoes `` ) , built of three beams , see figs .
[ fig : two](c ) and ( d ) .
the dependence of the power @xmath36 on the propagation constant @xmath38 shows that , although the half - vortex mode ( c ) is built of three beams , its power @xmath36 is larger ( near the first bloch band ) than that of the in - bulk vortex mode ( a ) , which is composed of four beams , see fig .
[ fig : two](e ) .
the linear stability analysis shows that there exists a common stability region at intermediate values of propagation constants @xmath38 for the in - bulk vortices and surface horseshoes '' , see fig .
[ fig : two](f ) . , @xmath68 ( b ) .
profiles of the 7-beam surface states : ( c ) at @xmath69 , ( d ) at @xmath68 .
( e ) power @xmath36 of the states versus @xmath38 .
( f ) the real part of the instability growth rate , @xmath37 , versus @xmath38 .
green triangles correspond to ( a ) and ( b ) , and red circles to ( c ) and ( d ) .
parameters are @xmath41 , @xmath42 and @xmath43 . ] on top of the simple few - beam self - trapped states , like the conservative 2d nonlinear systems @xcite , our @xmath0-symmetric systems can also support complex multi - beam ones built of up to 12 beams , see fig .
[ fig : three ] . due to the interaction between individual beams ,
their intensity is larger at the center of the structure , the intensity difference gradually vanishing with the increase of propagation constant @xmath38 . near the interface in the truncated system
, there are no beams located in the linear medium , while the near - interface beams become stronger , see fig .
[ fig : three](c , d ) .
power @xmath36 increases with propagation constant @xmath38 for both the in - bulk and surface self - trapped states , see fig .
[ fig : three](e ) . results of the linear - stability analysis for these states , displayed in fig .
[ fig : three](f ) , reveal a common stability region for the in - bulk and surface modes , at intermediate values of propagation constants @xmath38 . noise .
( a ) the density profile at @xmath70 evolves from the the in - bulk vortex self - trapped nonlinear waves in fig .
[ fig : two ] ( a ) .
( b ) the corresponding phase distribution for ( a ) .
( c ) the density profile at @xmath70 evolves from the in - bulk multi - beam self - trapped mode in fig .
[ fig : three ] ( b ) .
( d ) the density profile at @xmath70 evolves from the surface multi - beam self - trapped mode in fig .
[ fig : three ] ( d ) . ] by simulating the beam propagation with @xmath40 random noise , we have verified the stability of the vortex modes , as shown in fig .
[ fig : two ] , and of multi - beam ones , see fig .
[ fig : three ] . in particular , fig .
[ fig : four](b ) demonstrates that the phase distribution of the input vortex mode keeps the phase - winding structure .
thus , the direct simulations corroborate the predictions of the linear - stability analysis . in conclusion
, we have found several novel species of in - bulk and surface self - trapped states in 2d kerr - nonlinear optical systems with @xmath0-symmetric pls ( photonic lattices ) .
these include stable in - bulk localized vortices and surface half - vortices ( horseshoes " ) .
the self - trapped modes are related to truncated nonlinear bloch waves , the surface modes being linked with the truncated in - bulk ones .
along with the basic vortex and half - vortex states , which are built , respectively , of four and three constituent beams .
the stable multi - beam self - trapped states , composed of up to 12 constituents , have been found too .
the formation of these surface modes results from the interplay of the surface effects , nonlinearity , and the @xmath0-symmetry . due to the surface - enhanced reflection ,
the discrete diffraction is stronger in the direction perpendicular to the interface than in the direction parallel to it @xcite , therefore the surface modes feature stronger nonlinearity , which is necessary to balance the diffraction .
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in this supplementary section we present a list of virtual processes that contribute to the first term of eq.(9 ) in the main text .
table i corresponds to the single - level anderson model , whereas tables ii - iv dwell on the universal hamiltonian .
in addition , we present the dimensionless integral that gives rise to @xmath21 ( defined by eq .
15 in the main text ) : @xmath166 where @xmath167 and @xmath168 .
( [ eq : f ] ) can be derived by adding the @xmath169 transition amplitudes of tables ii - iv along with the second and third term of eq.(9 ) in the main text , with @xmath136 .
the derivation is simplified by exploiting time - reversal as well as particle - hole symmetry , although similar integrals may be derived in absence of particle - hole symmetry .
note that the sum and integral in eq .
( [ eq : f ] ) are uv - finite , even though numerous individual amplitudes in tables ii - iv are uv - divergent ; the delicate cancellation between different uv divergences adds considerable confidence on the veracity of our results .
moreover , we find that the main contribution to @xmath21 originates from states with @xmath170 and @xmath171 .
these observations together justify our assumption of energy - independent tunneling amplitudes and uniform energy - level spacings . in other words ,
our assumptions hold provided that the tunneling - amplitudes and the energy - level spacings vary slowly on energy scales of order @xmath14 , which is typically much smaller than the fermi energy .
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we develop a general method to evaluate the kondo temperature in a multilevel quantum dot that is weakly coupled to conducting leads .
our theory reveals that the kondo temperature is strongly enhanced when the intradot energy - level spacing is comparable or smaller than the charging energy .
we propose an experiment to test our result , which consists of measuring the size - dependence of the kondo temperature . _
introduction. _ the kondo effect , a many - body phenomenon that emerges from the interaction between localized and itinerant fermionic degrees of freedom , is characterized by a low - temperature infrared ( ir ) divergence in perturbative calculations of physical observables such as resistivity and magnetic susceptibility@xcite .
this ir divergence is controlled by an ultraviolet ( uv ) cutoff @xmath0 that appears in the expression for the kondo temperature via @xmath1 , where @xmath2 is the kondo coupling and @xmath3 is the fermi level density of states per spin for itinerant carriers . such expression for @xmath4
is generally valid for @xmath5 and can be derived perturbatively starting from the venerable kondo hamiltonian @xcite , @xmath6
. an accurate microscopic theory of @xmath2 and @xmath0 provides crucial guidance for experimental explorations of strongly correlated electron systems . a precise way to quantify
@xmath4 is to work with a `` first - principles '' microscopic model that reduces to @xmath7 at energy scales below @xmath0 .
quite generally this first - principles hamiltonian can be written as @xmath8 , where @xmath9 captures the hybridization between the localized and itinerant degrees of freedom .
a perturbation theory calculation of physical observables in @xmath9 then yields hallmark kondo - like divergences , with @xmath0 and @xmath2 unequivocally determined in terms of the microscopic parameters of @xmath10 .
perhaps the first author to successfully implement the aforementioned scheme was haldane@xcite , who evaluated the magnetic susceptibility for the single - level anderson hamiltonian to fourth order in the hybridization amplitude @xmath11 . in the local - moment regime and for an infinite bandwidth in the continuum he obtained @xmath12 and @xmath13 , where @xmath14 is the coulomb charging energy . over time , haldane s formula @xmath15 has remained as the norm for the interpretation of experimental studies of the kondo effect in quantum dots@xcite , even though its applicability in these devices is _ a priori _ unclear .
a primary concern regarding haldane s formula is that it makes no reference to the multiple energy levels present in real dots .
this concern was first addressed by inoshita et al .
@xcite , who suggested that the dense energy spectrum of quantum dots should enhance @xmath4 by several orders of magnitude .
nevertheless , no such giant enhancement has been observed@xcite .
more recently , aleiner _
et al .
_ @xcite argued that , in real quantum dots with an average single - particle spacing @xmath16 , the main modification from haldane s formula should consist of replacing @xmath17 by @xmath18 .
the conclusions of refs .
[ , ] rely on effective kondo hamiltonians , and are thus less rigorous than the `` first - principles '' approach described above . in this paper
we follow the spirit of ref . [ ] and construct a precise theory for @xmath4 in real quantum dots that are weakly coupled to conducting leads .
we adopt the _ universal hamiltonian _ @xcite as an appropriate `` first - principles '' model for real quantum dots , and reach results that differ qualitatively from those of refs .
[ , , ] . in the infinite bandwidth limit we conclude that @xmath19 for @xmath20 , where @xmath21 is a function of @xmath22 ( fig .
1 ) . this result predicts an unconventional dependence of @xmath4 on the size of the quantum dot .
is a lengthscale defined in the text .
_ inset _ : the function @xmath21 of eq .
( [ eq : l_dot ] ) for a quantum dot with infinite equally - spaced energy levels . ] _ method. _ our calculation centers on spin - flip matrix elements of an effective hamiltonian , @xmath23 where @xmath24 and @xmath25 are degenerate eigenstates of @xmath26 , and @xmath27 is the effective hamiltonian derived from degenerate perturbation theory in @xmath9 .
@xmath24 and @xmath25 are tensor products of a target ( i.e. the localized degrees of freedom ) and a projectile ( i.e. an itinerant particle that scatters off the target ) .
both the spin of the projectile and the spin of the target are flipped in the course of spin - flip processes .
the calculation of eq .
( [ eq : a ] ) is considerably simpler than that of the magnetic susceptibility in ref .
[ ] because it requires neither partition functions nor external magnetic fields .
in spite of its relative simplicity , eq .
( [ eq : a ] ) is closely connected to the scattering t - matrix and thus to a physical observable , namely the scattering rate . in view of the above connection
, our approach exploits the long - known fact@xcite that in the kondo model the spin - flip matrix elements of the t - matrix produce the `` running '' kondo coupling @xmath28 where @xmath29 is the temperature .
the computation of eq .
( [ eq : a ] ) from a `` first - principles '' model and its subsequent identification with eq .
( [ eq : kondo ] ) produces the desired explicit expression for @xmath2 and @xmath0 in terms of microscopic parameters .
_ effective hamiltonian. _ given a hamiltonian @xmath8 , where @xmath26 has a degenerate energy spectrum , there exists a perturbative green s function technique@xcite to construct its exact eigen energies . according to this approach ,
the key eigenvalue equation to be solved is @xmath30 where @xmath31 is the projection operator onto the degenerate subspace spanned by the eigenvectors of the unperturbed energy @xmath32 , @xmath33 are the eigenvalues of @xmath10 and @xmath34 are the projections of the corresponding eigenvectors onto the projected subspace .
hence @xmath35 , which is hermitian on the projected subspace , may be identified with an effective hamiltonian . also , @xmath36 and @xmath37 for non - negative integers @xmath38 . in eq .
( [ eq : un ] ) , @xmath39 if @xmath40 and @xmath41 if @xmath42 .
in addition , @xmath43 .
@xmath44 is extended over all sets of non - negative integers @xmath45 satisfying the conditions @xmath46 @xmath47 and @xmath48 . from eq .
( [ eq : eigen ] ) it follows@xcite that @xmath49 _ single - level anderson model. _ in order to verify that eq .
( [ eq : a ] ) produces the correct @xmath0 and @xmath2 , we employ the simplest first - principles model for which rigorous results have been long established@xcite . using the standard notation ,
the anderson hamiltonian is @xmath8 , where @xmath50 we evaluate @xmath51 to fourth order in the hybridization amplitude @xmath11 .
we choose @xmath52 as the initial and final scattering states .
@xmath53 is the fermi sea in the continuum and @xmath54 denotes the empty state of the localized level .
the momentum of the projectile is assumed to be close to the fermi surface ( i.e. @xmath55 ) . @xmath24 and
@xmath25 can be connected only via spin - flip processes ; this choice is convenient in that it filters out spin - independent scattering .
the effective hamiltonian may be evaluated using eq .
( [ eq : messiah ] ) and noting that @xmath31 projects onto a two - dimensional subspace spanned by @xmath24 and @xmath25 ; the outcome reads @xmath56 with @xmath57 and @xmath58 where we have exploited @xmath59 and defined @xmath60 .
our @xmath27 connects states with equal energy and thus contains less information than the effective hamiltonian derived from a fourth - order schrieffer - wolff@xcite transformation , with which it agrees when @xmath61 . at any rate
, this limitation has no practical consequences because all observable properties are determined by @xmath62 and @xmath63 located at the fermi surface . from eq .
( [ eq : h_eff ] ) the lowest order contribution to @xmath51 reads @xmath64 where @xmath65 denotes virtual intermediate states that satisfy @xmath66 . also , @xmath67 and @xmath68 ( @xmath69 as we focus on elastic scattering ) .
each time @xmath9 acts on a state it changes the number of particles by one both in the continuum and in the localized level , yet it conserves the total number of particles and the total spin . accordingly @xmath70 and @xmath71 .
next , we compute the 4th order contribution to the scattering amplitude using @xmath72 in eq .
( [ eq : h_eff ] ) : @xmath73 where @xmath74 . the second and third terms in eq .
( [ eq : t4 ] ) were derived by inserting @xmath75 between two subsequent @xmath31 operators in eq .
( [ eq : h_eff ] ) .
in particular , the second term in eq .
( [ eq : t4 ] ) is uv divergent and plays a crucial role in ensuring that @xmath51 remains uv finite even when the bandwidth of the continuum states is taken to infinity .
\{@xmath76,@xmath77,@xmath78 } label intermediate states , which are collected in table i of the supplementary material .
summing over all contributions and assuming an infinite bandwidth in the continuum we obtain @xmath79 where @xmath3 is the fermi surface density of states in the continuum and @xmath80 is the infrared energy cutoff .
for the present zero - temperature calculation @xmath81 .
@xmath82 can be identified ( modulo a factor @xmath83 ) with eq .
( [ eq : kondo ] ) , which yields @xmath84 and @xmath85 .
these expressions agree with those of ref .
[ ] .
_ connection with scattering theory. _ here we show that eq .
( [ eq : a ] ) is closely linked to a physical observable . according to standard scattering theory@xcite
, the spin - dependent scattering amplitude in the anderson model is given by @xmath86 where @xmath87 and @xmath88 label the spin of the projectile , @xmath89 and @xmath90 are eigenstates of the full hamiltonian @xmath10 in absence of projectiles and @xmath91 is the exact ground state energy , i.e. @xmath92 and @xmath93 . in the local moment regime and for a large system containing an odd number of electrons
@xmath89 and @xmath90 are spin 1/2 ground states . at @xmath94
the spin resides on the localized level but for @xmath95 the magnetization is spatially delocalized@xcite .
below we use @xmath96 and @xmath97 to denote the spin direction ( @xmath98 or @xmath99 ) of @xmath89 and @xmath90 , respectively .
we evaluate spin - flip matrix elements perturbatively for the real part of eq .
( [ eq : t_lan ] ) with @xmath55 .
it is immediate to see that the leading order contribution agrees with @xmath100 .
the fourth order term involves expanding @xmath89 , @xmath90 , @xmath10 and @xmath91 to second order in @xmath9 ; the result agrees with eq .
( [ eq : t4_and ] ) .
in particular , the @xmath101 ( re)normalization of @xmath89 and @xmath90 @xcite coincides with the last term in eq .
( [ eq : t4 ] ) .
this easily - overlooked term is essential for the correct evaluation of the t - matrix . in sum , @xmath102 for @xmath55 .
the su(2 ) symmetry of the anderson hamiltonian dictates @xmath103 , where @xmath104 is a vector of pauli matrices and we ignore spin - independent scattering .
the imaginary part of the t - matrix , which quantifies the electronic scattering rate off the localized level , can then be extracted by virtue of the optical theorem : @xmath105 ^ 2 \propto j^2 + 2 j^3 \log(\lambda/\omega)+ ... $ ] _ multilevel quantum dots. _ we are now ready to evaluate @xmath0 and @xmath2 for real quantum dots via eq .
( [ eq : a ] ) . the universal hamiltonian of a quantum dot that is weakly connected to a conducting lead can be written as @xmath8 , where @xmath106 @xmath107 labels the discrete single - particle energy levels in the dot , @xmath108 is the number operator for dot electrons , @xmath109 is the gate charge , @xmath14 is the charging energy , and we have neglected intradot exchange interactions . for simplicity we take @xmath110 and @xmath111 for @xmath112 .
these simplifications are partly justified because our theory is uv - finite ( see below and the suppl .
material ) .
the unperturbed initial and final scattering states are @xmath113 , where @xmath53 is the fermi sea in the lead , @xmath114 creates a projectile in the lead just above the fermi surface , @xmath115 is an eigenstate of the dot containing @xmath116 electrons and @xmath117 creates an electron in the dot at level `` 0 '' located immediately above the highest ( @xmath118-th ) doubly - occupied level ( @xmath119 ) .
the unperturbed energy is @xmath120 , where @xmath121 is the kinetic energy of the filled fermi seas ( herein @xmath122 ) and @xmath123 is the kinetic energy for the singly - occupied level `` 0 '' ( tunable by a gate voltage ) .
@xmath124 is the coulomb energy cost for adding @xmath125 electrons to the dot ; we have chosen @xmath126 without loss of generality by shifting all @xmath127 by a constant .
we begin by recognizing that @xmath128 and that eq .
( [ eq : h_eff ] ) remains valid .
therefore @xmath100 and @xmath129 are given by eqs .
( [ eq : t2_0 ] ) and ( [ eq : t4 ] ) , respectively . for the former
we find @xmath130 where we used @xmath131 and defined @xmath132 .
next , we focus on @xmath129 .
its computation requires considering numerous sets of intermediate states ; these are listed in tables ii , iii and iv of the supplementary material . for simplicity
we start by separating out the contribution from the @xmath133 ( singly occupied ) level in the dot . assuming an infinite bandwidth in the lead we arrive at @xmath134 where @xmath135 .
eqs .
( [ eq : a2d ] ) and ( [ eq : m0_3 ] ) are independent of @xmath16 and essentially identical to those of the single - level anderson model . finally , we sum the contributions from @xmath136 levels .
these depend on @xmath16 and encode the influence of the multilevel energy spectrum in the kondo physics . tables ii and iii show that individual virtual processes involving @xmath136 levels are plagued with ir and uv divergences . remarkably , different divergences end up cancelling one another , partly assisted by the last two terms in eq .
( [ eq : t4 ] ) . on one hand ,
the cancellation of @xmath136 infrared divergences corroborates that kondo correlations arise only from processes involving the singly occupied level in the dot . on the other hand
, the cancellation of @xmath136 ultraviolet divergences confirms that high - energy excited states in the dot and lead do not alter the physics of the kondo effect . in spite of being divergence free ,
the influence of @xmath136 levels is important and makes the kondo coupling @xmath16-dependent . in the infinite bandwidth limit and in proximity to the particle - hole symmetric point ( @xmath137 )
we obtain @xmath138,\ ] ] where @xmath21 is a dimensionless function of @xmath22 evaluated numerically ( fig . 1 and suppl .
material ) . when @xmath139 , @xmath140 and multilevel effects are negligible
; in the opposite limit @xmath141 and multilevel effects are important .
the sum of eqs .
( [ eq : a2d ] ) , ( [ eq : m0_3 ] ) and ( [ eq : a4 ] ) can be arranged as @xmath142 .
for @xmath143 we obtain @xmath144 where @xmath145 is the kondo coupling corresponding to a single - level dot and @xmath146 is the width of the energy levels in the dot .
eq .
( [ eq : l_dot ] ) is valid for @xmath147 , i.e. @xmath148 , and constitutes the main result of this paper .
we selected @xmath0 on physical grounds so that it sets the energy scale below which ( i ) the universal hamiltonian maps onto the kondo hamiltonian , ( ii ) the renormalization group flow for @xmath2 is that of the simple kondo model . _
experimental implications. _ from eq .
( [ eq : l_dot ] ) , the kondo temperature for a multilevel quantum dot is @xmath149 , for any @xmath22 insofar as @xmath148 ( this condition implies that the broadening of the many - body energy eigenvalues of the isolated dot is much smaller than the energy spacing between them ) .
fig .
1 displays @xmath4 as a function of the linear dot dimension @xmath150 . introducing a lengthscale @xmath151 such that @xmath152 and @xmath153
, it follows that @xmath154 and @xmath155 .
@xmath156 is kept fixed ( independent of @xmath150 ) and we take @xmath157 and @xmath158 ; these are reasonable extrapolations based on available experimental data .
clearly haldane s single - level formula is accurate for smallest dots with @xmath139 ; in contrast , the multilevel enhancement of the kondo temperature becomes important for larger dots with @xmath159 . for @xmath160
, @xmath21 is so large that @xmath20 is possible only for a very small value of @xmath161 , which in turn results in an unmeasurably low @xmath4 .
therefore eq .
( [ eq : l_dot ] ) is experimentally relevant for dots with @xmath162 , wherein the multilevel enhancement is more modest yet still noticeable ( @xmath163 ) . in conclusion ,
we have developed a method to evaluate the kondo temperature of real quantum dots with unprecedented precission .
our theory predicts an unconventional and potentially measurable size - dependence of @xmath4 in dots with @xmath164 .
our formalism is valid and our results readily generalizable for models that incorporate energy - dependence in the dot - lead tunneling amplitude as well as non - uniform distribution of energy levels in the dot . _
acknowledgements. _ we are indebted to a. andreev , o. entin - wohlman , j. folk and l. glazman for helpful conversations .
this research has been supported by nserc and cifar .
50 for a review see e.g. p. coleman , _ many - body physics _ , http://www.physics.rutgers.edu/@xmath165coleman/mbody.html .
f.d.m .
haldane , j. phys .
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d. goldhaber - gordon _ et al .
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van der wiel _
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science * 289 * , 2105 ( 2000 ) .
i.l . aleiner _
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phys . rep . *
358 * , 309 ( 2002 ) .
h. suhl , phys . rev . * 138 * , a515 ( 1965 ) ; h. suhl in _ theory of magnetism in transition metals _ , ed .
w. marshall ( academic press , new york , 1967 ) .
see e.g. a. messiah , _ quantum mechanics ( vol .
ii ) _ ( north - holland publishing co. , amsterdam , 1962 ) .
d.c .
langreth , phys .
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( addison - wesley , reading , ma , 1994 ) .
|
.volume fraction for @xmath98 , rlp . [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ]
according to the theory , the average voronoi volume for a packing with a distribution of radius @xmath109 , is given by the following self - consistent equation : where the different quantities are calculated as follow : @xmath111 @xmath112 to simplify we denoted @xmath113 , @xmath114 and @xmath80 the step - function .
@xmath115 @xmath116 @xmath117 @xmath118 @xmath119 @xmath120
|
we develop a model to describe the properties of random assemblies of polydisperse hard spheres .
we show that the key features to describe the system are _ ( i ) _ the dependence between the free volume of a sphere and the various coordination numbers between the species , and _
( ii ) _ the dependence of the coordination numbers with the concentration of species ; quantities that are calculated analytically .
the model predicts the density of random close packing and random loose packing of polydisperse systems for a given distribution of ball size and describes packings for any interparticle friction coefficient .
the formalism allows to determine the optimal packing over different distributions and may help to treat packing problems of non - spherical particles which are notoriously difficult to solve .
understanding the basic properties of spheres packings is a major challenge since this problem may provide valuable knowledge regarding low temperature phases in condensed matter physics @xcite .
the canonical example is perhaps the monodisperse sphere packing problem .
it has been mathematically proven that the optimum way to arrange monodisperse spheres is the face - centered cubic lattice ; a problem that has been solved recently by hales , @xmath0 400 years after the famous kepler conjecture on the issue . on the other hand , it is commonly observed that packings arrange in a random fashion at a lower density state called random close packing or rcp @xcite .
furthermore , packings are mechanically stable up to an even lower limit called random loose packing , rlp . in parallel with the large literature dealing with monodisperse sphere packings , a large body of experimental
, theoretical and numerical work has been devoted to the analysis of polydisperse systems ; the interest arising due to the simple fact that polydispersivity is omnipresent in most realistic systems and industrial applications @xcite . while previous approaches have focused on frictionless packings , an integrated analytical approach that brings together different observations for all packings from rlp to rcp and for any friction or coordination number is still lacking .
based on our previous statistical mechanics approach @xcite , here we build such a framework .
we show that the key aspect to solve this problem is the dependence of the various coordination numbers between the different species and the concentration of the species .
this is calculated here and shown to agree well with computer simulations .
this result is then incorporated into a statistical theory of volume fluctuations as in @xcite which calculates the free volume of a particle in terms of the coordination number .
the main result is the prediction of the rlp and rcp limiting densities for a given distribution of ball sizes as well as the prediction of densities for any packing in between those limits .
the formalism allows for a determination of the best packing fraction in terms of different distribution of ball sizes with specified constraints , as we show with a simple example .
we discuss possible generalization of the method to solve more difficult problems like the phase behavior of systems of non - spherical particles like rods or spherocylinders in any dimensions ; problems of long - standing history in condensed matter @xcite .
recent theoretical advances @xcite allow for the prediction of the density of rcp and rlp for equal - size ball packings using a relation between the average volume and the geometrical coordination number . following this approach ,
we here describe long - range spatial correlations through a mean - field background term .
this approximation makes the problem amenable to analytic calculations , and is shown to describe well our simulation results .
an explicit inclusion of such correlations is possible in our framework , but severely complicates any solution attempts .
thus , we believe that the present approach is accurate enough for many important properties , such as the volume fraction calculation . the above theoretical framework will guide the present formalism for polydisperse systems .
we first treat the case of binary mixtures of hard spheres of radius @xmath1 and @xmath2 in 3d and then generalize the problem to any distribution and dimension .
* calculation of @xmath3. * the key quantity to calculate is @xmath3 , denoting the mean number of contacts of a ball of radius @xmath4 with a ball of radius @xmath5 , versus the concentration of one of the species . we need a formula for @xmath3 as a function of @xmath6 , the later being the size ratio , @xmath7 is the fraction of small balls in the packing @xmath8 with @xmath9 the number of @xmath10 balls and @xmath11 is the global geometrical coordination number averaged over all the particles : @xmath12 where @xmath13 is the average coordination of each species .
the coordinations are determined by three equations : @xmath14 we assume that these coordinations are inversely proportional to the average solid angle extended by contacting balls @xmath1 and @xmath15 .
the average solid angles are denoted @xmath16 and are calculated in terms of the solid angle that a ball @xmath5 occupy on a ball @xmath4 according to @xmath17 with ( see fig .
[ voro]a ) @xmath18 thus , @xmath19 represents the mean occupied surface on a @xmath10 ball weighted by the concentrations @xmath7 and @xmath20 .
this represents an approximation since the real weights are @xmath21 and @xmath22 , respectively . then , @xmath23 leading to the following normalizations : @xmath24 thus , the system of eqs .
( [ z1221 ] ) is reduced to a system of three equations for four unknowns @xmath3 . to close the system
we assume proportional laws and deduce @xmath3 from @xmath13 by considering that @xmath3 is proportional to the number of contacts of the @xmath10 balls times the number of contacts of the @xmath25 balls : @xmath26 using the first equation in ( [ z1221 ] ) we find the constants @xmath27 and @xmath28 , leading to the solution : @xmath29 figure [ fig : zandp]a , compares this solution to numerical simulations of hertz packings jammed at rcp @xcite for @xmath30 and @xmath31 .
we find that the formulae are very accurate for size ratios below 1.5 and present small deviations up to size ratio 2 .
the results are also in agreement with @xcite . *
the voronoi cell. * the common way to divide a system into volumes associated with each particle is the voronoi tessellation . the voronoi cell for monodisperse particles @xcite is composed by all the points nearest to the center of the ball than to any other ball .
this definition has been extended in @xcite to the case of polydisperse systems and non - spherical particles : instead of considering the classical voronoi polyhedron defined by the center of the particle , one should consider all the points which are closer to the surface of a given particle .
such a construction is called the _
voronoi s region _ and tiles a system of nonspherical convex particles and polydisperse systems as can be seen in fig .
[ voro]b . following this approach
we calculate the average volume of a polydisperse voronoi cell , denoted @xmath32 .
the volume fraction is given by @xmath33 , where @xmath34 is the mean volume of a ball .
we first find the analytical formula of the voronoi s region .
the boundary of the voronoi cell in the direction @xmath35 of a @xmath10 ball next to a @xmath25 ball at position @xmath36 is ( fig .
[ voro]b ) : @xmath37 where @xmath35 and @xmath38 are unitary .
thus , the boundary of a voronoi cell of a ball @xmath10 in the direction @xmath35 is the minimum of @xmath39 over all the particles @xmath25 for any @xmath40 .
this leads to @xmath41 the volume of the cell of the ball @xmath10 is then given by @xmath42 .
we define the orientational voronoi volume , @xmath43 , along the direction @xmath35 by @xmath44 .
this leads to @xmath45 this definition leads to the results of @xcite when @xmath46 .
since the system is isotropic , the mean voronoi volume can be calculated as : @xmath47 * calculation of the mean voronoi volume. * having calculated the voronoi cell exactly in eq .
( [ voronoi ] ) , we now proceed to develop a probability theory of volume fluctuations in the spirit of the quasiparticle approximation used in @xcite to obtain the mean voronoi volume . for a given ball @xmath10
, the calculation of @xmath48 reduces to finding the ball @xmath49 that minimizes @xmath39 .
we call @xmath49 the voronoi ball for the ball @xmath10 .
we consider @xmath50 , @xmath51 and @xmath52 .
we have @xmath53 .
therefore , we just need to compute the inverse cumulative distribution function , denoted @xmath54 , to find all the balls @xmath25 with @xmath55 , and thus not contributing to the voronoi volume of the ball @xmath10 .
the average voronoi volume is then given by the expression @xmath56 we calculate the mean voronoi volume for the balls of radius @xmath1 and @xmath15 separately and then average them .
we denote @xmath57 and @xmath58 the inverse cumulative distributions respectively , and @xmath59 and therefore , @xmath60 * calculation of @xmath54. * there are three salient steps in the calculation of @xmath54 : _ ( i ) _ the separation of @xmath54 following eq .
( [ p_c ] ) . _
( ii ) _ the separation of each term @xmath61 , @xmath62 , into two contributions : a term taking into account the contact spheres , @xmath63 , and a bulk term , @xmath64 .
the contact term clearly depends on @xmath3 .
the bulk term averages over all spatial correlations of non - contact particles and , without significant loss of accuracy as shown below , we assume that it only depends on the average value of @xmath32 . in principle , it is possible to use a more realistic form for this term , but this would render the problem practically unsolvable . _ ( iii ) _ the separation of @xmath63 and @xmath65 into two terms @xmath66 and @xmath67 , @xmath68 , for each species .
an assumption of the theory ( to be tested a posteriori with computer simulations ) is that all of these terms are independent .
thus @xmath69 also , we work in the limit of large number of particles leading to boltzmann - like exponential distributions for each @xmath70 and @xmath71 @xcite .
four important quantities are then defined : _ ( i ) _ @xmath72 and _ ( ii ) _ @xmath73 : the excluded volume and surface on the ball , respectively , where no center of a ball @xmath25 can be located for a given ball @xmath10 and for a given @xmath74 . _ ( iii ) _ @xmath75 : the mean number of balls @xmath25 by the left free volume . _ ( iv ) _ @xmath76 : the mean number of balls @xmath25 by the left free surface on a ball @xmath10 .
these considerations lead to : @xmath77 next , we calculate these four quantities . to simplify we denote @xmath78 , @xmath79 and @xmath80 the step - function .
we obtain : @xmath81 where @xmath82 and @xmath83 .
the fourth quantity , @xmath76 , is the most difficult to calculate . in terms of the occupied areas
eqs .
( [ occ ] ) we have @xmath84
. however , for the contact terms , the analogy with the boltzmann - like exponential distribution of volumes is far from being exact .
this is because this form is based on the large number limit which in the case of contacting balls is no more than around 6 .
therefore , the exponential distribution is used as an ansatz with @xmath76 a variational parameter as in @xcite .
we denote @xmath85 the mean solid angle of the gaps left between the @xmath25 contacting neighbors of a @xmath10 ball ( fig .
[ voro]a ) .
we obtain : @xmath86 then , we perform numerical simulations to find @xmath87 .
we randomly generate balls of radius @xmath4 and @xmath5 with the proportion @xmath21 and @xmath22 respectively around a ball of radius @xmath4 and then evaluate the mean free surface @xmath88 and its inverse to obtain @xmath76 .
we find @xmath89 which is a generalization of the results of @xcite to polydisperse systems .
using eqs .
( [ pij ] ) , ( [ sstar ] ) and ( [ rhos ] ) into ( [ pc ] ) , @xmath54 can be calculated by solving the equations numerically .
figure [ fig : zandp]b depicts the comparison of the theoretical results of the probability of voronoi volumes @xmath54 with computer generated hertzian packings with @xmath31 for @xmath90 at rcp .
the calculated distribution is quite accurate for most of the range except for small deviations at high values of @xmath74 , which however , do not affect much the value of the average @xmath91 in eq .
( [ ww ] ) .
this shows that our mean - field approximation already captures the main contribution of the probability distribution function @xmath54 .
* calculation of @xmath32. * the above considerations lead to a final form to calculate @xmath92 using eq .
( [ pc ] ) into ( [ ww ] ) : @xmath93 notice that @xmath94 depends on @xmath32 , eq .
( [ sstar ] ) , and @xmath95 depends on the @xmath3 , eq .
( [ rhos ] ) , which in turn depends on the concentration @xmath7 and @xmath11 through eq .
( [ zs ] )
. therefore eq .
( [ final ] ) is a self - consistent equation to obtain @xmath96 , for a given @xmath97 . a numerical integration of eq .
( [ final ] ) is performed versus @xmath7 for a given value of @xmath11 . by considering the isostatic limits of @xmath31 and @xmath98 we predict the limits of rcp and rlp at zero friction and infinite friction between the spheres , respectively @xcite .
the results for the volume fraction at rcp versus @xmath7 are depicted in fig .
[ fig : jamcompare]a which also compares the results to numerically generated packings of hertz spheres @xcite .
we see a very good agreement indicating that the theory captures well the behavior of polydisperse packings and that the approximations used are reasonable . for size ratios larger than 2 deviations
are found indicating the limit of validity of the approach .
for any other value of interparticle friction between 0 and @xmath99 , the density is obtained by setting @xmath11 between the limiting isostatic values of 6 and 4 , respectively .
the resulting volume fraction is shown in fig .
[ fig : jamcompare]b .
the result for rlp for a given @xmath7 is a new prediction as this problem has not been investigated before .
our results promote new experiments to test the rlp limit of polydisperse systems shown in fig .
[ fig : jamcompare]b . the formalism can be extended to consider any distribution of sphere radius .
the main modification is that all the sums over the radius are replaced by integrations over the desired distribution of radius @xmath100 ( the binary case corresponds to two delta - functions at @xmath1 and @xmath15 ) . and
all the quantities are calculated for balls of internal radius @xmath101 and external @xmath102 and @xmath7 and @xmath20 are replaced by @xmath103 and @xmath104 , respectively , including the coordinations ( see supplementary information b ) .
this result allows to explore the space of radius distributions in search of the optimal packings for a given polydispersivity .
this analysis could be of industrial interest if one wishes to optimize the packing fraction by introducing different species in the mixture .
\(a ) versus @xmath7 for different values of @xmath97 as indicated .
error bars are std over 10 realizations of the packings with 10,000 balls .
( b ) three dimensional surface plot of @xmath105 as a function of @xmath11 and @xmath7 for @xmath106 .
the numerical results at rcp and rlp are provided in supplementary information a.,title="fig:",scaledwidth=25.0% ] ( b ) versus @xmath7 for different values of @xmath97 as indicated .
error bars are std over 10 realizations of the packings with 10,000 balls .
( b ) three dimensional surface plot of @xmath105 as a function of @xmath11 and @xmath7 for @xmath106 .
the numerical results at rcp and rlp are provided in supplementary information a.,title="fig:",scaledwidth=25.0% ] we calculate the volume fraction for several distributions @xmath100 constraint by ball radius in the range [ 1,2 ] , in search of the optimal packing .
we calculate the volume fraction for various @xmath100 ranging from uniform to two - peaked gaussian distributions of varied widths .
we find that the more small balls we add the better the packing until a certain point where the volume fraction starts to decrease .
this maximum can be rationalized assuming that the small balls always fill the gaps between the large ones as long as there are enough large balls .
further extensions of the theory to any dimension can be performed by replacing @xmath107 by d in eq .
( [ voronoi ] ) and developing a theory of volume fluctuation in d - dimensions .
we notice that many of the approximations employed in 3d may become exact for large d , thus we expect improved results in the mean field limit of infinite dimensions .
the method allows to treat more difficult problems .
for instance , the prediction of the volume fraction of a jammed system of non - spherical particles is a long - standing problem .
theoretical predictions of onsager @xcite are valid for large aspect ratios , like elongated rods .
experiments however , find interesting new physics for small aspect ratios . in this respect
, the present polydisperse theory could be mapped to the problem of ellipsoids , spherocylinders or rods .
a voronoi cell needs to be calculated as a function of the angles defining the orientation of the non - spherical particles in analogy of the calculation between two particles of different radii .
the integration over @xmath108 in eq .
( [ final ] ) is then replaced by integration over weighted orientational angles . the above analysis can also be extended to dimensions beyond three @xcite .
although many of the appproximations should work better in higher dimensions , some of the hypotheses ( for example , the contact term ansatz ) need to be reassessed .
thus , higher dimensions studies can not be addressed as trivial extensions and need to be handled with care . in summary
, a theoretical framework is presented that predicts the rlp and rcp limits of a system of polydisperse spheres and brings together distinct results into a common theoretical framework .
the formalism has the potential to solve other problems in condensed matter physics such as the mixing and phase behavior of systems of hard particles of different shapes and size . 99 a. coniglio , a. fierro , h. j. herrmann , m. nicodemi , eds , _ unifying concepts in granular media and glasses _
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the work has been supported by the european community - access to research infrastructure action of the improving human potential programme , by the daad exchange programme ( ppp - polen ) , by the polish state committe for scientific research ( grants no .
2p03b07123 and pb1060/p03/2004/26 ) and by the research centre jlich .
|
due to the high sensitivity of the @xmath0 reaction to the nucleon nucleon potential , bremsstrahlung radiation is used as a tool to investigate details of the nucleon nucleon interaction .
such investigations can be performed at the cooler synchrotron cosy in the research centre jlich , by dint of the cosy11 detection system .
+ the results of the identification of bremsstrahlung radiation emitted via the @xmath1 reaction in data taken with a proton target and a deuteron beam are presented and discussed .
+ the installation of a neutron detector at the cosy-11 facility [ 1,2 ] enables to study a plethora of new reaction channels .
it opens wide possibilities not only to investigate the isospin dependence of the meson production [ 3 ] , but also to measure the bremsstrahlung radiation created in the collisions of nucleons .
the study of the latter process is interested since it is highly sensitive to the kind of the nucleon - nucleon potential , and hence may serve as a tool to discriminate between various existing potential models [ 4,5 ] .
although bremsstrahlung radiation has been studied since many years , it is still the subject of interest of many theoretical and experimental groups [ 4,5,6,7 ] .
+ at the cosy11 experiment a signal from @xmath2quanta was observed in the time of flight distribution for the neutral particles measured between the target and the neutral particle detector [ 2 ] .
this encouraged us to analyse the data in view of the bremsstrahlung radiation in a free @xmath3 and a quasi
free @xmath4 reactions .
data have been taken using a proton target and a deuteron beam with a momentum close to the threshold of the @xmath5 process .
events corresponding to the @xmath3 and @xmath6 reaction have been identified by measuring the outgoing charged as well as neutral ejectiles .
the protons and deuterons are detected by means of drift chambers and scintillator hodoscopes while neutrons and photons are registered in a scintillator
lead sandwich type detector .
the momentum vectors of the charged ejectiles are reconstructed by tracking back the trajectories to the target point [ 1 ] . in case of a neutron the time of flight between the target and the neutral particle detector together with the known position of the hitted detection unit enables to determine its four momentum vector [ 8 ] . + in order to identify the @xmath3 reaction events with two tracks in the drift chambers and a simultaneous signal in the neutron detector have been selected . in fig.1 ( left ) the squared mass of one particle is plotted versus the squared mass of the other registered particle .
based on this figure the measured reactions can be grouped according to the type of ejectiles .
thus reactions with two protons , proton and pion , proton and deuteron , and pion and deuteron can be clearly selected .
next the distribution of the time of flight between the target neutron detector was determined with requirement that one of the charged particles was identified as a proton and the other as a deuteron . due to the baryon number conservation , gamma quanta
are the only one possible source of a signal in a neutron detector .
indeed a clear peak around the time corresponding to the time of light of the light is visible ( see fig.1 right ) .
the gamma quanta may originate from bremsstrahlung reaction or from the decay of produced mesons eg . via the @xmath7 .
it is possible to distinguish between these hypothesis calculating the missing mass produced in the @xmath8 reaction .
fig.2 left shows the distribution of the squared missing mass as obtained for the @xmath8 reaction . a significant peak around 0 mev@xmath9/c@xmath10 the squared mass of a gamma quanta
constitute an evidence for events associated to the deuteron
proton bremsstrahlung .
in addition a broad structure at higher masses originating from two pions emmited from reaction @xmath11 or two gamma quanta from @xmath12 reaction is visible .
the extraction of the total cross section of the @xmath3 reaction requires the luminosity and acceptance determination , which will be performed in the near future .
the analysis of the @xmath13 reaction is more complicated , however in this case all three baryons , namely two protons and neutron can be measured in the drift chambers and in the neutron detector , respectively .
fig.2 ( right ) presents the time of flight with the condition that in coincidence with a neutral particle also two protons were identified . a clear signal originating from gamma quanta
is seen at a time value of 24 ns which coresponds to the velocity of light . using the missing mass technique , it might be possible to verify if gamma quanta originate from the direct @xmath13 reaction . at present
the data analysis is in progress .
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we are grateful to the u.s .
department of energy for financial support .
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this paper demonstrates that complex @xmath0-symmetric periodic potentials possess real band spectra
. however , there are significant qualitative differences in the band structure for these potentials when compared with conventional real periodic potentials . for example , while the potentials @xmath1 @xmath2 have infinitely many gaps , at the band edges there are periodic wave functions but no antiperiodic wave functions . numerical analysis and higher - order wkb techniques
are used to establish these results .
.5 cm for a quantum mechanical model having a periodic potential the schrdinger equation is @xmath3 where the potential @xmath4 is periodic with period @xmath5 : @xmath6 in conventional treatments of eq .
( [ e1 ] ) @xcite the periodic potential @xmath4 is assumed to be real . imposing the condition that the wave function @xmath7 be bounded leads to a real spectrum consisting of continuous bands separated by gaps .
there is an infinite number of bands and gaps , except for the special family of so - called _ finite - gap _ potentials such as the lam potentials @xcite . in this note
we extend the conventional analysis to include the case of complex periodic potentials .
we find that complex periodic potentials having @xmath0 symmetry exhibit _ real _ band spectra , despite the non - hermitian character of the schrdinger equation ( [ e1 ] ) .
( here , @xmath8 represents parity reflection and @xmath9 represents time reversal . ) potentials having this symmetry satisfy @xmath10^*=v(x ) .
\label{e3}\end{aligned}\ ] ] examples of such potentials are @xmath11 , @xmath12 , and @xmath13 .
in addition to the property that these potentials have real spectra , the band structure displays several novel features that are strikingly different from the case of real periodic potentials .
the work reported here was motivated by recent investigations of non - hermitian @xmath0-symmetric hamiltonian models having real discrete spectra .
one such class of models is defined by the hamiltonian @xcite @xmath14 despite the lack of conventional hermiticity , the spectrum of this hamiltonian is real , positive , and discrete ; each of the energy levels increases as a function of increasing @xmath15 .
it has been observed that the reality of the spectrum is a consequence of @xmath0 symmetry , which is a weaker condition than hermiticity .
this observation has also been used to construct new classes of quasi - exactly solvable quantum theories @xcite and to study new kinds of symmetry breaking in quantum field theory @xcite .
there have been many other instances of non - hermitian @xmath0-invariant hamiltonians in physics .
energies of solitons on a _ complex _ toda lattice have been found to be real @xcite .
hamiltonians rendered non - hermitian by an imaginary external field have been used to study population biology @xcite and to study delocalization transitions such as vortex flux - line depinning in type - ii superconductors @xcite . in these last two cases ,
initially real eigenvalues bifurcate into the complex plane due to the increasing external field , indicating the growth of populations or the unbinding of vortices .
we begin by summarizing the standard floquet analysis of the schrdinger equation ( [ e1 ] ) for the case where @xmath4 is real and periodic @xcite .
we define a _ fundamental pair _ of linearly independent solutions @xmath16 and @xmath17 satisfying the initial conditions @xmath18 any solution @xmath7 to eq .
( [ e1 ] ) is a linear combination of @xmath16 and @xmath17 .
it is then a straightforward algebraic exercise to show that @xmath7 is bounded provided that the _ discriminant _ @xmath19 , which is defined by @xmath20 satisfies the constraint @xmath21 to illustrate the features of the discriminant we consider a typical periodic potential , @xmath22 , for which the period @xmath23 . in fig .
[ f1 ] we plot @xmath19 as a function of @xmath24 .
note that @xmath19 is oscillatory and is well approximated by the function @xmath25 for large @xmath24 .
the crucial feature of @xmath19 , which can not be seen from this plot , is that its graph crosses the lines @xmath26 an infinite number of times ; each of the maxima of @xmath19 lies above @xmath27 and each of the minima lies below @xmath28 .
the regions of energy for which @xmath29 are called _ bands _ and the regions of energy for which @xmath30 are called _ gaps_. the gap size decreases exponentially as a function of @xmath24 .
the band edges at which @xmath31 correspond to periodic solutions to eq .
( [ e1 ] ) , @xmath32 , and the band edges at which @xmath33 correspond to antiperiodic solutions @xmath34 . now consider the calculation of the discriminant for the case of a _ complex _ periodic potential @xmath4 .
in general , a complex periodic potential will have no bounded solutions because the discriminant is typically complex . however , for complex @xmath0-symmetric periodic potentials , one can easily show that the discriminant @xmath19 is real when @xmath24 is real .
the @xmath0 symmetry is crucial here ; for a potential that is not @xmath0 symmetric [ one that does not satisfy eq .
( [ e3 ] ) ] , the discriminant is complex for all values of @xmath24 .
having established that complex @xmath0-symmetric periodic potentials have real discriminants , we can then apply the criterion in eq .
( [ e7 ] ) to locate the real energy bands within which the corresponding wave function @xmath7 is a bounded function .
we have computed the discriminants for the class of complex @xmath35-symmetric periodic potentials @xmath36 in figs .
[ f2]-[f7 ] we plot the discriminants for the cases @xmath37 . while these plots superficially resemble fig .
[ f1 ] for large @xmath24 , they exhibit new and intriguing features that are significantly different from the case of a real periodic potential .
the most obvious new feature is the appearance for @xmath38 of a local minimum of @xmath19 between @xmath28 and @xmath27 .
such a dip is rigorously forbidden in the case of real periodic potentials @xcite . the most dramatic differences between the discriminants for the complex @xmath35-symmetric periodic potentials in eq .
( [ e8 ] ) and real periodic potentials can not be easily seen in the figures .
we have performed a careful numerical study of the local minima and maxima of @xmath19 .
our study reveals that _ none of the local minima lies below _
@xmath28 .
this shows that there are _ no antiperiodic solutions _
@xmath7 to the schrdinger equation ( [ e1 ] ) .
nevertheless , all of the local maxima of @xmath19 lie above @xmath27 .
hence , there are an infinite number of band gaps in the spectrum and the band - edge wave functions are periodic . @xmath39 @xmath39 to perform this numerical analysis it is necessary to locate the positions of the local minima and maxima of @xmath19 to extremely high accuracy
. this can be done using wkb methods @xcite .
we take the energy @xmath24 to be large @xmath40 and define a small parameter @xmath15 by @xmath41 then we make an exponential _ ansatz _ for the wave function @xmath7 : @xmath42 . \label{e10}\end{aligned}\ ] ] substituting @xmath7 in eq .
( [ e10 ] ) into the schrdinger equation ( [ e1 ] ) gives a recursion relation for the functions @xmath43 : @xmath44 ^ 2&=&0,\nonumber\\ iq_0''(x)-2q_0'(x)q_1'(x)&=&0,\nonumber\\ iq_1''(x)-2q_0'(x)q_2'(x)-[q_1'(x)]^2-v(x)&=&0,\nonumber\\ iq_{n-1}''(x)-\sum_{j=0}^nq_j'(x)q_{n - j}'(x)&=&0\quad(n\geq3 ) .
\label{e11}\end{aligned}\ ] ] the solution to these equations is @xmath45,\nonumber\\
q_4(x)&=&\pm{1\over8}\left(v'(x)-v'(0)-\int_0^xdt\,v^2(t)\right),\nonumber\\ q_5(x)&=&{i\over16}\left[v''(x)-v''(0)-2v^2(x)+2v^2(0)\right],\nonumber\\ q_6(x)&=&\mp{1\over32}\left(v'''(x)-v'''(0)-5v(x)v'(x)+5v(0)v'(0 ) + \int_0^xdt\,[2v^3(t)-v(t)v''(t)]\right ) , \label{e12}\end{aligned}\ ] ] and so on .
note that @xmath43 is normalized so that @xmath46 .
in general , the formula for @xmath47 is @xmath48\quad(n\geq3 ) .
\label{e13}\end{aligned}\ ] ] in order to obtain a wkb formula for the discriminant @xmath19 in eq .
( [ e6 ] ) we need to evaluate @xmath43 at @xmath49 .
the periodicity of the potential @xmath4 simplifies the results considerably ; when @xmath50 is odd , @xmath51 and when @xmath50 is even , only the integrals in eq .
( [ e12 ] ) remain : @xmath52 , \label{e14}\end{aligned}\ ] ] and so on .
the wkb formula for the discriminant is particularly simple when the potential @xmath4 is @xmath0 symmetric : @xmath53 . \label{e15}\end{aligned}\ ] ] one obtains the same formula for potentials that are real and symmetric under parity @xmath8 . for the complex @xmath0-symmetric potentials @xmath4 in ( [ e8 ] ) the wkb formula for the discriminant in eq .
( [ e15 ] ) is @xmath54 .
\label{e16}\end{aligned}\ ] ] a similar wkb formula exists for the real odd - parity potentials @xmath55 : @xmath56 . \label{e17}\end{aligned}\ ] ] we can illustrate the extreme accuracy of these wkb approximations by comparing them with numerical computations of the discriminant .
for example , for @xmath57 at @xmath58 ( which corresponds to @xmath59 ) we find numerically that @xmath60 .
the first three orders of the wkb approximation taken from eq .
( [ e16 ] ) give @xmath61 , @xmath62 , and @xmath63 .
similarly , for @xmath64 at @xmath58 we find numerically that @xmath65 .
the first three orders of the wkb approximation taken from eq .
( [ e17 ] ) give the same values : @xmath66 , @xmath62 , @xmath63 .
despite this impressive precision , the wkb formulas ( [ e16 ] ) and ( [ e17 ] ) can not be used directly to answer the crucial question of whether there are band gaps because these approximations to the discriminant @xmath19 never cross the values @xmath26 .
the reason for this inadequacy of the wkb approximation is that the differences @xmath67 and @xmath68 are exponentially small when @xmath69 .
therefore , these differences are subdominant with respect to the wkb asymptotic series and are not accessible to any order in powers of @xmath15 . indeed
, the wkb series can only provide information about quantities with an _ algebraically _ small error , and not an exponentially small error .
we emphasize that the wkb approximation has this shortcoming only at the maxima and minima of the approximation . at other points
any exponential discrepancy is completely negligible compared with algebraic errors .
the wkb series is still an extremely useful ingredient in the numerical search for zeros of @xmath70 .
( these zeros are the dividing points between bands and gaps . )
our procedure is first to find the energies at which there are maxima and minima of the wkb approximation to the discriminant and then to evaluate , with high numerical precision , the actual value of the discriminant in a tiny neighborhood of each of these points . by doing this
we can determine whether or not the discriminant @xmath19 crosses the lines @xmath26 .
for the real periodic potentials @xmath71 our procedure confirms the rigorous theoretical result that every maximum of the discriminant lies above @xmath27 and every minimum lies below @xmath28 . consider , for example , the potential @xmath22 . from fig .
1 , it is clear that the first maximum lies above @xmath27 .
the first minimum occurs at @xmath72 , where the discriminant has the value @xmath73 .
the second maximum occurs at @xmath74 , where the discriminant is @xmath75 .
similar behavior is found for the other potentials in the class @xmath71 . in stark contrast , for the potentials @xmath76 , while the maxima of the discriminant lie above @xmath77 , the minima of the discriminant lie above @xmath28 .
thus , for these potentials there are no antiperiodic wave functions . as an example
, lengthy and delicate numerical analysis verifies that for the potential @xmath78 the first three maxima of the discriminant @xmath19 are located at @xmath79 , @xmath80 , and @xmath81 .
the value of the discriminant @xmath19 at these energies is @xmath75 , @xmath82 , and @xmath83 .
the first two minima of the discriminant are located at @xmath84 and @xmath85 and at these energies @xmath19 has the values @xmath86 and @xmath87 .
similar behavior is found for the other potentials in the class ( [ e8 ] ) .
we conclude by pointing out that from the expressions for @xmath88 in eq .
( [ e14 ] ) the wkb series truncates if the potential is a polynomial in @xmath13 .
for example , for the complex @xmath0-symmetric periodic potential @xmath89 the wkb series in eq .
( [ e15 ] ) truncates after the first term because @xmath90 vanishes for @xmath91 . for this case
the wkb approximation is exact and the discriminant is given by @xmath92 one can verify this result directly by solving the schrdinger equation ( [ e1 ] ) for this potential exactly ; the solution is a bessel function : @xmath93 .
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j. bae is supported in part by the hanyang university fellowship and y. kwon is supported in part by the fund of hanyang university .
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it is questionable that grover algorithm may be more valuable than a classical one , when a partial information is given in a unstructured database . in this letter , to consider quantum search when a partial information is given , we replace the fourier transform in the grover algorithm with the haar wavelet transform .
we then , given a partial information @xmath0 to a unstructured database of size @xmath1 , show that there is the improved speedup , @xmath2 .
suppose that we have a problem of finding a desired one of unstructured @xmath1 items .
it is known that a classical search algrithm may take @xmath3 times , but the fast quantum search algorithm provides the quadratic speedup , @xmath4.@xcite the quantum algorithm whose central idea is amplitude amplification was first provided by grover in 1996.@xcite@xcite it is well - known that grover algorithm is 1)optimal in the context of applying unitary operators repeatdly and 2)efficient in searching a target of a unstructured database .
however , it is not evident that , when a partial information is given , the grover algorithm is still more valuable than a classical one . in this letter , we provide the quantum search algorithm which is able to benefit from a partial information .
the key building block in our construction is the ( haar ) wavelet transform instead of the fourier transform in the grover algorithm .
+ let us first consider the grover algorithm .
a bijection between a database and quantum states is necessary before applying the grover algorithm .
if a superposition of @xmath1 states is initially prepared , the grover algorithm amplifies the amplitude of the target state up to around one , while those of other states dwindle down to nearly zeros .
the amplitude amplification is performed by two inversion operations : inversion about the target by the oracle and inversion about the initial state by the fourier transform .
noting the fact that two simultaneous reflections about two mirrors crossing by an angle @xmath5 induce @xmath6 rotation , one may imagine that the inversions in the grover algorithm rotate the initial state around the target state.@xcite if the target state and the initial state are denoted by @xmath7 and @xmath8 respectively,(here the initial state is prepared by the fourier transform of a state @xmath9 , i.e. @xmath10 ) the inversion operators is expressed as @xmath11 since @xmath12 , the grover operator is written as @xmath13 then , after applying the operator @xmath4 times , the final state comes to @xmath14 , which is @xmath15 , @xmath16 .
the query complexity of this algorithm , the number of callings of the oracle , is therefore @xmath4 .
we here note that the running time has nothing to do with the choice of @xmath9 . + to consider the partial information , let us think about the following situation .
+ _ suppose that hanyang university library has @xmath1(@xmath17 , for some @xmath18 ) books .
each book has a code number identifying itself .
recently , the library decides to label new code numbers to each book since a better way to classify books has provided .
a labelling machine is applied to the tedious job .
each code is composed of @xmath19 numbers .
the first numbers must be @xmath20 , since they stands for the region .
and hanyang university is in south korea , so the second number is @xmath20.the other numbers stand for the category each book is included .
an earlier number means a larger category .
the rules of labeling a code number is that the @xmath21-th number , @xmath22 , is in @xmath23 $ ] for @xmath24 .
the following codes are a good example , _
@xmath25 the first code means @xmath26 and the next one @xmath27 .
the first two numbers @xmath28 stands for the region , and hanyang university . to aid the searching a book in the library
, the library also has implemented the grover algorithm in a quantum computer .
+ after the labelling is completed , one discovers a fault in the machine .
therefore , the library decides to do labelling again , fixing the fault in the labelling machine .
they then recommend to apply the grover algorithm to whom may be concerned to search a book , during labelling .
+ dr .
lee , a postdoc in physics , has to submit his research paper by tomorrow @xmath29 a.m. it is now 6 p.m. he wish to fill the reference section in the paper , but the library in mess blocks his work .
the director of the library recommends him to try the grover algorithm , but it was calculated , based on the number of books @xmath1 , that @xmath30 hours is the running time of the algorithm .
therefore , he concludes that the grover algorithm can not help him . at that moment
, he gets a call from the library that the only @xmath21-th number of all code numbers is correct .
+ it is sure that the grover algorithm can not complete the search task in time , since it takes over @xmath30 hours .
the only thing that dr .
lee can use in order to overcome his hopeless situation , is the fact that only the @xmath21-th number was correctly labelled .
the partial information may save his problem .
however , the grover algorithm can not benefit from the partial information of this problem . +
we here introduce the fast wavelet quantum search algorithm(wqsa ) , which is a modification of the grover algorithm by replacing the fourier transform with the haar wavelet transform , to resolve the situation of dr .
lee .
let us note that to apply the following operator @xmath31 is one iteration of the wqsa .
the haar wavelet transform @xmath32 is represented @xmath33 where @xmath34.\end{aligned}\ ] ] and @xmath35 the haar 1-level decomposition operator as follows ; @xmath36_{2^k \times 2^k}\end{aligned}\ ] ] we have used @xmath37 as the @xmath38 unit matrix and @xmath39 as the @xmath40 zero matrix .
it is clear that the wavelet transform @xmath32 is unitary since the operator @xmath41 is unitary .
+ since the operator is composed of the wavelet transform , consistently , the initial state is prepared by applying the inverse wavelet transform @xmath42 to a state @xmath9 , i.e. , the initial state is now @xmath43 .
the power of our wqsa appears in the initialization procedure .
it is quite remarkable that the state @xmath44 is a superposition of @xmath45 states , where @xmath46(@xmath21 is given by @xmath47 ) , while the state @xmath48 is a superposition of @xmath1 states .
then it is expected that the running time is @xmath2 .
choosing the initial state as @xmath44 , @xmath49 when the target state exists in the restricted domain of the @xmath45 states , we look forward to an improved speedup with the partial information .
since @xmath50 , by setting @xmath51 , @xmath52 and @xmath24 , and @xmath53 , the state @xmath44 is explicitly , @xmath54 the following diagram shows the set of states composing @xmath44 . .
the @xmath55axis runs from @xmath56 to @xmath57 .
when the initial state is @xmath58 or @xmath59 , the running time is the same to that of the grover algorithm since the lowest two retangulars include all states.,scaledwidth=40.0% ] we thus arrive at the following proposition .
suppose that we solve the problem of finding a desired one in the set @xmath60 .
given a partial information that the target state is in the subset @xmath61 , we complete the search task in @xmath62 times by choosing the initial state as @xmath63 .
proof ) let the target state @xmath64 . let us take the initial state as @xmath65 .
it suffices to show that it takes @xmath62 times for the wqsa to find the target state with the following setting .
+ let @xmath66 .
the wavelet quantum search operator is @xmath31 where @xmath32 is the haar wavelet transform .
applying the operator @xmath42 to the @xmath9 , we have the initial state @xmath67 and the state @xmath68 iterations of the operator @xmath69 create the following state , @xmath70 iterations is @xmath71 , where @xmath72 and @xmath73 .
hence , we have shown that the total number of iterations is @xmath62 . if we denote @xmath74 and @xmath46 , then the running time is written as @xmath2 q.e.d .
+ let us revisit the dr .
lee s problem.the partial information that the @xmath21-th number @xmath22 is correctly labelled leads dr .
lee to apply the wqsa so that the reference section is filled in time .
however note that there is no improvement in running time when the intial state is @xmath58 or @xmath59 since , in those cases , the initial state is still a superposition of @xmath1 states .
therfore , from the proposition , we know that he can complete the submission in time if the @xmath21 is larger than @xmath75 .
+ we have exerted how to utilize a partial information , in order to enhance quantum search .
our construction provides a way for quantum search to benefit from a partial information .
since the running time of the grover algorithm has nothing to do with the choice of unitary operator , the complexity of the wqsa is the same to the grover algorithm .
however , we have obtained the speedup @xmath2 by preparing the initial state as @xmath44 .
the running time of the wqsa depends on the choice of @xmath47 , while that of the grover algorithm does not .
this is because the state @xmath44 is a superpositin of states in the restricted domain of @xmath45 states .
the speedup is indeed originated in the initialization .
+ finally , let us discuss the haar wavelet basis .
although other wavelet transforms may be applied to the wqsa , we chose the haar wavelet transform .
it is observed that the first half of the haar wavelet basis differs with the second half of the wavelet basis on the phase @xmath76 .
this implies that destructive and constructive interference between states accepts a set of states containing the target and rejects the other states . in this sense , other known wavelet basis , e.g. daubechies s ,
are not appropriate to play the role of seclecting a subset of the @xmath1 states .
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the work of c.s.k . was supported by grant no . r02 - 2003 - 000 - 10050 - 0 from brp of the kosef .
y.d . is supported in part by nfsc under contract no.10305003 , henan provincial foundation for prominent young scientists under contract no.0312001700 and in part by the project sponsored by srf for rocs , sem .
this work is also supported by grant no .
f01 - 2004 - 000 - 10292 - 0 of kosef - nsfc international collaborative research grant .
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the pure penguin process @xmath0 is one of the most important probes of physics beyond the standard model .
recently babar and belle have measured the unexpectedly large transverse polarization in the decays @xmath0 , which may single out new physics effects beyond the standard model .
we study the possibility that the phenomenon could serve as an important probe of anomalous tensor interactions . we find that a spin flipped tensor interaction with a small strength and a phase could give a possible solution to the polarization puzzle . + * pacs numbers 13.25.hw , 12.60.jv * ( 0,0 ) looking for signals of physics beyond the standard model is one of the most important missions of high energy physics .
it is well known that flavor - changing neutral currents induced in @xmath1 decays are one of the best probes of new physics beyond the standard model because they arise only through loop effects in the standard model ( sm ) . to this end , the decays @xmath0 are of particular interests , since they are pure penguin processes and have interesting polarization phenomena as well as relatively clear experiment signature . within the sm
, it is expected that both @xmath2 and @xmath3 are mainly longitudinally polarized , while its transverse polarization is suppressed by the power of @xmath4 . however , last year both babar and belle had observed rather small longitudinal polarizations in the decays @xmath5 due to @xmath6 , both groups have measured unexpectedly large transverse polarizations in the @xmath7 decays . this summer babar collaboration has again reported their full angular analysis of the the decay @xmath8
@xcite @xmath9 which has confirmed their previous measurements and called urgent theoretical explanations . the final states @xmath2 and @xmath10 are fast moving in the @xmath1 meson frame and any spin flip of fast flying quark will be suppressed by power of @xmath11 .
the charge interaction currents structure of the sm is left - handed , therefore , will result in the dominance of longitudinal polarization .
such situation has been known to us for many years @xcite .
so that , the recent measurements of large transverse polarizations in @xmath0 are referred as a puzzle within the high energy physics community @xcite . the analysis of the decays within the sm can be performed in terms of an effective low - energy theory with the hamiltonian @xcite @xmath12 the amplitude for the decay within the sm can be written as @xmath13~.\end{aligned}\ ] ] since @xmath1 meson is a pseudoscalar , the final two vector mesons must have the same helicity . in the helicity basis
, the amplitude can be decomposed into three helicity amplitudes , which are @xmath14~,\nonumber \\ h_{\pm\pm}&=&i\frac{g_f}{\sqrt{2}}v_{tb}v_{ts}^ { * } a(\phi k^ * ) m_{\phi}f_{\phi } \left [ ( m_{b}+m_{k^*})a_1 \mp \frac{2m_b p_c } { m_b + m_{k^*}}v \right]~.\end{aligned}\ ] ] in naive factorization @xcite , @xmath15 then the branching ratio is thus read as @xmath16 and the longitudinal and the transverse polarization rates are @xmath17 using the wilson coefficients @xmath18 evaluated at scale of @xmath19 @xcite , and the decay constants @xmath20 gev , @xmath21 gev and the form factors of light - cone qcd sum - rules @xcite , one can get @xmath22 it must be reminded that a theoretical estimation of the branching ratios depend very strongly on the form factors from different hadronic models and the theoretical frameworks of @xmath1 meson nonletponic decays , even though most frameworks and form factors predict dominance of the longitudinal polarizations . for example , recent calculation of @xmath23 by cheng and yang by using qcd factorization @xcite gives @xmath24 for lcsr and @xmath25 for bsw form - factors @xcite , respectively , while pqcd @xcite calculation gives @xmath26 where form - factors are not inputs . however , both studies present the dominance of longitudinal polarization .
after babar and belle measurements of the abnormal large transverse polarization , there have been some theoretical explanations ; namely , through the final state interactions ( fsi ) contributions @xcite , large annihilation contributions and new physics from right - hand current interactions @xcite and transverse @xmath2 from the emitted gluon of @xmath27 which might be enhanced by new physics @xcite . in this letter , we investigate the possibility that the abnormally large transverse polarization may arise from a new tensor interaction beyond the sm ( bsm ) , @xmath28 where @xmath29 is the relative interaction strength normalized to that of @xmath30 in the sm and @xmath31 is the new physics phase . in principle , such a tensor operator could be produced even in mssm @xcite .
interestingly , the recent study of radiative pion decay @xmath32 at pibeta @xcite has found deviations from the sm in the high-@xmath33 kinematic region , which may indicate the existence of a tensor quark - lepton interaction @xcite .
we also note that kagan mentioned the case of tensor operator for resolving the puzzle @xcite .
our starting point arises from the observation that the tensor interaction only contributes to transverse polarization but not to longitudinal one .
the matrix element reads @xcite @xmath34 which is scaled as @xmath35 since @xmath36 for fast flying @xmath2 .
however , if the @xmath2 meson is produced instead from a vector interaction vertex , we will have @xmath37 , and it is easy to understand that the longitudinal polarization dominate over the transverse one by a large factor @xmath38 because of @xmath39 . using the form factors defined in ref .
@xcite , we can write down the amplitude of the tensor operator in eq .
( [ tensor ] ) in naive factorization approximation , @xmath40 from this equation , we can get the new physics contributions , @xmath41~.\end{aligned}\ ] ] compared with eq.9 , the tensor interaction contributions to @xmath42 are enhanced by a factor of @xmath43 . [ cols="^,^ " , ] numerical results are presented in fig .
1 . from fig .
1(a ) , we can find that the transverse polarization in @xmath44 is very sensitive to the presence of new tensor interactions .
for @xmath45 , we can easily find solutions to the polarization puzzle depending on the phase of the tensor interaction .
for example , to account for @xmath46 within @xmath47 , we get intervals @xmath48 , @xmath49 , @xmath50 for @xmath51 , respectively . of course , the branching ratio measurements could also give constraints on such a tensor interaction operator , which are presented in fig .
1(b ) . here
we can see the windows are very narrow because the longitudinal contribution estimated within the sm already saturate the experimental branching ratio .
however , it is well known that theoretical calculations of the branching ratios of hadronic @xmath1 decays suffer from large uncertainties .
it is believed that polarization fractions could be predicted more accurately than the branching ratios , because some of hadronic uncertainties could be cancelled in the former ones . in the future , if theoretical frameworks for hadronic @xmath1 decays could achieve @xmath52 accuracy and their predictions of longitudinal branching ratio still saturate the experimental measurement , the tensor interaction scenario could be ruled out . in such a case , we need not only new physics contributions to transverse part but also new contributions destructive to longitudinal part
. however , it would be very hard to account for the large branching ratio of @xmath53 because the similarity between the amplitudes of @xmath53 and the longitudinal amplitude of @xmath54 in the heavy @xmath55 limit . in conclusion
, we have studied the large transverse polarization puzzle in @xmath54 decays , which is taken as an important probe of an anomalous tensor interactions .
we find that a relatively weak tensor interaction could resolve the puzzle .
if we take the coupling @xmath56 , such a solution might be a signal of new physics with tensor interaction at tev scale . with the running of @xmath1 factories babar and belle ,
we have witnessed many challenging phenomena .
theoretically , we need more accurate and complete framework to clarify whether the sm could explain those abnormal phenomena or not . + _ note added : when we finished our work , we note the paper@xcite where the same tensor operator is also studied . _
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+ local - susceptibility measurements via the nmr shifts of @xmath0p and @xmath1v nuclei in the high - pressure phase of ( vo)@xmath2p@xmath2o@xmath3 confirmed the existence of a unique alternating antiferromagnetic chain with a zero - field spin gap of 34 k. the @xmath0p nuclear spin - lattice relaxation rate scales with the uniform spin susceptibility below about 15 k which shows that the temperature dependence of both the static and dynamical spin susceptibilities becomes identical at temperatures not far below the spin - gap energy .
magnetic excitations of a low - dimensional quantum antiferromagnet have been one of the current topics among the condensed matter physicists .
vanadyl pyrophosphate ( vo)@xmath2p@xmath2o@xmath3 had long been believed as a prototype of a spin-@xmath4 two - leg ladder which has a magnetic lattice intermediate between one and two spatial dimensions @xcite . the ladder model , however , has been rejected by an observation of a dominant magnetic interaction perpendicular to the supposed ladder axis via the inelastic neutron scattering ( ins ) measurements @xcite .
a dimerized ( alternating ) chain model has now been becoming accepted as an alternative starting point , although a mechanism of the major exchange interaction between distant pairs of v@xmath5 spins via po@xmath6 tetrahedra is still under study @xcite .
the ins experiments has also revealed the existence of the mode with a gap nearly twice the gap of the lowest excited triplet which can not be explained by a simple alternating - chain model .
this mode has first been assigned as a bound state of two magnons possibly formed via interchain couplings @xcite , but it was difficult to account for the intensity comparable to the fundamental mode .
recent nmr @xcite and high - field magnetization @xcite studies have suggested on this issue that the two structurally - distinguishable chains of v atoms , which were thought to be magnetically identical , have different spin - gap energies .
this gives a natural explanation for the existence of two distinct modes with almost equal spectral weight , and has been supported by the subsequent raman - scattering experiments @xcite and theoretical studies on relevant exchange interactions @xcite .
the above confusion concerning the modelling and interpretation of the experimental results of ( vo)@xmath2p@xmath2o@xmath3 comes not only from the unexpectedly strong v - v exchange via po@xmath6 tetrahedra , but also from the presence of structurally - inequivalent v chains @xcite .
more recently , azuma have found that ( vo)@xmath2p@xmath2o@xmath3 transforms into another phase with different symmetry under pressure @xcite .
all the v atoms occupies a unique crystallographic site in the high - pressure ( hp ) phase , so that the magnetic chains made of v@xmath5 spins are all equivalent .
therefore , hp-(vo)@xmath2p@xmath2o@xmath3 will be a better example of the alternating antiferromagnetic chain with quantum spin @xmath4 . in this letter
, we report microscopic characterization of the magnetic chains in the hp phase of ( vo)@xmath2p@xmath2o@xmath3 via nmr . a single spin component characterized by a zero - field gap of 34
k was found , presenting support for the double - chain scenario for the ambient - pressure ( ap ) phase .
single crystals of the hp phase of ( vo)@xmath2p@xmath2o@xmath3 were grown as described in @xcite . since the crystals were too small to observe an nmr signal , they were crushed into powders and the nmr measurements were made on these powders .
standard spin - echo pulse techniques were utilized for most of the experiments .
an example of the field - swept @xmath0p nmr spectrum in the hp phase of ( vo)@xmath2p@xmath2o@xmath3 is shown in figure [ fig:31pspectrum ] .
the spectrum in the ap phase @xcite is also shown for comparison .
the spectrum in the hp phase consists of a single line as expected from the unique crystallographic site of phosphor in the unit cell .
this is contrasted with the ap phase where the spectrum splits into two groups of lines owing to the presence of two kinds of v chains with different gap energies @xcite .
the line - shape analysis revealed that the symmetry of an nmr - shift tensor at the p site is almost uniaxial . assuming the exact uniaxial symmetry , we determined the two independent principal values @xmath7 and @xmath8 corresponding to the shift with the external field parallel and perpendicular to the local symmetry axis , respectively .
the results are shown in figure [ fig:31kvst ] as a function of temperature .
both @xmath7 and @xmath8 scale the bulk magnetic susceptibility @xmath9 which is corrected by subtracting the contribution of paramagnetic impurities .
following the standard @xmath10@xmath9 analysis , the tensor components of the hyperfine coupling at the p site were determined as @xmath11 t/@xmath12 and @xmath13 t/@xmath12 .
these values yield the isotropic and uniaxial components , @xmath14 t/@xmath12 and @xmath15 t/@xmath12 , respectively .
@xmath16 is larger than and different in sign from that due to the classical dipolar field of v@xmath5 spins @xmath17 t/@xmath12 , indicating that the v@xmath5 spins are transferred not only to the p-@xmath18 orbitals but also to the p-@xmath19 orbital .
the susceptibility of a one - dimensional ( 1d ) gapped spin system at temperatures well below the gap @xmath20 is proportional to @xmath21 @xcite . in order to determine @xmath20
, we fitted the @xmath22 dependence of the isotropic component of the nmr shift @xmath23 below 10 k to the form @xmath24 , where the reduction of @xmath20 by fields is explicitly written .
the result is shown in the inset of figure [ fig:31kvst ] .
the obtained parameters are @xmath25 % , @xmath26 k@xmath27 , and @xmath28 = 31 k which gives @xmath29 = 34 k with the use of the measured @xmath30 factor @xcite .
@xmath31 is in good agreement with that evaluated from the bulk @xmath9 but is larger than the values determined from the critical field of the magnetization process ( @xmath3223 k ) @xcite and the ins on polycrystals ( @xmath3225 k ) @xcite for unknown reasons .
a free - induction - decay ( fid ) signal of @xmath1v has also been observed below about 50 k. the spectrum was obtained by integrating the fid signal while sweeping the external field .
the @xmath22 dependence of the @xmath1v nmr shift @xmath33 determined from the peak position of the spectrum is shown in figure [ fig:51kvst ] .
also shown in the inset is a plot of @xmath23 versus @xmath33 with @xmath22 the implicit parameter .
a linear relation found between @xmath23 and @xmath33 demonstrates that the @xmath22 dependence of the local spin susceptibility is identical for both the sites .
this is a clear sign of hp-(vo)@xmath2p@xmath2o@xmath3 having only one independent spin component .
the @xmath22 dependence of @xmath33 was analyzed in the same way as that of @xmath23 using @xmath20 determined above .
the @xmath22-independent orbital ( van - vleck ) shift was then obtained to be 0.182 % .
the hyperfine coupling constant at the v site determined from the slope of the @xmath34 plot is @xmath35 t/@xmath12 , which is in a reasonable range as a core - polarization field of a @xmath36 transition - metal ion @xcite .
figure [ fig:31wvst ] shows the @xmath22 dependence of the @xmath0p nuclear spin - lattice relaxation rate @xmath37 .
@xmath38 above 8 k was determined as the time constant of the exponential recovery of @xmath0p magnetization @xmath39 . below 8 k where non - exponential recovery appears ,
we analyzed @xmath39 by fitting to the form @xmath40 which incorporates the relaxation rate @xmath41 due to paramagnetic impurities @xcite .
as shown in the inset of figure [ fig:31wvst ] , @xmath37 exhibits activated behavior below about 20 k. the exponential decrease of @xmath37 is , however , masked below @xmath328 k synchronizing the appearance of non - exponential recovery .
the asymptotic value of @xmath37 at low @xmath22 is suppressed by applying fields as expected for the impurity - limited relaxation rate .
@xmath37 depends on @xmath42 at higher temperatures as well where the recovery is exponential , but the @xmath42 dependence roughly follows the 1d diffusive form @xmath43 as observed in ap-(vo)@xmath2p@xmath2o@xmath3 @xcite .
details of the @xmath42 dependence of @xmath37 will be presented in a separated paper .
the activation energy was estimated as @xmath44 k by fitting the data between 8 and 20 k to the form @xmath45 .
as the interbranch ( @xmath46 ) transitions within the lowest excited triplet @xcite are expected to dominate the nuclear - spin relaxation due to the predominantly - isotropic hyperfine fields , the obtained @xmath47 would give an estimate of the zero - field gap .
@xmath47 indeed agrees well with @xmath29 evaluated from the nmr shift .
figure [ fig : t1tkvst ] shows the @xmath22 dependence of @xmath48 divided by @xmath49 .
one of the remarkable features of the result is that the ratio @xmath50 becomes @xmath22 independent below about 15 k. ( an upturn below @xmath327 k is due to the impurity contribution to @xmath37 and is extrinsic . )
it is well known that , while the nmr shift is proportional to the uniform static susceptibility @xmath51 , @xmath37 samples the dissipative part of the dynamical susceptibility @xmath52 at the nuclear larmor frequency @xmath53 @xcite ; @xmath54 here @xmath55 is the fourier transform of the hyperfine coupling . since @xmath55 has a maximum at @xmath56 = 0 , @xmath37 at the p site is most sensitive to @xmath57 which is dominant at low @xmath22 in a gapped 1d spin system @xcite .
the @xmath22-independent behavior of @xmath50 therefore indicates that the @xmath22 dependence of @xmath58 and @xmath51 at low @xmath22 is identical and should be described by a common energy gap .
such a characteristic of the magnetic excitations in a gapped 1d spin system has been predicted theoretically based on a picture of free magnons @xcite , but has rarely been observed experimentally @xcite . to our knowledge , this is the first experimental verification of @xmath48 and @xmath10 having identical @xmath22 dependence at low @xmath22 , not relying on any model - dependent form of these quantities . from the experimental viewpoint , it is worth noting that the scaling between @xmath48 and @xmath10 holds below @xmath59 .
this suggests nearly free propagation of magnons being realized at temperatures not far below @xmath20 .
it is therefore practical to use experimental data in the region @xmath60 for a reliable estimate of @xmath20 , although the activated behavior of physical quantities such as @xmath9 and @xmath37 is theoretically justified only for @xmath61 @xcite . above about 20 k , the scaling breaks down and @xmath62 increases gradually with @xmath22
this means that @xmath63 grows more rapidly than @xmath51 .
as the temperature is now comparable with or higher than @xmath20 , interactions between magnons and/or the @xmath64 component of spin fluctuations will become increasingly important and would enhance @xmath65 over @xmath51 . in conclusion ,
we have measured @xmath0p and @xmath1v nmr in the high - pressure phase of ( vo)@xmath2p@xmath2o@xmath3 .
it was found that the temperature dependence of the local static spin susceptibility at the p site is identical with that at the v site .
the dynamical spin susceptibility @xmath65 near @xmath66 also scales with the static susceptibility at low temperatures below about a half of the spin - gap energy which was estimated to be 34 k at zero field .
all of these observations provides microscopic evidence for a unique kind of magnetic chain existing in the high - pressure phase of ( vo)@xmath2p@xmath2o@xmath3 , as well as for coexistence of magnetically - inequivalent chains in its ambient - pressure phase .
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the authors would like to acknowledge fapesp , capes and cnpq for financial support .
the calculations were carried out at cce - usp , center for high performance computing at ufabc and cenapad / sp .
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nitrogen - doped carbon nanotubes can provide reactive sites on the porphyrin - like defects .
it s well known that many porphyrins have transition metal atoms , and we have explored transition metal atoms bonded to those porphyrin - like defects in n - doped carbon nanotubes .
the electronic structure and transport are analized by means of a combination of density functional theory and recursive green s functions methods .
the results determined the heme b - like defect ( an iron atom bonded to four nitrogens ) as the most stable and with a higher polarization current for a single defect .
with randomly positioned heme b - defects in a few hundred nanometers long nanotubes the polarization reaches near 100% meaning an effective spin filter .
a disorder induced magnetoresistance effect is also observed in those long nanotubes , values as high as 20000% are calculated with non - magnectic eletrodes . since their discovery by iijima in 1991 ,
carbon nanotubes@xcite ( cnt ) have become the subject of intense research due to their potential for applications,@xcite such as in novel electronic devices.@xcite furthermore , in a seminal paper by tsukagoshi _
et al _ cnts entered the realm of spintronics , whereby one envision the possibility of using the electron spin , instead of its charge , as information carrier.@xcite in that work the authors demonstrated that the spin coherence length of polarized electrons injected onto cnts is larger than 300 nm .
thus , carbon nanotube devices could be used to manipulate spins in a coherent manner .
the prototypical spintronics device uses spin - polarized electrons , which are injected from a source into an unpolarized region and analyzed by a polarized drain . within this arrangement the so - called giant magnetoresistance effect ( gmr)@xcite manifests itself by altering - via an external magnetic field - the relative orientations of the magnetic moments of the electrodes . from a practical point of view
this setup usually involves sandwiching different materials .
an alternative to this has been given by kirwan et al .
whereby initially unpolarized electrons are scattered by magnetic impurities adsorbed on the surface of a segment of a carbon nanotube .
this way , both the electrodes as well as the device itself are made of the same material .
an alternative to this has been given by kirwan _
et al . _
@xcite whereby initially unpolarized electrons are scattered by magnetic impurities adsorbed on the surface of a segment of a carbon nanotube.@xcite this way , both the electrodes as well as the device itself are made of the same material , thus avoiding issues related to surface matching at the interface consequently hindering spurious scattering .
however , one of the issues concerning the use of cnts as spintronics devices is the need to incorporate dopants or defects in order to change their electronic transport properties . closed shell species
do not interact very strongly with the pristine wall of a carbon nanotube.@xcite furthermore , transition metal atoms are more likely to form clusters when interacting with the pristine wall of the nanotube .
@xcite even linear chains of fe atoms are more energetically favorable than free standing iron atoms .
@xcite one possible path to circumvent this problem is to incorporate doping agents during the growth process . in that context , carbon nanotubes sythetized in a nitrogen - rich atmosphere - the so - called @xmath0 nanotubes - are potential candidates.@xcite it has already been demonstrated that these nanotubes could be , for example , used as gas sensors for a variety of chemical species.@xcite the most stable defect in these structures is a pyridine - like defect consisting of a 4 nitrogen divacancy ( 4nd).@xcite this defect ( shown in fig .
[ figure1]a ) is formed by two vacancies surrounded by four substitutional nitrogen atoms .
we have previously exploited the reactivity of this defect to attach ammonia molecules and study the behavior of the system as a sensor.@xcite interestingly , this defect is similar to molecules in the porphyrin class , in particular , to a molecule known as heme - b ( shown in fig .
[ figure1]b ) , which is found , for instance , in hemoglobin and myoglobin .
this heme b molecule has an iron atom bonded to the site with four nitrogens .
thus it is intuitive to assume that an iron atom - and other transition metal ( tm ) atoms - gets bonded to the 4nd defect of the carbon nanotubes .
the heme b - like defect has been recently synthesized by lee et , al .
@xcite their stability was studied by means of repeated cyclic voltammograms , and they have not observed significant differences after @xmath1 cycles , which is attributed to the stability of the covalent incorporation of the atoms . the use of carbon nanotubes - or any other long one dimensional system - with defects , however , poses an additional problem ; the position of the defects can not be controlled during growth .
the result is a device with a large number of randomly positioned defects .
thus , in order to obtain quantitatively meaningful theoretical predictions , one must take into consideration the effects of disorder .
it is generally assumed that disorder has a detrimental effect .
recently , however , we have shown that one - dimensional boron - doped graphene nanoribbons can present near - perfect conductance polarization due to disorder.@xcite such an effect should not depend on the system under consideration provided there was a difference in transmission probabilities for majority and minority spin cases for a single scatterer .
furthermore , the introduction of a large number of defects with non - zero magnetic moment leads not only to structural disorder , but also to a magnetic one .
in fact , the magnetic moments of the impurities might be pointing in random directions .
we can then consider that , in the absence of a magnetic field , approximately half of the moments are pointing up and half of them are pointing down .
a magnetic field would tend to align the magnetic moments leading to an analogous to the magnetoresistance effect without the need to rely on a multilayered material .
in this work we show that porphyrin - like cnts can exhibit a nearly - perfect polarization and extremely large disorder induced magnetoresistance ( as high as 20000% ) driven by disorder .
the disorder was simulated using cnts containing a large number of magnetic impurities randomly positioned along the tube .
for those calculations we used a combination of density functional theory@xcite coupled to recursive green s functions methods@xcite . we have initially performed _
ab initio _ calculations within density functional theory ( dft)@xcite for a segment of a ( 5,5 ) @xmath0 nanotube containing a 4nd defect ( figure [ figure1]a ) . as we have said before
, the porphyrin molecules can have different tm in the 4nd site , as shown in figure [ figure1]b .
we , thus , performed calculations using iron , cobalt , manganese and nickel atoms in the middle of the 4nd defect .
the final arrangement for the case of iron is presented in figure [ figure1]c .
as one can see , nine irreducible cells of the pristine system were used to describe the region containing the defect .
the computational code used was siesta @xcite which uses a linear combination of atomic orbitals ( lcao ) as basis set . in the particular case of this work
we have employed a double zeta basis set with polarization orbitals .
we used the generalized gradient approximation ( gga ) as parametrized by perdew - burke - ernzerhof @xcite ( pbe ) for the exchange correlation functional .
finally , the atomic coordinates have been optimized using a conjugated gradient scheme until the forces on atoms were lower than 0.03 ev / . furthermore , in order to assess whether the transition metal atoms in adjacent cells are magnetically coupled@xcite we simulated a supercell with 18 irreducible cells ( twice the initial size ) with two heme - b - like defects ; one case with ferromagnetic ordering ( @xmath2 ) , and another with a antiferromagnetic one ( @xmath3 ) .
the total energy difference ( e@xmath4-e@xmath5 ) is negligible for all transition metal atoms considered in this work , so we infer that there s no magnetic coupling in our system . for the electronic transport calculations , the system - following the procedure proposed by carolli _ _
et al__@xcite - is initially divided in three regions namely , the right and the left electrodes and a central scattering region .
the electrodes for our system are taken as semi - infinite repetitions of the pristine carbon nanotube . in the absence of spin - orbit interactions
one assumes the two spin fluid approximation , whereby one can calculate the electronic transport properties of the majority and minority spins independently of each other .
we then use the landauer - bttiker @xcite formula to calculate the transmission coefficients of the system . in order to access the electronic transport properties of a more realistic system with hundreds of nanometers we use a combination of dft and recursive green s function methods .
@xcite to do so , we split up a long nanotube into small pieces .
each piece is simulated using a separate dft calculation as already described previously , and the hamiltonian and overlap opperators respectively @xmath6 and @xmath7 are stored . for our ( 5,5 )
@xmath0 nt we also have to consider five different rotations for the position of the defect . with those smaller blocks we build up a long nanotube , ranging from 20 to 600 nanometers , by randomly placing the segments with defects together with pristine pieces , as shown in fig .
[ long ] .
one then recursively reduce the system to two renormalized electrodes coupled via an effective scattering potential that contains all the information about the central region . in the low bias limit
the differential spin - dependent conductance can be calculated by the landauer - buttiker formula for the current @xcite @xmath8 where @xmath9 is the fermi distribution function for a given temperature .
the total conductance is then given by the sum of the majority and minority conductances , @xmath10 .
we are interested in two quantities .
firstly , in order to quantify the spin filtering effect of this device , we calculate the degree of polarization@xcite @xmath11 secondly , as discussed earlier one also needs to take into consideration the relative orientations of the magnetic moments .
one can analyze the changes in conductance due to an external magnetic field that tends to align the local magnetization of each impurity .
the magnetic field in our calculations is taken into consideration only in the alignment of the magnetic moments , it has no other effect on the electronic structure of our system .
consequently , the value of this disorder induced magnetoresistance ( mr ) is given by @xmath12 where @xmath13 corresponds to the total conductance for the case where all of the magnetic moments are pointing along the same direction , and @xmath14 is the total conductance for the case where there is an equal distribution of positive and negative magnetic moments .
finally , in these long and disordered @xmath0 nanotubes , different defect distributions along the @xmath0 give different values of conductance . in order to get statistically meaningful values of conductance
we have calculated about 200 random arrangements for each concentration and length of the nanotubes .
for cn@xmath15 nanotube containing an iron atom ( heme - b - like defect ) .
the net magnetic moment of the system is localized in the iron atom.,scaledwidth=50.0% ] upon placing the different tm atoms in the 4nd , we observe that they strongly bind with a similar final structure in all situations .
table [ table1 ] shows the binding energies and the final magnetization of the system for each one of the tm atoms .
as can be seen , the binding energies are relatively high ( the reference is the isolated atom infinitelly separated from the nanotube ) .
one also sees a local magnetic moment in all cases , except for the nickel atom . a similar behavior was also observed by shang , _ et al .
_ .@xcite in fig .
[ rhoupdn ] we present the local magnetic moment , @xmath16 of the heme b - like defect . from this figure
we can notice a highly localized magnetic moment in the iron atom . .binding energy , magnetization and polarization for different transition metal atoms bonded to the 4nd defect . [ cols="^,^,^,^,^",options="header " , ] [ lengths100 ] in order to address the effect of a magnetic field applied to the system and the possibility that not all the magnetic moments are completely aligned we also considered that only 80% of the defects are magnetically aligned .
the results are shown in figure [ condpol2 ] , and the linear fits for the localization lengths also are presented in table [ lengths100 ] .
we can note a decrease ( increase ) in localization length for majority ( minority ) spin compared to the 100 % case .
this is to be expected since one is moving towards higher magnetic disorder , _ i.e. _ the 50 % case . from fig .
[ condpol2]c we can see a polarization near 100% for 700 nm long @xmath0 nanotubes .
most importantly , the magnetoresistance ( fig . [ condpol2]d ) presents values which are one order of magnitude lower than the fully aligned arrangement , but it is still in the 1000 % range .
thus , even in the case where not all spins are aligned , there is still an extremely large disorder induced magnetoresistance .
thus , this disorder - driven gmr effect is extremely robust toward fluctuations in the alignment of the magnetic moments .
we have used an ideal paramagnet model to estimate the needed magnetic field to obtain a 80% magnetization at 300k and 3k .
unfortunately the needed magnetic field at ambient temperature is about 200 t , making it impracticable for ambient temperature devices .
for a temperature of 3k the needed field will be about 2 t for an 80% magnetization and about 5 t for a 95% magnetization , so it s possible to observe the predicted effects in this paper in low temperature experiments .
in summary we have observed that transition metal atoms bind strongly to nitrogen defects in cn@xmath15 carbon nanotubes in a fashion similar to heme b molecules .
the end result is a scattering site with a localized magnetic moment on the transition metal atom that leads to a small conductance polarization in the case of a single impurity . for a large number of such impurities randomly distributed along carbon nanotubes a few hundred nanometers long
, it leads to near perfect polarization and a large magnetoresistance ( up to 20000% ) .
an interesting feature of the system proposed here is the fact that they do not need polarized electrodes as it is usually the case .
we estimate that , at low temperature , this effect could be measured experimentally .
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the experimental setup ( fig . [ fig1 ] ) consisted of a brass container filled with solid carbon dioxide ( dry ice ) .
a clean glass slide was placed over the brass container , where a drop ( @xmath2 ) of deionized and degassed water was deposited using a syringe pump . to increase contrast and observe the freezing front
, red food dye was added to the water .
the process was recorded from the side using a long distance microscope ( vzm1000 edmund optics ) mounted on a color camera , at a frame rate of 50 frames per second .
we used both backlight and bottom light illumination provided by optic fiber lamps .
the resolution obtained was @xmath3 pixels with approximately @xmath4 .
we measured the plate temperature near the droplet using a standard thermocouple .
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in this fluid dynamics video we show how a drop of water ( @xmath0 ) freezes into a singular shape when deposited on a cold surface ( @xmath1 ) .
the process of solidification can be observed very clearly due to the change in refraction when water turns into ice .
the drop remains approximately spherical during most of the process , with a freezing front moving upwards and smoothly following the interface .
however , at the final stage of freezing , when the last cap of liquid turns into ice , a singular tip develops spontaneously .
interestingly , the sharp tip of the ice drop acts as a preferential site for deposition of water vapour , and a beautiful `` tree '' of ice crystals develops right at the tip .
the tip singularity attracts the vapour in analogy to a sharp lightning rod attracting lightning .
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we acknowledge partial support of fondecyt ( chile ) projects 1060627 and 1060651 , conicyt / pbct proyecto anillo de investigacin en ciencia y tecnologa act30/2006 and u.s .
national science foundation grant dms 06 - 00037 .
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we establish rigorous upper and lower bounds for the speed of pulled fronts with a cutoff .
we show that the brunet - derrida formula corresponds to the leading order expansion in the cut - off parameter of both the upper and lower bounds . for
sufficiently large cut - off parameter the brunet - derrida formula lies outside the allowed band determined from the bounds . if nonlinearities are neglected the upper and lower bounds coincide and are the exact linear speed for all values of the cut - off parameter .
the reaction diffusion equation @xmath0 provides a simple description of phenomena in fields such as population dynamics , chemical reactions , flame propagation , fluids , qcd , among others @xcite .
it is one of the simplest models which shows how a small perturbation to an unstable state develops into a moving front joining a stable to an unstable state .
the reaction term @xmath1 satisfies different conditions depending on the physical problem of interest .
one of the first , and most studied cases , is the fisher reaction term @xmath2 for which the asymptotic speed of the propagating front is @xmath3 , a value determined from linear considerations .
a more general case was studied by kolmogorov , petrovskii and piscounov ( kpp)@xcite who showed that for all reaction terms which satisfy the kpp condition @xmath4 the asymptotic speed of the front joining the stable @xmath5 point to the unstable @xmath6 point is given by @xmath7 these fronts are called pulled since it is the leading edge of the front which determines the velocity of propagation . in the rest of this work
we assume that @xmath8 .
the evolution of localized initial conditions for general reaction terms , and rigorous properties of the fronts were studied by aronson and weinberger @xcite .
the asymptotic speed of the front for all reaction terms can be found from the integral variational principle @xcite @xmath9 where the supremum is taken over all positive monotonic decreasing functions @xmath10 for which the integrals exist and where @xmath11 .
the supremum is always attained for reaction terms which are not pulled .
two effects not included in the classical reaction diffusion equation ( [ rd ] ) , are the effect of noise and the effect of a finite number @xmath12 of diffusive particles .
it was shown by brunet and derrida that such effects can be simulated by introducing a cut - off in the reaction term . in the case of noise
the cut - off parameter measures the amplitude of the noise while in the case of finite number of @xmath12 diffusing particles the cut - off parameter @xmath13 .
there is substantial numerical evidence that introducing a cut - off in the reaction terms reproduces accurately the effect of noise and finiteness in the number of diffusing particles @xcite . by means of an asymptotic matching brunet and derrida showed that for a reaction term @xmath14 a small cut - off changes the speed of the front to @xmath15 in recent work it has been show that the brunet - derrida formula for the speed is correct to @xmath16@xmath17 for a wider class of reaction terms @xcite .
the purpose of this work is to show that for reaction terms of the form @xmath18 where @xmath19 satisfies the kpp condition eq .
( [ kppcondition ] ) and @xmath20 is the step function , the speed @xmath21 of the front with the cutoff satisfies @xmath22 with @xmath23 we see that for @xmath24 , @xmath25 .
the function @xmath26 depends on the nonlinear terms of the reaction function . for small @xmath27 the series expansion of the upper bound @xmath28
is @xmath29 the contribution of the nonlinearities , contained in the term @xmath26 , appears at @xmath16@xmath17 , so that the leading order terms in the expansion of the upper and lower bounds give the brunet - derrida formula .
if nonlinearities are neglected the value @xmath30 is the analog of the kpp value @xmath3 for reaction terms which satisfy the kpp condition , but with a cutoff . in what follows we derive the bounds and apply them to the fisher reaction term @xcite @xmath31 and to the reaction term studied by brunet and derrida @xmath32 .
the main tool to obtain the bounds is the variational principle for the speed . as shown in previous work @xcite
, we may perform the change variables @xmath33 where @xmath34 in eq.([vp1 ] ) and write the variational expression for the speed as @xmath35 where @xmath36 is an arbitrary parameter , @xmath37 and the supremum is taken over positive increasing functions @xmath38 such that @xmath39 , @xmath40 and for which all the integrals in ( [ newvp ] ) are finite .
therefore , for any suitable trial function @xmath38 we know that @xmath41 consider now reaction terms @xmath1 with a cut - off @xmath27 of the form @xmath42 where @xmath43 , the nonlinearity , is such that @xmath44 .
we find @xmath45 where @xmath46 assume now that @xmath1 satisfies the kpp criterion eq.([kppcondition ] ) .
since @xmath47 , it follows that @xmath48 where @xmath49 and therefore @xmath50 \equiv \sup_{u(s ) } 2 \,\frac { g(1)/s_0 + \int_0^{s_0 } g(u(s))/s^2 d\,s}{\int_0^{s_0 } \left ( d u /d s\right)^2 d\,s}.\ ] ] one can prove ( rigorous details will be given elsewhere ) that @xmath51 is bounded above and that there exists a function @xmath52 for which the supremum is attained .
this function is the monotonic increasing solution to the euler - lagrange equation for @xmath51 satisfying the boundary conditions @xmath53 .
one can also prove that the variational parameter @xmath54 is finite and @xmath55 .
in summary , the maximizing function for @xmath51 is the solution of @xmath56 @xmath57 subject to the boundary conditions @xmath58 with the function and its derivative continuous at @xmath59 .
the solution to this problem is given by @xmath60 with @xmath61 where @xmath62 is the first positive solution of @xmath63 the maximum of @xmath64 $ ] can be calculated easily .
we obtain after performing the integrals , @xmath65 = 4 \sin^2 ( \phi _ * ) \equiv c^2_{up}. \label{top}\ ] ] to obtain the lower bound we shall use the optimizing function @xmath52 as a a suitable trial function in eq .
( [ lowerbound ] ) .
we obtain @xmath66 \label{bottom}\ ] ] since @xmath67 is negative , we may combine eqs.([top ] ) and ( [ bottom ] ) and write our main result as given in eq .
( [ main ] ) . as an example
consider the reaction term studied by brunet and derrida , @xmath32 .
the lower bound can be written explicitly as @xmath68 the integral has a long analytic expression which we omit here . from the explicit expression above it
is not difficult to show that the contribution of the two last terms , which arise from the nonlinear terms , are of @xmath69 . in figures 1 and 2
we show the bounds together with the brunet - derrida formula as a function of @xmath27 .
the solid lines correspond to the upper and lower bounds .
the dashed line is the brunet - derrida formula . .
the solid lines correspond to the bounds , the dots to the brunet - derrida formula .
, height=188 ] as a second example we consider the fisher reaction term @xmath70 with a cut - off .
the lower bound becomes @xmath71 again , the integral can be done analytically and we do not show it here . in fig .
3 we show the upper and lower bounds and the brunet - derrida formula . in this case
the brunet - derrida formula leaves the allowed band at larger value of @xmath27 . in general for reaction terms @xmath72 , the gap between the upper and lower bounds becomes narrower and the brunet - derrida formula valid for a smaller range of @xmath27 . .
lines as in fig . 1 ,
height=188 ] in summary , we have studied the effect of a cut - off on reaction terms which satisfy the kpp condition eq.([kppcondition ] ) .
we have found upper and lower bounds valid for all values of the cut - off parameters , which allow to assess the accuracy of the brunet derrida formula .
if we consider only the linear terms , the upper and lower bounds coincide and give the exact linear value for the speed , of which the two leading order terms are the brunet - derrida formula .
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the authors are very much grateful to v. i. anisimov for helpful advise on the lsda+u method and for a fruitful discussion on the spin alignment of ndnio@xmath0 , and also to t. mizokawa for useful discussion about their work @xcite .
this work is supported by grant - in - aid for coe research `` spin - charge - photon '' .
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electronic structures of ndnio@xmath0 and ynio@xmath0 are calculated by using lsda+u method with rotational invariance .
the jahn - teller distortion is not allowed under the observed magnetic ordering .
no orbital order on ni sites can not be observed in the calculation of both systems but different types of charge ordering . in a small distorting system
ndnio@xmath0 , all ni ions are trivalent and oxygen sites have a particular ordering , both in charge and orbital . in a large distorting system ynio@xmath0 ,
the charge disproportionation occurs , 2ni@xmath1 ni@xmath2ni@xmath3 .
therefore , the charge ordering stabilizes the asymmetry of the arrangement of ni magnetic moments in both systems .
2 perovskite transition metal oxides are now of high interest because of the large variety and the possible controllability of their physical properties .
perovskite nickelates rnio@xmath0 ( r = an trivalent rare earth or y ions ) can be classified into three different categories , according to their tolerance factor ( @xmath4 ) .
@xcite one is those whose tolerance factor is much smaller ( @xmath5 ) and has a larger distortion , such as lunio@xmath0 or ynio@xmath0 .
they are antiferromagnetic insulators at low temperatures .
this class of rnio@xmath0 undergoes transition to paramagnetic insulator at the neel temperature @xmath6 and also has another phase transition from paramagnetic insulator to paramagnetic metal ( m - i transition ) at very high transition temperature .
second is those whose tolerance factor is intermediate ( @xmath7 ) , such as ndnio@xmath0 or prnio@xmath0 .
these materials are also antiferromagnetic insulators at low temperatures and above @xmath6 they are paramagnetic metals .
@xcite third is lanio@xmath0 where @xmath8 and is paramagnetic metal . low temperature phase of above mentioned first and second classes of rnio@xmath0 have the unique magnetic structures .
@xcite its magnetic diffraction peak is characterized by the propagation vector @xmath9 and the magnetic unit cell is identified as the @xmath10 supercell of the crystallographic lattice .
@xcite the magnetic order is specified to be alternating ferromagnetic ( fm ) and antiferromagnetic ( afm ) couplings @xmath11 along three crystallographic directions . on the contrary to the asymmetric arrangement of magnetic bonds ( fm / afm ) around each nickel site , ni - o distances around each ni ion are almost equal to each other .
for instance , the difference between the longest and the shortest ni - o bonds in one nio@xmath12 octahedron is at most 2% in rnio@xmath0 .
@xcite the jahn - teller distortion is absent in rnio@xmath0 , even in the largely distorted system ynio@xmath0 , whose tolerance factor is 0.88 .
experimentally there is only one crystallographic site for ni ions in ndnio@xmath0 and two different sites for ni ions in ynio@xmath0 .
therefore , one can expect trivalent ions ni@xmath13 in ndnio@xmath0 . on the contrary , ni ions in ynio@xmath0 are divalent and quadrivalent , _
i.e. _ the charge disproportionation 2ni@xmath1 ni@xmath2ni@xmath3 occurs . in this letter , we study the electronic properties of the afm insulating phase of two perovskite nickelates , ndnio@xmath0 and ynio@xmath0 , which are typical ones of above mentioned two respective classes .
we use the lsda+u method with the rotational invariance in conjunction with the lmto - asa method .
@xcite the lsda+u method counts the electron - electron interaction between localized orbitals by the hartree - fock type interaction term .
the ionic positions and lattice parameters used in the present calculations are imported from the diffraction experiments .
@xcite the unit cell here is the @xmath14 crystallographic supercell containing 16 nd or y ions , 16 ni ions , and 48 o ions and , in addition , 32 empty spheres , totally 112 atomic spheres .
sixty - four k - points in the brillouin zone are sampled in the calculation of the density of states .
ndnio@xmath0 is antiferromagnetic up to @xmath15 , but the nd spin moment vanishes at about 30k .
@xcite therefore , three 4f electrons of nd ion are counted in the frozen core .
we also calculated the electronic structure of ndnio@xmath0 with 4f electrons in valence states , and the results show no significant difference .
the coulomb and exchange parameters @xmath16 and @xmath17 of ni ions are fixed to be 7.0 ev and 0.88 ev , respectively , through all the lsda+u calculations .
these values are consistent with photoemission experiments and the results of the lsda calculations
. calculated results are not changed much over a large range of values of @xmath16 and @xmath17 .
the crystallographic space groups of ndnio@xmath0 and ynio@xmath0 are orthorhombic @xmath18 and monoclinic @xmath19 , respectively .
the group theoretical analysis shows that , using the projection operator method , the jahn - teller distortion of one nio@xmath12 octahedron can not be transfered over the whole enlarged @xmath20 supercell with @xmath21 .
@xcite therefore , the jahn - teller distortion actually can not appear in the observed spin ordered state .
ndnio@xmath0 : there are two possible spin configurations in @xmath22 satisfying the observed transfer vectors @xmath21 .
one is that the spin magnetic moments align ferromagnetically on the ( 101 ) plane and the planes are stacked with a doubled period as @xmath23 , whose magnetic space group is monoclinic @xmath24 .
@xcite this spin configuration is assumed in the present letter . another possible spin configuration with @xmath25 spin order could be the doubled period checkerboard stacking along crystallographic @xmath26-axis , and the magnetic space group is orthorhombic @xmath27 .
@xcite the state of @xmath27 shows orbital ordering of e@xmath28 states on ni sites , but the resultant calculated total energy is higher by 0.18 ev per @xmath29 cell than that of @xmath24 .
because there is no possibility of subsidiary jahn - teller distortion to lower the total energy , @xmath27 could not be the symmetry to be considered .
the total energy is also calculated in a fictitious antiferromagnetic state of @xmath30 whose magnetic unit cell is identical to the crystallographic one and is higher by 2.0 ev per @xmath31 cell .
figure [ fig - ndnio3-pdos ] shows the projected density of states on the ni 3d orbitals in ndnio@xmath0 .
the system is insulator with a gap @xmath32ev .
each nickel ion has the local magnetic moment @xmath33 within the atomic sphere of a radius @xmath34 , contrary to the observed value of @xmath35 .
@xcite it should be noticed that the local magnetic moment can not be uniquely defined .
furthermore , ndnio@xmath0 is not a simple antiferromagnetic insulator but a dynamical effect is essential , which may be an origin of this discrepancy .
@xcite there is no distinctive variation in partial spin density of states of ni site in each magnetic sublattice .
an e@xmath28 band lies on the energy range @xmath36 .
the numbers of states in the energy ranges @xmath37 and @xmath38 are 16 , respectively , which are the one occupied and one vacant e@xmath28 states per ni ion . actually , these e@xmath28 orbitals extend over surrounding oxygen sites from ni ions due to strong hybridization between ni e@xmath28 and o p orbitals .
the extended occupied e@xmath28 orbital has a 60% weight on p orbitals on surrounding six oxygens , a 10% weight on an individual oxygen .
therefore , one would establish a model where one occupied e@xmath28 state with majority spin locates at the top of the valence bands , and it hybridizes strongly with the p states on nearby o ions .
this is the molecular orbital @xmath39 state .
@xcite then one can assign all nickel ions in ndnio@xmath0 to be trivalent ni@xmath13 ( t@xmath40e@xmath41 ) , even though the ni ion is not truly ionized by + 3 charge . in the projected density of states of ni ion site
, one observes a large amount of e@xmath28 states at the bottom of the d bands , which are the bonding states between ni d and o p , corresponding to the @xmath42 states in the molecular orbital picture .
@xcite the e@xmath28 band in the range @xmath43 does not show any orbital ordering .
in fact , off - diagonal elements within the e@xmath28 subblock of the occupation matrix @xmath44 are zero and the diagonal elements are identical
. the absence of the orbital ordering may be consistent with the fact that the @xmath45 $ ] axis has three - fold rotational symmetry in the present spin configuration , once the distortion is neglected . due to this pseudo three - fold rotational symmetry , the basis orbitals of e@xmath28 representation of the trigonal group d@xmath46 is a good basis set and those derived from e@xmath28 orbitals are @xmath47 and @xmath48 . therefore , there is no difference between the occupancies of @xmath49 and @xmath50 .
figures [ fig - ndnio3-sd ] is the spatial profiles of spin densities @xmath51 in the energy range @xmath52 ( @xmath39 state ) .
only one spin component of o p orbitals on the fm bond is bridging between two ni e@xmath28 orbitals , while both spin components on the afm bond couple with ni e@xmath28 orbitals of respective spins .
consequently , oxygen ions on the afm bond have more charge in this energy range than oxygen on the fm bond .
this is the realization of oxygen - site charge - ordered state discussed by mizokawa _
et al .
_ in the framework of the hartree - fock calculation .
@xcite however , the charge difference between oxygen sites of ni@xmath53-@xmath54-ni@xmath53 and ni@xmath53-@xmath55-ni@xmath56 in @xmath39 states is mostly compensated by hybridized @xmath42 state at lower energies . besides , all oxygen ions have no local magnetic moment .
more significant results seen in fig . [ fig - ndnio3-sd ] may be the p orbital ordering on oxygen sites .
the magnetic space group is @xmath24 and its unitary part is @xmath57 .
the unit cell and lattice primitive vectors are not identical to those in @xmath18 .
the spin density in fig .
[ fig - ndnio3-sd ] is consistent with this magnetic group @xmath24 , neither higher nor lower than this .
the symmetry lowering of the unitary part @xmath57 is the origin of the opening the band gap at @xmath58 in the majority spin band , despite to the absence of the orbital ordering on ni sites ( _ i.e. _ symmetry driven band gap ) .
therefore , one can conclude that the origin of the insulator phase at low temperatures in ndnio@xmath0 is the characteristic spin density on the oxygen sites or , equivalently , the orbital ordering there .
the energy gap @xmath59 ev is due to the symmetry of charge order . in fictitious ideal cubic structure without distortion or tilting of nio@xmath12 octahedra ,
the system becomes metal whose valence and conduction bands touch at points with each other .
@xcite the structure of @xmath39 bands is insensitive to the value of @xmath16 .
the value of the band gap is unchanged down to @xmath60 ev .
this is because the gap is driven by the symmetry .
ynio@xmath0 : one should expect larger distortion in ynio@xmath0 because ynio@xmath0 is the typical system with the small tolerance factor .
@xcite there are two different crystallographic ni sites , and the distances from each ni ion to surrounding o ion are different by 3@xmath614% from one type of ni ion to the other type .
@xcite calculated self - consistent solution is that with the apparent charge disproportionation and no orbital polarization .
since ni@xmath3 site has no spin magnetic moment , the two spin configurations discussed in ndnio@xmath0 become identical and the spin configuration of ynio@xmath0 is uniquely determined .
the magnetic space group is @xmath62 if the spins of ni@xmath3 ions are non vanishing and , on the contrary , @xmath63 if the spins of ni@xmath3 ions are zero .
the latter symmetry @xmath63 is actually the case .
figure [ fig - ynio3-pdos ] shows the projected density of states at ni ion sites . the system is insulator with a gap @xmath64ev .
the resultant magnetic moments for half of ni ions are @xmath65 within the atomic sphere of a radius @xmath66 , namely divalent ions ni@xmath67
( t@xmath40e@xmath68 ) , of larger nio@xmath12 octahedron and zero for another half of ni ions within the atomic sphere of a radius @xmath34 , namely quadrivalent ni@xmath3 ( t@xmath40 ) , of smaller octahedron . the experimentally observed magnetic moments are @xmath69 for ni@xmath67 ions and
@xmath70 for ni@xmath3 ions .
@xcite the discrepancy may be due to a possible non - collinear spin order .
the number of states in the energy range @xmath71 is 16 and dominant weight on ni@xmath67 .
because the number of ni@xmath67 in the @xmath14 cell is 8 , these states are assigned as two e@xmath28 states mainly on ni@xmath67 and surrounding oxygens .
the e@xmath28 orbitals of ni@xmath3 is lifted in the higher energy range ( @xmath72 ) without spin polarization .
a large amount of e@xmath28 orbitals in ni@xmath67 ions locates at the bottom of the d bands , in the range @xmath73 for ni@xmath67 , and in the range @xmath74 for ni@xmath3 , stabilizes the bonding between ni@xmath3 ions and oxygen ions . from these facts
, one can establish a model that deep @xmath42 molecular orbitals ( one per both ni@xmath67 and ni@xmath3 ) stabilize the system , and other two e@xmath28 states ( @xmath39 states ) per one ni@xmath67 ion located at @xmath75 .
the charge disproportionation is mainly due to the crystal field effect .
the low spin state in ni@xmath3 or ni@xmath13 ion is energetically unstable in small @xmath76 case and the ground state multiplet of ni@xmath67 is @xmath77a@xmath78 ( t@xmath40e@xmath68 ) for all arbitrary values of @xmath76 .
@xcite therefore , the small tolerance factor causes two different ni sites , compressed ni@xmath3 ( large @xmath76 ) and dilated ni@xmath67 ( small @xmath76 ) , rather than uniformly dilated ni@xmath13 ionic states .
@xcite the standard values of @xmath79 ( @xmath80 is the racah parameter ) for ni@xmath67 is presumably around 1.0 . once one estimate the crystal field effects from the ni - o bond lengths @xmath81 , the difference of @xmath82 on two ni ion sites is presumably about 20% .
two narrow e@xmath28 bands can be seen in the energy range @xmath83 and @xmath84 in fig .
[ fig - ynio3-pdos]a .
two ni@xmath67 e@xmath28 states are spatially extending over wide area and not only hybridizing with the nearest neighbor o ions but also extending over the nearest @xmath85 ions .
this situation is well depicted in the spin density .
figure [ fig - ynio3-sd ] shows the isometric surfaces of the spin density in the range of @xmath83 .
the d - wavefunctions on ni@xmath67 are extending over the ( 101 ) plane , and antiferromagnetically coupled with other ni@xmath67 ions .
charge in each ni ion is compensated in ynio@xmath0 as in ndnio@xmath0 and the difference between total charges in the muffin tin spheres on ni@xmath67 and ni@xmath3 sites is very small , equals to 0.03 .
this variation is the same order as that of oxygen ions in ndnio@xmath0 .
however , charge disproportionation is coupled with the lattice distortion , where larger oxygen octahedron is surrounding ni@xmath67 , and stabilizes the lattice system in ynio@xmath0 .
therefore , diffraction experiment can detect the charge ordering in the yttrium system easier than in the neodymium system . in conclusion ,
we have studied two typical antiferromagnetic insulating phase of rnio@xmath0 , ndnio@xmath0 and ynio@xmath0 , by using the lsda+u method . a possibility of the jahn - teller distortion is excluded by the group theoretical consideration .
no orbital order on ni sites is observed in both systems , but two different types of ordering are observed . in small distorting rnio@xmath0 such as ndnio@xmath0 , oxygen sites shows the orbital ordering and this is the origin of the gap opening in ndnio@xmath0 .
in large distorting rnio@xmath0 such as ynio@xmath0 , the charge disproportionation 2ni@xmath86ni@xmath2ni@xmath3 occurs . the charge ordering mechanism
can explain the stabilization of the asymmetric alignment of the local magnetic moments around each nickel site in both systems .
we may add the final comment about effects of electron - electron correlation in rnio@xmath0 .
the widths of the calculated e@xmath28 bands @xmath87 by the lda are @xmath88(ynio@xmath0 ) , @xmath89(ndnio@xmath0 ) , and @xmath90(lanio@xmath0 ) . then , the ratio of @xmath91 may be estimated as @xmath92 , assuming the value of the coulomb repulsion @xmath16 is common for all , and one can see a large reduction of @xmath91 in lanio@xmath0 , which may be the key parameter for the difference of the ground states of these perovskite nickelates .
ndnio@xmath0 shows the anomalous m - i transition .
@xcite lanio@xmath0 is presumably an anomalous metal of strongly correlated electrons @xcite and could not be treated within the framework of the lsda+u method .
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the authors would like to thank w. vogel for his valuable comments and discussions .
this work was support by the villum kann rasmussen foundation ( denmark ) , the eu projects q - essence and malicia , and danish national research foundation .
ngj is grateful for generous support from danmarks nationalbank ( denmark ) .
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we provide a straightforward demonstration of a fundamental difference between classical and quantum mechanics for a single local system ; namely the absence of a joint probability distribution of the position @xmath0 and momentum @xmath1 . elaborating on a recently reported criterion by bednorz and belzig [ phys .
rev .
a * 83 * , 52113 ] we derive a simple criterion that must be fulfilled for any joint probability distribution in classical physics .
we demonstrate the violation of this criterion using homodyne measurement of a single photon state , thus proving a straightforward signature of the breakdown of a classical description of the underlying state .
most importantly , the criterion used does not rely on quantum mechanics and can thus be used to demonstrate non - classicality of systems not immediately apparent to exhibit quantum behavior .
the criterion is directly applicable any system described by the continuous canonical variables x and p , such as a mechanical or an electrical oscillator and a collective spin of a large ensemble .
the conceptual differences between classical and quantum physics have intrigued and sometimes bewildered the physics community since the early days of quantum mechanics .
this has led to a search for indisputable manifestations of the quantum world through observations of non - classical behavior in experiments .
a field of particular curiosity is that of identifying the quantum to classical cross - over for ever larger systems , thereby eventually identifying non - classical effects in macroscopic systems .
recently this has led to the observation of , e.g. , macroscopic entangled atomic ensembles @xcite , interference of large molecules @xcite and experiments pushing toward observing non - classical effects in mechanical oscillators @xcite . in parallel to this fundamental interest , non - classicality is of central importance to quantum information processing , the essence of which is to advance computation beyond what is classically possible @xcite .
however , in some instances quantum effects are claimed by demonstrating consistency with an appropriate quantum model . yet
any rigorous demonstration of genuine quantum behavior must exclude the possibility of classical explanations .
the importance of this is exemplified in ref .
@xcite , where a pair of coupled classical oscillators is shown to exhibit signatures easily mistaken for those of entanglement expected from a quantum model .
thus , a definite conclusion on the quantum nature of a system can only result from the breakdown of the classical description and not from verified agreement with quantum mechanics .
this approach is most rigorously demonstrated by the bell - inequalities , where the underlying model of the system is stripped of any physics and is reduced to the very basic assumptions of locality and realism , resulting in an indisputable non - classicality criterion .
the bell - inequalities can not however , by their very nature , be investigated by data obtained from a single system . in this letter
we provide a conceptually simple demonstration of one of the key discrepancies between classical and quantum mechanics , valid for systems of a single degree of freedom : classical systems can always be described by a joint probability distribution for @xmath0 and @xmath1 , the two canonically conjugated coordinates of a system , whereas such a description does not apply in quantum mechanics due to the heisenberg uncertainty principle .
this discrepancy is most evident when the phase space description of the state of a system is examined .
classically , the phase space distribution @xmath2 is the joint probability of finding the system in an infinitesimal area around @xmath3 , and hence it obeys all the requirements of a probability distribution including being a non - negative function . as mentioned , in the case of a quantum phase space formulation , introduced by wigner @xcite ,
the heisenberg uncertainty renders this definition meaningless , as a joint probability distribution for @xmath0 and @xmath1 does not exist .
the phase space distribution is only defined through the single coordinate ( marginal ) distributions , projected from the distribution function @xcite and this relaxation of constraints allows for negative values of the function in areas smaller than @xmath4 .
this negativity is not directly observable due to the vacuum fluctuations preventing simultaneous measurement of @xmath0 and @xmath1 .
however , one can still infer the phase space distribution from measurements of only a single observable at a time and detect such negativities , thereby illuminating the failure of classical theory .
the usage of these negativities as markers of non - classicality has been discussed and demonstrated in several quantum optics systems ( see , e.g. , @xcite ) , using tomographic techniques .
often such methods search for the quantum state most compatible with the experimental data using statistical inference or variational techniques@xcite and thus inherently rely on quantum mechanics .
these methods are therefore not applicable for demonstrating the absence of a classical description
. alternatively , given measurements of all the coordinate distributions , the underlying state can be uniquely determined , and the phase space distribution fully calculated using the inverse radon transformation @xcite without relying on quantum mechanics .
though such methods have been used in quantum optics for demonstrating various states , the mathematical transformation involved is highly complicated .
furthermore the numerical stability of the inverse transformation is problematic , leading to numerical uncertainty at high frequencies , and sometimes results in unphysical states @xcite .
these limitations are a drawback for using tomographic techniques for validating the breakdown of a classical description , and the application of these methods is usually cumbersome .
our simple , unambiguous demonstration of the absence of a classical probability distribution is based on recent theoretical work by bednorz and belzig @xcite that verifies the negativity of the wigner function based on moments . as discussed in detail below ,
their results lead to a hierarchy of inequalities , such that violation of any one inequality indicates negativity of the wigner function . full tomographic reconstruction with the associated numerical complexities is thereby avoided .
we extend this approach such that it can be applied to quadrature measurement of a single photon state , and use the experimental data from the heralded single photon generation to directly disprove the existence of a joint probability of the position and momentum for this system .
we start by re - iterating the key results of bednorz and belzig , through a reformulation that relies only on classical mechanics .
the phase space of a system with a single degree of freedom is fully characterized by a two - dimensional phase space distribution @xmath5 .
that is , given the phase space distribution , the ensemble averaged result of any measurable quantity @xmath6 can be obtained by @xmath7 where @xmath8 is the decomposition of the quantity @xmath6 in terms of the generalized coordinate @xmath0 and its canonically conjugated momentum @xmath1 . to disprove the existence of a classical probability distribution we examine the ensemble average of a non - negative test function @xmath9 over a classically explainable system , which must have a proper distribution function that results in the ensemble average of @xmath10 be non - negative :
@xmath11 violating this condition is a direct proof of the absence of a joint probability distribution .
the condition can , however , be violated in quantum mechanics , where @xmath5 is the wigner function that can contain negative values .
the objective therefore is to optimize a test function such that it will be dominant at the possible negative areas of the distribution function . for a rotationally invariant phase space both the phase space distribution and the test function
can be described solely by the phase space radius @xmath12 , defined by @xmath13 . for reasons to become clear later
, we choose a specific form for the test function @xmath10 , writing it as a square of an @xmath14th order , even polynomial @xmath15 with real coefficients @xmath16 ; @xmath17 minimizing the above expression for a given order @xmath14 is done by straight - forward linear optimization of the coefficients @xmath16 : @xmath18 for all @xmath19 .
notice that the linearity of the problem ensures that the obtained minimum of @xmath20 is global and therefore the most optimal indicator of a possible violation of eq .
( [ eq : require ] ) for a given polynomial order @xmath14 .
it is important to emphasize that this is only a sufficient criterion for non - classicality , and an optimized positive average for a chosen @xmath14 does not ensure a classical probability distribution , since the negativity may only be exhibited by the inclusion of higher order terms in @xmath15 .
however , it is clear that increasing the polynomial order @xmath14 can not increase the minimized value of @xmath20 , and we conjecture that the limit of @xmath21 will exhibit any negativity of the wigner function , as the polynomial can represent an arbitrarily ( analytical ) sharp peaked function @xmath10 focused at the negativity .
assuming the existence of all moments ( e.g. , due to an exponentially decaying tail of the phase space distribution at large @xmath12 ) , this then becomes a necessary criterion for the negativity of the distribution function .
we also note here , that similar polynomial expansion has been discussed @xcite in the context of the p - function distribution .
the p - function is , however , only defined within the framework of quantum mechanics , and hence can not be used to prove the absence of a classical description .
we assume that , as is the case for many systems , the system in question can only be experimentally accessed by measuring one of the canonically conjugated variables ( e.g. , @xmath0 or @xmath1 ) at a time .
since we are restricted to single coordinate measurement at a time , neither the intensity nor the phase space distribution function is directly accessible .
for this method to be applicable to such experimental data , the functional @xmath22 must be expressed in terms of the moments of the projected coordinates @xmath23 , where @xmath24 is a measureable rotated coordinate .
to do this we use the identity @xmath25 where @xmath26 this is where quantum and classical approaches diverge . while classically eq .
( [ eq : mathidentity ] ) represents a measurable physical quantity , it is missing the key vacuum uncertainty , allowing for the breakdown of the classical description .
it is interesting to note the implication of identity ( [ eq : mathidentity ] ) . for the @xmath27th moment of the radial distribution to be known ,
we need @xmath27 cuts in phase space ; i.e. , different coordinate measurements at equally distributed angles .
regardless of any assumption about the underlying state , the average of eq .
( [ eq : mathidentity ] ) directly gives @xmath28 in the special case of a symmetric distribution function these moments are all identical , and eq .
( [ eq : moments ] ) reduces to @xmath29 the radial moments can thus be indirectly calculated from the quadrature measurements .
substituting these radial moments into eq .
( [ eq : require ] ) using the functional form of @xmath9 given by eq .
( [ eq : functional ] ) , we get , for a given set of measured moments @xmath30 , a necessary condition for classicality of the underlying state . if eq .
( [ eq : require ] ) is violated by the solutions of eq .
( [ eq : optimazation ] ) , the underlying state can not be explained by a proper phase space probability distribution , and one can not assign a joint probability distribution to @xmath0 and @xmath1 . to demonstrate the absence of a joint probability distribution we are going to consider the phase space description of a single photon state . in phase space
this can be described by the first excited state of a harmonic oscillator , which is rotationally invariant and contain negative parts in the wigner functions .
fig .
[ fig : epsart2 ] shows the optimal functional forms obtained for the this state for low polynomial orders . as higher order terms
are included , the optimized test function is increasingly probing the negative part of @xmath31 , yielding a negative expectation value .
profiles of the test function @xmath10 minimizing the expectation value @xmath20 for the first excited state of a quantum harmonic oscillator , as a function of the phase space radius , for different orders @xmath14 ( see text ) plotted against the profile of the corresponding wigner function . as the order of the polynomial increases , the function becomes centered around the negativity , decreasing elsewhere . in this case ,
negative expectation values are obtained starting at @xmath32 .
inset shows the polynomial order required to observe negative expectation values , as a function of the single - photon fractional content in a mixture with vacuum . as the fraction of vacuum
is increased , the state approaches a classically describable state and higher moments are needed to observe the negativity . ]
we note that negative expectation values appear only from the fourth order onward .
this is because the peak of the test function at the position of the negativity must be narrower than heisenberg s uncertainty in order not to smear the negativity ; this is in full agreement with ref .
@xcite .
the experimental demonstration is achieved with single photons generated by an heralded cavity - enhanced non - degenerate parametric down - conversion .
the equivalence between a single mode electromagnetic field and an harmonic oscillator allows us to describe the em field by a phase space of a single degree of freedom .
the down - conversion process produces two photons , and as one is detected as a trigger , the result is a single photon state where the losses introduce a statistically mixed component of vacuum .
the projection measurements ( quadratures ) are obtained by measuring the statistics of the noise , using an optical homodyne detection scheme . in this scheme ,
the weak investigated optical field is overlapped with a strong laser pulse on a beam splitter , and the interference of the two fields is detected and subtracted .
the phase of the strong laser field determines the angle @xmath33 ( eq . ( [ eq : quadratures ] ) ) of the measured coordinate .
measurements were taken without fixing the phase of the local oscillator , thus smearing the resulting distribution .
this enables us to treat the results as rotationally invariant even if non - invariant features existed prior to smearing .
such measurements will generate a rotationally invariant reconstructed state for any underlying state , but this does not necessarily average out negativities in the wigner function ( see , e.g. , @xcite ) .
for details of the experimental setup and the charactarization of the resulting single photon see ref .
@xcite . the data set contained 180,000 measured quadratures .
we have here revised the optimization of the functional to also account for statistical uncertainties inherent to a limited data set .
this is done by optimizing @xmath34 where @xmath35 is the standard deviation of @xmath10 .
the results are shown in fig .
[ fig : epsart ] .
the fact that the expectation value for our test function is negative with certainty of almost twenty standard deviations , clearly demonstrates that the measured state in this experiment can not be explained by classical theory , unambiguously negating the possibility of existence of a joint probability distribution for @xmath0 and @xmath1 .
the appearance of negative values from the twelfth order polynomials and higher indicate the quantum mechanical description of this state in terms of a wigner function includes negative valued areas .
we note that the minimized function from eq .
( [ eq : functional ] ) is monotonically decreasing for increasing order @xmath14 , and the onset of negativity at a certain order therefore means that all higher orders will also be negative .
this suggests a sequential authentication procedure for an unknown state .
as mentioned above , for a pure single photon state , negative expectation values are observable from the @xmath36-th order polynomial onwards .
the twelfth order polynomial required here is due to the vacuum component of the field , requiring higher orders of the polynomial as shown in the inset of fig .
[ fig : epsart ] , and is in agreement with the results obtained in ref . @xcite reporting 62% fraction of single photon in the resulting mixed state .
expectation value for the square of a polynomial relative to its standard deviation , as a function of the polynomial s order for the experimental data .
negativity by almost @xmath37 standard deviations disproves the existence of a joint probability distribution for @xmath0 and @xmath1 .
the inset shows a histogram of the raw measured quadrature data ( arbitrary units ) . ] in conclusion , we have experimentally demonstrated the non - existence of a joint probability distribution of two canonical variables .
this is done by violation of an inequality derived without the assumptions of quantum mechanics , thus allowing for it as proof of the absence of a classical description in systems not immediately evident to display quantum behavior .
the procedure used here can thus provide a simple , practical tool for demonstrating the non - classicality of a state based on quadrature measurements , where the existence of a classical joint distribution of two conjugated variables can be negated . in this way ,
this procedure is closely linked to other criteria @xcite demonstrating contextuality of measurements , and thus disproving the classical local hidden variable view . unlike ref .
@xcite , which are applicable to discrete variables , the method demonstrated here applies for continuous variables such as position and momentum , collective spin operators @xcite and quadrature phase operators .
this makes it useful to systems containing many particles , where criteria based on counting particles are not easily implemented and interpreted .
this method complements the full tomographic reconstruction techniques in that it is simpler and avoids numerical complexities of inverse transformations .
these kinds of conceptual proofs , when extended to different detection schemes , can shed more light on the quantum to classical correspondence , especially where the control of claimed macroscopic quantum states is in question .
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support by european community ( fp7 eu - india grant athena and erc starting grant no.203523 bismuth ) is gratefully acknowledged .
x.h . thanks claude ederer and chung - yuan ren for their helpful advices and comments .
thanks j.m .
perez - mato for useful discussions .
the calculations have been performed on the vienna scientific cluster ( vsc ) and , partially , in caspur supercomputing center in rome .
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we present a study based on several advanced first - principles methods , of the recently synthesized pbnio@xmath0 [ j. am .
chem .
soc * 133 * , 16920 ( 2011 ) ] , a rhombohedral antiferromagnetic insulator which crystallizes in the highly distorted @xmath1 crystal structure .
we find this compound electrically polarized , with a very large electric polarization of @xmath2 100 @xmath3c/@xmath4 , thus even exceeding the polarization of well - known bifeo@xmath0 .
pbnio@xmath0 is a proper ferroelectric , with polarization driven by large pb - o polar displacements along the [ 111 ] direction .
contrarily to naive expectations , a definite ionic charge of 4 + for pb ion can not be assigned , and in fact the large pb 6@xmath5-o 2@xmath6 hybridization drives the ferroelectric distortion through a lone - pair mechanism similar to that of other pb- and bi - based multiferroics . _
introduction _ multiferroics are materials in which different ferroic orders such as ferromagnetism , ferroelectricity and/or ferroelasticity may coexist in one single phase @xcite . in the last few years , there has been a tremendous boom of interest in these materials , due to the potential applications in memory devices or in novel type of magnetic switching in magnetoelectric multiferroics , based on the cross - coupling between ferroelectric and magnetic channels @xcite .
furthermore , these materials offer a rich and fascinating playground for the complex physical mechanisms underlying the processes involved in the observed properties.@xcite it is obvious that the search and the prediction by material design of new multiferroics is of great importance for both fundamental physics and technological applications @xcite .
relatively few multiferroics have been identified so far .
@xcite without doubt , the most studied and well characterized multiferroic is bifeo@xmath0 @xcite .
it crystallizes in the polar space group _ r3c _ ( no . 161 ,
point group _ c_@xmath7@xmath8 ) , and is predicted to have a g - type antiferromagnetic ( afm ) alignment of the fe spins @xcite .
the _
r3c _ symmetry corresponds to the so - called linbo@xmath7-type structure , which can be viewed as a highly distorted double perovskite with rhombohedral symmetry .
the primitive unit cell contains two formula units ( 10-atoms ) , arising from counterrotations of neighboring o octahedra about the [ 111 ] axis ( see fig.[fig:1 ] ) .
the crystal symmetry allows the presence of a spontanous polarization along the [ 111 ] direction , which arise from the relative displacement of the bi sublattices with respect to the feo@xmath9 octahedra cages along [ 111 ] .
the origin of the large spontaneous polarization of bi@xmath10fe@xmath10o@xmath11@xmath7 , @xmath2 90 @xmath3c/@xmath4 @xcite , has been explained by first principles calculations within the density functional theory plus hubbard - u approach ( dft+u)@xcite and the `` modern theory of polarization '' ( mtp ) @xcite , in terms of the lone - pair electrons present at the bi sites which are ultimately responsible for the large displacement along the [ 111 ] direction @xcite . inspired by the recent reports of inaguma _ et al .
_ on the synthesis of a new antiferromagnetically ordered compound with linbo@xmath7-type structure , such as pbnio@xmath7@xcite , we explore here the possibility of multiferroic behavior in pbnio@xmath7 .
we first summarize the experimental findings .
inaguma and coworkers synthesized two high - pressure polymorphs of pbnio@xmath7 with a ( a ) perovskite - type structure and ( b ) the linbo@xmath7-type structure @xcite ; the latter ( hereafter called l - pbnio@xmath0 ) is thermodynamically more stable than the perovskite - type one at ambient pressure . with respect to the orthorhombic structure , in the acentric rhombohedral linbo@xmath7-type ( _ r3c _ ) structure ( fig .
[ fig:1 ] ) pb and o atoms are displaced against each other along the threefold [ 111 ] axis leading to a large distortion of the pbo@xmath12 and nio@xmath12 octahedra ( see fig.[fig:1 ] ) .
the pb atom is coordinated by six oxygens at 2.10 and 2.25 , while the ni o bond distance splits into two subgroups ( 2.07 and 2.11 ) . magnetic susceptibility and
resistivity measurements indicate that l - pbnio@xmath7 undergoes an afm transition at @xmath13=205@xmath14k , and exhibits semiconducting behavior .
the afm ordering in the acentric crystal structure suggests possible multiferroic behavior . in the following
we show that l - pbnio@xmath0 is a _ new _ room - temperature _ multiferroic _ , with an _ exceptionally large _ polarization ( @xmath15 ) of about 100 @xmath3c/@xmath16 , which is the highest polarization ever predicted for any bulk material so far , and @xmath17 10% larger than that of bifeo@xmath0 @xcite .
( color online ) schematic view of the linbo@xmath7-type pbnio@xmath7 within space group _ r3c _ structure built up from two cubic perovskite unit cells , the black ( large ) , gray ( medium ) and red ( small ) spheres denotes pb , ni and o atoms , respectively .
with respect to the centrosymmetric ( _ r3c _ ) paraelectric phase ( see text ) the cations are displaced along the [ 111 ] axis relative to the anions ( the arrows represent the displacement vectors ) , and the oxygen octahedra slightly rotate with alternating sense around the [ 111 ] direction @xcite . the crystal structure was drawn using the program vesta@xcite . ]
_ computational details _ we performed dft based calculations using the vienna _ ab initio _ simulation package ( vasp ) @xcite based on the projector - augmented - wave method@xcite , withing the perdew , burke and ernzerhof ( pbe ) parameterization scheme@xcite for the generalized gradient approximation ( gga ) .
to overcome the deficiencies of standard exchange - correlation approximations for localized ni _ d _ states , we made use of three beyond - dft approaches : ( i ) dudarev s gga+u @xcite , using u=4.6 ev in accordance with constrained dft calculation of ref . ;
( ii ) the renowned heyd - scuseria - ernzerhof ( hse ) screened hybrid functional @xcite , which has been shown to give an excellent account of materials properties for magnetic multiferroics@xcite . ( iii ) the recently introduced variational pseudo self - interaction - correction method vpsic@xcite ( adopting the hse optimized structure ) , implemented within plane - wave basis set and ultrasoft pseudopotential scheme in the pwsic code .
the cutoff energy was set as 600 ev and a 8@xmath188@xmath188 monkhorst - pack grid of _ k _ points was used . in hse
, the fraction ( 1/4 ) of fock exchange was sampled using the twofold reduced * k*-point grid , to reduce the computational load .
the lattice parameters and atomic positions were relaxed ( at gga+u and hse level ) until the total energy changed by less than 10@xmath19@xmath20 ev per unit cell and the residual force was smaller than 0.01 ev / .
.structural parameters , magnetic moment and electronic band gap for afm - g configuration of l - pbnio@xmath7 within gga , gga+u and hse method , compared with the available experimental data .
_ a _ , _ c _ are the lattice parameters in the hexagonal setting , whereas _
x _ , _ y _ , and _ z _ are the internal atomic positions of pb and o. ni ions sit in the ( @xmath21 ) ( 0,0,0 ) positions . bond lengths of pb o and ni o are also reported .
@xmath22 ( @xmath14 ) is the ni o
ni angle .
m _ indicates the magnetic moment of ni . [ cols="<,^,^,^,^,^ " , ] [ tab : born ]
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this work was supported in part by the national natural science foundation of china , and the education ministry of china .
k@xmath71hn , j. kaplan and e.g.o .
safiani , nucl . phys . *
b157 * , 125 ( 1979 ) ; b. guberina , j.h .
k@xmath72hn , r.d .
peccei and r. r@xmath71ckl , nucl . phys . * b174 * , 317(1980 ) ; p. cho and a.k .
. rev . * d53 * , 150(1996 ) .
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we study the production of @xmath0 in @xmath1 annihilation through two virtual photons .
the cross section is estimated to be 23 fb at @xmath2 gev , which is smaller by a factor of six than the calculated cross section for the same process but through one virtual photon . as a result , while the annihilation into two photons may be important for certain exclusive production processes , the big gap between the inclusive production cross section @xmath3 0.9pb observed by belle and the current nonrelativistic qcd prediction of @xmath40.15pb still remains very puzzling .
we find , however , as the center - of - mass energy increases ( @xmath5 gev ) the production through two virtual photons @xmath6 will prevail over that through one virtual photon , because in the former process the photon fragmentation into @xmath7 and into the charmed quark pair becomes more important at higher energies .
+ pacs number(s ) : 12.40.nn , 13.85.ni , 14.40.gx production of @xmath8 through two photons in @xmath1 annihilation + kui - yong liu and zhi - guo he + department of physics , peking university , beijing 100871 , people s republic of china + kuang - ta chao + china center of advanced science and technology ( world laboratory ) , beijing 100080 , people s republic of china and department of physics , peking university , beijing 100871 , people s republic of china charmonium production is one of the important processes to test quantum chromodynamics ( qcd ) both perturbatively and non - perturbatively .
because of the simpler parton structure involved , which will reduce the theoretical uncertainty , charmonium production in @xmath1 annihilation is expected to be more decisive in clarifying the production mechanisms of heavy quarkonia .
the two b factories with babar and belle are collecting huge data samples of continuum @xmath1 annihilation events , which will allow us to have a fine data analysis for charmonium production .
recently , charmonium production in @xmath1 annihilation has become more interesting and puzzling , because of the large gap between the belle measurements@xcite and the theoretical calculations for both inclusice@xcite and exclusive @xcite charmonium production via double @xmath9 pairs based on nonrelativistic qcd ( nrqcd ) .
for the inclusive processes , belle has reported a measurement on the @xmath7 production in @xmath1 annihilation at @xmath2 gev@xcite , and found that a very large fraction of the produced @xmath7 is due to the double @xmath9 production in @xmath1 annihilation@xcite @xmath10 which corresponds to @xcite @xmath11 in contrast , the predicted values for the cross section in nrqcd ( the color - octet contribution is negligible for this process ) are much smaller than the data@xcite . in a recent analytical calculation and numerical estimation for the inclusive charmonia production including all s - wave , p - wave and d - wave states via double @xmath9 in nrqcd@xcite ,
we find @xmath12 this value is consistent with other previous calculations , including those obtained based on the quark - hadron duality hypothesis@xcite , but smaller than the belle data by about a factor of six . for the exclusive processes , the belle measurement@xcite for the cross section of @xmath13 is about an order of magnitude larger than the nrqcd calculation for @xmath14@xcite .
in order to solve the problem , calculations for the exclusive double - charmonium production from @xmath1 annihilation into two virtual photons are performed@xcite , and it is pointed out that the cross section for @xmath15 is larger than that for @xmath16 by about a factor of 3.7 , despite of possible uncertainties due to the choice of input parameters@xcite . this is a interesting result since it indicates that the @xmath1 annihilation through two virtual photon fragmentation may make important contributions to certain processes , and it might substantially reduce the discrepancy between experiment and theory for the exclusive process @xmath17 ( note that it is essential to check experimentally whether @xmath18 is largely misidentified with @xmath13 ) .
in this situation , it is necessary to examine the contribution of @xmath1 annihilation through two virtual photons to the inclusive production of @xmath7 . in the following
we will calculate the complete o@xmath19 color - singlet inclusive cross section for @xmath20 , and compare the production rate through two virtual photons with that through one virtual photon , to see whether the annihilation through two virtual photons can decrease the discrepancy between the belle measurement on @xmath21 production and the calculations based on nrqcd .
.,width=529,height=604 ] following the nrqcd factorization formalism , color - singlet scattering amplitude of the process @xmath22 in fig .
[ feynman ] is given by @xmath23 where @xmath24 is the intermediate @xmath9 pair which is produced at short distances and then evolves into a specific charmonium at long distances .
@xmath25 , @xmath26 , @xmath27 are respectively the color - su(3 ) , spin - su(2 ) , and angular momentum clebsch - gordon coefficients for @xmath28 pairs projecting out appropriate bound states . for @xmath7
the coefficient @xmath29 can be related to the radial wave function of the bound state and reads @xmath30 the spin projection operator can be defined as@xcite @xmath31 we write the spin projection operator which will be used in the calculation as @xmath32 where @xmath33 is the mass of the charmonium , which equals to @xmath34 in the nonrelativistic approximation .
( solid line ) and @xmath35 ( dotted line ) as functions of the scattering angle of @xmath7.,width=453,height=377 ] the amplitude for the upper feynman graph in fig .
[ feynman ] can be written as @xmath36\bar{u}(p_3)\gamma_{\mu}v(p_4 ) \frac{1}{p^2(p_3+p_4)^2},\end{aligned}\ ] ] and the amplitude for the corresponding flipped graph is denoted as @xmath37 .
the amplitude for the lower feynman graph in fig . [ feynman ] reads @xmath38 and the amplitude for the flipped graph is denoted as @xmath39 .
the calculation of cross section for @xmath40 is straightforward .
the differential cross section can be written in the form @xmath41 where @xmath42 represents the elements of the quadruple integral related to the final state phase space ( see , e.g. ref .
@xcite ) , and @xmath43 is the unpolarized module squared of the amplitude . for simplicity we will not write down their lengthy expressions here
. with eq .
( [ cross ] ) we can evaluate the inclusive cross sections for @xmath7 production from @xmath1 through two virtual photons .
the input parameters used in the numerical calculations are the same as in ref.@xcite @xmath44 @xmath45 now we give the numerical result at the belle energy @xmath2 gev : @xmath46 it is well known that @xmath47 is a pure electromagnetic process ( except for the hadronization of the quark pair at long distances ) , while @xmath48 involves both electromagnetic and strong interactions . for a naive order of magnitude estimate , the ratio of the production rate of the former to the latter would be proportional to @xmath49 , but the photon fragmentation into @xmath7 and into the charmed quark pair can substantially enhance the former . to see the role of photon fragmentation , we plot the differential cross sections of @xmath50 and @xmath51 as functions of the scattering angle of @xmath7 at @xmath2gev in fig .
[ distribution ] .
( here for the one - photon process we choose @xmath520.26 . )
one can see that the small angle @xmath7 production in @xmath53 is dominant .
this indicates that most of the @xmath7 production comes from the photon fragmentation ( corresponding to the upper graph in fig .
[ feynman ] with one photon fragmenting to @xmath7 and the other fragmenting to a charm quark pair ) .
indeed , our calculation shows that at @xmath2gev , the contribution from fragmentation graphes is about seventy - two percent of the total cross section . in fig .
[ ratio ] we show the ratio of the fragmentation contribution to the total cross section as a function of the @xmath1 center - of - mass energy in @xmath54 .
it is clear that the photon fragmentation becomes more and more dominant as the center - of - mass energy increases .
this is in agreement with the observation in the @xmath55 exclusive production through two virtual photons@xcite . from the above discussions we have seen the importance of the photon fragmentation to @xmath7 as well as to the charm
quark pair in the two - photon process @xmath50 .
a even more crucial result is that at high @xmath1 center - of - mass energies the contribution through two virtual photons will prevail over that through one virtual photon in the production of @xmath21 .
the reason lies simply in the fact that the virtuality of the photon in the two - photon process can be as small as @xmath56 , whereas it is as large as @xmath57 , the center - of - mass energy squared in the one - photon process . in fig .
[ num ] we show the cross sections of @xmath54 ( solid line ) and @xmath58 ( dotted line ) as functions of the center - of - mass energy @xmath59 .
we see clearly that the cross section for @xmath54 decreases very slowly , whereas that for @xmath60 decreases rapidly as @xmath59 increases . at @xmath61 gev ,
the two photon process becomes to prevail over the one photon process .
however , unfortunately , at the belle energy @xmath2 gev , since the enhancement effect due to the factor @xmath62 is not large enough as compared with the suppression factor @xmath49 , we find @xmath63 fb , which is still much smaller than @xmath64 fb@xcite , and therefore is negligible .
( solid line ) and @xmath60 ( dotted line ) as functions of the center - of - mass energy , width=453,height=377 ] in summary , we have calculated the complete @xmath65 color - singlet inclusive cross sections for @xmath21 production from @xmath66 annihilation into two photons . due to the suppression factor of @xmath49 ,
at the @xmath1 center - of - mass energy @xmath2 gev the cross section of this process is smaller by about a factor of six than that from @xmath66 annihilation into one photon .
we then conclude that while the @xmath66 annihilation into two photons could be helpful in solving the puzzle for the exclusive @xmath67 production , it can do very little to reduce the big gap between the observed inclusive production cross section of @xmath68 pb and the current nrqcd predictions of about 0.15 pb .
this puzzle still needs to be explained with new theoretical considerations .
we find , however , as the center - of - mass energy increases(@xmath6920 gev ) the production through two photons @xmath50 will prevail over that through one photon , because in the former case the photon fragmentation into @xmath7 and into the charmed quark pair becomes more important at higher energies .
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+ + * * +
we observe that + + + + + + 2 . for a given initial ratio , except for the light quarks and anti - quarks
, the output does not depend much on the initial baryon number density .
for the light quarks and antiquarks this variation is due to the presence of the exponentiated non - zero chemical potentials .
+ 3 . for a given initial baryon number density we can recast the equation for baryon number density in the form @xmath25 + @xmath26 ... (27 ) where c is a constant for a given temperature , light quark fugacity and given baryon number density . here
d is the light quark to antiquark initial fugacity ratio . for a fixed light quark fugacity and temperature , as d falls clearly the rhs of the above equation increases which indicates a rise in the chemical potential .
again for a system of higher chemical potential , the temperature has to drop at a slower rate due to the constraint imposed .
hence we observe that + + i ) the temperature falls at a slower rate for a smaller value of the light quark to antiquark initial fugacity ratio .
+ ii ) as the temperature falls at a slower rate , it would imply a lesser expenditure of energy to produce particles in general .
this would show up in the slower rise of all fugacity values except the light antiquark , which would show a higher growth rate .
this is due to the exponentiated chemical potential part .
+ 4 . for inclusion of the quark - flavour - changing process
we observe the following : + + due to the additional production of s - quarks from massless quarks via qfcp , we see an additional increase in fugacity of the strange quark while , the rate of equilibration falls for non - strange fermions .
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parton equilibration studies for a thermally equilibrated but chemically non - equilibrated quark - gluon plasma ( qgp ) , likely to be formed at the relativistic colliders at bnl and cern is presented .
parton equilibration is studied enforcing baryon number conservation .
process like quark - flavour interchanging is also taken into consideration .
the degree of equilibration is studied comparatively for the various reactions / constraints that are being considered .
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for certain hierarchical structures , one can study the percolation problem using the renormalization - group method in a very precise way .
we show that the idea can be also applied to two - dimensional planar lattices by regarding them as hierarchical structures .
either a lower bound or an exact critical probability can be obtained with this method and the correlation - length critical exponent is approximately estimated as @xmath0 .
the percolation problem is a question about how a global connection can be made possible by randomly filling local components by a certain probability @xmath1 .
while it can be explained in purely geometric terms without any interaction , when a global connection actually appears , the macroscopic behavior of the system exhibits all the characteristic features of a continuous phase transition with a diverging correlation length , just as we observe in other interacting spin systems such as the two - dimensional ( 2d ) ising model @xcite .
this analogy is given a precise meaning by the fortuin - kasteleyn representation of the @xmath2-state potts model @xcite , where the percolation turns out to be equivalent to the limit of @xmath3 .
since the percolation transition at a critical probability @xmath4 has a diverging correlation length , every microscopic length scale becomes irrelevant with respect to the critical phenomena , and the system behaves as if it does not have any specific length scale .
this is a qualitative explanation of the reason why a percolating cluster connecting two opposite sides of a 2d plane has a fractal dimension at @xmath5 .
the lack of a specific length scale implies that the system remains statistically invariant even if we zoom the system up or down , and this scale invariance readily lends itself to a renormalization - group ( rg ) study of the percolation problem @xcite . in certain cases where the underlying structure itself is fractal , it is possible to carry out the rg calculation to a good approximation or exactly , exploiting this fractal property @xcite .
such fractal structures usually contain groups of bonds which connect longer and longer distances in a regular fashion .
for this reason , one can sometimes arrange the groups of bonds in a hierarchical way according to their connection lengths .
( a ) is an example of a hierarchical structure called the enhanced binary tree , which is obtained by adding horizontal bonds to the simple binary tree .
it is hierarchical in the sense that filling a horizontal bond is comparable to a very long connection along the bottom layer and the connection length is dependent on the level of the horizontal bond @xcite .
that is , a horizontal bond in the highest level can connect two points at distance 7 along the bottom layer at maximum .
for a horizontal bond at the next highest level , this maximum connection distance is only as large as 3 lattice spacings .
an rg scheme for the enhanced binary tree is described in @xcite as shown in ( b ) : we calculate the probability for any of the leftmost points to connect to any of the rightmost points within the cell as a function of the bare coupling @xmath1 and a coarse - grained effective coupling @xmath6 , and then replace this probability by a new effective bond with strength @xmath7 .
the resulting expression for @xmath7 is written as @xmath8.\ ] ] by asking when @xmath9 becomes 1 , we obtained a lower bound of the percolation threshold as @xmath10 @xcite , which is consistent with the conclusion in @xcite that @xmath11 .
note that we get a lower bound since in iterating @xmath6 to @xmath7 , there is a small chance to regard a layer as percolated when it is actually not [ see , e.g. , ( c ) ] , whereas the opposite is not possible . .
( c ) this layer is not connected from left to right even though the cells inside it appear as filled according to the recursion scheme in ( b ) .
the solid and dotted lines represent filled and empty bonds , respectively . , title="fig:",scaledwidth=35.0% ] .
( c ) this layer is not connected from left to right even though the cells inside it appear as filled according to the recursion scheme in ( b ) .
the solid and dotted lines represent filled and empty bonds , respectively .
, title="fig:",scaledwidth=28.0% ] + .
( c ) this layer is not connected from left to right even though the cells inside it appear as filled according to the recursion scheme in ( b ) .
the solid and dotted lines represent filled and empty bonds , respectively .
, title="fig:",scaledwidth=41.0% ] although the above rg scheme is devised to investigate a hierarchical structure , we show in this work that it can be applied to non - hierarchical planar lattices as well . in ( a )
, we present a variation of the rg scheme shown above .
the similarity is obvious : we have taken away only one bond out of those in ( b ) , and this is meant to describe the triangular lattice .
it leads us to the following recursion , @xmath12 ^ 2 .
\label{eq : rec1}\ ] ] again , the bond connection in _ lower _ levels , composed of @xmath1 and @xmath6 , is converted to a single bond with @xmath7 at a _ higher _ level .
this distinction of levels might look arbitrary since the bonds in the plane do not have any hierarchy .
however , the important point is that all the argument above to find a lower bound remains still legitimate from this viewpoint . solving for @xmath13
, we find that @xmath14 and consequently , @xmath15 at @xmath16 . comparing this to the exact bond - percolation threshold in the triangular lattice , @xmath17 @xcite , we see that our method indeed yields a lower bound .
we now extend the cells to be renormalized by adding one more level .
that is , let us denote the width of the cell as @xmath18 and consider the case of @xmath19 .
for the triangular lattice , the shape of such a larger cell is given in ( b ) . by enumerating all the possible cases ,
the recursion relation is obtained as @xmath20 and we find its limiting value as @xmath21 with @xmath22 , @xmath23 , and @xmath24 .
the solution of @xmath25 is found at @xmath26 , which is an improved lower bound compared to the previous one , @xmath27 , even though the convergence turns out to be rather slow . .
( b ) a larger cell with @xmath19 for the triangular lattice . the rg scheme for the honeycomb lattice with ( c ) @xmath28 and ( d ) @xmath19 can be constructed in the same way , as well as that for the square lattice with ( e ) @xmath28 and ( f ) @xmath19 .
the double lines represent coarse - grained effective bonds and the dotted lines in ( c ) to ( f ) mean that the connections do not correspond to any bare interaction .
, title="fig:",scaledwidth=28.0% ] .
( b ) a larger cell with @xmath19 for the triangular lattice . the rg scheme for the honeycomb lattice with ( c ) @xmath28 and ( d ) @xmath19 can be constructed in the same way , as well as that for the square lattice with ( e ) @xmath28 and ( f ) @xmath19 .
the double lines represent coarse - grained effective bonds and the dotted lines in ( c ) to ( f ) mean that the connections do not correspond to any bare interaction . , title="fig:",scaledwidth=28.0% ] + .
( b ) a larger cell with @xmath19 for the triangular lattice . the rg scheme for the honeycomb lattice with ( c ) @xmath28 and ( d ) @xmath19 can be constructed in the same way , as well as that for the square lattice with ( e ) @xmath28 and ( f ) @xmath19 .
the double lines represent coarse - grained effective bonds and the dotted lines in ( c ) to ( f ) mean that the connections do not correspond to any bare interaction . , title="fig:",scaledwidth=28.0% ] .
( b ) a larger cell with @xmath19 for the triangular lattice . the rg scheme for the honeycomb lattice with ( c ) @xmath28 and ( d ) @xmath19 can be constructed in the same way , as well as that for the square lattice with ( e ) @xmath28 and ( f ) @xmath19 .
the double lines represent coarse - grained effective bonds and the dotted lines in ( c ) to ( f ) mean that the connections do not correspond to any bare interaction .
, title="fig:",scaledwidth=28.0% ] + .
( b ) a larger cell with @xmath19 for the triangular lattice .
the rg scheme for the honeycomb lattice with ( c ) @xmath28 and ( d ) @xmath19 can be constructed in the same way , as well as that for the square lattice with ( e ) @xmath28 and ( f ) @xmath19 .
the double lines represent coarse - grained effective bonds and the dotted lines in ( c ) to ( f ) mean that the connections do not correspond to any bare interaction . , title="fig:",scaledwidth=28.0% ] .
( b ) a larger cell with @xmath19 for the triangular lattice . the rg scheme for the honeycomb lattice with ( c ) @xmath28 and ( d ) @xmath19 can be constructed in the same way , as well as that for the square lattice with ( e ) @xmath28 and ( f ) @xmath19 .
the double lines represent coarse - grained effective bonds and the dotted lines in ( c ) to ( f ) mean that the connections do not correspond to any bare interaction .
, title="fig:",scaledwidth=28.0% ] if we also regard the honeycomb lattice as hierarchical , we can consider an rg scheme as depicted in ( c ) . by calculating the probabilty for any of the leftmost points to connect to any of the rightmost points within the cell ,
we find @xmath29 ^ 2 .
\label{eq : honey}\ ] ] by a little algebra as above , we find @xmath30 , which becomes one at @xmath31 . again
, this is lower than the exact value @xmath32 @xcite .
one may expect an improved estimate by considering a larger cell shown in ( d ) , which leads to @xmath33 the limiting solution is @xmath34 where @xmath35 , @xmath36 , and @xmath37 .
we find that @xmath38 at @xmath39 .
using the duality relation @xmath40 @xcite , we may turn this result to an _ upper _ bound of the bond - percolation threshold for the triangular lattice .
that is , our method gives a possible region of the threshold as @xmath41 , or equivalently , @xmath42 .
a more interesting case is found by considering the horizontal bonds in ( a ) and ( b ) as fictitious [ ( e ) and ( f ) ] .
this corresponds to the square lattice , and the interaction in the horizontal direction will appear only as an effective one mediated by shorter bonds
. then we can simplify as @xmath29 ^ 2,\ ] ] which happens to be the same as .
therefore , we find @xmath31 once again , but this value is identical to the exact value for the bond - percolation problem in the square lattice @xcite . since this method is supposed to give a lower bound , it should not be possible to improve this result further , so it will be worth checking whether this value really remains unchanged for a larger cell . from a larger cell depicted in ( f ) , we obtain a recursion @xmath43 and find its limiting value as @xmath44 with @xmath45 , @xmath46 , and @xmath47 .
the critical value making @xmath15 is also @xmath31 , as expected .
the fact that @xmath48 does not change with @xmath18 could be an evidence that the bond - percolation threshold is located exactly at @xmath49 for the square lattice .
in addition , we can argue that the connection probability over distance @xmath50 would be roughly determined by @xmath51 near the critical point .
in other words , the correlation length would be written as @xmath52 .
the slope of @xmath53 around @xmath5 does not vanish in every case considered above , so it generally behaves as @xmath54 where @xmath55 at @xmath56 with positive @xmath57 .
therefore , we see that @xmath58}\\ & \approx & ( p_c - p)^{-1},\end{aligned}\ ] ] by using @xmath59 . since the correlation length is assumed to diverge as latexmath:[$\xi \sim critical exponent as @xmath0 , which is an underestimate compared to the exact value , @xmath61 @xcite .
it is worth noting that this rg scheme does not make use of any explicit scaling transformation : we do not zoom up or zoom down the system at criticality as usually found in rg studies @xcite . in arguing the value of @xmath62 , therefore , we evaluate it directly in units of the given lattice spacing instead of any zooming ratio . by setting @xmath63 , in a sense ,
it is the translational invariance that we are actually exploiting in this study . in summary
, we have shown that the rg scheme devised for a hierarchical structure can be also applied to the 2d lattices even though they are not hierarchical .
it generally yields a lower bound , but correctly predicts the bond - percolation threshold for square lattice .
we have also approximately estimated @xmath0 .
this method is more related to the translational invariance rather than to the scaling invariance at criticality .
we are grateful for support from the swedish research council with grant no . 621 - 2008 - 4449 .
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99 g. marchesini et al . ,
_ comput . phys
. commun . _ * 67 * ( 1992 ) 465 .
g. corcella and m.h .
seymour , hep - ph/9908335 .
d collaboration , b. abbott et al .
lett . _ * 80 * ( 1998 ) 5498 .
cdf collaboration , t. affolder et al .
, fermilab - pub-99/220-e .
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we show vector boson transverse momentum distributions at the tevatron , obtained by running the herwig monte carlo event generator with matrix - element corrections .
we compare our results with some recent d and cdf data .
ur1589 + er/40685/939 + hep - ph/9911536 + november 1999 vector boson production at hadron colliders is a fundamental process to test quantum chromodynamics and the standard model of the electroweak interactions .
the lowest order processes @xmath0 and @xmath1 are not sufficient to perform reliable phenomenological predictions , but the initial - state radiation has to be taken into account . a possible way to deal with such multiple emissions consists in running a monte carlo event generator .
standard monte carlo algorithms @xcite describe parton cascades in the soft / collinear approximation , with ` dead zones ' in the phase space which can be filled by the using of the exact first - order matrix element .
in @xcite we implemented matrix - element corrections to the herwig simulation of drell yan interactions : we filled the missing phase space using the exact @xmath2 matrix element ( hard corrections ) and corrected the shower in the already - populated region using the exact amplitude for every hardest - so - far emission ( soft corrections ) . for @xmath3 production at the tevatron , about 4% of the events
are generated in the dead zone , about half of which are @xmath4 events .
similar results hold for @xmath5 production as well . an interesting observable
to study is the vector boson transverse momentum @xmath6 , which is the object of many theoretical and experimental analyses .
while in the parton shower approximation it has to be @xmath7 , after matrix - element corrections a fraction of events with larger values of @xmath6 is to be expected . in fig .
[ wqt ] ( a ) , we plot the @xmath3 @xmath6 spectrum at the tevatron according to herwig 5.9 , the latest public version , and to herwig 6.1 , the new version including matrix - element corrections to drell yan processes .
herwig has the option to vary the intrinsic transverse momentum @xmath8 of the incoming partons , which is set to zero , its default value , in the distributions shown in fig .
[ wqt ] ( a ) .
we observe a remarkable impact of the corrections : after some @xmath6 the 5.9 version does not give events anymore , while herwig 6.1 still has some events generated via the exact matrix element . in fig .
[ wqt ] ( b ) we compare some recent d data @xcite on @xmath3 production at the tevatron with the herwig 6.1 results , which we corrected in order to take the detector smearing effects into account .
we find good agreement overall ; we also consider the option of @xmath9 gev , but do not see any relevant effect after the detector corrections . as far as @xmath5 production is concerned ,
we have some preliminary cdf data @xcite , already corrected for detector effects , which we compare with herwig 6.1 in fig .
[ zqt ] , where the options @xmath8=0 , 1 and 2 gev are investigated .
the agreement is acceptable and the role of the implemented matrix - element corrections is crucial in order to succeed in fitting in with the data for @xmath10 gev . at very low @xmath6 ,
the best fit is obtained by setting @xmath11 gev . in fig
. [ ratio ] , we plot the ratio of the @xmath3 and the @xmath5 transverse momentum spectra , both normalized to unity , for different values of @xmath8 .
although it can be seen from fig .
[ zqt ] ( b ) that the @xmath5 @xmath6 spectrum depends strongly on @xmath8 at low @xmath6 , the ratio of the @xmath3 and @xmath5 spectra is insensitive to it .
this is good news for the @xmath3 mass measurement in hadron collisions , as this ratio is one of the main theory inputs that is needed .
a strong dependence on unknown non - perturbative parameters like @xmath8 could limit the accuracy of the @xmath3 mass measurement at the tevatron and , ultimately , at the lhc .
we have added matrix - element corrections to herwig s treatment of vector boson production in hadron collisions .
they make an enormous difference at high transverse momentum @xmath6 , but little at low @xmath6 .
although the dependence of the results on the non - perturbative intrinsic @xmath6 of partons in the proton ( @xmath8 ) is quite strong at low @xmath6 , it is very similar in the @xmath3 and @xmath5 cases , so that the ratio of the two @xmath6 spectra is almost independent of @xmath8 .
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1 . comparison of power - balance prediction ( solid curve ) and bessel - function prediction ( dashed curve ) with numerical integration result ( diamonds ) , at fundamental frequency ( @xmath42 ) .
parameters : ( a ) @xmath44 ; ( b ) @xmath45 .
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the inverse ac josephson effect involves rf - induced ( shapiro ) steps that cross over the zero - current axis ; the phenomenon is of interest in voltage standard applications .
the standard analysis of the step height in current , which yields the well - known bessel - function dependence on an effective ac drive amplitude , is valid only when the drive frequency is large compared with the junction plasma frequency or the drive amplitude is large compared with the zero - voltage josephson current . using a first - order krylov - bogoliubov power - balance approach we derive an expression for the threshold value of the drive amplitude for zero - crossing steps that is not limited to the large frequency or large amplitude region .
comparison with numerical solutions of the rsj differential equation shows excellent agreement for both fundamental and subharmonic steps .
the power - balance value for the threshold converges to the bessel - function value in the high - frequency limit .
the term ` inverse ac josephson effect ' was coined by levinsen _
et al .
_ @xcite to describe shapiro steps @xcite in the current - voltage characteristic of a small josephson junction that cross over the zero - current axis .
the phenomenon was recognized as being potentially important for voltage - standard applications inasmuch as the elimination of the dc bias current would eliminate one possible pathway for the entry of noise into the system .
it was studied extensively experimentally , analytically , and computationally by a number of authors ( see , _
e.g. _ , kautz @xcite plus references therein ) .
the ` standard ' analysis that emerged from these studies expressed the height in current of phase - locked shapiro steps in terms of bessel functions of an effective drive amplitude ; this analysis was shown to be appropriate when the frequency of the ac drive is large compared with the junction plasma frequency or when the amplitude of the ac drive is large compared with the zero - voltage josephson current .
our analysis , instead , is based on the power - balance formalism developed by krylov and bogoliubov @xcite ; it is not limited to the large frequency or the large amplitude regions .
comparison of the predictions of the power - balance approach with results obtained by direct numerical integration of the model differential equation typically show agreement to at least three significant digits . in the high - frequency limit
they approach those predicted by the bessel - function expressions .
the starting point of our analysis is the usual shunted - junction model of the small josephson junction , subjected to a dc - bias current and an ac - driving current ; the corresponding differential equation , in normalized form , is @xmath0 where @xmath1 and @xmath2 are , respectively , the dc- and ac - driving currents . the corresponding unperturbed equation , _
i.e. _ , with @xmath3 , @xmath1 , and @xmath2 set to zero , has two types of solutions , _ viz .
_ , oscillatory and rotary . since we are interested in non - zero - voltage states of the junction we focus on the latter , which have the form @xmath4 where am ( ) is the jacobian elliptic amplitude function of modulus @xmath5 , and @xmath6 is an arbitrary constant . the instantaneous junction voltage in this case is thus given by @xmath7 we now return to the original model given by eq
( 1 ) .
an obvious physical requirement for the junction voltage to be a stationary , periodic function similar to that given by eq .
( 3 ) is that the average power dissipated , _
i.e. _ , the conductance times the mean - square voltage , be equal to the average power furnished by the drive currents , _
i.e. _ , the mean - value of the drive current times the voltage . assuming that eq .
( 2 ) can be used as a first approximation to the solution of eq .
( 1 ) in the presence of dissipation and drive , we can write , using results from , the time - average power dissipated as @xmath8 where @xmath9 and @xmath10 are , respectively , the complete elliptic integrals of first and second kinds . assuming first the presence of only a dc - bias current , _
i.e. _ , setting @xmath11 , we can write the time - average input power as @xmath12 equating eq . ( 4 ) to eq .
( 5 ) gives an expression for the mccumber branch of the current - voltage characteristic of the junction in the following parametric form @xmath13 @xmath14 in the zero - bias configuration , instead , _
i.e. _ , with @xmath15 and @xmath16 , we can write the input power as @xmath17 which , by expressing the jacobian dn ( ) function in terms of its fourier - series expansion , we can write as @xmath18\ ! > \ , , \
] ] where @xmath19 , in which @xmath20 is the complementary modulus .
from eq .
( 9 ) we see that @xmath21 only if @xmath22 for some integer @xmath23 , and , assuming eq . ( 10 ) to be satisfied , that @xmath24 can vary smoothly from zero to @xmath25 depending on the value of the phase - shift term @xmath6 ( we have @xmath26 when @xmath27 ) .
the threshold value , @xmath28 , of the drive amplitude is the minimum value for which it is possible to satisfy the equation @xmath29 ; this yields the value @xmath30 for @xmath31 , the constant @xmath6 adjusts itself according to the relation @xmath32=\frac{\gamma_{thr}}{\gamma } \ , .\ ] ] finally , in the general case in which both @xmath33 and @xmath34 , the dissipation defined by eq .
( 4 ) is balanced by both dc and ac inputs .
this permits , _
e.g. _ , calculating the minimum bias - current value to which a phase - locked step extends : assuming eq .
( 10 ) to be satisfied , we find @xmath35 which , for @xmath36 [ see eq . ( 11 ) ] , is a negative number .
the ` standard ' bessel function expression for the height in current of a phase - locked shapiro step is @xcite @xmath37 where @xmath38 ( ) is the bessel function of order @xmath39 .
thus , from eqs .
( 6 ) and ( 13 ) , the threshold value of the drive amplitude in this approach is @xmath40 at this point it must be mentioned that the integer @xmath39 in eq . (
14 ) is _ not _ the same as the integer @xmath23 in eq .
( 10 ) : in fact , the bessel function index @xmath39 refers to a _ super_harmonic number whereas the power balance index @xmath23 refers to a _ sub_harmonic number . to be perfectly clear on this point , the corresponding shapiro steps in the current - voltage characteristic of the junction occur at normalized voltages @xmath41 thus , the only place where the two expressions , eq . ( 11 ) and eq . ( 15 ) , can be compared directly is at the fundamental frequency , where @xmath42 .
our results are summarized in fig .
1a , b . in these figures ,
the solid curve is the power - balance threshold , calculated from eq . ( 11 ) , the dashed curve is the bessel - function threshold , calculated from eq . (
15 ) , and the diamonds are values obtained from the direct numerical integration of eq .
( 1 ) . except for the highest point in fig . 1b , where the numerical result begins to diverge slightly from the power - balance prediction
, these two typically agree to at least three significant digits .
a similar agreement between power - balance prediction and numerical experiment is obtained at the second subharmonic ( @xmath43 ) . for higher frequencies
than those shown in fig .
1a , the two predicted threshold values , eq .
( 11 ) and eq .
( 15 ) , converge to a common asymptotic limit . in order to calculate numerically the points indicated by diamonds in fig . 1
, we often found it necessary to ` tune ' fairly precisely the initial conditions used for eq .
( 1 ) .
this fact suggests that , at least in some cases , the basin of attraction of the phase - locked state is rather small , perhaps indeed vanishing , a fact that would have important implications for the stability of the locked state , _
e.g. _ , against thermal fluctuations .
the technique of cell mapping @xcite might be a useful tool for exploring this question .
braiman _ et al .
_ @xcite have extended the bessel function approach to subharmonic frequencies .
in fact , their eq .
( 2 ) has just the form of the fourier - series expansion of the jacobian am ( ) function of our eq .
( 2 ) .
this approach gives expressions for the step height in current corresponding to eq .
( 14 ) for superharmonic steps in terms of sums of products of bessel functions .
the approach , however , clearly becomes rather unwieldy if more than the first few terms in the fourier expansion are employed .
the power - balance approach can also be extended beyond the lowest level of approximation that we have employed here @xcite ; this would presumably permit a description of superharmonic steps , which presently is lacking from our analysis . however , superharmonic steps occur at progressively higher voltages , which implies progressively higher frequencies , where the simple bessel - function expression , eq .
( 15 ) , is known to give a reasonable description of the situation .
thus , the reward to be obtained here might not be worth the effort .
finally , we must offer one _ caveat _ : as is apparent from fig .
1a , the power - balance prediction becomes progressively better than the bessel - function prediction as the ac drive frequency is reduced below unity ( which is the plasma frequency , with our normalization ) .
this , however , is a region where , in addition to simple periodic solutions of eq .
( 1 ) , exemplified by the ansatz of eq . ( 2 ) , there are known to exist also complicated quasi - periodic and chaotic trajectories @xcite . consequently , whereas eq .
( 11 ) does give a good estimate of the threshold value of the drive amplitude _ if _ a simple step exists , it does _ not _ guarantee the existence of such a step .
we wish to thank roberto monaco for a critical reading of the manuscript and niels grnbech - jensen for illuminating discussions and technical assistance .
b.a.m .
thanks the physics department of the university of salerno for hospitality during the visit that originated this work . financial support from the ec under contract no .
sc1-ct91 - 0760 ( tsts ) of the `` science '' program and from the human capital and mobility " program , from murst ( italy ) , and from the progetto fianalizzato `` tecnologie superconduttive e criogeniche '' del cnr ( italy ) is gratefully acknowledged .
99 m. t. levinsen , r. y. chiao , m. j. feldman , and b. a. tucker , appl .
. lett .
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s. shapiro , phys .
rev .
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r. l. kautz , j. appl .
phys . * 52 * ( 1981 ) 3528 .
see , _
e.g. _ , a. scott , active and nonlinear propagation in electronics ( wiley - interscience , new york , 1970 ) , chap .
iii .
p. f. byrd and m. d. friedman , handbook of elliptic integrals for engineers and scientists , second edition ( springer - verlag , berlin , 1971 ) .
y. braiman , e. ben - jacob , and y. imry , in : squid 80 , h. d. halhbohm and h. lbbig , eds .
( de gruyter , berlin , 1980 ) , pp .
783795 .
m. p. soerensen , a. davidson , n. f. pedersen , and s. pagano , phys .
rev .
a * 38 * ( 1988 ) 5384 .
see , _
e.g. _ , n. minorsky , nonlinear oscillations ( van nostrand , new york , 1962 ) , chap . 15 .
r. l. kautz and r. monaco , j. appl .
phys .
* 57 * ( 1985 ) 875 .
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we examine the out - of - equilibrium dynamical evolution of density profiles of ultrasoft particles under time - varying external confining potentials in three spatial dimensions .
the theoretical formalism employed is the dynamical density functional theory ( ddft ) of marini bettolo marconi and tarazona [ j. chem .
phys . * 110 * , 8032 ( 1999 ) ] , supplied by an equilibrium excess free energy functional that is essentially exact .
we complement our theoretical analysis by carrying out extensive brownian dynamics simulations .
we find excellent agreement between theory and simulations for the whole time evolution of density profiles , demonstrating thereby the validity of the ddft when an accurate equilibrium free energy functional is employed .
density functional theory ( dft ) is a very powerful tool for the quantitative description of the equilibrium states of many - body systems under arbitrary external fields .
it rests on the exact statement that the helmholtz free energy of the system , @xmath0 $ ] , is a unique functional of the inhomogeneous one - particle density @xmath1 .
moreover , the equilibrium profile @xmath2 minimises @xmath0 $ ] under the constraint of fixed particle number @xmath3 @xcite .
the task is then to approximate the unknown functional @xmath0 $ ] from which all equilibrium properties of the system follow .
much more challenging is the problem of studying _ out - of equilibrium dynamics _ of many - body systems , for which analogous uniqueness and minimisation principles are lacking . in this paper , we present results based on a recently - proposed dynamical density functional theory ( ddft ) formalism and we demonstrate that the latter is capable of describing out - of - equilibrium diffusive processes in colloidal systems at the brownian time scale .
we are concerned with the dynamics of typical soft - matter systems , such as suspensions of mesoscopic spheres and polymer chains in a microscopic solvent @xcite .
the enormous difference in the masses of the suspended particles and the solvent molecules implies a corresponding separation in the relaxational time scales of the two . at times of the order of the fokker - planck scale , @xmath4 , the solvent coordinates are long relaxed to thermal equilibrium . on the brownian diffusive time scale , @xmath5
, the momentum coordinates of the solute particles relax to equilibrium with the heat bath of the solvent molecules and thus a statistical description involving only the positions of the colloids is feasible @xcite . in this regime ,
the evolution of the coordinates @xmath6 of the @xmath3 colloidal particles is described by the set of stochastic langevin equations : @xmath7 + { \bf w}_i(t ) .
\label{bd : eq}\ ] ] in eq .
( [ bd : eq ] ) above , @xmath8 is the pair ( effective ) interaction potential between the mesoscopic particles @xcite , @xmath9 is the externally acting potential and @xmath10 $ ] is a stochastic term representing the random collisions with the solvent molecules and having the properties : @xmath11 where the averages @xmath12 are over the gaussian noise distribution and @xmath13 , the cartesian components . the constants @xmath14 and @xmath15 stand for the mobility and diffusion coefficients of the particles , respectively , and the einstein relation gives @xmath16 . applying the rules of the it stochastic calculus , marconi and tarazona
@xcite recasted the above equations into the form @xmath17 \\ & + & \nabla_{\bf r}\left[\int{\rm d}^3 r'\ , \langle \hat\rho({\bf r } , t ) \hat\rho({\bf r ' } , t)\rangle \nabla_{\bf r } v({\bf r } - { \bf r'})\right ] .
\label{exact.eq}\end{aligned}\ ] ] here , @xmath18 is the usual one - particle density operator and @xmath19 is the noise - average of this quantity . up to this point , all is exact .
now , the following _ physical assumption _ ( a ) is introduced : as the system follows its relaxative dynamics , the instantaneous two - particle correlations can be approximated with those of a system in thermodynamic equilibrium with the same , _ static _
one - particle density @xmath1 as the noise - averaged dynamical one - particle density @xmath20 .
then , eq . ( [ exact.eq ] ) can be cast into a form involving exclusively the _ equilibrium _ density functional @xmath0 $ ] as @xcite @xmath21}{\delta \rho({\bf r } , t)}\right ] , \label{ddft.eq}\ ] ] with the free energy functional @xmath22 = k_bt\int { \rm d}^3 r\rho({\bf r})\left\ { \ln\left[\rho({\bf r})\lambda^3\right ] - 1 \right\ } + f_{\rm ex}[\rho ] + \int{\rm d}^3 r v_{\rm ext}({\bf r } , t)\,\rho({\bf r } ) .
\label{f : eq}\ ] ] the dynamical equation of motion ( [ ddft.eq ] ) was in fact first derived in a phenomenological way by dieterich , frisch and majhofer @xcite . in carrying out concrete calculations with the theory put forward above and in comparing them with brownian dynamics ( bd ) simulation results based on the microscopic equations of motion , eq .
( [ bd : eq ] ) , two sources of possible discrepancies exist : first , the fundamental assumption ( a ) and second the approximate nature of the equilibrium density functional @xmath23 $ ] of eq .
( [ f : eq ] ) .
in this work we focus our attention to _ ultrasoft particles _ for which a very accurate and simple functional @xmath0 $ ] is known , namely the _ mean - field _ or _ random - phase approximation _ ( rpa ) functional given by eq .
( [ mfa : eq ] ) below .
this guarantees that one can explore the accuracy of the fundamental assumption ( a ) under well - defined external conditions .
consider a one - component system of ultrasoft particles .
it has been demonstrated that for such systems the following rpa - functional is quasi - exact @xcite : @xmath24 = \frac{1}{2}\int\int{\rm d}^3r\,{\rm d}^3r ' v(|{\bf r } - { \bf r'}|)\rho({\bf r})\rho({\bf r ' } ) .
\label{mfa : eq}\ ] ] eq .
( [ ddft.eq ] ) takes now with the help of eqs .
( [ f : eq ] ) and ( [ mfa : eq ] ) the form @xmath25 given an initial density field @xmath26 and a prescribed external potential @xmath27 , eq .
( [ explicit.eq ] ) can be numerically solved to yield @xmath28 . in this work
we apply an ultrasoft gaussian pair potential between the interacting particles that has been shown to describe the effective interaction between the centres of mass of polymer chains in athermal solvents @xcite @xmath29.\end{aligned}\ ] ] we set @xmath30 providing the energy unit for the problem , whereas @xmath31 , which corresponds to the gyration radius of the polymers , will be the unit of length henceforth .
accordingly , the natural time scale of the problem , providing the unit of time in this work , is the brownian time scale @xmath32 .
eq .
( [ explicit.eq ] ) is solved using standard numerical techniques , and for a variety of time - dependent confining external potentials @xmath33 to be specified below .
brownian dynamics simulations of eq .
( [ bd : eq ] ) are also straightforward to carry out .
the langevin equations of motion including the external field are numerically solved using a finite time - step @xmath34 in all simulations , and the technique of ermak @xcite . in order to obtain the time - dependent density @xmath28 we perform a large number @xmath35 of independent runs , typically @xmath36 , and average the density profile over all configurations for a fixed time @xmath37 .
we focus to external fields that correspond to a sudden change , i.e. , @xmath38 .
these force the system to relax from the equilibrium density @xmath39 , compatible to the external potential @xmath40 , to the new equilibrium density @xmath41 , corresponding to the external potential @xmath42 .
important questions related to such processes are what is the typical relaxation time @xmath43 for such a procedure and how does the system cross over from one equilibrium density to the other .
we consider two kinds of confinements : spherical ones , @xmath44 , where @xmath45 , and planar ones between two walls perpendicular to the @xmath46-cartesian direction , @xmath47 . in these cases
we obtain @xmath48 and @xmath49 , correspondingly , and the solution of eq .
( [ explicit.eq ] ) is greatly simplified since the integrals take the form of one - dimensional convolutions that can be evaluated very rapidly by use of fast fourier transform techniques .
three different external confinements have been specifically investigated .
two spherical ones @xmath50 ; \label{v1:eq } \\ v_{\rm ext}^{(2)}(r , t ) = \phi_0\left[(r / r_{1})^{10}\theta(-t ) + ( r / r_{2})^{10}\theta(t)\right ] ; \label{v2:eq } \label{spherical : eq}\end{aligned}\ ] ] and one slab confinement @xmath51 .
\label{v3:eq}\end{aligned}\ ] ] the energy scale @xmath52 sets the strength of the confining potential and is fixed to @xmath53 for all three confinements .
the only difference between the external potential for times @xmath54 and for @xmath55 lies in the different length scales @xmath56 for eqs .
( [ v1:eq ] ) and ( [ v2:eq ] ) , and @xmath57 for eq .
( [ v3:eq ] ) . for each confinement
we consider two cases that give rise to two different dynamical processes : @xmath58 ( @xmath59 ) , enforcing an _ expansion _ of the system and @xmath60 ( @xmath61 ) , bringing about a _
compression _ of the same . for the spherical confinements
an additional parameter is the particle number @xmath62 which is a conserved quantity , as is clear from eq .
( [ ddft.eq ] ) that has the form of a continuity equation .
@xmath3 enters the formalism through the normalisation of the density field at @xmath63 . for both spherical confinements
the particle number is @xmath64 . in the slab confinement ,
eq .
( [ v3:eq ] ) , the conserved quantity is the density per unit area @xmath65 .
we choose @xmath66 . in all cases
examined , the typical relaxation time was found to be of order @xmath67 ; after typically @xmath68 , the system fully relaxes into the new equilibrium profile . in the spherical confining potential @xmath69 with ( a ) @xmath70 and @xmath71 and ( b ) @xmath72 and @xmath73 .
the shown profiles are for the times @xmath74 , @xmath75 and @xmath76 , all in @xmath67-units .
the last time is practically equivalent to @xmath77 , since the system there has fully relaxed into equilibrium . in all figures ,
red curves denote the initial and blue ones the final static profile.,title="fig:",width=264 ] in the spherical confining potential @xmath69 with ( a ) @xmath70 and @xmath71 and ( b ) @xmath72 and @xmath73 .
the shown profiles are for the times @xmath74 , @xmath75 and @xmath76 , all in @xmath67-units .
the last time is practically equivalent to @xmath77 , since the system there has fully relaxed into equilibrium . in all figures ,
red curves denote the initial and blue ones the final static profile.,title="fig:",width=264 ] in fig . [
harmon : fig ] we show the results for the harmonic confining potential of eq .
( [ v1:eq ] ) .
it can be seen that the theory reproduces the time evolution of the density profile , both for the expansion [ fig .
[ harmon : fig](a ) ] and the compression [ fig .
[ harmon : fig](b ) ] processes .
an asymmetry in the two processes can be already discerned : the compression is not the ` time reversed ' of the expansion and this effect will be much stronger in the examples to follow .
though the profiles of the system are considerably different from those of an ideal gas , i.e. , effects of the interparticle interaction are present , the confining potential is smooth enough , so that the profiles are devoid of pronounced correlation peaks . in the spherical confining potential @xmath78 with ( a ) @xmath70 and @xmath79 and ( b ) @xmath80 and @xmath73 .
the shown profiles are for the times @xmath74 , @xmath81 and @xmath82 ( in units of @xmath67).,title="fig:",width=340 ] in the spherical confining potential @xmath78 with ( a ) @xmath70 and @xmath79 and ( b ) @xmath80 and @xmath73 .
the shown profiles are for the times @xmath74 , @xmath81 and @xmath82 ( in units of @xmath67).,title="fig:",width=340 ] the situation is different for the external potential of eq .
( [ v2:eq ] ) . here
, the power - law dependence is much more steep , so that the gaussian fluid develops correlation peaks close to the ` walls ' of the confining field .
the dynamical development of the profiles for the expansion and compression processes are shown in fig .
[ r10:fig ] . here , the asymmetry between the expansion and the compression processes is evident . in the former case , seen in fig .
[ r10:fig](a ) , the expansion of the confining potential leaves behind a density profile that has very strong density gradients close to the boundary of the initial confinement . since the latter ceases to act at @xmath83 , this leaves at @xmath84 instantaneously a region @xmath85 that is devoid of particles but in which the new external potential is essentially zero .
this leads to a collective diffusion of the particles towards the boundaries set by the new potential .
correspondingly , the high density peaks decrease rapidly and leak outward . in the inner region , @xmath86 of the profile ,
the dynamics is much slower and the relaxation to the final plateau there takes place at the end of the process , causing thereby the final development of the new , weaker correlation peaks close to the location of the boundary , @xmath87 . the compression process , depicted in fig .
[ r10:fig](b ) runs very differently .
there , the initial ` closing ' of the potential from @xmath88 to @xmath89 leaves at @xmath84 a region of high density at @xmath85 that now finds itself within a strongly repulsive external field .
there is an extremely rapid shrinking there , accompanied by the development of very high correlation peaks that actually ` overshoot ' in height with respect to the final equilibrium profile .
initially , the region in the centre of the sphere remains unaffected and only as the high peaks start diffusing does material flow toward the centre and at the latest stage of the dynamics the profile at @xmath90 reaches its new equilibrium value . , defined in eq .
( [ moment ] ) plotted against the time @xmath37 for the spherical confinement @xmath78 .
circles correspond to radii @xmath70 and @xmath79 ( expansion ) and squares show the resulting curve for the inverse process , @xmath80 and @xmath73 ( compression ) .
the lines are the analytical fits shown in the text .
solid line : eq .
( [ m2plus : eq ] ) ; long - dashed line : eq .
( [ m2minus : eq ] ) .
the arrows mark the characteristic time scales defined in these two equations.,width=302 ] in order to quantify better this asymmetry and also extract characteristic time scales for the two dynamical processes , we consider the second moment of the density , @xmath91 , defined through @xmath92 the quantity @xmath91 is a quantitative measure of the spread of @xmath93 around the centre of the external field and its time evolution is shown in fig .
[ m2:fig ] .
let the superscript ` @xmath94 ' denote the expansion and the superscript ` @xmath95 ' the compression process .
obviously , it holds @xmath96 .
we notice that for both processes @xmath91 is a monotonic function of @xmath37 but some interesting differences arise when one fits the two curves by analytic functions , shown as lines in fig .
[ m2:fig ] .
the expansion can be very accurately described by a single exponential : @xmath97 [ 1-\exp(-t/\tau^{+ } ) ] , \label{m2plus : eq}\ ] ] with the characteristic time scale @xmath98 . however , a double - exponential fit is necessary to parameterise the compression process , namely @xmath99\exp(-t/\tau_2^{- } ) , \label{m2minus : eq}\ ] ] with the fit parameter @xmath100 and the _ two _ characteristic time scales @xmath101 and @xmath102 . since @xmath103
, it follows that the compression process is at any rate faster than the expansion one .
the occurrence of the two distinct time scales @xmath104 in the compression requires some explanation .
the fast process that takes place at times @xmath105 corresponds to the abrupt shrinking of the profile at the wings of the distribution and is caused exclusively by the ` closing ' of the external field .
this is the same mechanism that brings about the overshooting of the density peaks .
once this is over , diffusion within the now already confined system takes place and the second characteristic time scale , @xmath106 , is solely determined by the interaction potential @xmath107 and the average particle density . in the expansion process , the first mechanism is absent thus a single time scale , @xmath108 , shows up , which is of intrinsic origin exclusively . since stronger density gradients occur during the compression than during the expansion process , even the larger of the two time scales of the compression , @xmath106 , is smaller than @xmath108
. the denser the system is , the faster the collective diffusion towards equilibrium . in a slab
confining potential @xmath109 with ( a ) @xmath110 and @xmath111 and ( b ) @xmath112 and @xmath113 .
the shown profiles are for the times @xmath114 and @xmath115 ( in units of @xmath67).,title="fig:",width=340 ] in a slab confining potential @xmath109 with ( a ) @xmath110 and @xmath111 and ( b ) @xmath112 and @xmath113 .
the shown profiles are for the times @xmath114 and @xmath115 ( in units of @xmath67).,title="fig:",width=340 ] finally , we turn our attention to the slab confinement .
the results from theory and simulation are shown in fig .
[ slab : fig ] .
once more it can be seen that the ddft offers an excellent description of the dynamics of the system .
the same asymmetry between expansion and compression that was seen in the spherical confinement also shows up for the case of the slab , including the overshooting of the peaks during the compression process .
in addition , the density profiles develop during their evolution secondary oscillations that are also very well reproduced by the theory . in summarising
, we have demonstrated that the dynamical density functional theory of marini bettolo marconi and tarazona @xcite , when supplemented by an accurate equilibrium density functional , can provide an excellent description of out - of - equilibrium dynamics of colloidal systems at the brownian time scale .
the accuracy of the ddft formalism has already been successfully tested for the system of one - dimensional hard rods @xcite , for which the exact density functional @xmath0 $ ] is known . to the best of our knowledge ,
this is the first study of the validity of ddft in three dimensions . as the phenomenology in 3d is much richer than in 1d , including the possibility of phase transitions , many intersecting ways for future applications open up .
discussions with andrew archer , bob evans , wolfgang dieterich and pedro tarazona are gratefully acknowledged .
this work has been supported by the deutsche forschungsgemeinschaft through the sfb tr6 .
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fig . 1 . : : flow chart of the present procedure for gamow factor and/or coulomb wave function , which includes momentum resolution by generating the gaussian random numbers . fig . 2 . : : results of @xmath64 fit for 8 mkm target of @xmath66 reaction with @xmath31 mev / c by eq .
( [ eqn : a6 ] ) .
3 . : : results of the @xmath64 fit for ref .
@xcite of @xmath66 reaction with @xmath31 mev / c by eq .
( [ eqn : a9 ] ) .
4 . : : results of the @xmath64 fits for @xmath67 reactions with @xmath31 mev / c by eq .
( [ eqn : a9 ] ) : @xmath68 8 mkm target ; @xmath69 1.4 mkm target . fig . 5 .
: : physical picture of the obtained source size based on the assumed resonance effects .
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we propose a new method for the coulomb wave function correction including the momentum resolution for charged hadron pairs and apply it to the precise data on @xmath0 correlations obtained in @xmath1 reaction at 70 gev / c .
it is found that interaction regions of this reaction ( assuming gaussian source function ) are @xmath2 and @xmath3 fm for the thicknesses of the target @xmath4 and @xmath5 microns , respectively .
the physical picture of the source size obtained in this way is discussed .
= -1 cm = 0 cm = 0 cm = 23.7 cm = 16 cm * 1 .
introduction.*recently we have obtained the new formulae for the coulomb wave function correction for charged hadron pairs @xcite . in particular
we have applied them ( in @xcite ) to data on @xmath0 correlation obtained in @xmath6 reaction at 70 gev / c @xcite ( which were originally corrected by usual gamow factor only ) . however , as it was pointed out to us by one of the author of @xcite , we did not take into account their published finite momentum resolution @xcite .
in fact , our formulae can not be applied directly to experimental data in which such momentum resolution is accounted for .
therefore in the present letter we would like to extend our method for the coulomb wave correction provided in @xcite to include also the momentum resolution case and to re - analyse data of @xcite and also to analyse the new , preliminary data of @xcite obtained with two kinds of thickness of ta target : 8 microns ( 8 mkm ) and 1.4 microns ( 1.4 mkm ) .
+ in the next paragraph we first reconstruct ( for the sake of completeness ) the analysis performed in ref .
@xcite and then , in the third paragraph , we propose our new method for the coulomb wave correction including this time also the momentum resolution .
the final part contains our concluding remarks .
+ * 2 . reconstruction of the analysis of @xmath0 correlation data performed by using the gamow factor with momentum resolution.*it has been stressed in @xcite that relative momenta of @xmath0 pairs observed by them have some finite resolutions .
the averaged correlation function , defined as @xmath7 depends therefore on this momentum resolution , where @xmath8 stands for non - coulomb correlation factor . to account for it
the following random number method has been proposed in ref .
@xcite in order to obtain the corresponding averaged quantities in analysis of the correlation data .
+ first of all , the relative momentum of the measured pair , @xmath9 , is decomposed into its longitudinal and transverse components : @xmath10 and @xmath11 , respectively , by making use of the uniform random number @xmath12 ( it is worthwhile to notice at this point that the transverse components @xmath11 in data of @xcite are smaller than @xmath13 mev / c ) .
one uses the following scheme here : @xmath14 at the next step , the gaussian random numbers for @xmath10 and @xmath11 are generated in the following way : @xmath15 in the above equations @xmath16 stands for the standard gaussian random number ] ] whereas @xmath17 and @xmath18 are longitudinal and transverse setup resolutions for the corresponding components , which are equal to ( values used in @xcite ) : @xmath19 mev / c ; @xmath20 mev / c ( for target of the thickness @xmath4 mkm ) and @xmath21 mev / c ( for the 1.4 mkm target ) . + finally , using the randomized number @xmath22 one calculates the corresponding randomized gamow factor correction : @xmath23 where @xmath24 . the full flow chart for this procedure is shown in fig .
1 . calculating now the average value of @xmath25 in 100 k events one can estimate the gamow factor with this finite momentum resolution , @xmath26 where @xmath27 is a free parameter .
it is understood ( or , rather , implicitly assumed ) that essentially all unlike sign pions one deals with here originate from decays of long lived particles like @xmath28 , @xmath29 , @xmath30 , and so on .
figure 2 shows the results of analysis of new data ( for 8 mkm target ) @xcite for region @xmath31 mev / c using this method .
+ * 3 .
proposal of the new method.*we would like to propose now a new method of coulomb wave function correction with a source function @xmath32 instead of the gamow factor , in order to analyse the same data . as usual
we decompose the wave function of unlike charged bosons with momenta @xmath33 and @xmath34 into the wave function of the center - of - mass system ( c.m . ) with total momentum @xmath35 and the inner wave function with relative momentum @xmath36 .
this allows us to express coulomb wave function @xmath37 in terms of the confluent hypergeometric function @xmath38 @xcite : @xmath39 where @xmath40 and the parameter @xmath41 .
assuming factorization in the source functions , @xmath42 ( here @xmath43 ) , we obtain the expression for coulomb correction for the system of @xmath0 pairs identical ( modulo the sign ) as in @xcite : @xmath44\ : , \label{eqn : a8}\end{aligned}\ ] ] where @xmath45 for the specific choice of gaussian source distribution , @xmath46 , we have @xmath47 whereas exponential source function , @xmath48 , leads to @xmath49 using now the same method of gaussian random numbers as in the previous paragraph , we can analyse the old and the new data on @xmath0 pairs @xcite using the following formula : @xmath50 figs .
3 and 4 show results of our analysis of the old and the new data , respectively .
table [ tbl : a1 ] show our results obtained using eq .
( [ eqn : a8 ] ) applied to old and new data with @xmath51mev / c . + * 4 . concluding remarks.*we have proposed the new method for the coulomb wave function correction with momentum resolution and applied it to the analysis of the precise data provided by @xcite .
authors of ref .
@xcite have analysed their @xmath0 correlation data using gamow factor for coulomb corrections together with the random numbers method to account for final momentum resolution .
we have repeated this analysis replacing gamow factor by the coulomb wave function but following the same method for correction for the momentum resolution effect ( cf .
eq .
( [ eqn : a8 ] ) ) . as a result
we have obtained the following ranges of interaction for the gaussian source function : @xmath52 to get a correct physical picture of the source size , we should calculate the root mean squared size , which is equal to : @xmath53 the present study of @xmath0 pair correlations has shown therefore that one can estimate the interaction region even from the @xmath0 correlation data .
it can be compared with the size of the ta nucleus , which is given by : @xmath54 as one can see , @xmath55 is significantly bigger than @xmath56 .
we attribute this difference to a physical picture shown in fig . 5 , i.e. , to the fact that unlike - sign pions are mostly ( if not totally ) emerging from the long - lived resonances shown there .
( in a future one should consider also a possibility of more direct estimation of the parameter @xmath27 and its role in determining the source size parameter correlation data , a similar function @xmath57 is introduced : @xmath58\:,\ ] ] where @xmath59 and @xmath60 are the degree of coherence and an exchange function due to the bose - einstein effect .
@xmath57 is attributed the resonances effect ; @xmath61 depends on the monte carlo programs @xcite . ] ) . + for completeness we have also tried to analyse the same data using exponential source function instead of gaussian .
as is shown in table [ tbl : a2 ] this leads to errors on @xmath62 of the order of @xmath63% , i.e. , with this type of source function we can not estimate the source size ( therefore it has to be discarded ) . +
* acknowledgements:*authors are grateful to dr .
l. g. afanasev for his kind correspondences and for providing us with the new data on @xmath0 correlation prior to publication .
this work is partially supported by the grant - in - aid for scientific research from the ministry of education , science and culture of japan ( # 06640383 ) , and the japan society of promotion of science ( jsps ) .
one of authors ( i. a. ) is also partially supported by russian fund of fundamental research ( grant 96 - 02 - 16347a ) . 99 m. biyajima , t. mizoguchi , t. osada and g. wilk , _ phys .
lett . _
* b353 * ( 1995 ) 340 .
m. biyajima , t. mizoguchi , t. osada and g. wilk , _ phys .
lett . _ * b366 * ( 1996 ) 394 .
l. g. afanasev et al .
, _ sov .
j. nucl .
phys . _ * 52 * ( 1990 ) 666 ( russian ed . : _ jad .
fiz . _
* 52 * ( 1990 ) 1046 ) .
l. g. afanasev , private communications ( oct . , 1995 ) .
l. g. afanasev , private communications ( dec . , 1995 ) ; l. g. afanasev et al .
, in preparation .
m. g. bowler , _ phys .
lett . _
* b270 * ( 1991 ) 69 .
l. i. schiff , _ quantum mechanics _ , 2nd ed .
( mcgraw - hill , new york , 1955 ) , p. 117 .
e. a. de wolf , proceeding of xxivth int .
sym . on multiparticle dynamics ( vietri sul mare , sept .
1994 ) .
[ tbl : a1 ] .results of the @xmath64 fits of @xmath65 for gaussian source by eqs .
( [ eqn : a6 ] ) and ( [ eqn : a9 ] ) . [ cols="^,^,^,^,^",options="header " , ]
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99 heyman p and garcia - molina h 2006 collaborative creation of communal hierarchical taxonomies in social tagging systems _ technical report infolab 2006 - 10 .
department of computer science , stanford university , stanford , ca , usa _
schmitz c , grahl m , hotho a , stumme g , cattuto c , baldassarri a , loreto v and servedio vdp 2007 network properties of folksonomies _ proceedings of the sixteenth international world wide web conference ( www2007 ) _
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we analyze _ citeulike _ , an online collaborative tagging system where users bookmark and annotate scientific papers . such a system can be naturally represented as a tripartite graph whose nodes represent papers , users and tags connected by individual tag assignments .
the semantics of tags is studied here , in order to uncover the hidden relationships between tags .
we find that the clustering coefficient reflects the semantical patterns among tags , providing useful ideas for the designing of more efficient methods of data classification and spam detection .. the recent development of the world wide web is characterized by a growing number of online social communities . in many such cases ,
individuals provide bits of information - about either their tastes , opinions or interests - and software applications gather and organize them into a database , allowing the browsing of the whole information collected so .
a class of such collaborative systems focusses on collecting users online bookmarks with either a general approach or a more specialized one . in particular , some websites have been recently born to store user generated scientific bibliographies . in these systems ,
the elementary contribution , the `` post '' , is made of three ingredients : a user , an article and an annotation of it by a number of tags chosen by users . in exchange for this voluntary contribution
, a user can browse others bibliographies and annotations .
tags are an alternative classification method with respect to traditional taxonomies , where items belong to `` taxa '' represented as a tree like set of categories : here , each category contains in turn a number of more specialized sub categories , and so on until the desired resolution of classification is been reached . instead , in tagging systems items are tagged by users characterized by diverse tagging strategies depending on a number of individual variables .
the set of tag resource relations in such a community is called a `` folksonomy '' .
such communities are now extremely popular , storing hundreds of thousands posts and more .
the tagging system we analyze here , _ citeulike _
@xcite , has been built , at the time of our survey , by ca .
180000 references annotated by ca .
48000 tags supplied by ca .
6000 users .
our dataset includes about 550000 `` tag assignments '' : each assignment is a t - uple ( user , resource , tag ) .
the sequence of chronologically ordered tags , in particular , can be interpreted as a stream of words , to which one can applies the traditional statistical text analysis to uncover how human behavior affects it .
the statistical analysis of word occurrences in a written text has shown that word frequencies are power law distributed according to the zipf s law , according to which a large number of words appears in a text only a few times , while a few words occur orders of magnitude more often @xcite .
such feature has been modeled by many models based on the preferential attachment principle , that is , the assumption that authors employ already used words with a probability proportional to the current word frequency .
moreover , it has been observed that the rate of new words decrease with the text
length @xcite , that is , the number of distinct word @xmath0 in a text of length @xmath1 scale as @xmath2 with @xmath3 .
however , models in literature assume that new words are introduced at a constant growth rate , so that their total number , i.e. the vocabulary , is a linear constant of the total number of words ( both new or repeated ones ) used so far @xcite . yet , to discover the semantical properties of _ citeulike _ , one rather represents it by means of the network formalism , which proved fruitful in the analysis of many natural and social phenomena involving unsupervised interacting units : in a network perspective , elementary interacting agents or objects are represented by nodes , interactions by edges connecting them
. the widespread success of such approach has been triggered by the discovery that many networks instances one encounters in reality share common statistical properties with no external tuning .
for example , the degree @xmath4 , that is , the number of edges pointing to a node , follows in many cases a broad distribution @xmath5 with long tail decaying algebraically as @xmath6 with @xmath7 .
if edges have varying intensities , each of them is attached a weight @xmath8 representing its intensity ; accordingly , a node is characterized by its @xmath9 , equal to the sum of the weights of edges pointing to it .
the distribution of @xmath10 , too , is power
law distributed in many real weighted network instances .
furthermore many such networks exhibit a strong transitivity , i.e. with high probability , the neighbors of a node are themselves connected by an edge , with respect to purely random realization of a network with equal number of nodes and links .
networks sharing the above properties are currently named `` complex networks '' @xcite .
the network approach has also been recently adopted to analyze the semantical structure of tagging systems @xcite .
tags can be represented by networks in different ways , in order to study how the behavior of users maps into the dynamical or topological features .
a more recent stream of research deals with the organization of tags , which are implicitly linked by hierarchical and logical associations emerging despite the diversity of users as their number is large enough .
the underlying semantical organization of tags reveals the dominant trends within a tagging community and allows to improve its navigability .
recently , algorithms have been introduced in order to infer a taxonomy of tags from a folksonomy @xcite .
the statistical properties we observed in the _ citeulike _ data are consistent with the findings obtained in similar surveys , confirming that tags in collaborative systems form complex networks indeed .
moreover , we have investigated how the underlying semantics of tags reflects on the topology of the network . as a matter of facts ,
tags provided by users come with no explicit hierarchy beside the chronological ordering , leading authors to analyze the stream of tags as a text like sequence of words .
interestingly , the time ordered sequence of tags displays statistical properties already observed in written texts , such as the fat tails in the word frequency distributions or the sub - linear vocabulary growth .
our analysis confirms the sub - linear vocabulary growth observed in written texts .
the number of distinct tags @xmath11 introduced by users after @xmath12 assignments grows approximately as @xmath13 , as shown in figure [ sublinear ] although the pace is slightly smaller than in other collaborative tagging systems already surveyed @xcite .
the frequency of tags , too , reported in [ wordsfreq ] , reminds that of words observed in written texts , algebraically decaying according to the zipf s law @xcite .
however , the sole frequency of tags as a function of time does not convey much information about the semantics , although it reflects the different centrality of associated concepts in the underlying knowledge organization . to investigate tag pair relations
, one has to represent the unit elements of a tagging system as nodes of a network .
the dataset we focus on can be naturally represented as a tri partite network , where each node represents either a user @xmath14 , a resource @xmath15 or a tag @xmath12 ; if a tag assignment @xmath16 exists , an edge is drawn from @xmath14 to @xmath15 , and from @xmath15 to @xmath12 . since in a single user s
post a resource can be tagged more than once , one post can correspond to multiple tag assignments @xcite .
although efficient algorithm have been developed to analyzed such tri partite network @xcite , the heterogeneity of nodes discourages in general the application of traditional network methods , mainly conceived to deal with network connections representing peer - to - peer relationships .
thus , to study how tags are organized we chose to project the tri partite networks on the tag space . as a result
, the tag co occurrence network we study is composed by nodes representing tags only , between which an undirected edge of weight @xmath8 is drawn if @xmath8 distinct resources are labeled by both tags .
the resulting network displays some of the typical features of weighted scale free networks .
we have measured the distribution of the sum @xmath17 of the weights of edges pointing to a given node , or the _ strength _ of the node : such distribution @xmath18 , plotted in figure [ strength - dist ] , exhibits a clear power
law decay @xmath19 , with @xmath20 for large values of @xmath17 .
interestingly , the heterogeneity of the observed nodes weights does not necessarily reflects the centrality of corresponding concepts , that are supposed to be assigned together with a wider range of more specialized concepts , in the underlying hierarchy of tags .
as it has been already shown @xcite , reshuffling the tag assignment in order to destroy the logical association among words does not change dramatically the shape of @xmath18 , which proves that the @xmath21 heterogeneity is more a consequence of frequency distribution broadness than of the varying roles of concepts in the semantical organization of the whole vocabulary .
nevertheless , the tag co occurrence network unveils some semantical feature of the underlying ontology if , instead of focussing on the properties of single nodes , one turns to the inspection of quantities involving its environment .
an example of such is represented by the analysis of the neighbor average degree @xmath22 of nodes with degree @xmath4 , where the degree is the number of incoming edges of a node . in our study
we examined instead the clustering properties of the tag co occurrence network through the clustering coefficient .
such coefficient @xmath23 counts the average density of triangles involving nodes with degree @xmath4 or , in other words , the probability that the nearest neighbors of a node with degree @xmath4 are in turn connected one to each other .
this reads @xmath24 where @xmath25 is 1 if a link exists between @xmath26 and @xmath27 and 0 otherwise , and @xmath28 is the frequency of nodes with degree @xmath4 .
this quantity has been found to characterize most complex networks found in nature and society , where it takes substantially larger values with respect to a purely random networks @xcite .
the properties of the clustering coefficient are often associated to the hierarchical organization of nodes @xcite .
indeed , the clustering coefficient appears to encode a signature of semantical relations between words . as represented in figure [ clustering - real ]
, the clustering coefficient @xmath23 in _ citeulike _ decays algebraically for large values of the degree @xmath4 , according to @xmath29 .
however , the clustering value displays an apparent fluctuation at @xmath30 . by inspecting the nodes corresponding to such value
, one discovers that the sharp rise taking place at al @xmath30 corresponds to a non existing resource labelled by @xmath31 distinct uncorrelated randomly chosen tags , which mimics the a spam contribution to the collaborative systems .
one is led thus to conjecture that the overall semantical organization of concept represented by tags is encoded in a characteristic behavior of the clustering coefficient @xmath23 , so that tags assigned in a semantically inconsistent way fall far away from this behavior . to verify such conjecture ,
we have performed the same statistical analysis after removing from the data set the tag assignments related to the spam like page . as shown in figure
[ clustering - real ] , after the removal the clustering coefficient follows a more regular behavior , confirming that the strong fluctuation observed above was due indeed to the presence of a single meaningless set of assignments involving a single resource .
tags assigned only to the spam resource form a complete co occurrence network , so that their clustering coefficient is equal to @xmath32 .
thus , the behavior of clustering coefficient of the tag co occurrence networks can be used as a test for models representing the tag semantical organization or , equivalently , how users choose tags when annotating a resource . as noted in literature @xcite
, users typically use tags hierarchically , labelling a resource by tags related to the same topics but with different generality , adding more specialized tags as the number of collected resources grows . on a very basic level , we have tested how such hierarchical tagging , affects the topology of the tag co occurrence network by a simple toy model defined in the following .
let us assume that tags are organized on a taxonomy , that is , a tree like structure stemming from a seed node , where each node corresponds to a tag and is an offspring of another tag belonging to the same branch of knowledge with higher generality . at discrete time steps
, a new post is added to the system , with a new resource and 2 tags .
the first tag can be a new one , with probability @xmath33 : in such case , the new tag is an offspring of a tag randomly chosen among the already employed ones .
otherwise , the first tag is chosen at random among the already employed ones .
the second tag is either chosen at random from within the whole set of used tags or , with probability @xmath34 , it is chosen according to hierarchy : in such case , the second tag is drawn at random among the nodes that lie on the shortest path length from the first tag to the seed node on the tree like taxonomy .
the tag co occurrence network resulting from the above algorithm share some features of the _ citeulike _ one , if we assume a time dependent @xmath33 which reproduces the sub - linear vocabulary growth observed in reality , and by a suitable choice of the parameter @xmath34 , which mimics the relevance of hierarchy in tagging activity . to reproduce the growth rule ,
we have set @xmath35 and imposed that @xmath36 and @xmath37 where the number of resources @xmath38 , the number of tag assignments @xmath39 , the number of tags @xmath40 and @xmath41 are set to the same values they take in _
citeulike_. as a result , this yields @xmath42 and @xmath43
. as shown in figure [ strength - dist ] , the strength distribution @xmath18 of tags is a scale free one with a good agreement with reality in the decaying exponent for large values of @xmath17 if one sets @xmath44 . for such choice of the parameter , the clustering coefficient reproduces qualitatively the algebraic decay observed in _
citeulike _ , as shown in figure [ summarymodel ] , although the absolute value differs of orders of magnitude .
we have found , thus , a simple model that captures the complex features of a tag co occurrence network issued from the dataset describing an online collaborative tagging system , _
citeulike_. in particular , by assuming that users label resources by hierarchically associated tags , the probability distribution of nodes strength is reproduced for a suitable choice of the parameters .
moreover , the model reproduces qualitatively the decaying asymptotic behavior of the clustering coefficient @xmath23 .
such quantity encodes a signature of the semantical organization of concepts represented by tags , so that malicious or meaningless tag assignments can be detected by inspecting the perturbation to the clustering coefficient they generate . establishing a relationship between clustering and semantics may suggest tools and algorithms for technological tasks such as automatic categorization of resources , recommendation and spam detection techniques .
the authors acknowledge useful discussions with francesca colaiori , stefano leonardi , ciro cattuto , vito d.p .
servedio and andrea baldassarri .
the authors acknowledge the european project delis for support and r. cameron for providing the data . as a function of time @xmath12 , where time is measure in chronologically ordered tags assignments ( plus symbols ) .
solid line represents @xmath45 for a comparison.,scaledwidth=80.0% ] of tag frequencies @xmath46 ( plus symbols ) .
solid line represents @xmath47 for a comparison.,scaledwidth=80.0% ] of the node strengths @xmath17 in the tag co occurrence network .
the best fitting power law exponent , represented by the solid curve , yields @xmath48 .
, scaledwidth=80.0% ] of the tag co occurrence network as a function of the nodes degree @xmath4 before ( circles ) and after ( plus symbols ) the removal of a spam post from the dataset .
the solid line represents a decay @xmath49 and the dashed vertical ruler is set at @xmath30.,scaledwidth=80.0% ] ( plus symbols ) ; the solid line represents the decay @xmath50 for a comparison with real data .
inset : the clustering coefficient in the co occurrence network derived from the model with @xmath51 ( plus symbols ) .
the solid line represents the decay @xmath49 for a comparison with real data.,scaledwidth=80.0% ]
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fig . 1 . : : \(a ) analyses of data for s + pb @xmath45 reaction by eq .
( [ eq : ratio ] ) .
( b ) the same but with coulomb corrections switched off for @xmath46 limit .
( results for @xmath47 pair production looks similar with only change being in the value of @xmath48 which depend on the value of radius parameter @xmath38 . ) fig . 2 . : : flow chart for our procedure of `` seamless fitting ( sf ) '' . fig . 3 . : : results of sf for bec for kaons produced in s + pb collisions ; ( a ) for @xmath47 pairs ; ( b ) for @xmath49 pairs . fig .
4 . : : results of sf for bec for kaons produced in @xmath1 + pb @xmath45 reaction .
5 . : : results of sf for bec for pions produced in s + pb @xmath50 reaction .
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we applied an improved coulomb correction method developed by us recently to data on identical @xmath0-pairs production in s + pb and @xmath1 + pb reactions at 200 gev / c obtained by na44 collaboration . to analyse
the whole range of the momentum transfers measured the method of `` seamless fitting '' has been proposed and used together with the asymptotic expansion formula for the coulomb wave function .
we found that such coulomb corrections lead sometimes to different than previously reported ( by na44 collaboration ) interaction region and strongly influence the long range correlations .
+ preprint * dpsu-95 - 4 * ( july , 1995 ) = -45pt = 0 cm = 0 cm = 23.7 cm = 16 cm * introduction : * recently na44 collaboration has reported their data on the bose - einstein correlations ( bec ) of @xmath2 pairs produced in s + pb and @xmath1 + pb reactions at 200 gev / c @xcite . in our previous work
@xcite we have analysed these data by making use of the various source functions with the long range correlation ( without , however , invoking any sort of coulomb corrections ) .
the data @xcite have been corrected for coulomb interactions by applying only the gamow factor . as was pointed out some time ago by bowler
this is , however , not sufficient @xcite . in our recent works @xcite the still improved method of coulomb corrections was presented but not yet applied to any concrete data set . in the present letter , we apply it therefore to the analysis of data for @xmath2 pairs production in s + pb and @xmath1 + pb reactions mentioned above ( in the whole measured momentum transfer region ) and compare our results with those obtained before in @xcite . to be able to analyse the whole range of measured momentum transfer @xmath3 and to avoid wild oscillations developing at large @xmath3 s ( cf .
fig .
1a ) we have to use the the asymptotic expansion of the coulomb wave function together with the procedure of `` seamless fitting '' ( sf ) explained below ( cf . figs .
1b and 2 ) . +
* theoretical formula of bec with coulomb wave function : * to write down an amplitude @xmath4 satisfying bose - einstein statistics it is convenient to decompose the wave function of identical ( charged in our case ) bosons with momenta @xmath5 and @xmath6 into the wave function of the center - of - mass system ( c.m . ) with total momentum @xmath7 and the inner wave function with relative momentum @xmath8 .
it allows us to express @xmath4 in terms of the confluent hypergeometric function @xmath9 @xcite : @xmath10\ : , \label{eq : a}\\ \psi({\mbox{\bf q}},{\mbox{\bf r } } ) & = & \gamma(1+i\eta)e^{-\pi \eta/2 } e^{i{\mbox{\scriptsize \bf q}}\cdot{\mbox{\scriptsize \bf r } } } \phi(-i\eta;1;iqr(1 - \cos \theta))\:,\nonumber\\ \psi_s({\mbox{\bf q}},{\mbox{\bf r } } ) & = & \gamma(1+i\eta)e^{-\pi \eta/2 } e^{-i{\mbox{\scriptsize \bf q}}\cdot{\mbox{\scriptsize \bf r } } } \phi(-i\eta;1;iqr(1 + \cos \theta ) ) \:,\nonumber\end{aligned}\ ] ] where @xmath11 and the parameter @xmath12 .
assuming factorization in the source functions , @xmath13 ( here @xmath14 ) , one obtains the following expression for theoretical bec formula @xcite including the improved coulomb correction @xcite : @xmath15 \label{eq : bec}\\ & = & ( 1 + \delta_{{\mbox{\scriptsize 1c } } } ) + ( \delta_{{\mbox{\scriptsize ec } } } + e_{{\mbox{\scriptsize 2b}}}),\label{eq : result}\end{aligned}\ ] ] where @xmath16 denotes gamow factor and the first and the second parentheses in eq .
( [ eq : result ] ) correspond to the first and the second terms in eq .
( [ eq : bec ] ) and @xmath17 see @xcite .
] and @xmath18 to analyse data corrected only by the gamow factor using our formulae we should use the following ratio : @xmath19 ( 1 + \gamma q).\hspace{1 cm } \label{eq : ratio}\end{aligned}\ ] ] it should be noted that the normalization and an effective degree of coherence , i.e. , the denominator of the ratio @xmath20 , are related to each other .
notice also that other parameters like the additional normalization factor @xmath21 , the long range correlation @xmath22 and @xmath23 are introduced by hand .
+ * source function : * to obtain an explicit expression , we have to decide on some form of the source function . in the present letter , we shall use the gaussian source distribution , @xmath24 . for this type of source function
we have the following formulae for the elements of eqs .
( [ eq : result ] ) and ( [ eq : ratio ] ) @xcite : @xmath25
to analyze data corrected by the coulomb wave function as was done in @xcite , we should modify the formula ( [ eq : ratio ] ) replacing it by the following one : @xmath26 ( 1 + \gamma q ) .
\label{eq : a21}\end{aligned}\ ] ] for the sake of reference , we write down here also the conventional formula ( i.e. , a kind of standard formula without corrections due to the final state interactions ) : @xmath27 ( 1 + \gamma q ) .
\label{eq : a22}\ ] ] * asymptotic expansion of the coulomb wave function : * first of all , it should be noted that the expansion in eq .
( [ eq : result ] ) has to be the limited to @xmath28 only due to mathematical properties of the confluent hypergeometric function used @xcite .
this can be seen as wild oscillation developing in fig . 1 ( a ) where the eq . ( [ eq : ratio ] ) has been simply used .
if we , instead , set the coulomb correction to zero in the region @xmath29 limit , a small step appears as seen in fig . 1 ( b ) . therefore in order to analyse in a consistent way the whole region of the momentum transfer measured we have to use the following asymptotic expansion of the coulomb wave function : @xmath30 \ : , \label{eq : asym}\\ \psi^{{\mbox{\scriptsize asym } } } ( { \mbox{\bf q } } , { \mbox{\bf r } } ) & \!\!\ ! = & \!\!\ ! \exp\{i(qz - \eta\ln(r - z))\}\times \left(1+\frac{\eta^2 } { iq(r - z)}\right)\nonumber\\ & \!\!\ ! & \!\!\ !
+ f(\theta ) \frac{\exp\{i(qr - \eta \ln(2qr))\}}r\ : , \label{eq : asym1}\\ \psi_{{\mbox{\scriptsize s}}}^{{\mbox{\scriptsize asym } } } ( { \mbox{\bf q } } , { \mbox{\bf r } } ) & \!\!\ ! = & \!\!\ ! \exp\{i(-qz - \eta \ln(r+z))\}\times
\left(1+\frac{\eta^2}{iq(r+z)}\right ) \nonumber\\ & \!\!\ ! & \!\!\ !
+ f(\pi - \theta ) \frac{\exp\{i(qr - \eta \ln(2qr))\}}r\ : , \label{eq : asym2}\end{aligned}\ ] ] where @xmath31 and @xmath32 in analyses we should assure a smooth connection between both regions of @xmath33 . to avoid the divergence of denominators @xmath34 , we have to introduce a cutoff parameter @xmath35 ( of the order of @xmath36 ) such that @xmath37 always depends on the magnitude of the interaction region ( @xmath38 ) . ] .
this procedure , shown as a flow chart in fig . 2 ,
is called the `` seamless fitting ( sf ) '' .
+ * analyses of data by sf method : * our results obtained in terms of the new formula ( [ eq : ratio ] ) are shown in table i and figs . 3 and 4 . whereas the parameter @xmath22 ( representing here the influence of long range correlations ) increases now noticeably in comparison with that obtained previously in @xcite , we find that the interaction region represented by @xmath39 remains s + pb reactions almost the same . only in @xmath1 + pb collisions the estimated @xmath38 becomes larger with inclusion of coulomb correction than that obtained in @xcite .
these facts suggest that we should be careful in interpreting any data ( at least those for kaons ) which were corrected for coulomb interactions only by the gamow factor . + * concluding remarks : * we have proposed the possible method of applying the coulomb correction for the bec in the whole region of momentum transfer , the seamless fitting " ( sf ) . we confirm that this method works well when analysing data corrected only by the gamow factor .
+ our analyses of data of @xmath0 pairs in s + pb reaction @xcite shows ( cf . table i ) that @xmath40 = 4 fm and @xmath41 = 3 fm ( i.e. , they differ substantially ) , contrary to the estimation provided in @xcite that @xmath42 @xmath43 3 fm .
this is an important result for the study of signals of the quark gluon plasma ( qgp ) ( see refs .
@xcite ) .
moreover , we found that the long range correlations are strongly affected by coulomb corrections ( most probably because of the long range character of coulomb interactions ) .
+ for the sake of reference , we show also in table ii and fig .
5 results of our analysis of data for s + pb @xmath44 reaction @xcite ( which were corrected by the coulomb wave function method @xcite ) performed by using both eq .
( [ eq : a21 ] ) and the `` standard '' formula ( eq . ( [ eq : a22 ] ) ) . as one can see there are no significant differences between parameters estimated by means of these two formulae , in particular the magnitude of the interaction region is in both cases almost the same . +
* acknowledgements : * the authors would like to thank s. esumi , t. nishimura , and s. d. pandey ( members of na44 collaboration ) for useful conversations and correspondences .
this work is partially supported by japanese grant - in - aid for scientific research from the ministry of education , science and culture ( # 06640383 ) .
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fmi acknowledges the support by conacyt ( mexico ) grant no . 34668-e .
w. wang , f.m . izrailev and g. casati ,
phys . rev . *
e 57 * , 323 ( 1998 ) ; f. borgonovi , i. guarneri and f.m .
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a novel approach is suggested for the statistical description of quantum systems of interacting particles .
the key point of this approach is that a typical eigenstate in the energy representation ( shape of eigenstates , se ) has a well defined classical analog which can be easily obtained from the classical equations of motion .
therefore , the occupation numbers for single - particle states can be represented as a convolution of the classical se with the quantum occupation number operator for non - interacting particles .
the latter takes into account the wavefunctions symmetry and depends on the unperturbed energy spectrum only . as a result ,
the distribution of occupation numbers @xmath0 can be numerically found for a very large number of interacting particles . using the model of interacting spins
we demonstrate that this approach gives a correct description of @xmath0 even in a deep quantum region with few single - particle orbitals .
2 in many physical systems such as complex atoms , heavy nuclei and interacting spins , highly excited eigenstates in the unpertured many - particles basis can be treated as a chaotic superposition of a very large number of components , for a relatively strong interaction among particles ( see , e.g. @xcite ) .
this fact has been used in @xcite in order to develop a statistical description of closed systems with a finite number of fermi - particles . in particular
, it was analytically shown that for a strong enough interaction , a smooth dependence of occupation numbers on the energy occurs , which is related to global properties of chaotic eigenstates . as is known , the direct numerical computation of excited eigenstates is strongly restricted due to a very large number of many - particles states , which grows extremely fast with the number of particles . however , the mean values of occupation numbers turn out to depend on the average shape of chaotic eigenstates in the unperturbed basis , not on exact , specific values of their components @xcite . in this letter
we develop a novel approach to quantum systems with chaotic behaviour in the classical limit .
this approach takes into account both the chaotic properties of the classical system , and the specific features of the unperturbed single - particle spectrum . as a result
, one can avoid diagonalization of hamiltonian matrices of a huge size which may be practically unfeasible .
this kind of approach can be applied to generic hamiltonian systems with two - body interaction of the type : @xmath1 here @xmath2 describes @xmath3 _ non - interacting _ particles with @xmath4 as single - particle hamiltonians , and @xmath5 stands for a long - range two - body interaction between the particles . in what follows , we assume that the single - particle spectrum is determined by a finite number @xmath6 of single - particles energies @xmath7 , @xmath8 ; however , the approach is valid for more generic systems with an infinite spectrum .
the unperturbed hamiltonian @xmath2 determines the many - particle states @xmath9 ( with @xmath10 , @xmath11 as creation - annihilation operators ) , that form the basis in which the exact eigenstates of @xmath12 are represented . as usual , we assume that the basis is ordered according to increasing unperturbed energy values @xmath13 .
the distribution of occupation numbers ( don ) for single - particle states is defined by the relation , @xmath14 where @xmath15 is the occupation number operator giving the occupation numbers @xmath16 equal to @xmath17 or @xmath18 for fermi - particles , and to @xmath19 for bose - particles .
these numbers @xmath20 indicate how many particles in a many - particle _ basis _ state @xmath21 , occupy a particular single - particle state @xmath22 . correspondingly
, the occupation numbers @xmath23 give the probability that one of @xmath3 particles in a many - particle _ exact _ state with the _ total _ energy @xmath24 , occupies a particular single - particle state @xmath22 .
the total number @xmath25 of many - body states equals @xmath26 for fermi and @xmath27 for bose - particles .
one should note that while in the above expression for the don , the eigenfunctions @xmath28 depend on the total hamiltonian @xmath12 , the term @xmath20 depends on the unperturbed spectrum only .
this fact is crucial for our semiquantal approach . due to the chaotic structure of exact eigenstates
, one can make an average of the don over a small energy window @xmath29 around the fixed value @xmath24 .
this averaging procedure is similar to that used in the conventional statistical mechanics developed for systems with a finite number of particles in contact with a heat bath , or for isolated systems of an _ infinite _ number of _ non - interacting _ particles .
expression ( [ nqu ] ) for the mean values @xmath23 can be considerably simplified by introducing the so - called _ shape of eigenfunctions _ , se ,
( envelope of eigenstates in energy representation ) .
the form of the se has been studied in details both in the model with random two - body interaction @xcite , and in dynamical models of interacting particles @xcite .
the introduction of the average quantity se ( thus neglecting correlations between different components @xmath28 ) represents the key point of our approach .
we assume that the unperturbed many - body energy spectrum has an intrinsic degeneracy .
this situation is typical for spin systems , and is more complicated in comparison with those studied before @xcite .
below we show how this difficulty can be overcome .
let us redefine the state @xmath30 by means of a indexes pair @xmath31 , where @xmath32 labels the `` unperturbed energy '' @xmath33 of the many - body state , while @xmath34 labels its degeneracy @xmath35 .
if there are @xmath36 different `` unperturbed '' energies , one can write , @xmath37 , therefore , one has , @xmath38 according to standard definition@xcite , the se is given by : @xmath39 by substituting @xmath40 with @xmath41 as an average over @xmath42 , we obtain an approximate expression for the don in terms of the se , @xmath43 needless to say , if an unperturbed spectrum has no degeneracy , eq.([crus ] ) can be written in a similar way by taking an average over a small window of energy around @xmath33 .
as one can see , expression ( [ crus ] ) depends on two terms of different nature .
the first one , @xmath44 , refers to the unperturbed many - particle spectrum and reflects the specific properties of a single - particle spectrum , as well as quantum features related to fermi - dirac or bose - einstein statistics .
in contrast , the second term , @xmath45 , refers to global properties of eigenstates and describes _ interaction _ effects .
therefore , the basic idea of our semiquantal " approach is to substitute the latter term ( se ) by its classical analog which can be easily found from classical equations of motion .
classical analogs of the se have been studied in different models , see , for example , @xcite ) . in practice
, one has to derive the distribution @xmath46 for the probability to find the unperturbed energy @xmath47 for @xmath2 , given the conserved total energy @xmath24 .
this can be obtained by generating many different initial classical configurations on the energy surface @xmath48 or sampling the @xmath49 values generated by one single trajectory onto the energy surface and computing the correspondent distribution of @xmath50 @xcite .
the two procedures have been found to give the same results in the chaotic region@xcite . in order to facilitate the comparison with the quantum se , in our numerical simulations the bin size of @xmath47
equals the energy distance between neighbour values of @xmath33 . in the same way
one can define the classical distribution of occupation numbers , @xmath51 ( see also @xcite ) . for the quantum - classical comparison ,
the bin size of @xmath52 is taken to be equal to the spacing between close single - particle energy levels @xmath7 .
let us stress that in this semiquantal approach ( sa ) it is possible to study specific systems of , let us say , @xmath53 interacting particles occupying @xmath54 single - particle levels .
surely , one expect this approach to be valid for highly excited chaotic states in a deep semiclassical region .
however , by direct numerical simulations we have found that the sa gives correct results even for energy values close to the ground state .
the model that we have studied , consists of @xmath3 3-d interacting spins placed in a magnetic field @xmath55 directed along the @xmath56-axis . in order to have a proper many - body operator one should require a coupling between _ all spins _ ( not only between neighbors ) .
the hamiltonian thus reads , @xmath57 using the well known relations @xmath58 , one can write , @xmath59 + & \\
+ \frac{1}{4}(j_x - j_y ) \sum\limits_{i=1}^{n-1 } \sum\limits_{j = i+1}^{n } [ s_i^+ s_j^+ + s_i^- s_j^- ] + & \\ \frac{1}{4 } ( j_x+j_y ) \sum\limits_{i=1}^{n-1 } \sum\limits_{j = i+1}^{n } [ s_i^+ s_j^- + s_i^- s_j^+ ] \label{h00 } \end{array}\ ] ] in the mean field approximation , we put @xmath60 .
the interaction can be further simplified by the particular choice @xmath61 .
thus , our hamiltonian @xmath62 has the following form , @xmath63 .
\label{h0}\ ] ] for simplicity , the @xmath3 classical constants of motion @xmath64 have been set to 1 , therefore , the only free classical parameters are the total conserved energy @xmath24 and the interaction @xmath65 .
the classical model has been studied in @xcite and it was numerically found to be chaotic and exponentially unstable in a wide energy range .
more precisely , in order to have strong chaos , one needs the interaction @xmath65 between particles to be strong enough .
a convenient choice is to take the interaction strength @xmath66 .
one should stress that this situation is the most difficult for theoretical studies , see discussion in @xcite .
quantization follows the standard rules , which gives , @xmath67 and @xmath68 with @xmath69 .
the action of creation and annihilation operators is defined by : @xmath70 where @xmath71 are the non - symmetrized states ( first quantization states ) .
there are @xmath72 single - particle energy levels @xmath73 with @xmath74 .
therefore , the unperturbed many - particle energy spectrum consists of a number of degenerate levels with the spacing equal to @xmath75 .
note , that both the ground state @xmath76 and the upper level @xmath77 are non - degenerate .
the classical limit is recovered when spins are allowed to have any possible orientation , that is @xmath78 , and @xmath79 .
the choice we have done ( @xmath61 ) allows to reduce the dimension of the hilbert space by , approximately , one half .
this happens because the operator @xmath5 in ( [ h0 ] ) connects only those unperturbed many - body states that are separated by the spacing @xmath80 . in what follows
, we consider the subset of the many - body states containing the ground state . from these states
we construct the completely symmetrized states @xmath81 ) , in the second quantization , can be written as : @xmath82 with @xmath83 here @xmath84 , with @xmath85 and @xmath86 the creation - annihilation operators satisfying the standard relation @xmath87= \delta_{hk}$ ] .
as to the coefficients @xmath88 , @xmath89 , they have quite complicated expressions ; however , they can be easily computed numerically .
the procedure we have used in our numerical simulations consists of the following steps : a ) compute the classical values @xmath90 , and @xmath91 as described above ; b ) compute the quantum values @xmath90 and @xmath91 by diagonalization of the total hamiltonian ( [ cal0 ] ) ; c ) compute @xmath90 and @xmath91 by using the semiquantal approximation according to expression ( [ crus ] ) .
the results for the don are summarized in fig.[n4 ] . for the convenience of comparison , symbols refer to the quantum and sa results , while the classical data are presented as hystograms , .
note , that each particle can have energy within the interval @xmath92 $ ] . in order to make the quantum - classical comparison as close as possible
, we took the classical bin size equal to @xmath75 .
all distributions have been normalized in such a way that @xmath93 . in cases (
a ) and ( b ) we choose the same energy , close to the ground state and two different @xmath94 values . in ( c ) and
( d ) instead , we choose a higher energy value .
the latter case ( ( c ) and ( d ) ) should describe a situation more classical than the former ( higher energy should correspond to higher temperature ) .
one can see that , while classical and quantum data only disagree in the _ deep _ quantum region ( a ) ( energy close to the ground state and small @xmath94 ( _ big _
@xmath75 ) ) , there is a very nice correspondence between quantum and sa data in all cases ( including region ( a ) ) .
this nice correspondence is far from trivial since sa does not take into account quantum correlations inside exact eigenstates , together with possible correlations between the two terms , @xmath95 and @xmath96 in expression ( [ nquan ] ) , both expected to be strong in the deep quantum region .
it is very interesting to explore the occurence of the bose - einstein ( be ) distribution in our model .
a similar problem has been studied in detail for the model of two - body random interaction @xcite where the conditions for the appearence of fermi - dirac distribution have been found for few interacting fermi - particles . by assuming , a priori , the validity of the be - distribution @xmath97^{-1}$ ] in our closed system
, one can find the temperature " @xmath98 and the chemical potential " @xmath99 via the standard relations , @xmath100 here @xmath101 is the numerically computed energy obtained from the single - particle quantum distribution ( see details in @xcite ) , and @xmath3 is the number of particles . notice that , due to interaction , @xmath102 .
the comparison between be and sa distributions are shown in fig.2 .
as one can see , even for relatively small @xmath103 , the distribution @xmath90 can be closely approximated by be - distribution .
this confirms the expectation that a strong enough interaction between particles can play the role of an internal heat bath @xcite .
therefore , the standard quantum distributions can be used , with a corresponding renormalization of the energy @xmath101 ( see details in @xcite ) .
thus in our model there is no need to increase further the number of particles , since even for @xmath103 a remarkable agreement with be distribution has been found .
we would like to stress that an exact quantum treatment of the last example calls for a diagonalization of a huge matrix of size @xmath104 , while , with the semiquantal approach , all computations required few minutes on a standard portable pc . in conclusion , we suggest an effective semiquantal approach to closed systems of interacting particles , based on the chaotic structure of eigenstates . in this approach
, the computation of the distribution of occupation numbers can be easily performed by making use of the classical analog of the shape of eigenstates in the unperturbed many - particle basis .
we demonstrate the effectiveness of this approach using the model of 3-d spins with anisotropic ising interaction .
the data show that semiquantal computations give results which are very close to the exact ones .
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a variety of detectors has been proposed for dark matter direct detection , but most of them by the fact are still at r&d stage . in many cases , it is claimed that the lack of an adequate detectors radio - purity might be compensated through heavy uses of montecarlo simulations , subtractions and handlings of the measured counting rates , in order to claim higher sensitivity ( just for a particular scenario ) .
the relevance of a correct evaluation of systematic effects in the use of montecarlo simulations at very low energy ( which has always been safely discouraged in the field so far ) and of multiple subtractions and handling procedures applied to the measured counting rate is shortly addressed here at some extent .
many other aspects would also deserve suitably deep investigations . in this paper
some arguments presented at the taup09 conference will be shortly summarized .
more details , tables and figures can be found in the slides at the conference site@xcite .
let us firstly comment the possibility of reliable evaluations of the background contributions at the kev energy region in the field of dark matter searches . as well known , it has been generally discouraged this procedure in the field of dark matter over more than twenty years .
in fact , the estimation by a montecarlo simulation of the background component in the counting rate from the residual radioactivity requires a detailed knowledge of : i ) the exact set - up geometry ( detector or detectors matrix , all materials , details of the assembling , of the shield layers , of the site , etc . ) ; ii ) the detector response function ( e.g. energy resolution , @xmath0 ratio , channeling , etc . ) ; iii ) the nature , the position and the concentration of all the existing radioactive contaminants ; iv ) etc .. unfortunately , apart from the geometrical layout of the set - up that are generally well known by people inside the experimental group , all the other quantities necessary in the montecarlo simulation require dedicated measurements .
moreover , there are some quantities ( such as concentration of residual contaminants , etc . ) that can be poorly known and just upper / lower limits are available ; in some cases these quantities can be even totally unknown .
as an example , the experimental energy resolution as a function of the energy and the energy scale should be measured / verified down to the energy threshold , as done e.g. by the dama / libra experiment where they are continuously measured by external / internal known sources from mev down to the energy threshold @xcite . on the contrary , in other experiments these quantities
are instead extrapolated from calibrations at much higher energy ( as done e.g. by liquid noble gas set - ups , where the energy threshold and the few kev energy scale are generally unproven , also because of position dependence , of non - uniform signal collection , etc . ; see e.g. @xcite ) .
regarding the presence of residual contaminants in the set - up , generally only limits on the contributions of the `` standard '' contaminants are given ; these limits forbid any reliable estimation of the background ( being unknown the exact values ) and can not be obviously exceeded ( see also later ) .
moreover , possible presence of many non - standard contaminants should be also included .
in addition , the montecarlo simulation also depends on the precise location of all the contaminants that is generally unknown even for the `` standard '' ones in complex set - ups .
the situation is more complex for multi - detectors set - up and when the energy distribution refers to events where each detector has all the others in anticoincidence ( _ single - hit _ events ) .
thus , it is trivial to conclude that a reliable precise simulation of the background counting rate in particular at kev energy region is not univocally determined and is a quite impossible task .
in addition , beyond the fore - mentioned arguments , we need to take into account that a montecarlo code can not manage all the possible low energy atomic physical processes .
this argument is still subject of improvements ; in fact , as an example , non - negligible differences are also obtained by different versions of the same montecarlo code @xcite .
some instructive examples are given by the trails in montecarlo simulations in ref .
@xcite .
as shown there , these simulations noticeably differ from the measured energy distributions in the cases of xenon-10 and of zeplin - iii .
in fact , they predict twice the measured rate for xenon-10 near 200 kev , and more than one order of magnitude the measured rate for zeplin - iii in the mev range @xcite .
the same approach has been also pursued by the same authors @xcite trying a montecarlo simulation of the background in the dama / libra set - up .
in particular , apart from errors in the details of the set - up geometry reconstruction and in the multiple - hit definition , many crude and arbitrary approximations in the nature and in the location of the residual contaminants have been arbitrarily assumed ; in fact , e.g. : i ) only standard contaminants , ii ) only unbroken chains , iii ) only uniform location of contaminants in the detectors , etc . have been taken into account . as a result of this rough , partial and arbitrary approach ,
the predicted rate has been estimated within a factor 10 lower than the measured one . instead of refining the quality of the simulation or reasonably recognizing the impossibility of precise determination , the authors just pursued the exercise of arbitrarily increasing `` by hand '' the assumed values of the contaminants at levels much larger than the measured experimental limits @xcite .
this also implies an overestimate of the background in higher energy region with respect to the measured experimental rate . in conclusion , although the arbitrary and the erroneous adopted procedures , these authors do not succeed in reproducing either the low or the high energy spectra .
nevertheless , as a conclusion of this arbitrary exercise @xcite , this artificially - boosted simulated spectrum has been subtracted by the measured one , attempting to obtain a limit for the unmodulated dark matter signal component .
this example shows how subtraction procedures using montecarlo simulations in the few kev energy region can give rise to erroneous conclusions ; thus , any constraint on dark matter signal on this basis would be an artefact .
furthermore , let us also note that the measured spectra e.g. of the existing / past nai(tl ) detectors ( such as e.g. anais , frejus , naiad , elegant , etc ... ) do not support even the shape presented in ref .
@xcite .
in addition , well different counting rates at kev energy region are present even for detectors of the same experimental group , as e.g. the case of naiad in 1996 and in 2003 @xcite . in conclusion
, it does not exist an unique recipe for a precise and reliable montecarlo simulation of whatever set - up , and for nai(tl ) in particular .
let us finally remind that a safer approach has been presented in this conference by dama collaboration ; this shows that enough space is present in the measured counting rate of the dama / libra kev energy spectrum for the unmodulated component of dark matter signal @xcite .
in the second part of this contribution @xcite , problems related to the application of multiple subtraction procedures of the measured counting rate , as pursued by experiments trying to identify the presence of recoil nuclei in the measured energy spectrum , have been summarized . in fact , many of the existing dark matter candidates also in the wimp class can give rise to signals that either have totally an electromagnetic nature ( see e.g. @xcite ) or involve electromagnetic signals associated to nuclear recoils ( see e.g. @xcite ) ; obviously , approaches that are based on multiple subtraction procedure of the electromagnetic component of the counting rate are blind to similar scenarios .
moreover , well known side processes exist for recoils ( such as recoils induced by neutrons , fission fragments , end - range alphas , surface electrons , etc . ) .
this approach is generally pursued when the detectors suffer from a not - suitable radiopurity level in the sensitive target - material and in the surroundings .
those activities generally apply a large number of cut procedures to the data ; each one is affected by non - negligible systematic errors which are usually not suitable quantified . as an example , the xenon-10 experiment applies more than 10 different cuts to the data ; the experiment collects @xmath1 events but only 10 are claimed to survive to the cuts and handling procedures @xcite .
very high reduction factors following applied multiple cuts are dangerous because of the difficult precise estimate of all the involved systematics . for example , it has been shown in ref .
@xcite that zeplin - i has claimed a sensitivity 3 orders of magnitude larger than the one properly obtained when accounting for systematics .
thus , the robustness of some results appeared in the `` race for the best exclusion - plot '' ( valid just in a single set of assumptions for a certain kind of wimp ) should be considered `` cum grano salis '' .
for example , for some liquid noble gas set - ups , apart from the robustness of the applied cuts themselves , the very low energy scale and the energy threshold are determined by extrapolating from calibrations at much higher energy and applying some kind of corrections for relevant non - uniformity of the detector s response ; with a light yield of about 2.2 photoelectrons / kev and not specific calibrations even @xmath2 1.5 - 2 kev electron equivalent is claimed as energy threshold @xcite .
considering the energy threshold dependence of the exclusion plots , up to several orders of magnitude differences can be present between claimed and realistic evaluation of the experimental sensitivity .
another crucial aspect is the proper accounting for the existing experimental and theoretical uncertainties in the calculation and in the comparison of experiments using different target materials and approaches .
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financial support from the dfg is gratefully acknowledged by jk and tp . financial support from the australian research council
is gratefully acknowledged by alo and ar .
alo also thanks the institut fr theoretische physik at the technische universitt clausthal .
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a self - interacting polymer with one end attached to a sticky surface has been studied by means of a flat - histogram stochastic growth algorithm known as flatperm .
we examined the four - dimensional parameter space of the number of monomers up to 91 , self - attraction , surface attraction and force applied to an end of the polymer . using this powerful algorithm the _ complete _ parameter space of interactions and
force has been considered .
recently it has been conjectured that a hierarchy of states appears at low temperature / poor solvent conditions where a polymer exists in a finite number of layers close to a surface .
we find re - entrant behaviour from a stretched phase into these layering phases when an appropriate force is applied to the polymer .
we also find that , contrary to what may be expected , the polymer desorbs from the surface when a sufficiently strong critical force is applied and does _ not _ transcend through either a series of de - layering transitions or monomer - by - monomer transitions .
new experimental methods in the physics of macromolecules @xcite have been used to study and manipulate single molecules and their interactions .
these methods make a contribution to our understanding of such phenomena as protein folding or dna un - zipping ; one can push or pull a single molecule and watch how it responds .
it is possible to apply ( and measure ) forces large enough to induce structural deformation of single molecules .
one can monitor the mechanism of some force - driven phase transition occurring at the level of a single molecule .
theoretical understanding of this behaviour has attracted much attention @xcite .
the response of a single polymer to an external force under good solvent conditions was considered some time ago@xcite .
the response under poor solvent conditions ( below the @xmath0-point ) , where the self - attraction and an external force compete with each other , was examined later @xcite .
another phenomenon commonly studied in polymer physics is the adsorption of a polymer tethered to a `` sticky '' wall .
the response of such a polymer to a force perpendicular to the wall has also recently been considered @xcite .
however , when both the self - attraction ( ie .
monomer - monomer attraction that leads to polymer collapse ) and the surface attraction ( ie monomer - wall attraction that leads to adsorption ) compete the response to an external force has not yet been elucidated ( some interesting results can be found in @xcite ) .
certainly , the full phase diagram has not been considered .
making such a study now is all the more timely because of the very recent discovery @xcite of a new low temperature phenomenon of layering transitions ( without a force ) .
it is this layering phenomenon that raises the intriguing question about the response a low - temperature polymer may have to an external force . in the layering state a polymer is tightly confined within a fixed number of layers above the wall .
it may be therefore be especially interesting experimentally to examine such a situation .
we demonstrate for the _ first _ time how the full two - dimensional phase diagram of surface and self - attraction changes as the force is increased .
the desorbed extended regime , which changes its scaling behaviour as soon as the force is made non - zero , simply grows as the force is increased .
the second - order phase transitions of adsorption and collapse become first order .
the rest of the phase diagram remains relatively unaffected as long as the force is small .
after the force passes a critical value , which depends on the zero temperature force required to pull a polymer from a wall , a re - entrant behaviour occurs at low temperatures . for different values of the force
, this re - entrant behaviour occurs for both the adsorption and collapse of polymers , including the layering phases mentioned above .
we provide a full force - temperature diagram for all ratios of surface attraction to self - attraction .
all this is achieved with the use of a recently developed algorithm , flatperm @xcite , that is specifically designed to obtain information about the whole phase diagram in one simulation run : it is effectively a stochastic enumeration algorithm that estimates the complete density of states .
the model we have considered is a self - avoiding walk on a three - dimensional cubic lattice in a half - space .
the self - avoiding walk is attached at one end to the wall with surface energy per monomer of @xmath1 for _ visits _ to the wall .
the self - avoiding walk self - interacts via a nearest - neighbour energy of attraction @xmath2 per monomer - monomer _ contact_. ( note that the attractive energies @xmath2 and @xmath1 are taken to be positive . ) a force @xmath3 is applied in the direction perpendicular to the boundary of the half - space ( wall ) .
the total energy of a configuration @xmath4 of length ( number of monomers ) @xmath5 is given by @xmath6 and depends on the number of non - consecutive nearest - neighbour pairs ( contacts ) along the walk @xmath7 , the number of visits to the planar surface @xmath8 , and the height @xmath9 in the direction perpendicular to the boundary ( wall ) of the half - space .
figure 1 shows a diagram of the two - dimensional analogue . for convenience ,
we define @xmath10 , @xmath11 and @xmath12 where @xmath13 is the temperature and @xmath14 is the boltzmann constant .
the partition function is given by @xmath15 with @xmath16 being the density of states .
it is this density of states that is estimated directly by the flatperm simulation .
our algorithm grows a walk monomer - by - monomer starting on the surface .
we obtained data for each value of @xmath5 up to @xmath17 , and all permissible values of @xmath7 , @xmath8 , and @xmath9 . when @xmath18 the phase diagram of the model contains several phases and transitions between them @xcite .
[ cols="^,^ " , ] for small @xmath19 and @xmath20 there is a desorbed extended ( de ) phase with the polymer behaving as a free flexible polymer in solution ( ie . swollen or extended in three dimensions ) . for @xmath19
fixed and small , increasing @xmath20 leads to a second - order phase transition ( adsorption ) to a state in which the polymer is adsorbed onto the wall and behaves in a swollen two - dimensional fashion ( ae ) .
alternately , if @xmath19 is increased at small @xmath20 a second - order collapse transition occurs to a state resembling a dense liquid drop .
this phase is known as desorbed collapsed ( dc ) on the assumption that it has little contact with the wall @xcite .
however , it has been subsequently argued @xcite that for larger @xmath19 there is also a polymer - surface transition to a surface - attached globule ( sag ) phase , where the polymer behave as a liquid drop partially wetting the wall .
this transition will not be seen directly by studying thermodynamic polymer quantities as it occurs as a singularity in the surface free energy and not the bulk free energy of the polymer . when @xmath20 is large ,
so that the polymer is adsorbed onto the wall , increasing @xmath19 will result in a two - dimensional ( second - order ) transition to a adsorbed and collapsed phase ( ac ) . in recent work @xcite this ac phase
was also referred to as the _
1-layer _ phase because for very large @xmath19 and @xmath21 there exist meta - stable @xmath22-layer phases where the polymer is two - dimensionally collapsed and more - or - less restricted to @xmath22 layers parallel to the wall ( for small @xmath22 ) .
a series of first - order transitions between adjacent values of @xmath22 occur as @xmath20 is varied at fixed @xmath19 .
all these transition lines can be seen in the figure 2 ( a ) which shows a plot of the logarithm of the maximum eigenvalue of the ( @xmath23 ) matrix of second derivatives in the variables @xmath19 and @xmath20 of @xmath24 for fixed @xmath25 .
the local maxima indicate transitions .
the transition to the ac phase is expected to be first order in the thermodynamic limit .
using the evidence available in the literature @xcite , let us now consider what we can expect when @xmath26 . the first important feature to note is that the isotropic de phase is replaced by an anisotropic phase in which the height of the end point of the polymer scales linearly with @xmath5 ; we denote this phase as the _ stretched _ phase .
consequently the transition from stretched to adsorbed phases becomes first - order @xcite .
likewise , at least in three dimensions @xcite , the transition from the vertically stretched phase to the collapsed phase also becomes first - order .
this implies that the multicritical point ( where for @xmath18 the de , ae and dc phases meet ) is now a triple point : the meeting of three first order lines .
the transition from the ae to ac phases should not be effected by the application of a small force as it acts in a direction perpendicular to the driving phenomenon of planar collapse . to a polymer in
the 2-layer adsorbed collapsed phase._,width=302 ] finally , it is intriguing to ask what happens to the layering phases observed in @xcite .
one can imagine that the force simply extends a vertical ` tail ' from a layered block ( see figure 3 ) and that as the force is increased the monomers are peeled off one at a time with corresponding micro - transitions @xcite for each monomer pulled until a vertical rod is achieved .
instead we see at some point a sharp first order transition between the highly stretched vertical rod and a layered system with short tail . in figures 2(b ) and 2(c ) we show plots of the logarithm of the maximum eigenvalue of the matrix of second derivatives in the variables @xmath19 and @xmath20 of @xmath24 at fixed @xmath27 ( as in figure 2(a ) ) but at values of @xmath27 being @xmath28 and @xmath29 .
it is clear that as @xmath27 is increased the stretched phase that occurs for small @xmath19 and @xmath20 expands while the positions of the other phases and transitions move little .
we immediately note that these plots _ do not _ tell the whole story since physically one is usually interested in fixing the force @xmath3 rather than @xmath27 : fixing @xmath27 implies that the force applied goes to zero at low temperatures .
it is for this reason that the re - entrant behaviour for absorbing polymers @xcite is not seen directly in these plots .
however , re - entrant behaviour does occur and occurs for _ any _ ratio of surface to bulk energies .
let us now consider the more traditional force - temperature diagram and return to this point . in figure 4 a plot of the force @xmath30 needed to pull a polymer from the wall for a ranges of temperatures and a parameter @xmath31 which measures the relative strength of the surface ( wall ) interaction to the self - interaction .
needed to pull a polymer from the surface against temperature @xmath13 and a parameter @xmath31 .
the parameter @xmath31 controls the relative strength of wall attraction and self - attraction with @xmath32 and @xmath33 .
the limiting cases of surface desorption and of pulling a collapsed polymer are easily visible in the plot for @xmath34 and @xmath35 , respectively._,width=325 ] we have parameterised the energies of surface and self - attraction as @xmath32 and @xmath36 respectively .
using this parameterisation for @xmath37 gives the whole range of attractive activities : the ratio of surface to bulk activities is given as @xmath38 and so is constant for fixed @xmath31 .
for @xmath35 we have @xmath39 and @xmath40 which corresponds to pure self - attraction while the other boundary of the parameter space with @xmath34 gives @xmath41 and @xmath42 which is the pure surface adsorption case .
this extends the diagrams given in @xcite in which only adsorption is considered . if a force smaller than @xmath43 is applied the polymer is in the phase appropriate to the value of @xmath31 : either collapsed or adsorbed or both .
on the other hand for forces larger than @xmath43 the polymer is in the ` stretched ' phase .
we immediately note that the reentrant behaviour observed in the adsorption - only case @xcite persists for all @xmath31 . fixing the force to be at a value slightly larger than the zero temperature critical force and then increasing the temperature leads to transitions from the stretched state to a non - stretched phase and back again to the stretched state .
this arises due to the entropy of the zero temperature state ; one can easily extend the arguments in @xcite to demonstrate that re - entrant behaviour can occur even when the zero - temperature configuration of the non - stretched state is a hamiltonian ( fully compact ) walk in a cube rather than a totally adsorbed polymer .
let us discuss the re - entrant behaviour returning to the ( @xmath19,@xmath20,@xmath27)-parametrisation .
for fixed @xmath2 , @xmath1 , and @xmath3 , changing the temperature @xmath13 implies moving on a ray in the ( @xmath19,@xmath20,@xmath27)-space . at high temperatures the system is in the stretched phase near the origin . at low temperatures ,
the state of the system depends on the choice of @xmath2 , @xmath1 , and @xmath3 . for very large @xmath3
the system remains stretched at all temperatures , whereas for very small @xmath3 and low temperatures , the system is in a layered phase .
the re - entrant behaviour manifests itself in the following way : there is a range of @xmath3 for which the system , when moving along a ray in the ( @xmath19,@xmath20,@xmath27)-space ( ie . for fixed @xmath2 , @xmath1 , and @xmath3 ) , changes from a stretched state near the origin to a layered one at intermediate temperatures and then changes back to a stretched state as @xmath19 , @xmath20 , and @xmath27 become larger .
if the critical force is zero then the curve in the @xmath44 plane corresponds to the phase boundary of the de phase with the apex of the curve around @xmath45 being the location of the multicritical point . on the other hand for @xmath46
there is a kink in the function @xmath47 at exactly @xmath48 which is a consequence of the first order point coming from the transition from sag / layer phases from small @xmath31 to the ac phase for larger @xmath31 .
there is the appearance of a kink joining the multicritical point to the zero temperature transition which is presumably a finite temperature effect of the transition to the ac phase . in this paper
we have studied how the phase diagram of a self - attracting polymer that is also attracted and tethered to a flat wall changes as a vertical force is applied to the un - tethered end of the polymer .
we have accomplished this using a flat histogram monte carlo simulation that is capable of studying the whole range of microscopic energies , temperature and polymer length up to a maximum of @xmath49 monomers .
we demonstrate that re - entrant behaviour occurs at low temperature and for a range of forces for _ all _ relative strengths of self - attraction and surface attraction . for small forces
we have found that only the transition boundary of the `` stretched '' phase moves with increasing force while the rest of the phase diagram is relatively unchanged . in contradiction to what may be expected we have found that the novel layering meta - phases found for large but finite polymer length are unaffected by small forces .
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fig . [ fig1 ] .
comparison of experimental and model results for coherence length variation with observation angle for ( a ) a tuned rcled and ( b ) a detuned rcled . fig . [ fig2 ] .
coherence length variation as a function of numerical aperture generated using empirical model data in the coherence model for tuned and detuned rcleds .
markers at na extrema denote experimental points determine by measurements at low na and from an integrating sphere .
predictions of maximum and minimum coherence as a function of microcavity finesse and detuning with respect to the emission source across the useful emission na ( see text ) .
the diagram in the top left corner shows highlights the definition of useful na .
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an analytical expression for the self coherence function of a microcavity and partially coherent source is derived from first principles in terms of the component self coherence functions .
excellent agreement between the model and experimental measurements of two resonant cavity leds ( rcleds ) is evident .
the variation of coherence length as a function of numerical aperture is also described by the model .
this is explained by a microcavity s angular sensitivity in filtering out statistical fluctuations of the underlying light source .
it is further demonstrated that the variable coherence properties of planar microcavities can be designed by controlling the underlying coherences of microcavity and emitter whereby coherence lengths ranging over nearly an order of magnitude could be achieved .
the last two decades have seen widespread use of optical microcavities , both for experimental physics and commercial applications .
microcavities redistribute emission from an underlying source and depending on the ensuing radiation pattern allow light collection for use elsewhere . commercially available microcavity devices such as
resonant cavity light emitting diodes ( rcleds ) use the planar microcavity geometry to increase the extraction efficiency of spontaneous emission from materials with high dielectric constants .
@xcite more recently microcavities have been used to spectrally and spatially isolate quantum dot emitters to increase the efficiency of single photon production .
@xcite recent work on planar microcavities has identified the dependence of numerical aperture ( na ) on emission properties such as spectral linewidth @xcite and coherence length,@xcite the latter of which is the focus of the following paper . note that , migration of these results to the spectral domain are trivial due to the implicit fourier relationship with the coherence domain .
in addition to the choice of domain , the name _ coherence _ has been used , instead of _
time_. coherence highlights the statistical properties of light and not necessarily to the time dependence of the emission process that produces the light .
this distinction between the light and the emission event that produced it is necessary for the descriptions presented in this letter .
the coherence length is a relevant attribute of any optical device as it defines the length scales over which mutual interference occurs .
applications such as low coherence interferometry , for non invasive medical imaging @xcite ( also known as optical coherence tomography ) and optical time domain reflectometry ( otdr ) , for ranging measurements in optical components @xcite and surface mapping in integrated circuits @xcite , rely on the coherence of source being both small enough to eliminate coherent reflections from distant objects and large enough to examine detail on the relevant length scale .
the correlations of low coherence sources are also applicable to schemes of all - optical routing.@xcite the variable coherence properties of rcleds @xcite make them ideal as a light source for applications where a range of length scales need to be analysed where previously , multiple light sources would have been required .
this letter examines the physics of variable emission coherence from planar microcavities and indicates methods by which this aspect of commercial devices such as rcleds could be engineered .
although experimental results of rcleds are used in this paper , the results are applicable to planar microcavities in general .
the coherence domain picture of microcavity emission requires the definition of the self coherence properties of a light source .
the self coherence function of an emitter , @xmath0 , defined in eqn .
( [ eqn0 ] ) is synonymous with the spectral distribution function by the wienner - kinchin theorem .
@xmath1 a derivation for @xmath2 , the self coherence function of a combined emitter and microcavity system , is too involved to be described here from first principles . instead
, a key result from the spectral domain is used in combination with the wienner - kinchin theorem . the emission intensity from an emitter microcavity system is given by the spectral overlap of the optical transfer function of the bare cavity , @xmath3 , which is a function of the angle relative to the emission region , @xmath4 , and the field distribution of the underlying source @xmath5 , which is assumed to be isotropic .
@xcite @xmath6 therefore the coherence function for the microcavity can be written as a convolution expression : @xmath7 @xmath8 is the maximum microcavity enhancement for an emitter tuned to the cavity resonance at an angle @xmath9 , @xmath10 here @xmath11 and @xmath12 are the transmission and reflectivity of the front cavity mirror and @xmath13 is the reflectivity of the back mirror ; all function of @xmath4 , the internal emission angle .
@xmath2 can be greatly simplified by evaluating the definite integral . to do this ,
the underlying spectral distribution of emission is assumed to be lorentzian in nature corresponding to exponential decay in the time domain .
the transfer function of the cavity , @xmath3 can also be approximated by lorentzians in the immediate vicinity of the cavity modes , @xmath14 as the free spectral range of longitudinal modes in microcavities is usually much greater than the spectral width of the source .
@xmath15 here @xmath16 and @xmath14 are the frequencies of the emission and cavity resonances respectively and @xmath17 and @xmath18 are the coherence times of the underlying source and cavity resonances respectively .
note that @xmath17 and @xmath16 are assumed to be constant with emission angle . since @xmath19 is not necessarily symmetric ,
consider the expansion of the integral in eqn .
( [ eqn9 ] ) for @xmath20 : @xmath21 here , @xmath22 and @xmath23 . evaluating the integrals of eqn .
( [ eqn10 ] ) gives the self coherence function for the microcavity at an internal angle @xmath4 , @xmath24 the real and imaginary parts of this expression are evaluated to be : @xmath25 \right\ } \end{aligned}\ ] ] the expansion of the integral in eqn .
( [ eqn9 ] ) for @xmath26 gives @xmath27 .
such a symmetry is necessary in this case so that the fourier transform gives the real - valued spectral density function for cavity emission .
examination of eqns .
( [ eqn11 ] ) and ( [ eqn12 ] ) highlight the correct asymptotic response : if @xmath28 then @xmath29 , the coherence function of the cavity . for this condition
@xmath30 such that : @xmath31 similarly , if @xmath32 then @xmath33 , the coherence function for the underlying emission .
for this condition @xmath34 such that : @xmath35 these two limiting cases are also of particular interest to commercial applications . for @xmath28
is typically required for lasing applications , here , notice that the self coherence function , @xmath36 is a function of angle , however is solely dependent on the cavity .
engineering the cavity allows direct control of the lasing mode .
the second limit , @xmath32 , has only recently become applicable in semiconductor devices with the advent of devices that have isolated single quantum dot light sources .
@xcite the spectrally pure emission from single quantum dot sources within a microcavity will have coherence properties that vary weakly with angle since @xmath37 .
the control over emission direction is still available by engineering the cavity s tuning . for rcleds @xmath38 , however , the difference is not large enough to reduce the dynamics to one of the cases described above by eqns .
( [ eqn13 ] ) and ( [ eqn14 ] ) .
the coherence model has been compared with experimental results for tuned and detuned rcleds operating near @xmath39 nm , the details of which can be found in refs . and .
angle resolved spectra were sampled through a pin hole subtending a half angle @xmath40 at the device using a monochromator with a resolution @xmath41 nm . these were converted to the coherence domain by inverse fourier transform . the coherence length , @xmath42 , measured with respect to the angle , @xmath43 in air , is defined in eqn .
( [ eqn15 ] ) is used to analyse the correspondence of spectral domain measurements and the coherence model developed here .
@xmath44 the fitting parameters for the coherence model are the coherence times of the cavity mode and underlying source , @xmath45 and @xmath17 respectively and the relative detuning , @xmath46 . here
, it is assumed that the underlying cavity coherence is approximately constant with angle , a fair approximation for high index devices .
figure [ fig1 ] shows the variation of coherence length , @xmath42 for the tuned ( fig .
[ fig1 ] .
( a ) ) and detuned ( fig .
[ fig1 ] .
( b ) ) samples .
the broken lines represent the limiting case of @xmath47 which is clearly an insufficient description for both rcled samples .
the solid line is a least squares fit of the model and the three unknown model parameters .
the correspondence of experiment and theory is excellent for the lower angles . at large angles
, it should be noted that optical loss from the doped regions of the rcleds may cause the deviation from the theoretical trend .
this is supported by the observation that the deviations occur at the same spectral position , i.e. a relative shift in the deviation s angular position of @xmath48 due to the relative device tunings .
table [ tab1 ] shows the fitting parameter values for the rcled samples in terms of coherence time ( and spectral wavelength ) .
this has been done as it is more natural to specify the detuning in terms of wavelength .
.table of fitting parameter values given for both coherence ( time ) and spectral domain . [ cols="<,>,>",options="header " , ] good agreement is apparent between parameters of the tuned and detuned rcleds .
in addition , the values for the detuning closely match the design values of @xmath49 nm and @xmath50 nm for tuned and detuned devices respectively .
the trend itself is also of significant importance ; clearly interference between the cavity mode and underlying emission only resolves to a peak in coherence at the tuning angle .
this is only visible in the case of the detuned rcled at approximately @xmath51 , corresponding to a detuning of @xmath52 nm . in the previous study , the density of emission states within the numerical aperture of emission was taken into account . here
, the density of off - axis states increases as @xmath53 .
in addition , the differential change in solid angle with respect to air @xmath54 and the cavity @xmath55 must be taken into account . by integrating eqn .
( [ eqn12 ] ) over solid angle , the variation of the self coherence function as a function of na is given by : @xmath56 figure [ fig2 ] .
shows the variation of coherence length as a function of na for the two rcleds under investigation evaluated using the coherence model with the parameters shown in tab .
[ tab1 ] .
although experimental results for these trends are not available , they do follow the generic trend observed recently in ref . .
the square markers show the coherence length extrema calculated from measured spectra at normal incidence and @xmath57 in an integrating sphere offering further confirmation of the result .
the coherence properties of planar microcavities are modelled well by the coherence formula of eqn .
( [ eqn12 ] ) .
consider therefore , using this formula to predict how large a range of coherence length variation can be engineered over useful numerical apertures .
the reader will notice , that a low nas , when only a few of the transverse cavity modes are sampled , the coherence is near to that of the cavity .
statistical fluctuations within the cavity are small since the cavity samples the emission source over a long time than its coherence time resulting in a sharper spectral line and enhanced coherence properties .
the behaviour is also seen for all other emission angles into air as the cavity lifetime is approximately constant , however , the emission frequency does change with angle . at large nas ,
each cavity mode still samples the underlying emission over a longer time scale than the coherence time of the source , however , the sampled emission is reconstituted spectrally , reproducing to some extent the underlying statistical fluctuations .
this appears to be a fair description given the observations .
therefore , in order to create a larger range of coherence variation in these device , the finesse of the cavity must be increased and the coherence of the emission source must be minimised . most relevant , however , will be to increase the cavity finesse , over which most control can be achieved . to test this concept ,
consider varying the cavity finesse in the model for the tuned device discussed above .
fig .
[ fig3 ] shows the maximum and minimum coherence lengths for useful nas for the rcled device as a function of finesse and for a range of device tunings . here ,
_ useful _
na must be stressed : the maximum coherence corresponds to the na through which at least @xmath58 % of the device power emitted into air is sampled while the minimum coherence is limited by the limitation of coupling optics set here at an angle of @xmath59 .
these limits are illustrated at the top left corner of fig .
[ fig3 ] . despite the restrictions on useful na ,
a large variation in coherence length is apparent in this extrapolated example as is evident from fig .
[ fig3 ] . increasing the finesse by an order of magnitude , increases the maximum coherence length by just over a factor of @xmath60 .
the minimum na is increased by nearly a factor of @xmath61 .
the range of coherence lengths , on the other hand goes from @xmath62 m to @xmath63 m such that it almost spans an entire order of magnitude .
it was evident in the result of figs .
[ fig1 ] a ) and [ fig1 ] b ) that the maximum coherence length occurs at the tuning angle or wavelength .
this behaviour is also seen in the results of fig .
[ fig3 ] where the maximum coherence length occurs for a tuned sample and indeed gives the greatest variation over useful nas .
quantitatively , this amounts to a @xmath61 fold increase in maximum coherence for a tuned device compared with a @xmath64 nm detuned , which would be used for greater power extraction into large nas .
clearly , high finesse , tuned microcavity devices provide the greatest coherence variation over useful nas .
the self coherence function for planar microcavities has been modelled using an analytical formula based on the underlying self coherence functions of the emitter and microcavity .
the correspondence of the measured coherence of two rcled samples and the coherence model described by eqns .
( [ eqn12 ] ) and ( [ eqn16 ] ) is compelling .
firstly , a good function fit to the coherence lengths of fourier transformed spectra sampled at discrete angles was found .
furthermore , the same fitting parameters reproduced closely , the coherence variation as a function of na , by integrating over the whole set of emission states.@xcite the central result of this analysis is the observation that the coherence of the device is dependent on the underlying coherence of the cavity and emitter .
indeed the underlying statistical fluctuation of the light source are exposed when varying na . at low nas ,
the coherence length is close to that of the microcavity . at large nas ,
the coherence length is closer to that of the underlying emitter .
a key aspect of the modelling shows that the coherence properties of planar microcavities can be engineered through microcavity design and selection of an emitter with suitable coherence properties . here
, the model was used to predict a larger range of coherence variation by increasing the finesse of the microcavity .
coherence lengths across an order of magnitude could be achieved if these design consideration were considered carefully .
these observations suggest that variable coherence is a result of a filtering mechanism ; this is also the spectral domain picture of microcavity emission .
however , in the coherence domain , it is the statistical fluctuations of the light source that are filtered by the slow microcavity sampling time .
this is the only robust interpretation that can reconcile the difference in coherence between planar microcavity emission viewed through small and large nas and raises questions about the relationship between the emission lifetime and coherence time in any type of microcavity device .
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this work was sponsored by the oak ridge national laboratory , managed by ut - battelle , llc , for the u.s.department of energy under contract de - ac05 - 00or22725 .
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a novel approach that enables the study of parallel transport in magnetized plasmas is presented .
the method applies to general magnetic fields with local or nonlocal parallel closures .
temperature flattening in magnetic islands is accurately computed . for a wave number @xmath0 , the fattening time scales as @xmath1 where @xmath2 is the parallel diffusivity , and @xmath3 ( @xmath4 ) for non - local ( local ) transport .
the fractal structure of the devil staircase temperature radial profile in weakly chaotic fields is resolved . in fully chaotic fields ,
the temperature exhibits self - similar evolution of the form @xmath5 $ ] , where @xmath6 is a radial coordinate . in the local case
, @xmath7 is gaussian and the scaling is sub - diffusive , @xmath8 . in the non - local case , @xmath7 decays algebraically , @xmath9 , and the scaling is diffusive , @xmath10 .
the study of transport in magnetized plasmas is a problem of fundamental interest in controlled fusion , space plasmas , and astrophysics research .
three issues make this problem particularly challenging : ( i ) the _ extreme anisotropy _ between the parallel ( i.e. , along the magnetic field ) , @xmath11 , and the perpendicular , @xmath12 , conductivities ( @xmath13 may exceed @xmath14 in fusion plasmas ) ; ( ii ) magnetic _ field lines chaos _ which in general complicates ( and may preclude ) the construction of magnetic field line coordinates ; and ( iii ) _ nonlocal parallel transport _ in the limit of small collisionality . as a result of these challenges ,
standard finite difference and finite elements numerical methods suffer from a number of ailments .
chief among them are the pollution of perpendicular dynamics due to truncation errors in the discrete representation of the parallel heat flux , and the lack of a discrete maximum principle ( to enforce temperature positivity ) . despite the severity of these issues
, recent studies have succeeded in partially addressing some of them , and important progress has been made in the study of parallel transport .
reference @xcite discussed a finite element numerical implementation of nonlocal heat transport with applications including temperature flattening across magnetic islands and tokamak disruptions .
the use of high - order discretizations has been shown to mitigate numerical pollution of the perpendicular dynamics in finite - difference @xcite and finite - element methods @xcite .
a maximum principle has been shown to be enforceable by the use of limiters at the discrete level in finite differences @xcite , and finite elements @xcite , albeit with the effect of rendering both spatial discretizations formally first - order accurate . in ref .
@xcite a second order finite - difference iterative krylov method was used to find the steady state solution of the heat transport equation in a weakly chaotic magnetic field . motivated by the strong anisotropy typically encountered in magnetized plasmas ( @xmath15 ) , we study parallel heat transport in the extreme anisotropic regime ( @xmath16 ) .
in addition to the previously mentioned numerical difficulties , this regime presents potentially insurmountable issues regarding the algorithmic inversion of the discretized transport equation .
in particular , the potentially degenerate null space of the parallel transport operator might render the use of state - of - the - art scalable iterative inversion methods ( e.g. multigrid methods ) impractical . to overcome these numerical challenges , we present a novel lagrangian green s function approach .
the proposed method bypasses the need to discretize and invert the transport operators on a grid and allows the integration of the parallel transport equation without perpendicular pollution , while preserving the positivity of the temperature field .
the method is applicable to local and non - local transport in integrable , weakly chaotic , and fully chaotic magnetic fields . beyond the integration method ,
this letter presents novel physics results .
the difference between local and non - local parallel temperature mixing and flattening inside magnetic islands is studied .
of particular interest is the dependence of the temperature relaxation time on the wave number of the temperature perturbation .
understanding this problem is of significant interest to assess the impact of magnetic islands on magnetic confinement .
parallel transport is also studied in the case of weakly chaotic and fully chaotic @xmath17-d magnetic fields . in the case of weakly chaotic fields ,
the fractal hamiltonian structure of the magnetic islands implies that , as the ratio @xmath18 increases , the radial temperature profile approaches a devil - staircase .
going beyond previous studies @xcite , we unveil the fractal structure of the devil - staircase in the previously unaccessible @xmath19 regime .
these results open the possibility of a deeper understanding of the role of cantori which have been observed to act as partial transport barriers in numerical studies @xcite and experiments @xcite .
another problem of considerable interest in fusion and astrophysical plasmas is the understanding of electron heat transport in fully chaotic fields .
the case of local parallel transport has been extensively studied since the pioneering work in refs .
@xcite .
however , most studies have restricted their attention to local transport . here , in addition of the study of the local transport , we present novel results on the self - similar , non - gaussian spatio - temporal scaling of radial heat transport in the non - local case .
our starting point is the heat transport equation in a constant density plasma [ eq_1 ] _ t t = - , where @xmath20 is the heat flux . for local transport , in the limit @xmath21 , [ eq_2 ] * q*= - _ , where @xmath22 .
substituting ( [ eq_2 ] ) into ( [ eq_1 ] ) , we get [ eq_3 ] _ t t = -_s q _ , q_=-__s t , where @xmath23 is the derivative along the field line , and we have assumed the tokamak ordering @xmath24 . in the case of non - local transport , following ref .
@xcite , we consider [ eq_5 ] q_= _ 0^ dz , with @xmath25 . the case @xmath26 reduces to the collisionless heat flux characterized by the free - streaming of electrons along magnetic field lines @xcite , and the case @xmath27 reduces to the local diffusive case
the model in eq .
( [ eq_5 ] ) allows the interpolation between these two regimes .
in particular , eq .
( [ eq_5 ] ) allows the incorporation of parallel transport processes with underlying non - gaussian ( levy @xmath28-stable ) stochastic processes @xcite .
temperature mixing and homogenization in a magnetic island .
panel ( a ) shows the initial condition @xmath29 , and panel @xmath30 the final relaxed state , @xmath31 , where @xmath32 is the magnetic potential , for both local and non - local transport .
panels ( c ) and ( d ) show the solution at @xmath33 in the local and non - local cases respectively . ] substituting eq .
( [ eq_5 ] ) into eq .
( [ eq_1 ] ) , and assuming @xmath11 constant , the transport equation can be written in a compact form as [ eq_6 ] _ t t = _ ^_|s| t , where the operator @xmath34 denotes the symmetric fractional derivative of order @xmath28 , defined as @xmath35=-|k|^\alpha \hat{t}$ ] . here
, @xmath36 denotes the fourier transform @xcite . as expected , in the limit @xmath4 , eq .
( [ eq_6 ] ) reduces to the diffusion equation in eq .
( [ eq_3 ] ) .
the proposed method is based on the green s function solution of eq .
( [ eq_6 ] ) along the lagrangian trajectories of the magnetic field . the unique magnetic field line trajectory ( parametrized by the arc length ) that goes through a point @xmath37 is given by the solution of [ eq_9 ] = , ( s=0)=*r*_p .
thus , given an initial condition in the whole domain , @xmath38 , the initial condition along the field line is @xmath39 and the temperature at @xmath40 at time @xmath41 is given by [ eq_8 ] t(*r*_p ,
t ) = _ s_1^s_2 t_0 g_(s , t ) ds , where @xmath42 is the green s function of eq .
( [ eq_6 ] ) . for unbounded field lines , @xmath43 and the green s function
is given by [ greens ] g_(s , t)= _
-^ e^- t | k |^-
i k s d k . for @xmath4 , eq . ( [ greens ] ) gives the gaussian distribution [ eq_10 ] g_2(s , t ) =( _ t)^-1/2 , and in the non - local free streaming case , @xmath3 , it gives the cauchy distribution [ eq_11 ] g_1(s , t ) = .
for general @xmath28 , @xmath44 $ ] where @xmath45 is the symmetric @xmath28-stable levy distribution , see for example ref .
@xcite . in the case of
closed ( periodic ) field lines , the integration domain in eq .
( [ eq_8 ] ) is a finite interval @xmath46 with @xmath47 , and one has to use the periodic green s function , @xmath48 , obtained by mapping the unbounded green s function , @xmath42 , into the periodic domain .
poincare plot of the weakly chaotic magnetic field used in solution of the parallel heat transport equation shown in fig .
[ fig_3 ] . ] radial temperature profile of the time - asymptotic solution of the parallel heat transport equation for the weakly chaotic magnetic field in fig .
[ fig_2 ] .
the zooms in the successive panels unveil the fractal structure of the devil - staircase profile . ] at this point it is important to indicate a fundamental difference between the work in refs @xcite and the method proposed here . in refs .
@xcite , the flux is calculated throughout the computational domain by integrating along the field lines . however , in these references , once the flux is computed , the lagrangian approach is abandoned and the flux is mapped to gaussian quadrature points for the finite - element standard integration of the temperature evolution equation on a grid . on the other hand ,
the method proposed here is fully lagrangian and completely bypasses the use of finite differences or finite elements integration schemes , circumventing the numerical limitations discussed before .
another clear advantage of the use of the green s function is that the solution at an arbitrary time @xmath49 is obtained directly from the integral in eq .
( [ eq_8 ] ) .
this is to be contrasted with the standard explicit and implicit time discretization schemes that require the knowledge of the solution at previous times @xmath50 before the solution at @xmath49 can be constructed . a similar decoupling occurs in space . according to eq .
( [ eq_8 ] ) , the evaluation of @xmath51 at @xmath37 is fully decoupled from the solution at @xmath52 , yielding naturally to a massively parallel approach . in
what follows , we use the lagrangian green s function method to study local and non - local parallel transport in magnetic fields of increasing levels of complexity .
we start with the study of transport in magnetic islands , which we model with a @xmath53-d magnetic field with vector potential @xmath54 , where @xmath55 and @xmath56 are radial and angular coordinates respectively .
of particular interest is the difference in the temperature relaxation properties in the presence of local and nonlocal transport .
figure 1 shows the solution of the parallel transport of an initial condition of the form @xmath57 with @xmath58 .
as expected , for large enough times , the temperature relaxes to a unique solution of the form @xmath31 in both the local and non - local cases .
however , the dynamic process leading to the final relaxed state is different .
as the figure shows , temperature mixing in the non - local case is less homogeneous and slower than in the local case . in general , from eq .
( [ greens ] ) it follows that the homogenization time of a temperature perturbation with mode number @xmath0 scales as @xmath59 with @xmath4 ( @xmath3 ) in the case of local ( non - local ) transport . for the study of transport in @xmath17-d magnetic fields ,
we assume cylindrical geometry periodic in @xmath60 with period @xmath61 where @xmath62 is a constant .
the magnetic field consists of a perturbed screw - pinch of the form = ( r b /)/[1+(r/)^2 ] _
+ b_0 _ z + * b*_1(r,,z ) , with @xmath63 , @xmath64 , and @xmath65 constants .
the magnetic potential of the perturbation , @xmath66 , consists of a superposition of modes [ eq_20 ] a_z(r,,z)=_m , n a_mn(r ) ( m - n z / r + _ mn ) , with [ eq_21 ] a_mn= a(r ) ( ) ^m . for each @xmath67 , the values of @xmath68and @xmath69 are chosen so that the safety factor satisfies @xmath70 and @xmath71 .
the prefactor @xmath72 is included to guarantee the regularity of the radial eigenfunction near the origin , @xmath73 .
the function , @xmath74/2 $ ] , is introduced to guarantee the vanishing of the perturbation for @xmath75 and the existence of well - defined flux surfaces at the plasma boundary .
self - similar spatio - temporal evolution of the radial temperature profile for local , @xmath4 , transport in a fully chaotic magnetic field . in this case
the scaling function is gaussian and the scaling exponent is @xmath8 .
] self - similar spatio - temporal evolution of the radial temperature profile for non - local , @xmath3 , transport in a fully chaotic magnetic field . in this case
the scaling function is strongly - non - gaussian and , as the dashed - line fit shows , exhibits algebraic decay of the form @xmath76 .
the scaling exponent is @xmath10 . ] in the study of transport in weakly chaotic fields , only two modes were included .
as the poincare plot in figure [ fig_2 ] shows , the magnetic field in this case exhibits a rich fractal - like structure resulting from the existence of higher - order resonances .
figure [ fig_3 ] shows the time asymptotic , final radial temperature solution along the @xmath77 horizontal line in fig .
[ fig_2 ] , corresponding to the initial condition @xmath78 .
as expected , the same state is reached in the presence of local and non - local transport .
however , as mentioned before , the relaxation time is longer in the non - local case .
the fractal structure of the magnetic field gives rise to a devil - staircase temperature profile in which horizontal sections ( resulting from high - order resonances ) are mixed with vertical sections ( resulting from kam invariant circles and cantori ) . to study transport in a fully chaotic magnetic field we considered a set of @xmath79 strongly overlapping modes . in this case , the poincare plot ( not shown ) is fully hyperbolic and does not exhibit any structure .
the initial condition consists of a narrow cylindrical shell " of the form @xmath80 ^ 2 $ ] , with @xmath81 and @xmath82 .
figures [ fig_4 ] and [ fig_5 ] show the time evolution of the radial profile of the temperature averaged in @xmath56 and @xmath60 in the local and the non - local free streaming cases . in both cases
, the temperature exhibits an asymptotic self - similar evolution of the form t _ , z ( , t ) = ( t ) ^-/2 l ( ) , where the similarity variable is defined as @xmath83 with @xmath84 the scaling exponent . from here
, it follows that the second moment scales as @xmath85 . as fig .
[ fig_4 ] shows , in the local transport case the scaling function is approximately gaussian , @xmath86 .
consistent with ref .
@xcite , in this case @xmath87 .
this subdiffusive scaling results from the combination of the diffusive transport along the field line and the quasilinear diffusion of the field line itself due to the magnetic field line chaos . on the other hand , in the nonlocal case
, the scaling function is strongly non - gaussian and exhibits an algebraic decay of the form @xmath88 .
interestingly , in the non - local case @xmath89 .
this diffusive scaling results from the coupling of the quasilinear diffusion of the field line due to chaos and the free - streaming transport along the field line .
summarizing , this letter presented a lagrangian green s function method for the accurate and efficient computation of purely parallel ( @xmath21 ) local and non - local transport in arbitrary magnetic fields with constant @xmath11 .
because of the parallel nature of the lagrangian calculation , the formulation naturally leads to a massively parallel implementation , suitable for today s supercomputers .
the method was applied to compute temperature mixing in magnetic islands , for which the ratio of the non - local free - streaming , @xmath90 , and diffusive , @xmath91 , transport relaxation times scales as @xmath92 . radial transport in @xmath17-d magnetic fields in cylindrical geometry was also studied . in the case of weakly chaotic fields ,
the fractal structure of the devil staircase of the radial temperature profile was resolved . in the case of fully chaotic fields ,
the radial transport exhibited self - similar spatio - temporal behavior .
contrary to the well - known diffusive case , the non - local case exhibits a non - gaussian , algebraic decaying temperature profile with scaling exponent @xmath10 .
current work includes the implementation of the lagrangian green s function method as integral part of a more general multiscale framework that incorporates finite @xmath12 , non - constant @xmath11 , and heat sources .
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the coefficients in the zero mode hamiltonian ( [ hzero ] ) are @xmath148\big],\nonumber\\ b&=&\frac{1}{2n_0 ^ 2}\left[\varepsilon(1+m^2 ) -8c_2{\cal o}_{\phi\theta}m^2\right]\nonumber\\ c&=&\frac{1}{3n_0 ^ 2}(\varepsilon-4c_2{\cal o}_{\phi\theta}),\nonumber\\ \alpha&=&\frac{\sqrt{2}}{6n_0 ^ 2}(1 + 3m^2 ) ( \varepsilon-4c_2{\cal o}_{\phi\theta}),\nonumber\\ \beta&=&-\frac{2m}{\sqrt{3}n_0 ^ 2}(\varepsilon -4c_2{\cal o}_{\phi\theta}),\nonumber\\ \gamma&=&-\frac{2m}{n_0 ^ 2}\sqrt{\frac{2}{3}}\left [ \varepsilon - c_2{\cal o}_{\phi\theta}(1 + 3m^2)\right],\nonumber\\ \eta&=&-3c_2n_0 ^ 2{\cal o}_{\phi\phi}/\hbar^2.\nonumber\end{aligned}\ ] ]
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we discuss the quantum phases and their diffusion in a spinor-1 atomic bose - einstein condensate . for ferromagnetic interactions ,
we obtain the exact ground state distribution of the phase fluctuations corresponding to the total atom number ( @xmath0 ) , the magnetization ( @xmath1 ) , and the alignment ( or hypercharge ) ( @xmath2 ) of the system .
the mean field ground state is shown to be stable against these fluctuations , which dynamically recover the two continuous symmetries associated with the conservation of @xmath0 and @xmath1 as in current experiments .
since the observation of bose - einstein condensation of trapped atomic clouds @xcite , the coherence properties of the condensate has become the focus of many theoretical studies @xcite . within the mean - field theory ,
it is commonly assumed that the condensate can be described by a @xmath3 symmetry breaking field @xcite , equivalent to a coherent state assumption of the ground state .
( see refs .
@xcite for discussions of @xmath3-symmetric approaches ) .
although quite successful in providing theoretical understanding to many experimental observations , such a coherent state assumption is not necessarily consistent with real experimental situations , where the fluctuations of the atom numbers are difficult to control @xcite .
a coherent state leads to a poisson distribution of atoms . for a ground state with average of @xmath0 atoms ,
the associated number fluctuations are of the order @xmath4 . as was initially pointed out in refs .
@xcite , this number fluctuation of a coherent state condensate leads to the diffusion " ( or spreading ) of its initial phase . in a scalar condensate
, this diffusion , a dynamic attempt to restore the @xmath3 symmetry of the interacting atomic system , can be studied in terms of a zero mode , or the goldstone mode of the condensate @xcite .
more physically meaningful discussions in terms of the relative phase of two condensates were studied soon afterwards @xcite . experimentally , starting with the remarkable direct observation of the first order coherence in an interference experiment @xcite , direct correlations between number and phase fluctuations were observed with a condensate in a periodic potential @xcite , and more recently , in the remarkable mott insulating state @xcite obtained by loading a superfluid condensate into an optical lattice @xcite . the emergence of spinor-1 condensates @xcite ( of atoms with hyperfine quantum number @xmath5 ) has created new opportunities to understand quantum coherence and the associated number / phase dynamics in a three component condensate @xcite . in this paper , we investigate the quantum phase dynamics of a spinor-1 condensate due to atom number fluctuations .
we will focus on ferromagnetic interactions , when the condensate wave functions for the three spin components share the same spatial mode @xcite ; in this case , the phase fluctuations translate into fluctuations of the direction of the macroscopic condensate spin .
we consider a system of @xmath0 spin-1 bosonic atoms interacting via only s - wave scattering @xcite . a weak magnetic field @xmath6 ( as always exists in an experiment )
fixes the quantization z - axis such that quadratic zeeman effect can be neglected . in the second - quantized form ,
the hamiltonian is @xcite @xmath7\psi_i(\vec r)\nonumber\\ & & + \frac{c_0}{2}\sum_{i , j}\int d\,\vec r\psi_i^\dag(\vec r)\psi_j^\dag(\vec r ) \psi_i(\vec r)\psi_j(\vec r)\nonumber\\ & & + \frac{c_2}{2}\sum_{i , j , k , l}\int d\,\vec r\psi_i^\dag(\vec r ) \psi_j^\dag(\vec r)\vec{\mathbf f}_{ik } \cdot\vec{\mathbf f}_{jl}\psi_l(\vec r)\psi_k(\vec r),\hskip 18pt \label{ham}\end{aligned}\ ] ] where @xmath8 ( @xmath9 ) denotes the annihilation operator for the @xmath10-th component of a spinor-1 field .
the trapping potential @xmath11 is assumed harmonic and spin - independent .
the larmor precessing frequency is @xmath12 with @xmath13 the magnetic dipole moment for state @xmath14 .
the pseudo potential coefficients are @xmath15 and @xmath16 , with @xmath17 ( @xmath18 ) the s - wave scattering length for two spin-1 atoms in the combined symmetric channel of total spin @xmath19 ( @xmath20 ) .
@xmath21 is the mass of atom , and @xmath22 is the spin 1 operator @xcite .
hamiltonian ( [ ham ] ) is invariant under u(1 ) gauge transformation @xmath23 and so(3 ) spin rotations @xmath24 ( for @xmath25 ) @xcite .
a non - zero @xmath6 or the conservation of magnetization @xmath26 reduces the so(3 ) to its subgroup so(2 ) generated by @xmath27 .
following the bogoluibov theory we assume there exist three ` large ' condensate components @xmath28 around which we study the small quantum fluctuations ( off - condensate excitations ) via @xmath29 where @xmath30 is the number of condensed atoms in component @xmath10 . at near zero temperatures , higher than quadratic terms of @xmath31 in the hamiltonian ( [ ham ] ) are neglected .
the assumed mean field ground state @xmath32 breaks both continuous symmetries u(1 ) and so(2 ) , thus we expect to observe multiple zero energy goldstone modes @xcite . in the first order
, we obtain the usual coupled gross - pitaevskii equation ( gpe ) for the condensate modes @xmath33 .
the quantum fluctuations @xmath34 obey the usual bogoluibov - de gennes equations ( bdges ) @xcite .
the condensate number fluctuations , can be studied through the number fluctuation operators @xcite @xmath35\nonumber\\ & \approx & { 1\over \sqrt{n}}\int d\vec r [ \psi_j^\dag(\vec r ) \psi_j(\vec
r)-n\phi_j^*(\vec r)\phi_j(\vec r)].\end{aligned}\ ] ] using the gpes and the bdges @xcite their equations of motion are @xcite , @xmath36 from which we find that both @xmath37 and @xmath38 are constants of motion , an obvious outcome since the hamiltonian ( [ ham ] ) commutes with operators of both the total number of atoms and magnetization .
we define the phase operator @xmath39,\end{aligned}\ ] ] with @xmath40 the associated phase mode functions .
denote @xmath41 , the canonical quantization condition @xmath42=i\hbar$ ] is satisfied if the constraint @xmath43 is enforced , where @xmath44 $ ] .
as for a scalar or binary condensate @xcite @xmath45 we note that @xmath46=i\hbar\delta_{jk}j_j$ ] , @xmath47=\delta_{jk}\phi_j(\vec r)$ ] , and @xmath48=-i\hbar\delta_{jk}\theta_j(\vec r)$ ] .
these lead to @xmath49 @xcite , which helps to define the complete dynamic equations for the number and phase fluctuations @xcite .
including both the initial and time phases , the ground state wave functions can be generally expressed as @xmath50 with a common chemical potential @xmath51 , a lagrange multiplier enforcing the conservation of atom numbers .
minimization of the total energy eq .
( [ ham ] ) leads to @xmath52 @xcite .
it was further proven in ref .
@xcite that the steady state solution takes the same spatial mode for each of its three spin components ( for ferromagnetic interactions ) , i.e. @xmath53 where the real - valued mode function @xmath54 is normalized to unity , and governed by an equivalent scalar gpe @xmath55\phi(\vec r ) = \mu\phi(\vec r ) , \label{gpe1}\end{aligned}\ ] ] of a scattering length @xmath18 ( note @xmath56 ) .
define @xmath57 and @xmath58 as the relative atom numbers and relative magnetization , @xmath59 and @xmath60 in the ferromagnetic ground state @xcite .
similarly , the phase functions share the same spatial mode and can be generally expressed as @xmath61 with @xmath62 governed by the following equation @xmath63\theta(\vec r ) = n\tilde{u}\phi(\vec r ) .
\label{eqphase}\end{aligned}\ ] ] the normalization @xmath64 determines the goldstone parameter @xmath65 , which in the thomas - fermi ( tf ) limit is @xmath66(a_{\rm ho}/15 a_2)^{3/5}n^{-3/5}$ ] , with @xmath67 the ground state size @xmath68 of a harmonic trap with a frequency @xmath69 . equations ( [ gpe1 ] ) and ( [ eqphase ] ) are identical to their counterparts for a scalar condensate @xcite . in the ferromagnetic ground state , individual atomic spins align along the same direction , thus they collide only in the symmetric total spin @xmath70 channel . to conserve the magnetization with respect to the @xmath6-field direction , the condensate spin direction is simply tilted at an angle @xmath71 .
if this direction is taken as the quantization axis , then the spinor condensate behaves essentially as a scalar one @xcite when the @xmath6-field is zero . with the mode functions for atom number and phase fluctuations
, we find the zero mode dynamics can be expressed in terms of an associated hamiltonian . after laborious calculation
we arrive at @xcite @xmath72 with @xmath73 , @xmath74 , redefined as canonically conjugated variables @xmath75 and @xmath76 .
@xmath77 & @xmath78 are two hermitian and positive - definite matrices involving only parameters of the system .
thus @xmath79 , the assumed ground state , a mean field symmetry breaking state with coherent condensate amplitudes @xmath32 is stable .
the associated quantum fluctuations of atom numbers can be studied with the linearization approximation eq .
( [ bog ] ) .
matrices @xmath77 & @xmath78 are simplified with @xmath80 and @xmath81 .
we also find @xmath82 with @xmath83\theta(\vec r)$ ] a non - negative quantity .
we note that matrix @xmath84 is diagonalized by an orthogonal transformation @xmath85 , @xmath86 , and @xmath87 @xcite .
the corresponding @xmath88 , @xmath89 , and @xmath90 are respectively fluctuations of the total number of atoms , the magnetization , and the alignment . with these new collective operators ,
the zero mode hamiltonian becomes @xmath91 where all the coefficients are listed in the appendix .
replacing @xmath92 by @xmath93 , hamiltonian ( [ hzero ] ) leads to the ground state distribution of fluctuations @xmath94 with a width @xmath95 and the zero point energy @xmath96^{1/2}$ ] .
we find @xmath97 and @xmath98 in the tf limit ( and also taking the small @xmath99 ) .
figure [ fig2 ] shows selected results for the @xmath0-dependence of @xmath100 and @xmath101 . while @xmath100 increases monotonically with @xmath0 , the ground state width @xmath101 for @xmath102 peaks at some intermediate values of atom numbers and eventually saturates in the limit of large @xmath0 ( or strong interactions ) . and @xmath103 for a @xmath104rb condensate in a cylindrically symmetric trap with @xmath105 ( hz ) and @xmath106 .
@xmath107 ( solid line ) , @xmath108 ( dashed line ) , and @xmath109 ( dash - dotted line ) . , width=312 ] the above ground state distribution of @xmath110 can easily be understood . for a condensate with fixed total number of atoms and magnetization , the conservations of @xmath0 and @xmath1 require @xmath111 , which lead to @xmath112 , i.e. , completely diffused phases
; the ferromagnetic interaction , nevertheless , prepares a correlated ground state such that both distributions for @xmath90 and @xmath102 take a gaussian form with @xmath113 and @xmath114 .
such a distribution in @xmath90 and @xmath102 is in fact the minimal uncertainty coherent state consistent with the symmetry breaking ground state .
however , it is experimentally difficult to produce such a condensate having fixed atom numbers and phase fluctuations .
for any initial state , the dynamic solution @xmath115 as governed by the hamiltonian eq .
( [ hzero ] ) for @xmath116 $ ] can be used .
this solution has a simple structure @xcite ; in addition to oscillating terms of the forms @xmath117 and @xmath118 , the phase fluctuations of @xmath119 and @xmath120 also contain diffusion terms proportional to @xmath121 , which indicates the linearization approximation of eq .
( [ bog ] ) is valid only for a finite duration . finally , let s consider the diffusion of the direction of the condensate spin @xmath122 . because all three spin components share the same spatial mode function @xcite , individual spins of bose condensed atoms are parallel for ferromagnetic interactions , i.e. they act as a macroscopic magnetic dipole pointing along the same direction ( independent of the spatial coordinates ) : @xmath123 while undergoing precessing with respect to the z - axis . @xmath124 .
as the phase dynamics attempts to restore the u(1 ) and so(2 ) symmetries of the system , their respective initial values become irrelevant .
the fluctuation becomes @xmath125 $ ] with a zero average .
using @xmath126 , @xmath127 , and @xmath128 to denote the initial variances of @xmath0 , @xmath1 , and @xmath2 , we find in spherical coordinates @xmath129 ( note @xmath130 ) , @xmath131 and @xmath132 . both are fixed constants @xmath133 ^ 2\right\rangle&= & 3\sigma_n^2,\\ \left\langle\left[\delta f_\theta(t)\right]^2\right\rangle&=&\frac{1}{1-m^2 } \left ( 3m^2\sigma_n^2 + 2\sigma_m^2\right),\end{aligned}\ ] ] ( due to conservations of @xmath0 and @xmath1 ) when respective fluctuations in @xmath0 , @xmath1 , and @xmath2 are uncorrelated . in the azimuthal @xmath134 direction
a simple phase diffusion results @xmath135 with @xmath136/2\hbar,$ ] and a diffusion rate @xmath137 , proportional to @xmath138 , ( which can be explained in terms of the single axis twisting of @xmath139 in the isospin subspace of a spior-1 condensate @xcite ) .
the @xmath0- and @xmath140-dependence of the diffusion parameter @xmath141 are shown in fig .
[ fig3 ] for a @xmath104rb condensate . in the tf limit and
applying the results of ref .
@xcite , we find that @xmath142 , generally a very small quantity .
@xmath143 is the tf radius .
typically , the dephasing rate is a fraction of @xmath144 for a condensate of @xmath145 atoms and @xmath146 , i.e. its macroscopic spin direction is lost in a few cycles of trap oscillation .
figure [ fig3 ] also indicates that the faster a ferromagnetic condensate diffuses its pointing direction along the precession direction , the larger and more tightly confined a condensate is .
but for @xmath141.,width=312 ] when @xmath25 , zero mode dynamics becomes much simpler if the condensate spin direction is taken as z - axis .
although there still exist more than one goldstone mode in this case , the mean field ground state corresponds to all atoms in state @xmath147 @xcite .
thus to leading order , the only relevant fluctuation is that of the total number of atoms ( corresponding to the u(1 ) symmetry ) , and the phase diffusion dynamics becomes the same as in a scalar condensate @xcite .
our theory as developed in this paper does reduce to this simple limit . in conclusion ,
we have studied in detail quantum phase diffusions of a spinor-1 condensate with ferromagnetic interactions .
the condensate ground state is very simple , all three spin components have the same spatial mode function and their associated phase functions are also identical .
we have constructed the zero mode hamiltonian for the condensate number and phase fluctuations , and solved for the ground state distribution of these fluctuations when both @xmath0 and @xmath1 are conserved .
furthermore , we have obtained analytically the dynamic number and phase fluctuations relating to both the quantum phase diffusion and the initial distribution of these fluctuations .
we have identified a quantum phase diffusion coefficient for the pointing direction of the condensate spin and recovered its small - time quadratic t - dependent spreading .
this work is supported by the nsf grant no .
phys-0140073 and by a grant from nsa , arda , and darpa under aro contract no .
daad19 - 01 - 1 - 0667 .
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this research was supported in part by the u.s .
department of energy under grants no . da - ac02 - 76-er02289 task b and no .
de - fg02 - 95er40896 and in part by the university of wisconsin research committee with funds granted by the wisconsin alumni research foundation .
the research of ts is supported in part by the u.s .
department of energy under grant no .
de - fg02 - 91er40626 .
one of us ( mmb ) would like to thank the aspen center for physics for its hospitality during the preparation of this manuscript .
m. m. block _ et al .
_ , photon - proton and photon - photon scattering from nucleon - nucleon forward amplitudes , e - print archive : hep - ph/9809403 , phys . rev .
d,*60 * 054024 , 1999 .
r. m. baltrusaitis _
lett . * 52 * , 1380 , 1984 .
et al . _ ,
lett . * 70 * , 525 , 1993 . m. aglietta _
et al _ , proc 25th icrc ( durban ) * 6 * , 37 , 1997 .
r. s. fletcher _
et al . _ ,
rev . d*50 * , 5710 , 1994 .
see http://sunshine.chpc.utah.edu/research/cosmic/hires/ the pierre auger project design report , fermilab report ( feb . 1997 ) .
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-.35 in we use the high energy predictions of a qcd - inspired parameterization of all accelerator data on forward proton - proton and antiproton - proton scattering amplitudes , along with glauber theory , to predict proton
air cross sections at energies near @xmath0 30 tev .
the parameterization of the proton - proton cross section incorporates analyticity and unitarity , and demands that the asymptotic proton is a black disk of soft partons . by comparing with the p - air cosmic ray measurements , our analysis results in a constraint on the inclusive particle production cross section .
@=11
@=12 .65 in cosmic ray experiments measure the penetration in the atmosphere of particles with energies in excess of those accelerated by existing machines interestingly , their energy range covers the energy of the large hadron collider ( lhc ) and extends beyond it .
however , extracting proton proton cross sections from cosmic ray observations is far from straightforward @xcite . by a variety of experimental techniques , cosmic ray experiments
map the atmospheric depth at which cosmic ray initiated showers develop .
the measured shower attenuation length ( @xmath1 ) is not only sensitive to the interaction length of the protons in the atmosphere ( @xmath2 ) , with @xmath3 but also depends on the rate at which the energy of the primary proton is dissipated into electromagnetic shower energy observed in the experiment .
the latter effect is parameterized in eq.([eq : lambda_m ] ) by the parameter @xmath4 ; @xmath5 is the proton mass and @xmath6 the inelastic proton - air cross section .
the value of @xmath4 depends on the inclusive particle production cross section in nucleon and meson interactions on the light nuclear target of the atmosphere and its energy dependence .
we here ignored the fact that particles in the cosmic ray `` beam '' may be nuclei , not just protons .
experiments allow for this by omitting from their analysis showers which dissipate their energy high in the atmosphere , a signature that the initial energy is distributed over the constituents of a nucleus .
the extraction of the pp cross section from the cosmic ray data is a two step process .
first , one calculates the @xmath7-air total cross section from the measured inelastic cross section @xmath8 next , the glauber method@xcite is used to transform the measured value of @xmath6 into a proton proton total cross section @xmath9 ; all the necessary steps are calculable in the theory . in eq.([eq : spa ] ) the measured cross section for particle production is supplemented with @xmath10 and @xmath11 , the elastic and quasi - elastic cross section , respectively , as calculated by the glauber theory , to obtain the total cross section @xmath12 .
the subsequent relation between @xmath6 and @xmath9 involves the slope of the forward scattering amplitude for elastic @xmath13 scattering , @xmath14 , @xmath15_{t=0 } \,,\ ] ] and is shown in fig.[fig : p - air ] , which plots @xmath16 against @xmath9 , for 5 curves of different values of @xmath6 .
this summarizes the reduction procedure from @xmath6 to @xmath9 @xcite .
also plotted in fig.[fig : p - air ] is a curve of @xmath16 _ vs. _ @xmath9 which will be discussed later .
a significant drawback of the method is that one needs a model of proton air interactions to complete the loop between the measured attenuation length @xmath1 and the cross section @xmath6 , _
i.e. , _ the value of @xmath4 in eq .
( [ eq : lambda_m ] ) . a proposal to minimize the impact of theory is the topic of this letter .
we have constructed a qcd - inspired parameterization of the forward proton proton and proton antiproton scattering amplitudes@xcite which is analytic , unitary and fits all data of @xmath17 , @xmath16 and @xmath18 , the ratio of the real - to - imaginary part of the forward scattering amplitude ; see fig.[fig : ppcurves ] .
we emphasize that all 3 quantities are simultaneously fitted .
using vector meson dominance and the additive quark models , it accommodates a wealth of data on photon - proton and photon - photon interactions without the introduction of new parameters@xcite .
because the model is both unitary and analytic , it has high energy predictions that are essentially theory independent . in particular
, it also _ simultaneously _ fits @xmath9 and @xmath16 , forcing a relationship between the two .
specifically , the @xmath16 _ vs. _ @xmath9 prediction of the model is shown as the dashed curve in fig.[fig : p - air ] .
the dot corresponds to our prediction of @xmath9 and @xmath16 at @xmath19 = 30 tev .
it is seen to be slightly below the curve for 490 mb , the lower limit of the fly s eye measurement , which was made at @xmath20 30 tev .
the percentage error in the prediction of @xmath9 at @xmath21 tev is @xmath22% , due to the statistical error in the fitting parameters ( see references @xcite,@xcite ) . in fig.[fig :
sigpp_p - air ] , we have plotted the values of @xmath9 _ vs. _ @xmath6 that are deduced from the intersections of the @xmath16-@xmath9 curve with the @xmath6 curves of fig.[fig : p - air ] .
figure [ fig : sigpp_p - air ] allows the conversion of the measured @xmath6 to @xmath9 .
the percentage error in @xmath6 is @xmath23 % near @xmath24 mb , due to the error in @xmath9 from the model parameter uncertainties .
our prediction for the total cross section @xmath9 as a function of energy is confronted with all of the accelerator and cosmic ray measurements@xcite in fig.[fig : sigtodorpp ] . for inclusion in fig.[fig : sigtodorpp ] , we have calculated the cosmic ray values of @xmath9 from the _ published _ experimental values of @xmath6 , using the results of fig.[fig : sigpp_p - air ] .
we note the systematic underestimate of the cosmic ray points , roughly about the level of one standard deviation .
it is at this point important to recall eq.([eq : lambda_m ] ) and consider the fact that the extraction of @xmath6 from the measurement of @xmath1 requires a determination of the parameter @xmath4 .
the measured depth @xmath25 at which a shower reaches maximum development in the atmosphere , which is the basis of the cross section measurement in ref .
@xcite , is a combined measure of the depth of the first interaction , which is determined by the inelastic cross section , and of the subsequent shower development , which has to be corrected for .
the position of @xmath25 also directly affects the rate of shower attenuation with atmospheric depth which is the alternative procedure for extracting @xmath6
. the model dependent rate of shower development and its fluctuations are the origin of the deviation of @xmath4 from unity in eq.([eq : lambda_m ] ) .
its values range from 1.5 for a model where the inclusive cross section exhibits feynman scaling , to 1.1 for models with large scaling violations@xcite .
the comparison between data and experiment in fig.[fig : sigtodorpp ] is further confused by the fact that the agasa@xcite and fly s eye@xcite experiments used different values of @xmath4 in the analysis of their data , _
i.e. , _
agasa used @xmath26 and fly s eye used @xmath27 .
we therefore decided to match the data to our prediction and extract a common value for @xmath4 .
this neglects the possibility that @xmath4 may show a weak energy dependence over the range measured . by combining the results of fig.[fig : ppcurves](a ) and fig.[fig : sigpp_p - air ] , we can plot our prediction of @xmath6 _ vs. _ @xmath19 . to obtain @xmath4 ,
we leave it as a free parameter and make a @xmath28 fit to the rescaled high energy cosmic ray data for @xmath6 . in fig.[fig : p - aircorrected2 ] we have replotted the published high energy data for @xmath6 , against our prediction of @xmath6 _ vs. _ @xmath19 , using the common value of @xmath29 , obtained from a @xmath28 fit .
the error of @xmath30 is the statistical error of the @xmath28 fit , whereas the error of @xmath31 is the systematic error due to the error in the prediction of @xmath6 .
clearly , we have an excellent fit , with good agreement between agasa and fly s eye .
the analysis gives @xmath32 for 6 degrees of freedom ( the low @xmath28 is probably due to overestimates of experimental errors ) .
of course , the improved fit of the cosmic ray data has no effect on the fit to the accelerator data .
our result for @xmath4 is interesting it is close to the value of @xmath33 obtained using the sibyll simulation@xcite for inclusive particle production .
this represents a consistency check in the sense that our model for forward scattering amplitudes and sybill share the same underlying physics .
the increase of the total cross section with energy to a black disk of soft partons is the shadow of increased particle production which is modeled by the production of ( mini)-jets in qcd .
the difference between the @xmath4 values of 1.20 and 1.33 could be understood because the experimental measurement integrates showers in a relatively wide energy range , which tends to increase the value of @xmath4 .
we predict @xmath34 mb for the total cross section at lhc energy ( 14 tev ) , where the error is due to the statistical errors of the fitting parameters . in the near term
, we look forward to the possibility of repeating this analysis with the higher statistics of the hires @xcite cosmic ray experiment that is currently in progress and the auger @xcite observatory .
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we wish to thank the kekb accelerator group for the excellent operation of the kekb accelerator .
we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of industry , science and resources ; the national science foundation of china under contract no .
10175071 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea and the chep src program of the korea science and engineering foundation ; the polish state committee for scientific research under contract no .
2p03b 17017 ; the ministry of science and technology of the russian federation ; the ministry of education , science and sport of the republic of slovenia ; the national science council and the ministry of education of taiwan ; and the u.s .
department of energy .
the fox - wolfram moments were introduced in g. c. fox and s. wolfram , phys .
lett . * 41 * , 1581 ( 1978 ) .
the fisher discriminant used by belle is described in k. abe _ et al . _
( belle collab . )
lett . * 87 * , 101801 ( 2001 ) and k. abe _ et al . _
( belle collab . ) , phys . lett . *
b511 * , 151 ( 2001 ) .
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we report results on the decay @xmath0 and its charge conjugate using a data sample of 85.4 million @xmath1 pairs recorded at the @xmath2 resonance with the belle detector at the kekb asymmetric @xmath3 storage ring
. ratios of branching fractions of cabibbo - suppressed to cabibbo - favored processes are determined to be @xmath4 , @xmath5 and @xmath6 where the indices 1 and 2 represent the cp=+1 and cp=@xmath71 eigenstates of the @xmath8 system , respectively .
we find the partial - rate charge asymmetries for @xmath0 to be @xmath9 and @xmath10 .
the extraction of @xmath11 @xcite , an angle of the kobayashi - maskawa triangle @xcite , is a challenging measurement even with modern high luminosity @xmath12 factories .
recent theoretical work on @xmath12 meson dynamics has demonstrated the direct accessibility of @xmath11 using the process @xmath13
@xcite . if the @xmath14 is reconstructed as a cp eigenstate , the @xmath15 and @xmath16 processes interfere .
this interference leads to direct cp violation as well as a characteristic pattern of branching fractions .
however , the branching fractions for @xmath17 meson decay modes to cp eigenstates are only of order 1 % . since cp violation through interference
is expected to be small , a large number of @xmath12 decays is needed to extract @xmath18 . assuming the absence of @xmath19 mixing , the observables sensitive to cp violation that are used to extract the angle @xmath18 @xcite are , @xmath20 where the ratios @xmath21 and @xmath22 are defined as @xmath23@xmath24@xmath25 and @xmath26 are cp - even and cp - odd eigenstates of the neutral @xmath17 meson , @xmath27 is the ratio of the amplitudes of the two tree diagrams shown in fig . 1 and @xmath28 is their strong - phase difference
. the ratio @xmath29 corresponds to the magnitude of cp asymmetry and is suppressed to the level of @xmath30 due to the ckm factor @xmath31 and a color suppression factor @xmath32 .
note that the asymmetries @xmath33 and @xmath34 have opposite signs .
the ratio of the cabibbo - suppressed decay @xmath35 to the cabibbo - favored decay @xmath36 has been reported by cleo @xcite to be @xmath37 while belle finds @xmath38 @xcite .
assuming factorization , the ratio @xmath22 is expected to be @xmath39 in the tree - level approximation , where @xmath40 is the cabibbo angle , and @xmath41 and @xmath42 are meson decay constants .
the measurements are in good agreement with this theoretical expectation . previously , belle reported the observation of the decays @xmath43 and @xmath44 with @xmath45 @xcite . this paper reports more precise measurements of these decays with a data sample of @xmath46 , containing 85.4 million @xmath1 pairs , collected with the belle detector at the kekb asymmetric - energy @xmath47 ( 3.5 on 8 gev ) collider operating at the @xmath2 resonance . at kekb , the @xmath2
is produced with a lorentz boost of @xmath48 nearly along the electron beamline .
the belle detector is a large - solid - angle magnetic spectrometer that consists of a three - layer silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of silica aerogel threshold erenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter ( ecl ) comprised of csi(tl ) crystals located inside a super - conducting solenoid coil that provides a 1.5 t magnetic field .
an iron flux - return located outside of the coil is instrumented to detect @xmath49 mesons and to identify muons ( klm ) .
the detector is described in detail elsewhere @xcite .
we reconstruct @xmath14 mesons in the following decay channels . for the flavor specific mode ( denoted by @xmath50 ) , we use @xmath51 @xcite . for cp
= + 1 modes , we use @xmath52 and @xmath53 while for cp = @xmath71 modes , we use @xmath54 , @xmath55 , @xmath56 , @xmath57 and @xmath58 .
the charged track , @xmath59 and @xmath60 selection requirements have been described in ref . @xcite . for each charged track , information from the acc , tof and specific ionization measurements from the cdc are used to determine a @xmath61 likelihood ratio @xmath62 = @xmath63 , where @xmath64 and @xmath65 are kaon and pion likelihoods . for kaons ( pions ) from the @xmath66 mode we used the particle identification requirement of @xmath67 . for kaons from the @xmath68 mode
we require @xmath69 while for pions from @xmath70 mode we require @xmath71 .
the @xmath72 mesons are reconstructed from @xmath73 combinations in the mass window @xmath74 with the charged pion particle identification requirement @xmath75 .
to reduce the contribution from the non - resonant background , a helicity angle cut @xmath76 is applied where @xmath77 is the angle between the normal to the @xmath72 decay plane in the @xmath72 rest frame and the @xmath72 momentum in the @xmath14 rest frame . to remove the contribution from @xmath78
, we require the @xmath79 invariant mass to be greater than @xmath80 from the @xmath81 nominal mass .
the @xmath82 mesons are reconstructed from two oppositely charged kaons in the mass window of @xmath83 with @xmath84 .
we also apply the @xmath82 helicity angle cut @xmath76 where @xmath77 is the angle between one of the @xmath82 daughters in the @xmath82 rest frame and the @xmath82 momentum in the @xmath14 rest frame .
we form candidate @xmath85 and @xmath86 mesons using the @xmath87 and @xmath88 decay modes with mass ranges of @xmath89 and @xmath90 , respectively .
the @xmath85 momentum is required to be greater than @xmath91 .
both @xmath85 and @xmath86 candidates are kinematically constrained to their nominal masses .
the @xmath14 candidates are required to have masses within @xmath922.5@xmath93 of their nominal masses , where @xmath93 is the measured mass resolution which ranges from @xmath94 to @xmath95 depending on the decay channel . a @xmath14 mass and ( wherever possible)vertex constrained fit is then performed on the remaining candidates .
we combine the @xmath14 and @xmath96/@xmath97 candidates ( denoted by @xmath98 ) to form @xmath12 candidates . we apply tighter particle identification cuts , @xmath99 for prompt kaons ( pions ) , to identify @xmath100 events .
the signal is identified by two kinematic variables calculated in the center - of - mass ( c.m . )
frame .
the first is the beam - energy constrained mass , @xmath101 , where @xmath102 and @xmath103 are the momenta of @xmath14 and @xmath104 candidates and @xmath105 is the beam energy in the c.m
. frame .
the second is the energy difference , @xmath106 , where @xmath107 is the energy of the @xmath14 candidate , @xmath108 is the energy of the @xmath104 candidate calculated from the measured momentum and assuming the pion mass,@xmath109 . with this definition , real @xmath110 events peak at @xmath111 even when they are misidentified as @xmath35 , while @xmath112 events peak around @xmath113 mev .
event candidates are accepted if they have @xmath114 and @xmath115 . in case of multiple candidates from a single event
, we choose the best candidate on the basis of a @xmath116 determined from the differences between the measured and nominal values of @xmath117 and @xmath118 . to suppress the large combinatorial background from the two - jet like @xmath119 ( @xmath120 , @xmath121 , @xmath122 or @xmath123 ) continuum processes , variables that characterize the event topology are used .
we construct a fisher discriminant @xmath124 , from 6 modified fox wolfram moments @xcite .
furthermore , @xmath125 , the angle of the @xmath12 flight direction with respect to the beam axis is also used to distinguish signal from continuum background .
we combine these two independent variables , @xmath124 and @xmath125 to make a single likelihood ratio variable ( @xmath126 ) that distinguishes signal from continuum background .
we apply a different requirement for each sub - mode based on the expected signal yield and the backgrounds in the @xmath118 sideband data . for @xmath110 where @xmath127 , @xmath128 we require @xmath129 whereas for @xmath130 , @xmath131 , @xmath55 , @xmath56 , @xmath57 and @xmath58 we require @xmath132 . to give an example of the performance of this selection , the @xmath129 requirement keeps 87.5 % of the @xmath133\pi^{-}$]signal while removing 73 % of the continuum background .
the signal yields are extracted from a fit to the @xmath134 distribution in the region @xmath135 .
the @xmath110 signal is parameterized as a double gaussian with peak position and width floated . on the other hand , we calibrate the shape parameters of the @xmath112 signal using the @xmath36 data .
this accounts for the kinematical shifts and smearing of the @xmath134 peaks caused by the incorrect mass assignments of prompt hadrons .
the peak position and width of the @xmath136 signal events are determined by fitting the @xmath110 distribution using the kaon mass hypothesis for the prompt pion , where the relative peak position is reversed with respect to the origin .
the shape parameters for the feed - across from @xmath110 are fixed by the fit results of the @xmath110 enriched sample .
the continuum background is modeled as a first order polynomial function with parameters determined from the @xmath134 distribution for the events in the sideband region @xmath137 .
backgrounds from other @xmath12 decays including contributions from @xmath138 and @xmath139 are modeled as a smoothed histogram from monte carlo simulation .
the fit results are shown in fig . 2 . [ cols="<,^,^,^,^ " , ]
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for the reader s convenience , we present here analytical form of the expressions used in the main part of the paper . for @xmath5 given by ( [ eq : gauss ] ) and @xmath17 given by ( [ eq : nontrivialtransmission ] )
, we obtain the integral @xmath41
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in the context of the sensational results concerning superluminal velocities , announced recently by the opera collaboration , we have proposed a classical model yielding a statistically calculated measured velocity of a beam , higher than the velocity of the particles constituting the beam .
the two key elements of our model , necessary and sufficient to obtain this curious result , are a time - dependent `` transmission '' function and statistical method of the maximum - likelihood estimation . opera neutrino anomaly , superluminal neutrinos 06.30.gv velocity , acceleration , and rotation , 06.20.dk measurement and error theory , 07.05.kf data analysis : algorithms and implementation ; data management , 13.15.+g neutrino interactions , 14.60.lm ordinary neutrinos , 03.30.+p special relativity inspired by the amazing results of the opera collaboration @xcite , claiming that the velocity of light @xmath0 has been beaten by a beam of neutrinos , we propose a simple classical model , which yields the curious effect of a seeming increase of `` effective '' velocity ( see @xcite , for an earlier independent approach , and also @xcite , for a wave version ) .
there are dozens of papers , which have recently appeared , adopting various attitudes towards the results presented by the opera collaboration , for example @xcite ( see also an example of an older work on superluminal velocities @xcite ) .
the attitude of our paper is definitely skeptical . in view of the fact that we have found a ( `` mathematical '' ) model providing an `` artificial '' increase of real velocity , we are forced to put the results announced by the opera collaboration in doubt .
moreover , we have also proposed a simple working example , such that appropriately fitting its parameters one can simulate the controversial results of the opera collaboration .
strictly speaking , our paper does not indisputably invalidates the conclusions drawn by the opera collaboration , but it seriously weakens their argumentation , indicating a logical gap in their reasoning .
our model is purely classical and dynamics free .
no new physics , nor quantum mechanics , nor even ( classical ) wave mechanics is involved .
only standard classical kinematics notions as well as the statistical method of the maximum - likelihood estimation ( mle ) are used in our approach .
the assumptions of our model are the following .
a spatially homogeneous , lasting the period @xmath1 , beam of classical particles ( `` extraction '' ) moving with a constant speed @xmath2 ( @xmath3 ) travels from a source to a detector .
the distance between the source and the detector is @xmath4 .
the probability density function ( pdf ) of the time of emission of the particles within the duration @xmath1 of production of the beam is given by the function @xmath5 .
in an ideal situation ( none of the particles is lost ) we would obtain , according to classical kinematics , the measured data waveform @xmath6 , where @xmath7 .
now , let as suppose that the fraction of the particles emitted , measured by the detector , due to some physical mechanism , is given by the ( non - negative ) `` transmission '' function @xmath8 where @xmath9 is the number of the particles emitted at the time instant @xmath10 and @xmath11 is the number of the particles detected , which have been sent at the same time instant @xmath12 .
in other words , in general , not all particles emitted are detected ( @xmath13 ) obvious , and moreover @xmath14 conceivable . for simplicity
, we will assume that the numbers of the particles , @xmath15 and @xmath16 , are so large that we are allowed to use a continuous approximation .
then , the transmission function @xmath17 is a ( continuous ) function satisfying the condition @xmath18 . to be able to draw conclusions from experimental data
, we should implement some statistical methodology .
to be so precise as possible , in our approach , we adopt the method of the mle , as the opera collaboration has done @xcite . in the framework of the mle
, we introduce the likelihood function @xmath19 .
the logarithm @xmath20 of @xmath19 is given by the formula @xmath21\equiv\sum_{j}\log\left[w(t_{j}+\delta t)\right],\label{eq : genlikelihood}\ ] ] where @xmath22 are the time instants corresponding to the measurement events at the detector , and the time deviation @xmath23 we are interested in ( see @xcite , for details ) provides the maximum of @xmath20 ( and also of @xmath19 ) .
as the numbers of the measured events @xmath22 are large , in our continuous approach , instead of summation in ( [ eq : genlikelihood ] ) , we should use integration with an appropriate integration measure @xmath24 , where @xmath25 represents the time distribution of the experimental events detected by the opera .
in fact , @xmath25 is determined by product of two factors .
the first factor , @xmath26 , is proportional to the number of the particles sent , i.e. @xmath27 , and the second one is proportional to the transmission function @xmath17 .
then , @xmath28 finally , @xmath29f(t)w(t)dt.\label{eq : conlikelihood}\ ] ] to demonstrate that our idea actually works , we propose a specific example .
the parameters of the example are so fitted that it yields the time deviation @xmath30 ( for comparison , the opera result is @xmath31 ) . for calculational simplicity ,
we assume the gaussian form of the pdf ( see the solid line in fig.[fig:2curves ] ) , @xmath32 as well as the gaussian form of of the transmission function , @xmath33,\label{eq : nontrivialtransmission}\ ] ] where the one tenth in front of the exponent is reminiscent of `` @xmath34 variation '' in @xcite .
the time distribution of the `` detected experimental data '' @xmath25 , corresponding to @xmath5 of the form ( [ eq : gauss ] ) and to the transmission function @xmath17 of the form ( [ eq : nontrivialtransmission ] ) is presented in fig.[fig:2curves ] by the dashed curve . ] in this ( doubly ) gaussian case it is even possible to solve the problem analytically ( see appendix ) , but for our purposes a numerical value will do .
it appears , that the maximum of ( [ eq : conlikelihood ] ) is attained for the time deviation @xmath35 where s.d .
means the standard deviation .
one can easily translate the dimensionless ( [ eq : numdelt ] ) into a dimensionfull entity . in the opera experiment @xmath36 , whereas in our example we can reasonably assume , for definiteness , that @xmath1 equals ( in dimensionless units , or in the units of the standard deviation ) twice the two standard deviations , i.e. @xmath37 ( see two vertical intervals in fig.[fig:2curves ] ) .
then , the dimensionfull time deviation corresponding to ( [ eq : numdelt ] ) is @xmath38 in particular , our analysis confirms the findings of @xcite that the time deviation @xmath23 is independent of the distance @xmath4 , but depends on the shape of the beam .
obviously , the numerical coincidence ( [ eq : unitdelt ] ) has only an illustrative purpose .
the non - zero time deviation @xmath23 directly implies a modified velocity @xmath39 according to the elementary formula @xmath40 in our model the sign of @xmath23 ( positive in our example ) depends on details of the form of the transmission function @xmath17 .
e.g. , reversing the sign in front of the exponent in ( [ eq : nontrivialtransmission ] ) reverses the sign of @xmath23 .
we would like to stress that our specific example is not supposed to mimic a real situation in the opera experiment , but only to demonstrate that it is , in principle , possible to easily come to its conclusions without any reference to superluminal particles and/or some exotic phenomena .
any considerations concerning a possible physical mechanism governing the time - dependence of the `` transmission '' function @xmath17 are outside the scope of our paper .
we can only speculate that @xmath14 could be a property of the source , or of the detector or it could follow from interactions in the earth s crust .
moreover , measurement errors , always encountered in real experiments , do not enter our considerations as they have nothing to do with the discussed effect .
it is also possible to intuitively explain the non - zero value of @xmath23 .
namely , deforming the shape of the pdf @xmath5 with an appropriate ( time asymmetric ) transmission function @xmath17 shifts a portion ( its `` upper '' part ) of the `` waveform '' @xmath5 forward or backward , yielding positive or negative @xmath23 , respectively , which is next erroneously interpreted by the method of the mle as a `` real '' time deviation . in a sense ,
our approach is a `` corpuscular '' analog of the well - known situation of `` superluminal '' velocity of light in material media @xcite , where to some extent , the role of @xmath17 plays dispersion . in conclusion , we would like to emphasize that there are two elements of our model , responsible for the curious effect of the superluminal velocity : the time - dependent transmission function @xmath17 , which should `` favor '' leading edges and `` disfavor '' trailing edges of the beam , and statistical methodology , erroneously assuming the method of the mle .
this is why it is conceivable that the paradoxical results of the opera collaboration could be possibly avoided upon another approach , e.g. an approach giving preference to the front velocity or to another statistical method .
thus , no superluminal particles are necessary to explain the `` superluminal '' velocities derived by the opera collaboration . moreover , independently of the further fate of the conclusions drawn by the opera collaboration ( acceptance or refutation ) , we have demonstrated that standard statistical methods , e.g. mle , should be used with great care , as they can provide erroneous output . therefore , the results of our paper do not rely on the final solution of the opera paradox .
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following eq . 6 in the main text ,
the time - ordered bare susceptibility is a @xmath49 matrix @xmath110 defined as , ^0_i , j(,)=_i(,)^_j(,0 ) , where the density operators are defined in eq .
3 in the main text .
first , we perform the wicks decomposition and then convert the @xmath111 operators to the band basis , c_r(,)=_u_r_()a _ ( , ) .
after we perform a fourier transformation to matsubara space , sum over the internal matsubara frequency , and do the analytic continuation , we obtain the final expression of retarded bare susceptibility as
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we investigate the charge excitations of a weyl semimetal in the axionic charge density wave ( axionic cdw ) state .
while it has been shown that the topological response ( anomalous hall conductivity ) is protected against the cdw state , we find that the long wavelength plasmon excitation is radically influenced by the dynamics of the cdw order parameter . in the normal state , we show that an undamped collective mode should exist at @xmath0 if there is an attractive interaction favoring the formation of the cdw state .
the undamped nature of this collective mode is attributed to a gap - like feature in the particle - hole continuum at @xmath0 due to the chirality of the weyl nodes , which is not seen in other materials with cdw instability . in the cdw state ,
the long wavelength plasmon excitations become more dispersive due to the additional interband scattering not allowed in the normal state . moreover , because the translational symmetry is spontaneously broken , umklapp scattering , the process conserving the total momentum only up to @xmath1 with @xmath2 an integer and @xmath3 the ordering wave vector , emerges in the cdw state .
we find that the plasmon excitation couples to the phonon mode of the cdw order via the umklapp scattering , leading to two branches of resonant collective modes observable in the density - density correlation function at @xmath4 and @xmath0 .
based on our analysis , we propose that measuring these resonant plasmon - axion excitations around @xmath4 and @xmath0 by the momentum - resolved electron energy loss spectroscopy ( m - eels ) could serve as a reliable way to detect the axionic cdw state in weyl semimetals .
_ introduction_ the weyl semimetal@xcite is a new gapless state of matter attracting a lot of attention in the past few years due to its rich topological properties.@xcite in the wake of the study on topological insulators@xcite , the search for semimetals exhibiting linear band dispersion near the fermi energy has been one of the most active fields in condensed matter physics .
one typical example is the dirac semimetal@xcite in which the conduction and valence bands become degenerate at certain points in the brillouin zone , namely the dirac nodes , and the low energy single particle states near the dirac nodes can therefore be described by dirac equations with a four - component spinor .
if the dirac semimetal has a zero mass , the dirac equation can be further reduced to two independent sets of weyl equations with two - component spinors . as a result ,
the weyl semimetal is a special case of the dirac semimetal with zero mass , which could be obtained by breaking either the inversion or the time - reversal symmetries in a dirac semimetal .
in addition , it can be proved@xcite that weyl nodes must appear in pairs with opposite chirality if the weyl equations are imposed on a lattice .
therefore , if a crystalline material is a weyl semimetal , the low energy single particle excitations can be described by a minimal model of the weyl fermions around two weyl nodes obeying the weyl equations : h_0= v_fd _ = 1 - _ , [ h0 ] where @xmath5 is the fermi velocity , and @xmath6 is the chirality of the weyl nodes .
very recently , the experimental realization of the weyl semimetal has been found in taas.@xcite the spontaneous symmetry breaking phases in the weyl semimetal induced by the electron - electron correlation have also been investigated.@xcite one intriguing aspect unique to the weyl semimetal is the chiral anomaly accompanying the charge - density - wave ( cdw ) state.@xcite if the pair of the weyl nodes described in eq . [ h0 ] is located at @xmath7 in the brillouin zone , the electron - electron interaction could result in a cdw state with an ordering wavevector @xmath8 .
since in the cdw state the electron occupation number at @xmath9 is different , the chiral symmetry is also broken .
it can be further shown that the phase of the cdw order parameter can be identified as the axion field which couples to the electromagnetic fields and exists only in the presence of the chiral anomaly.@xcite in this paper , we study the charge excitations of the weyl semimetal in the axionic cdw state . in the normal state
, we find an undamped collective mode at @xmath0 if there is an attractive interaction favoring the formation of the cdw state . in the cdw state , we find that the nature of the long wavelength plasmon excitation is dramatically changed .
the phonon mode of the cdw order , or equivalently the collective mode of the axion field , couples to the plasmon excitation via the umklapp process , resulting in two resonant collective modes which can be seen in the density - density correlation function at both @xmath4 and @xmath0 .
this resonant feature does not emerge in the case where the chiral symmetry is broken by the application of a pairing of the electric @xmath10 and the magnetic @xmath11 fields.@xcite we propose that measuring these resonant plasmon - axion excitations around @xmath4 and @xmath0 by the momentum - resolved electron energy loss spectroscopy ( m - eels)@xcite could serve as a reliable way to detect the cdw state as well as the nature of the axion field in a weyl semimetal . _ hamiltonian and formalism _ we start from the effective hamiltonian @xmath12 describing the weyl fermions around two weyl nodes at @xmath9 given in eq.[h0 ] together with electron - electron interactions : h&=&h_0 + h_cdw + h_lc , + h_cdw & = & v_cdwd ( + _ cdw)(--_cdw ) , + h_lc & = & d v_()(- ) . where @xmath13 is the 3d coulomb interaction in momentum space .
it has been shown in previous study that the cdw instability is one of the leading instabilities in the weyl semimetal induced by repulsive interactions@xcite , which justifies using @xmath14 to stduy the cdw fixed point .
the charge density operator can be expanded around the proximity of the weyl nodes , leading to & & ( ) _ ^d c^_r ( + ) c_r()+c^_l ( ) c_l ( - ) , + & & ( --_cdw ) _ ^d c^_l ( - ) c_r ( ) [ den ] where @xmath15 are small momenta compared to @xmath16 , @xmath17 is the cut - off in the integration over the momentum , and @xmath18
. since we are only interested in the long - wavelength plasmon excitations and the charge fluctuations near the cdw ordering wave vector @xmath3 , the integration over @xmath19 is limited to the region of small @xmath20 we perform the standard mean - field theory to decouple @xmath14 with the axionic cdw order parameter defined as @xmath21 , and consequently the mean - field hamiltonian in the cdw state can be written as@xcite @xmath22 , where ( ) & = & ( cc - _ + & ^ + & - - _ - ) , + & = & ( cc m_1 - im_2&0 + 0&m_1 -im_2 ) , , and @xmath23 $ ] .
@xmath24 and @xmath25 are the creation operators near @xmath26 and @xmath27 respectively , and they are related via @xmath28 , where @xmath8 is the cdw ordering wave vector .
@xmath29 can be diagonalized as @xmath30 , where @xmath31 , and @xmath32 $ ] are the corresponding eigenvectors which are related to @xmath33 via a unitary matrix @xmath34 such that @xmath35 .
the cdw order parameter is self - consistently obtained by solving the mean - field equation : = d _
u^*_r,()u_l , n_f(e _ ) , which supports a non - zero order parameter if @xmath36 is negative and smaller than a critical value ( @xmath37 ) . clearly , in the normal state
, the density fluctuations at @xmath38 are independent of those at @xmath19 , thus they do not affect the long wavelength plasmon excitations at all .
if the system is in the cdw state , a non - zero order parameter induces a coupling between the density fluctuations at @xmath19 and @xmath38 .
such a coupling can be described by the umklapp scattering in which the total momentum is conserved only up to @xmath1 with @xmath2 an integer , and significant changes in the properties of the long wavelength plasmon excitations occurs .
note that in principle all the density fluctuations in different umklapp channels ( i.e. , different @xmath2 ) are coupled , but for the demonstration of principle , it is sufficient to consider the two channels given in eq . [ den ] . to take the umklapp process into account
, we introduce a matrix form to represent the density - density correlation function written as @xmath39 , where @xmath40 and @xmath41 refer to @xmath42 and @xmath43 respectively .
in general , @xmath44 refers to the charge excitations inside each weyl nodes , and @xmath45 refers to those between different weyl nodes . performing a generalized rpa which has been widely used in studying the coupling between different collective excitations@xcite , we obtain ^grpa ( , ) = ^-1 where @xmath46 , @xmath47 , and @xmath48 are @xmath49 matrices , ^0_i , j ( , ) & = & - _ , = 1 ^ 4 ^d m_i , j^ , ( , ) + & & ( ) , + [ chi0 ] @xmath50 is the matrix element which is given in the supplementary materials , and @xmath51 is the interaction kernel . for the calculations , we adopt the lorentz cut - off , i.e. , @xmath52 and @xmath53 .
since we only focus on the excitations near the weyl nodes , @xmath54 .
furthermore , we choose @xmath55 and @xmath56 to be the units of momentum and energy in all of our calculations .
( a ) real ( blue solid ) and imaginary ( red dashed ) parts of @xmath57 .
the parameters used are @xmath58 , @xmath59 and @xmath60 .
the particle - hole excitations from the intraband scatterings are greatly suppressed , resulting a gap - like feature in @xmath61 .
the intraband and interband scatterings in @xmath62 and @xmath63 are schematically illustrated in ( b ) and ( c ) respectively.,title="fig:",width=278 ] ( a ) real ( blue solid ) and imaginary ( red dashed ) parts of @xmath57 . the parameters used are @xmath58 , @xmath59 and @xmath60 .
the particle - hole excitations from the intraband scatterings are greatly suppressed , resulting a gap - like feature in @xmath61 .
the intraband and interband scatterings in @xmath62 and @xmath63 are schematically illustrated in ( b ) and ( c ) respectively.,title="fig:",width=278 ] _ collective charge excitation in normal state _
in the normal state without cdw , @xmath64 . as a result , @xmath65 and @xmath66
are decoupled .
@xmath65 has an undamped delta - function - like peak corresponding to the plasmon excitations , which has been studied previously@xcite . as for @xmath66 , we find that an undamped delta - function - like peak could emerge if @xmath36 is negative .
the occurrence of this undamped mode is due to a gap - like feature appearing in the bare susceptibility @xmath62 shown in fig .
[ fig : chi022](a ) .
because the weyl nodes have opposite chirality , the intraband scattering is largely suppressed while the interband scattering is robust in @xmath62 as demonstrated in fig .
[ fig : chi022](b ) .
this unusual chirality effect leads to @xmath67 for @xmath68 , where @xmath69 is the particle - hole continuum edge for the interband scattering .
in contrast , @xmath63 is dominated by the intraband scattering as sketched in fig .
[ fig : chi022](c ) , and consequently @xmath70 is finite for @xmath71 , where @xmath72 is the particle - hole continuum edge for the intraband scattering , and almost zero for @xmath73 . by kramers
kronig relations , it can be proved that @xmath74 increases monotonically from @xmath75 until @xmath76 . if @xmath36 is negative , a collective mode with the frequency @xmath77 satisfying @xmath78 exists , which is corresponding to the undamped delta - function - like peak obtained in @xmath66 .
@xmath77 decreases with the increase of @xmath79 , and the normal state remains stable until @xmath80 where cdw state starts to emerge .
@xmath81 as a function of @xmath79 is plotted in fig .
[ fig : chirpa22 ] .
we notice that the critical interaction strength for cdw can be obtained by @xmath82 which is roughly proportional to @xmath83 .
this dependence of @xmath17 comes from the diverging density of states for the interband scattering in @xmath62 , which has also been observed in previous study.@xcite we emphasize that the gap - like feature in @xmath84 is a direct consequence of the chirality of the weyl nodes . as a result ,
the undamped delta - function - like peak in the @xmath81 in the normal state is a unique feature for the weyl semimetal , and it is rarely seen in other materials with cdw instability in which the peak is significantly damped by the particle - hole continuum . the undamped collective mode in @xmath85 associated with the cdw instability in the normal state .
the peak moves toward zero frequency as @xmath36 approaches @xmath86 .
the parameters used are @xmath58 , @xmath87 and @xmath60.,width=278 ] _ resonant plasmon - axion excitation in cdw state _ in the cdw state , we find two new effects in the bare susceptibilities .
first , since the cdw order parameter mixes up the states with different chiralities , all the scatterings have finite probability . as a result , @xmath70 develops new spectral weights within the interband particle - hole continuum ( @xmath73 ) .
these extra contributions in @xmath88 push the plasmon frequency toward the lower edge of the particle - hole continuum , resulting in the plasmon frequency @xmath89 being more dispersive in the cdw state than in the normal state@xcite .
second , because @xmath90 is non - zero in the cdw state , the frequencies of the collective excitations are now determined by ( + ^0 ( , ) _ ) = 0 , [ plas ] which has _ two _ solutions as explained below .
eq .
[ plas ] can be rewritten as : k_11(,)k_22 ( , ) - v_v_cdw^0_12(,)^2 = 0 , where [ k1122 ] k_11(,)&= & 1+v _
^0_11(,)a(1- ) , + k_22(,)&= & 1+v_cdw ^0_22(,)b(1-),and @xmath89 ( @xmath91 ) is the plasmon ( cdw phonon ) frequency without @xmath90 .
note that we have expanded @xmath92 and @xmath93 in the region without uncorrelated particle - hole excitations , i.e. , @xmath94 .
this is also the region where the undamped collective excitations could exist . substituting eq .
[ k1122 ] into eq .
[ plas ] , it can be proved straightforwardly that two collective excitations exist with frequencies near @xmath95 and @xmath96 respectively .
since these two collective excitations are mixtures of the plasmon excitation and the cdw phonon mode , they should result in prominent peaks in both @xmath97 and @xmath98 , emerging as resonant modes between these two channels .
fig .
[ fig : plascdw ] plots the frequencies of these two resonant modes solved from eq .
[ plas ] for different @xmath19 together with @xmath99 and @xmath100 .
note that because magnitude of the spectral weight due to the interband scattering generally scales with @xmath101 ( @xmath102 ) , we could not draw a quantitative conclusion on the amount of the spectral weight for each resonant mode .
nevertheless , the qualitative features of the collective excitations are robust regardless of the @xmath17 chosen . the frequencies of the two branches of collective excitations due to the mixture of the plasmon excitation and cdw phonon ( axion ) mode .
the parameters used are @xmath58 , @xmath103 , @xmath104 , @xmath105 , and @xmath60 .
the solid lines represent the particle - hole continuum edges for the intraband ( @xmath99)and interband ( @xmath100 ) scatterings respectively .
the branch marked by the circle has more plasmon characters while the one marked by the triangle has more cdw phonon ( axion ) characters .
both branches should show up in the density - density correlation functions around @xmath4 and @xmath0 , which in principle can be detected by m - eels.,width=278 ] it is remarkable to see that different mechanism to break the chiral symmetry in a weyl semimetal could result in different behaviors of the plasmon excitation .
zhou _
et al .
_ studied the plasmon excitation of a weyl semimetal with the chiral symmetry broken by the application of a pair of the electric @xmath10 and the magnetic @xmath11 fields.@xcite in that case , the chirality - dependent chemical potential @xmath106 is introduced , which leads to a @xmath19-dependent damping effect of the plasmon excitation .
however , since @xmath106 does not provide any coupling between different weyl nodes , the plasmon excitation does not have the resonant nature as discussed in the cdw state .
it can be seen that the cdw phonon mode has a dispersion of @xmath107 as predicted by the goldstone theory . however , the reason why we obtain the gapless goldstone mode is that the effective hamiltonian in eq .
[ h0 ] has the continuous translational symmetry .
if the lattice effect is included , we will obtain a gapped cdw phonon mode .
nevertheless , the weyl semimetals discovered recently@xcite have quite small @xmath108 , thus the wavelength of the cdw state @xmath109 is much longer than the lattice constant .
the lattice effect can therefore be argued to be quite small , and the cdw phonon mode should have a very small gap . moreover , because the phase of the cdw order parameter is shown to be the axion field@xcite , the cdw phonon mode is in fact a collective excitation of the axion field . as a result , measuring these resonant plasmon - axion excitations around @xmath4 and @xmath0 could serve as a reliable way to detect the axion field .
_ conclusion _ in this paper , we have investigated the charge excitations of a weyl semimetal in the charge - density - wave ( cdw)state . we have found that in the normal state , an undamped collective mode could emerge in the density - density correlation function at the cdw ordering wavevector @xmath3 if there is an attractive interaction favoring the formation of the cdw state . in the cdw state , we have found that the plasmon excitation becomes more dispersive compared to the behavior in the normal state .
moreover , we have shown that due to the umklapp scattering enabled by the cdw state , the plasmon excitation couples to the cdw phonon ( axion ) mode , leading to two branches of collective excitations .
both collective excitations can be seen in the density fluctuations at @xmath4 and @xmath0 simultaneously , exhibiting an interesting resonant nature between two distinct channels .
we have proposed that measuring the resonant plasmon - axion excitations around @xmath4 and @xmath0 by m - eels could serve as a reliable way to detect the cdw state as well as the nature of the axion field in a weyl semimetal .
_ acknowledgement _ this work is supported by a start up fund from binghamton university .
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author acknowledges hospitality of the institut fr halbleiter physik at the universitt linz , where the work was partly performed .
author appreciates valuable discussions with g. bauer , g. brunthaler , m.v .
entin , v. kravtsov , i.m .
suslov , e.i .
rashba , and v. volkov .
the work was supported by the russian foundation for basic research ( grant 97 - 02 - 17387 ) , by the programs on `` physics of solid - state nanostructures '' and `` statistical physics '' and by grant from nwo the netherlands .
kravchenko , w.e .
mason , g.e .
bowker , j.e .
furneaux , v.m .
pudalov , and m.diorio , phys .
b * 51 * , 7038 ( 1995 ) .
kravchenko , d. simonian , m.p .
sarachik , w. mason , and j.e .
furneaux , phys .
lett . * 77 * , 4938 ( 1996 ) .
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we present a model that explains two phenomena , recently observed in high - mobility si - mos structures : ( 1 ) the strong enhancement of metallic conduction at low temperatures , @xmath0 k , and ( 2 ) the occurrence of the metal - insulator transition in 2d electron system .
both effects are prescribed to the spin - orbit interaction anomalously enhanced by the broken inversion symmetry of the confining potential well .
* introduction . * recently , in experiments with high mobility si - mos structures , a strong drop in resistivity @xmath1 has been found @xcite as temperature decreases below @xmath2 k. this effect is evidently in disagreement with the conventional interpretation of the one - parameter scaling theory ( opst ) @xcite , according to which all states in 2d system at zero magnetic field should be localized in the limit of @xmath3 .
the subsequent scaling analysis of the temperature and electric field dependencies of the conductivity @xcite has revealed a critical behavior , typical for the metal - insulator transition .
finally , convincing evidence for the existence of the extended states in 2d system at zero field has been obtained in experiments _ in magnetic field _ , in studies of the quantum hall effect to insulator transitions @xcite . the extended states which in high magnetic field are located in the centre of the corresponding landau bands , at decreasing field were found to remain in a finite energy range , giving rise to a mobility edge .
the experimental results thus suggest the existence of a true metallic state and of the metal - insulator ( m - i ) transition in 2 dimensions .
these results are in apparent contradiction with the conventional opst , and the origin of the metallic state remains puzzling . in this work , both experimental findings are explained as a consequence of the spin - orbit interaction enhanced by the broken inversion symmetry .
the suggested model provides a good agreement with the experimental data on the temperature dependence of the resistivity @xmath1 .
* analysis of the experimental results . *
fig .
1 shows a set of the curves @xmath4 vs @xmath5 typical for the high - mobility samples @xcite , at different electron densities @xmath6 . at @xmath7
k , the resistivity , @xmath4 , increases slowly as temperature decreases , the latter is characteristic for the weakly localized regime . at lower temperature
, @xmath4 drops sharply for all curves belonging to the `` metallic '' range of densities , @xmath8 .
the resistance drop is observed at densities in the range from @xmath9 to @xmath10 . the critical density , @xmath9 is sample dependent and is equal to @xmath11 @xmath12 for the sample shown in fig . 1 .
the drop in @xmath13 diminishes with decreasing sample mobility , and is almost replaced by a conventional rise in @xmath4 at @xmath3 in the sample with 8 times lower mobility , @xmath14 @xmath15/vs .
the latter behavior is consistent with that reported in earlier studies on low - mobility samples @xcite .
* empirical fit of the data . * the @xmath13- curves in fig . 1 may be fitted well by an empirical dependence which summarizes the scattering probabilities of two processes : @xmath16 the first term is independent of temperature , while the second one describes a scattering through an energy gap , @xmath17 .
the curves shown in fig .
1 by thick continuous lines were obtained using two fitting parameters for each density , @xmath18 and @xmath19 .
* possible microscopic mechanisms . * as seen from fig . 1 , the characteristic temperature , @xmath19 , is of the order of @xmath20 k in high mobility samples . searching for a proper microscopic mechanism , we find two small energy gaps intrinsic to si - mos structures at @xmath21 : the valley splitting @xmath22 k and the zero - field spin - gap @xmath23 k @xcite .
it seems attractive to link the resistance drop to the transitions between the two electron valleys , located close to the x - points in the brillouin zone @xcite .
however , the intervalley `` um - klapp '' scattering would hardly occur , since it requires a combination of the reciprocal lattice vectors of the very high order . on the other hand ,
the phonon - induced intervalley transition would require participation of high energy phonons , @xmath24k , and is therefore unlikely at low temperatures .
electron tunneling , as the intervalley transition mechanism , would not lead to a strong temperature dependence of scattering .
* spin - orbit splitting and interaction effects . * in the one - electron approximation , the spin - orbit interaction is described by the hamiltonian @xcite : @xmath25 { \bf \sigma},\ ] ] where @xmath26 and @xmath27 are the momentum and spin operators , correspondingly . for the 2d electron system in si
, the contribution of the bulk crystal potential in @xmath28 is small ( @xmath29 ) , and the lack of inversion symmetry of the triangular confining potential @xmath30 plays the major role .
this lifts the spin - degeneracy at zero magnetic field , and leads to the appearance of a linear term in the energy spectrum of 2d electrons @xcite : @xmath31 the corresponding spin - gap @xmath32 can be viewed as the difference in energy for the electron states with spin directed in the plane but to the left and right side with respect to @xmath33 , or , equivalently , along and opposite to the effective magnetic field @xmath34 $ ] in the frame related to electrons moving in 2d plane with fermi velocity .
we suggest that the empirically determined energy gap , @xmath35 originates from @xmath36 , and is equal to @xmath37 ( where @xmath38 ) , whereas the temperature independent contribution to the resistivity , @xmath39 , is related to the spin - independent scattering .
finite quantum relaxation time @xmath40 and the corresponding level broadening @xmath41 should reduce the effective gap : @xmath42 * comparison of the model with the experimental results . *
the spin splitting @xmath43 k for the same si - mos structures was determined from the magnitude of the quantum oscillations of the chemical potential @xcite , extrapolated to zero field .
it is clear from eqs .
( 3 ) and ( 4 ) , that this value corresponds to @xmath44 . with this result
we obtain @xmath45kcm and the total energy spectrum becomes known . in order to estimate
the quantum level broadening , @xmath46 , the temperature dependence of the diagonal resistance was measured in the quantum hall effect regime with fermi energy adjusted to the zeeman energy gap at @xmath47 in the field of @xmath48 t. as a result , we have obtained an estimate , @xmath49 k for @xmath50@xmath12 ( here we presumed the gaussian level broadening and @xmath51 to be the full width ) . comparing the model effective energy gap , @xmath52 with the empirical value , @xmath53 k obtained in the fit at @xmath54 ( the 5th curve from the bottom in fig . 1 )
, we eventually find @xmath55 @xcite .
the empirical energy gap , @xmath56 , decreases to zero at @xmath57 , as seen in fig . 1 . in the above model ,
eq .
( 5 ) , this occurs because ( i ) the level broadening increases , @xmath58 and ( ii ) the spin splitting , @xmath59 , diminishes @xmath60 .
the empirical fitting parameter @xmath51 shown in fig .
2 vs electron density , rises indeed as density decreases .
the total fit in fig .
1 is in surprisingly good agreement with the experimental data , despite the very simplified character of the above model .
as @xmath6 decreases and approaches @xmath9 , the resistance drop starts at lower temperatures .
the weak decrease in @xmath13 noticeable at @xmath61 k is presumably due to the weak localization corrections @xmath62 @xcite .
this effect was ignored in the above model and the @xmath13 points corresponding to the negative @xmath63 were not fitted ; these points are connected by dashed lines in fig .
1 . fig .
2 shows also two relaxation times : @xmath64 calculated from the mobility in the @xmath65 limit , and @xmath66 .
it is noteworthy that both @xmath67 and @xmath40 diminish almost linearly as density decreases ( but at @xmath8 ) and independently of each other .
the momentum relaxation time @xmath64 provides the necessary resistivity value , @xmath68 at the critical density @xcite , whereas @xmath40 provides the effective spin - gap equal to 0 at @xmath69 . as a result
, @xmath64 _ decays faster and becomes smaller than _ @xmath40 with decreasing @xmath6 . at the critical density
, @xmath70 is of the order of 0.1 m@xmath71/vs and @xmath72k , that corresponds to @xmath73s , and @xmath74s .
the conclusion on interception of the two curves , @xmath75 and @xmath76 is model independent , whereas the numerical values of @xmath77 and the interception point depend on the model chosen for level broadening .
for instance , the interception occurs at 11.5 or @xmath78 @xmath12 for gaussian or lorenzian broadening correspondingly .
* metal - insulator transition at @xmath79 : spin - orbit interaction and symmetry effects . *
it is not only the low - temperature resistance drop but the total scaling behavior strongly depends on the symmetry .
the corresponding universality classes of the symmetry for random systems were established by dyson @xcite . in the presence of the spin - orbit ( so ) interaction ,
the orthogonal symmetry of the system is replaced by the symplectic symmetry . correspondingly , the level - repulsion exponent in the random matrix statistics @xcite changes from @xmath80 to @xmath81 .
it appears therefore , that states are less easily localized in systems with large @xmath82 .
the effect of the spin - orbit interaction on weak localization was studied both theoretically and experimentally @xcite . for the strong localization regime ,
there have been suggestions @xcite that m - i transition can occur in the presence of a strong so interaction .
the scaling function in 2d was found to behaves asymptotically like @xmath83 in the high conductance limit @xmath84 , with @xmath85 in the orthogonal and @xmath86 in the symplectic case @xcite .
the @xmath87-function in the symplectic case may thus become positive at sufficiently large @xmath88 . as disorder increases and conduction @xmath89 decreases
, all states will be localized even in the symplectic case
. the critical level of disorder and the critical conduction @xmath90 correspond to the point at which @xmath91 .
the behavior of the symplectic @xmath92 function in 2d is qualitatively consistent with the experimental data presented in fig . 1 , in the vicinity of @xmath9 .
as temperature , or broadening increases , the energy relaxation time @xmath93 appears as a cut - off parameter and @xmath94 may become larger than the inverse spin relaxation time @xmath95
. then the system would again behave as in the orthogonal symmetry case .
the natural measure of the so interaction strength is the spin - orbit gap @xmath96 given by eqs .
( 4 ) and ( 5 ) , whereas as an estimate for disorder we adopted @xmath97 .
thus , one may expect the m - i transition would manifest in those samples where @xmath98 , which is also consistent with occurrence of the transition in the samples with peak mobility larger than 5000 @xmath15/vs @xcite .
* discussion .
* the above model explains why the resistance drop is seen only in low - disordered samples with large @xmath40-values @xcite .
the effect is dependent also on the symmetry of the potential well .
this provides a key for testing the driving mechanism . as for other systems , the zero field spin - gap in gaas / al(ga)as is smaller by a factor of @xmath99 , due to the smaller @xmath100-factor value and much smaller @xmath101 @xcite .
thus , even in ideal samples with zero broadening , the resistance drop may occur at temperatures @xmath102 lower than those for the si - mos structures . in accord with this
, no signatures of the resistance drop were revealed in recent measurements on gaas / al(ga)as heterojunctions at temperatures down to 20mk @xcite .
recently , there have been suggestions on other possible collective mechanisms , such as coulomb interaction @xcite , spin - triplet pairing @xcite , and non - fermi - liquid behavior @xcite .
however , the corresponding models are not developed yet to provide a comparison with the experimental data .
* summary .
* it seems likely that the recently observed metal - insulator transition in high - mobility si - mos structures is the first experimental manifestation of the spin - orbit interaction induced transition in 2d .
the enhancement of the metallic conduction in these samples at low temperatures fits the same framework .
strong so interaction energy relative to the level broadening , and broken inversion symmetry are favorable for the 2d metallic state .
the coulomb interaction in this model provides the small level broadening at density down to @xmath9 . in the recent experiments @xcite ,
the 2d metallic phase was found to be easily destroyed by the in - plane magnetic field ; this is a strong evidence for the spin - related origin of the 2d metal .
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i would like to thank f.e.low for his collaboration in deriving eq .
many fruitful discussions i would like to thank u.g .
meiner , n.kaiser , b.holstein , and u.b.vankolk . for critical help with the manuscript
i would like to thank m.distler and m.pavan .
s. weinberg , transactions of the n.y .
academy of science series ii 38 ( 1977 ) , 185 ( i.i .
rabi festschrift ) and article in the proceedings of the workshop on chiral dynamics : theory and experiment , springer - verlag , july 1995 , a.m. bernstein and b. holstein editors .
meiner and s. steininger , phys .
b419 ( 1998 ) 403 .
n. fettes , u.g .
meiner , and s. steininger , hep - ph/9803266 ( to be published in nucl . phys . ) .
meiner , plenary talk at the `` quark - lepton physics 1997 '' , osaka , japan , may 1997 , hep - ph/9706367 .
d. sigg et .
al . , phys .
lett . 75 ( 1995 ) 3245 ,
a609 ( 1996 ) 269 , a617 ( 1997 ) 526 .
a. badertscher , in proceedings of the workshop on chiral dynamics , mainz .
sept 1 - 5 , 1997 , a.m. bernstein , d. drechsel , and th .
walcher , editors , springer - verlag , in press . , and private communication .
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it is demonstrated that there is a dynamic isospin breaking effect in the near threshold @xmath0 reaction due to the mass difference of the up and down quarks , which also causes isospin breaking in the @xmath1 system .
the photopion reaction is affected through final state @xmath1 interactions ( formally implemented by unitarity and time reversal invariance ) .
it is also demonstrated that the near threshold @xmath2 reaction is a practical reaction to measure isospin breaking in the @xmath1 system , which was first predicted by weinberg about 20 years ago but has never been experimentally tested . since the discovery of a large mass difference between the up and down quarks ( @xmath3 ) there has been considerable interest in the possibility of observing dynamical isospin breaking in the pion - nucleon system @xcite .
the theoretical consensus is that the magnitude of isospin breaking is not @xmath4 , but instead is @xmath5 @xcite , where @xmath6 0.2 gev . in this paper
it will be shown for the first time that the @xmath7 reaction is an excellent candidate to measure dynamic isospin violations due to the up , down quark mass difference .
weinberg first showed that there is an isospin violating effect in the s wave @xmath1 scattering length @xmath8 due to the up , down quark mass difference @xcite .
this predicted effect , which occurs for @xmath9 scattering or charge exchange but not for @xmath10 scattering , has more recently been calculated by chiral perturbation theory ( chpt ) @xcite .
the predicted magnitude of this effect is the same ( to within a factor of @xmath11 in @xmath12 scattering and charge exchange reactions .
however , since the magnitude of @xmath13 is small , the relative magnitude of the isospin violating term is @xmath14 30 % @xcite .
by contrast , for charge exchange where the isospin conserving amplitude is larger , the relative isospin violation is estimated to be @xmath14 2 to 3 % @xcite . in this paper
the ( dynamic ) isospin breaking effect of the up and down quark mass difference is shown to be present in the near threshold @xmath15 n @xmath16n reaction .
this is in addition to the well known ( static ) isospin breaking effect due to the threshold difference between the @xmath17 and @xmath18 channels . of this @xmath14
2/3 is due to the coulomb effect and @xmath14 1/3 is caused by the up , down quark mass difference .
the separation of the two threshold energies leads to the prediction of a unitary cusp in the @xmath19 reaction near the @xmath20 threshold due to the two step @xmath21 reaction @xcite .
the magnitude of this unitary cusp is due to the large ratio of the electric dipole multipole amplitudes , @xmath22 @xcite .
the first derivations of the unitary cusp @xcite used a k matrix approach to calculate the effect of charge exchange in the final state . due to their single scattering approximation , these calculations violate unitarity as will be discussed below . the chiral perturbation theory ( chpt )
calculations are basically isospin conserving but the biggest isospin non - conserving effect due to the pion mass difference ( which is almost purely electromagnetic in origin ) is inserted by hand @xcite .
these calculations also violate unitarity due to their truncation at the one loop ( single rescattering ) level .
there has been recent progress in performing chpt calculations in @xmath1 scattering which take the isospin violations due to the coulomb interaction into account @xcite .
the approximations made in the k matrix @xcite and published chpt papers @xcite can be overcome by using a 3 channel s matrix approach in which unitarity and time reversal invariance are satisfied ( a preliminary version of this work has been presented previously @xcite ) .
this has the advantage that both static and dynamic isospin breaking , and final state multiple scattering ( to all orders ) , are taken into account .
the s matrix for the 3 open channels @xmath23 can be written as : @xmath24 where @xmath25 represent elastic @xmath26 , @xmath27 , @xmath20 scattering with phase shifts @xmath28 , and @xmath29 respectively , @xmath30 represents the @xmath31 charge exchange amplitude , @xmath32 is a real number , and @xmath33 represents the inelasticity due to charge exchange . for convenience ,
the off diagonal matrix elements for the photo - production of the @xmath27 and @xmath20 channels are written as @xmath34 and @xmath35 , where @xmath36 and @xmath37 are proportional to the multipole amplitudes .
for the important case of production of s wave pions these are the @xmath38 transverse ( longitudinal ) multipoles . although not explicitly written here , the s matrix elements are for a fixed value of w , the total cm energy , and the quantum numbers l and j , the @xmath39 orbital and total angular momenta .
the notation of using 0 ( c ) for the neutral ( charged ) channel is conveniently generalized to neutron targets where the three channels are @xmath40 , and @xmath41 .
time reversal invariance requires that the s matrix be symmetric ( @xmath42 ) and unitarity requires that @xmath43 .
the form of the 2x2 @xmath1 portion of the s matrix has been chosen to be separately unitary and time reversal invariant .
eq .
[ eq1 ] is symmetric by construction .
applying the unitary constraint and assuming the weakness of the electromagnetic interaction by dropping terms of order e@xmath44 , one obtains @xcite : @xmath45 \\
m'_{c } & = e^{i(\delta_{\gamma}+\delta_{c } ) } [ a'_{c } \cos \frac{\phi}{2 } + i a'_{0 } \sin \frac{\phi}{2 } ] \end{array}\label{eq2}\ ] ] where @xmath46 and @xmath47 are smoothly varying , real functions , of the cm energy , and can be identified as proportional to the multipole matrix elements @xmath48(for @xmath17 ) and @xmath49 ( for @xmath50 ) in the absence of the final state @xmath1 interaction .
these equations show the connection between electromagnetic pion production and @xmath1 interactions .
note that the phase shifts @xmath28 , and @xmath29 appear as @xmath51 in photopion reactions compared to @xmath52 in elastic scattering because the @xmath53 interactions take place in the final ( initial ) state only .
eqs .
[ eq1 ] and [ eq2 ] are valid for both photo- and electro - production , i.e. for both real and virtual photons . in general @xmath54 , and @xmath47
are functions of @xmath55 ( the invariant four momentum transfer ) as well as w , l , and j. to order @xmath56 the @xmath1 sector parameters @xmath57 , and @xmath32 are functions of w , l , and j only . in this treatment terms of order @xmath56
have been neglected with the exception of @xmath58 which is small compared to both @xmath59 and @xmath29 ( except at the @xmath60 threshold ) , and the mass difference between the charged and neutral pions which is put in by hand .
this is the largest @xmath61 effect found in @xmath1 scattering @xcite and in photoproduction @xcite .
the effect of the @xmath61 terms have been worked out , although not presented here . in general
they produce a small coupling between the electromagnetic and @xmath1 sectors , so that e.g. the 2x2 @xmath1 sector of the s matrix is no longer separately unitary .
when the experimental situation reaches a sufficient level of accuracy these corrections should then be implemented .
eq .
[ eq2 ] is a generalization of the final state interaction theorem of fermi and watson @xcite who first pointed out the connection between photo - pion production and @xmath1 scattering . in their derivation , as in this one , time reversal and unitarity ( to order @xmath56 )
were assumed .
however they made the additional assumption that if one could neglect @xmath56 ( and higher order terms ) isospin would be conserved ( this was before the time of quarks ) .
this reduces the dimensionality of the s matrix to 2x2 which gives the simple solution @xmath62 where i=1/2,3/2 is the isospin of the @xmath1 system .
if isospin were conserved , so that the two thresholds are degenerate , eq .
[ eq2 ] reduces to the fermi - watson theorem . for the near threshold region in which the cm pion momentum q @xmath63 , the s wave phase shifts are @xmath64 q/3 = a_{\pi^{0}p } q , \delta_{c } = [ a_{3}+2a_{1}]q/3= a_{\pi^{+}n } q$ ] , and @xmath65 q/3 = a_{cex}q$ ] where @xmath66 is the s wave @xmath1 scattering lengths in the isospin 2i = 1 , 3 , or designated charge states .
however isospin conservation is badly violated in the threshold region due to the threshold energy difference between the @xmath67 and @xmath68 channels and the additional dynamic isospin violating effect due to the up and down quark mass difference .
this generalization of the fermi - watson theorem removes the approximation of isospin conservation .
note the interesting feature that below the @xmath20 threshold there are only two open channels and eq .
[ eq2 ] reduces to the form of the fermi - watson theorem , i.e. @xmath36 = real number @xmath69 or , equivalently , that @xmath70 .
it is of interest to show the connection between the s matrix formulation which takes the final state scattering into account to all orders and the original k matrix derivations @xcite which only took a single final state scattering into account .
this single scattering limit can be taken by expanding @xmath71 , and neglecting second order terms like @xmath72 , thus obtaining @xmath73 . by examing the region below the @xmath20 threshold where @xmath74 is continued as @xmath75
, it is observed that @xmath76 this means that the single scattering approximation does not obey unitarity , although in this particular case the numerical effects are not large .
the chpt calculations to one loop have a similar unitarity problem as was discussed by the chpt authors @xcite .
the connection between @xmath77 to the multipole amplitudes @xmath78 is @xmath79 .
note that the multipoles ( m ) have the dimensions of length , while the s matrix elements ( m ) are dimensionless .
the s wave photoproduction cross section is @xmath80 charge exchange reaction cross section is @xmath81 from which one obtains @xmath82 @xcite . defining @xmath83 one can then write eq .
[ eq2 ] in terms of multipole amplitudes as : @xmath84 \\
m_{c } & = e^{i(\delta_{\gamma}+\delta_{c } ) } [ a_{c } \cos \frac{\phi}{2 } + i a_{0 } q_{0 } | f_{cex } |/\cos \frac{\phi}{2 } ] \end{array}\label{eq3}\ ] ] in the threshold region eqs .
[ eq2 ] and [ eq3 ] lead to a unitary cusp . in this case ,
@xmath85 , and @xmath86 for the transverse multipole , or @xmath87 for the longitudinal multipole .
below the @xmath20 threshold @xmath74 is continued as @xmath88 .
thus the contribution of the two step @xmath89 reaction is in @xmath90 below ( above ) the @xmath20 threshold .
the magnitude of the unitary cusp , @xmath91 , is defined as the coefficient of @xmath74 : @xmath92 the value of @xmath91 can be calculated from the experimental value of @xmath93 @xcite .
assuming isospin is conserved @xmath94 @xcite . for @xmath95
the chpt prediction of @xmath96 @xcite is used .
this is in agreement with the preliminary value of @xmath97 from a recent experiment @xcite .
combining the two experimental ( theoretical ) numbers gives @xmath98 @xmath99 @xcite .
although it has not been previously noticed , the same physical mechanism that causes the isospin breaking in @xmath1 scattering and charge exchange also enters into @xmath91 .
this follows from unitarity and time reversal ( eqs .
[ eq2 ] and [ eq3 ] ) , combined with the prediction of isospin breaking for the s wave scattering lengths for charge exchange reactions @xmath100 , that there is an isospin violating contribution to @xmath91 , @xmath101 in the second line it has been assumed that there is no quark mass effect on @xmath102 .
it is estimated that @xmath103 2 to 3% @xcite .
a better theoretical calculation of this effect should be performed . the electric dipole amplitude @xmath104 with its unitary cusp is presented in figs . [ fig1 ] and [ fig2 ] .
the recent mainz / taps @xcite and saskatoon @xcite results for @xmath105 are presented in fig .
[ fig1 ] , where only the statistical errors are shown .
the small deviations between the data sets suggest the magnitude of the systematic errors . considering this
, the agreement between the calculated curves and the data is satisfactory .
the curves are the chpt calculation with three empirical low energy parameters used in fitting the data @xcite , and a unitary fit to the data @xcite using eq .
[ eq3 ] which has @xmath106 @xcite and a linear function of photon energy for @xmath107 with two parameters which were fit to the data . for this case @xmath108
@xcite and the effect of the two step charge exchange reaction is dramatic in the real ( imaginary ) part below(above ) the @xmath20 threshold .
approximately 1 mev above the @xmath20 threshold the @xmath109 .
the negative sign in @xmath110 makes @xmath111 have the opposite sign from @xmath112 .
it is interesting to note that both the power counting rules of chpt and the constraints of unitarity lead to pion rescattering in the final state as a critical dynamical ingredient .
although the unitary and chpt curves both agree with the data for @xmath105 there is an important difference between them .
the value of @xmath91 = 2.78 @xcite calculated for the @xmath17 reaction is smaller then the one calculated using the separately predicted values of @xmath113 and @xmath114 of @xmath115 .
this difference can be clearly seen in fig .
[ fig2 ] .
the reason for this discrepancy , which was discussed by the chpt authors @xcite , is due to the fact that the chpt calculation is carried out to one loop which is not sufficient for @xmath111 .
this is a general feature of chiral perturbation theory in which the imaginary part of the amplitude is not calculated as accurately as the real part , and thus unitarity is only approximately satisfied at a given order @xcite . as will be discussed below the difference between the unitary and one loop chpt values of @xmath91
can be observed in future experiments in which @xmath111 will be measured directly . to accurately measure the magnitude of the unitary cusp and to exploit the connection between electromagnetic pion production and low energy @xmath1 interactions
, one must measure @xmath111 . in photoproduction
this requires experiments with polarized beams and/or targets @xcite .
for the sake of brevity the formulas connecting the cross sections and polarization observables to the multipoles @xcite will not be quoted here . to briefly demonstrate the power of polarized photo - pion experiments ,
two asymmetries are shown in fig .
[ fig3 ] : @xmath116 for linearly polarized photons with an unpolarized target ; and t for unpolarized photons but with a target polarized normal to the reaction plane .
the results presented in fig . [ fig3 ] use the p wave predictions of chiral perturbation theory @xcite and the unitary fit to @xmath117 discussed above @xcite .
@xmath116 is primarily sensitive to the p wave multipoles and since the unitary fit has essentially the same p wave multipoles as chpt , the curves for chpt and the unitary fit are almost identical . by contrast , t is sensitive to a linear combination of p wave multipoles times @xmath111 , and shows its rapid rise above the @xmath20 threshold . for t ,
the large difference in @xmath111 between the unitary fit @xcite and chpt to one loop @xcite should be straightforward to distinguish experimentally .
a proposed experiment with tagged photons using an active , polarized proton target at mainz estimates that @xmath91 can be measured to @xmath118 to 2% @xcite .
similar results could be obtained using a laser backscattering source @xcite or small angle electron scattering with polarized , internal , targets in a storage ring facility @xcite .
there are several possible strategies to test isospin conservation from measurements in the near threshold @xmath119 reaction .
a definitive demonstration of this effect would involve precision measurements of the real and imaginary parts of @xmath117 for the unitary cusp contributions to the @xmath120 and @xmath17 reactions .
although the predicted effect of @xmath14 2 to 3% in @xmath91 is larger than the usual @xmath141% effect expected on the basis of electromagnetic effects , it represents a serious experimental challenge to accomplish this goal , particularly since free neutron targets are not available to make this measurement . a simpler strategy would be to perform a precision measurement of the unitary cusp for the @xmath121 reaction , extract the value of @xmath91 , and compare the result of with the value obtained using unitarity and isospin conservation quoted above .
an equivalent strategy is to obtain @xmath122 from the measured value of @xmath91 and compare it to the accurately measured value of @xmath123 @xcite , obtained from the width of the 1s state in pionic hydrogen , to see if there is any deviation from the pure isospin prediction that these values are equal and opposite @xcite .
it is important to note that there is a psi proposal @xcite to improve the pionic hydrogen measurement so that the error in @xmath123 will be reduced to @xmath14 1% .
it is therefore of interest to discuss the information that might be obtained from measurements of the s wave @xmath1 scattering lengths , @xmath8 , from photopion reactions with the values that have been obtained using conventional pion beams .
the values for @xmath8 for several physical channels are presented in table [ tab1 ] ( the appropriate isospin relations @xcite were used when required ) .
the most accurate and direct measurements come from pionic hydrogen experiments @xcite . in order to test isospin symmetry
the psi group has also measured pionic deuterium in order to get at the @xmath124 scattering length @xcite .
unfortunately the two body corrections in deuterium are large and add significantly to the error which makes the isospin conservation test uncertain at the required level of accuracy , so that these values are not quoted here .
the results of the two most recent empirical analyses of the @xmath1 data for pion kinetic energies greater than @xmath14 30 mev , extrapolated to threshold , are also presented @xcite in table [ tab1 ] . in the said study
the scattering lengths come from a dispersive analysis , the errors are hard to ascertain and are not quoted @xcite .
the chiral perturbation theory results @xcite are consistent with experiment .
if isospin symmetry is exact there are two independent scattering lengths in the i = 1/2 , 3/2 states , and six possible physical @xmath125n elastic scattering and charge exchange reactions , so that there are many possible tests if three or more channels can be measured precisely .
this is precisely the gap that can be filled by measurement of the final state @xmath125n interactions in photopion production , e.g. @xmath27 or @xmath126 in the @xmath127 reaction .
.s - wave @xmath1 scattering lengths @xmath8 for several channels in units of @xmath128 ( n = n or p ) [ cols="^,^,^,^,^,^ " , ] another test of isospin conservation has been made for medium energy @xmath125n scattering ( pion kinetic energy from 30 to 100 mev ) @xcite . in both cases
the scattering amplitudes in the i = 1/2 and 3/2 states were obtained from the data for @xmath129 elastic scattering . from these , the prediction for the @xmath130 charge exchange amplitude made on the assumption of isospin conservation was compared to the empirical charge exchange amplitude .
an isospin violation at the @xmath131 level has been found by both analyses @xcite , primarily in the s wave amplitude .
these analyses depend on the quality of the data and on the accuracy of the coulomb corrections , which have been criticized as being inconsistent with the strong interaction calculations that were employed @xcite . if isospin is indeed violated at the 7% level , it is not clear how to relate this to the isospin breaking predictions for the s wave scattering length @xcite which is applicable at lower energies .
what is required is an extension of the isospin breaking calculations to medium energies .
a test of the existing predictions requires experiments at lower energies which would measure the s wave scattering lengths .
in addition to the measurement of @xmath111 above the @xmath20 threshold discussed above , one could contemplate the more difficult measurement when only the @xmath27 channel is open . as was discussed previously , from a measurement of both the @xmath132 and @xmath133 parts of @xmath117 ( or @xmath134 ) one obtains @xmath59 , for which there is a predicted @xmath135 isospin breaking effect .
however , because of the small expected size of @xmath136 in the energy region below the @xmath20 threshold , this is a very difficult task .
it may be at the limits of feasibility using either a laser backscattering source , where an intense photon flux is concentrated in a small energy interval @xcite , or with internal target , small angle scattering with polarized internal targets @xcite . in conclusion ,
the connection between the @xmath137 reaction and @xmath138 scattering has been presented in a rigorous , model - independent way , which is unitary and time reversal invariant , and where the isospin breaking due to the threshold difference between the @xmath68 and @xmath67 channels , and the mass difference between the up and down quarks , is taken into account .
this leads to a predicted unitary cusp due to the two step @xmath139 charge exchange reaction .
the magnitude of the unitary cusp is given by @xmath140 .
this unitary cusp has been recently observed in photoproduction experiments @xcite .
it has been shown for the first time that there is a dynamical isospin breaking effect in the value of @xmath91 , due to the mass difference of the up and down quarks , in electromagnetic pion production .
this is linked by unitarity and time reversal invariance to a predicted quark mass effect in @xmath141 scattering and pion charge exchange @xcite . at present
there are accurate measurements for @xmath142 and @xmath143 from pionic atoms performed at psi @xcite .
to check isospin conservation requires at least one more precision measurement in another charge channel .
this is more readily performed in electromagnetic meson production for two reasons : 1 ) in order to accurately measure the s wave scattering length , it is important to work at very low energies ( e.g. @xmath144 10 mev ) at which @xmath1 experiments are hard to perform since the low energy charged pions decay and also since @xmath60 beams can not be made at any energy ; and 2 ) in electromagnetic pion production one can access charge states that can not be reached with conventional pion beams ( this is important for isospin checks ) .
it is shown that photoproduction experiments with polarized targets can lead to a precise measurement of the small but interesting isospin violating effects .
in addition to the polarization observables in photoproduction discussed above , similar information can be obtained in threshold @xmath60 electroproduction from the combination of the transverse- longitudinal ( tl and tl ) structure functions @xcite . this possibility will be discussed in a future publication . finally we stress that the important program of precisely measuring the @xmath1 and photopion reactions that are required to test the predicted isospin violation , is difficult but feasible .
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this work is financially supported by a grant - in - aid for specially promoted research ( no .
20001006 ) from the ministry of education , culture , sports , science and technology .
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standing waves near the zigzag and armchair edges , and their berry s phases are investigated .
it is suggested that the berry s phase for the standing wave near the zigzag edge is trivial , while that near the armchair edge is non - trivial .
a non - trivial berry s phase implies the presence of a singularity in parameter space .
we have confirmed that the dirac singularity is absent ( present ) in the parameter space for the standing wave near the zigzag ( armchair ) edge .
the absence of the dirac singularity has a direct consequence in the local density of states near the zigzag edge .
the transport properties of graphene nanoribbons observed by recent numerical simulations and experiments are discussed from the point of view of the berry s phases for the standing waves .
the electronic property of a graphene nanoribbon differs greatly from that of a carbon nanotube .
a metallic carbon nanotube exhibits a high mobility , while a graphene nanoribbon shows a transport gap .
@xcite the high mobility observed in metallic carbon nanotubes indicates that the scatterers are not effective in producing backward scattering .
there should be a mechanism which suppresses the backward scattering in metallic carbon nanotubes .
suppose that an electron with momentum @xmath0 is coming into the impurities which are represented by the circles in fig .
[ fig : abs](a ) .
the electron is scattered by the impurities and changes its momentum direction .
let us consider the probability amplitude that the electron is scattered in the backward direction , as shown in fig .
[ fig : abs](a ) . in this case , the final wave function , @xmath1 , is given by rotating the wave vector of the initial state , @xmath2 , by @xmath3 , so that we have the relationship between the initial state and final state as @xmath4 where @xmath5 is a rotational operator with angle @xmath6 .
note that this particular path shown in fig .
[ fig : abs](a ) is not the unique path that an electron can follow .
there is an another path that an electron can follow , which we denote it by the lines in fig .
[ fig : abs](b ) .
the new path relates to the original path through the `` time reversal '' .
we denote the final state in this `` time reversal '' path by @xmath7 .
because the final state is given by rotating the wave vector by @xmath8 , we have the relationship , @xmath9 between the initial state and the final state . by eliminating the wave function of the initial state from eqs .
( [ eq : path1 ] ) and ( [ eq : path2 ] ) , we get @xmath10 now , the total backward scattering amplitude is given by the sum of @xmath1 and @xmath11 as @xmath12 \phi_{\bf -k}$ ] . because the wave function gets an extra phase shift of @xmath3 ( called the berry s phase @xcite ) through a rotation of the wave vector around the dirac point , that is , @xmath13 is equivalent to @xmath14 ( @xmath15 ) , a `` time - reversal '' pair of backward scattered waves cancels with each other , i.e. , @xmath16 @xcite .
, combines with the final state of the scattering path shown in ( b ) , @xmath11 , to suppress the total backscattering amplitude , @xmath17 , due to the berry s phase .
the two scattering paths , ( a ) @xmath18 and ( b ) @xmath19 , are related with each other by `` time - reversal '' . ]
this , absence of backward scattering mechanism provides us with a simple solution explaining the high mobility , and the existence of a singularity at the dirac point is essential to the nontrivial phase shift of @xmath20 .
however , it is not obvious whether the wave functions realizing in graphene nanoribbons can acquire a nontrivial berry s phase or not .
the eigen state near the edge is the standing wave resulting from the interference between an incident wave and the edge reflected wave .
if the berry s phase for the standing wave is trivial , then the scatters , such as a potential disorder created by charge impurities , may give rise to backward scattering . as a result ,
the mobility of a graphene nanoribbon decreases considerably than that of a carbon nanotube .
@xcite in the present paper , we study the berry s phases of the standing waves near the zigzag and armchair edges , and their effects on the local density of states and the transport properties of graphene nanoribbons . in fig .
[ fig : unit ] , we consider the zigzag edge parallel to the @xmath21-axis , by which translational symmetry along the @xmath22-axis is broken .
thus , the incident state with wave vector @xmath23 is elastically scattered by the zigzag edge , and the wave vector of the reflected state becomes @xmath24 . by contrast , the armchair edge parallel to the @xmath22-axis breaks translational symmetry along the @xmath21-axis , so that the wave vector of the reflected state is @xmath25 .
since the brillouin zone ( bz ) is given by rotating the hexagonal lattice by 90@xmath26 , one can see in fig .
[ fig : unit ] that for the incident state near the k point , the reflected state by the zigzag edge is also near the k point , while the reflected state by the armchair edge is near the k@xmath27 point .
hence , the scattering by the zigzag edge is intravalley scattering , while that by the armchair edge is intervalley scattering .
coordinate system is fixed as shown .
( right ) the hexagon represents the first bz of graphene , and the corners of the hexagon are k and k@xmath27 points . ] to begin with , let us consider the scattering problem for the zigzag edge . since the zigzag edge is not the source of intervalley scattering , we focus on only the electrons near the k point . the incident and reflected waves are represented by the bloch states .
the bloch state in the conduction energy band is written as @xmath28 where @xmath29 is the area of sample , @xmath0 is the momentum measured from the k point , and @xmath30 is the polar angle between the vector @xmath0 and the @xmath31-axis .
note that the bloch function acquires a nontrivial berry s phase of @xmath3 by a rotation of the wave vector around the k point as @xmath32 the standing wave modes arise from the interference between the incident waves and the reflected waves .
let the wave vector of the incident wave be @xmath33 .
then the vector of the elastically reflected wave is given by @xmath34 , and the standing wave near the zigzag edge is written as @xmath35 where the phase @xmath36 should be determined in such a manner that @xmath37 [ @xmath38 satisfies the boundary condition @xcite .
we take the following boundary condition for the zigzag edge @xcite , @xmath39 which describes a situation in which the b - atoms located slightly above the horizontal dashed line at @xmath40 in fig .
[ fig : unit ] are separated from the lower semi - infinite graphene for @xmath41 .
the boundary condition of eq .
( [ eq : phib0 ] ) is satisfied when @xmath42 , and the standing wave is written as @xmath43 where @xmath44 is the length of the zigzag edge , and @xmath45 for @xmath46 and @xmath47 otherwise .
the value of @xmath48 is determined by the normalization condition .
we note that the standing wave of eq .
( [ eq : zigwfsum ] ) reproduces the result of tight - binding lattice model ( see appendix b of ref . ) .
first , we examine the berry s phase of the standing wave . for simplicity ,
let us consider the case @xmath49 for eq .
( [ eq : s - sol ] ) .
this corresponds to a normal incident process to the zigzag edge [ @xmath50 and @xmath51 .
the amplitude for this process does not vanish as @xmath52 this standing wave includes a backward scattering amplitude , and the existence of the backward scattering amplitude indicates that the berry phase of the standing wave vanishes .
indeed , since @xmath53 , the berry s phase of the reflected state is given by @xmath8 as @xmath54 while that of the incident state is @xmath3 [ see eq .
( [ eq : berry- ] ) ] .
the berry s phase of @xmath3 for the incident state is canceled by the berry s phase of @xmath20 for the reflected state , and hence the berry s phase for the standing wave vanishes in total .
the trivial berry s phase can be understood from eq .
( [ eq : s - sol ] ) , in which the bloch function of the standing wave is real @xcite .
next , we study the standing wave near the armchair edge .
the k and k@xmath27 points need to be considered simultaneously in the case of armchair edge since the armchair edge is the source of intervalley scattering .
suppose that the wave vector of the incident wave measured from the k point is @xmath33 .
thus , the wave vector of the elastically reflected wave measured from the k@xmath27 point is given by @xmath55 , and the standing wave is written as @xmath56 it can be shown that the bloch state near the k@xmath27 point is given by @xmath57 @xcite .
thus , we may write @xmath58 where @xmath59 for @xmath60 and @xmath61 otherwise . the relative phase @xmath62 can be determined by the boundary condition for armchair edge .
the value of @xmath62 must be @xmath63 , which will be derived elsewhere .
@xcite note that the bloch functions for the k and k@xmath27 points are the same .
thus , the berry s phase of the standing wave near the armchair edge is given by @xmath3 .
as a result , we can expect that the absence of backward scattering mechanism is valid near the armchair edge .
note , however , that the absence of backward scattering mechanism for the armchair edge is not identical to that discussed so far for carbon nanotubes .
this is because the notion of a `` time - reversal pair '' of scattered waves in the original argument @xcite should be replaced to a true time - reversal pair of scattered waves near the armchair edge .
the time - reversal state of eq .
( [ eq : armwf ] ) is given by @xcite @xmath64 a magnetic field breaks time - reversal symmetry , so that it invalidates the new absence of backward scattering mechanism and leads to weak anti - localization .
it is also interesting to point out that the berry s phase for the standing wave near the armchair edge is robust against an ordinary mass term , while it is not robust against a topological mass term @xcite .
the appearance of a nontrivial berry s phase is related to a singularity in parameter space @xcite . here , we show the relationship between the singularity at the dirac point ( @xmath65 ) in the momentum space and the standing wave . for the zigzag edge
, @xmath31 is a continuous variable , and therefore the spectrum with @xmath66 may cross the dirac singularity .
however , eq . ( [ eq : s - sol ] ) shows that the standing wave with @xmath67 does not exist , that is @xmath68 .
thus , the dirac singularity seems to be separated from the spectrum of the standing wave . to explore this point further
, we consider a zigzag nanoribbon by introducing another zigzag edge at @xmath69 , in addition to the zigzag edge at @xmath40 .
suppose that the edge atoms at @xmath69 are b - atoms , which imposes the boundary condition on the wave function at @xmath70 as @xmath71 .
this leads to the constraint equation for @xmath23 , @xmath72 where @xmath73 is an integer .
we note that this equation provides the solutions of @xmath74 which was obtained by brey and fertig in ref .
[ the minus sign in front of @xmath31 is a matter of notation ] .
note that @xmath73 must be a nonzero integer because the equation does not possess a solution when @xmath75 . in fig .
[ fig : sflow ] , we plot the solutions of eq .
( [ eq : const ] ) for the cases @xmath76 and 2 . the solution for @xmath77
is given by shifting the curve for @xmath78 by @xmath79 , and only the solution with @xmath80 shows an anomalous feature .
the curve with @xmath80 approaches a point on the @xmath31-axis .
this behavior can be checked by setting @xmath81 in eq .
( [ eq : const ] ) with @xmath80 .
it is appropriate to refer to this state with @xmath82 and @xmath66 as the critical state @xcite because this state is neither the standing wave nor the edge state @xcite .
the fact that no curve crosses the dirac singularity is consistent to the absence of a nontrivial berry s phase for the standing wave .
in other words , an electron can not approach the dirac singularity due to the presence of the zigzag edge . as a result
, an energy gap appears in the spectrum of the standing wave .
the minimum energy gap is determined by the critical state as @xmath83 where @xmath84 is the fermi velocity .
in contrast to the boundary condition for the zigzag edge , the boundary condition for the armchair edge does not forbid an electronic state cross the dirac singularity point , and therefore the electron can pick up a nontrivial berry s phase . and
@xmath85 which are allowed by the boundary condition for a zigzag nanoribbon .
the dirac singularity is not located on the curves , which shows that there is no singularity at the parameter space of the standing waves .
the inset shows the corresponding spectral flow in the energy dispersion relation . ] here , we consider pseudospin of the standing wave in order to examine the local density of states ( ldos ) near the zigzag edge .
the pseudospin for an eigenstate @xmath86 is defined by the expectation value of pauli matrices as @xmath87 ( @xmath88 ) where @xmath89 is a pseudospin density defined by @xmath90 .
the boundary condition of eq .
( [ eq : phib0 ] ) means that the pseudospin density is polarized into the positive @xmath91-axis locally near the zigzag edge , that is , @xmath92 and @xmath93 .
actually , by putting @xmath94 into eq .
( [ eq : s - sol ] ) , we see that the standing wave near the zigzag edge has amplitudes only on a - atoms .
this polarization of the pseudospin causes an anomalous behavior to appear in the ldos near the zigzag edge .
to show this , let us first review the ldos for a graphene without an edge .
assuming that electrons are non - interacting , the bulk ldos is given by @xmath95 where @xmath96 is proportional to @xmath97 , which results from the dirac cone spectrum .
the ldos near the zigzag edge can be calculated by using the standing wave given in eq .
( [ eq : s - sol ] ) .
the ldos has the form , @xmath98 where @xmath99 is defined as @xmath100 by performing the integral with respect to the angle @xmath6 in eq .
( [ eq : rfac ] ) , we obtain the analytical result for @xmath99 as @xmath101 where @xmath102 is a bessel function of order @xmath103 .
the ldos is symmetric with respect to @xmath104 .
the bulk ldos in eq .
( [ eq : ldos_bulk ] ) can be reproduced by setting @xmath105 in eq .
( [ eq : ldos ] ) since @xmath106 . in fig .
[ fig : pspin ] , we plot the ldos at @xmath107 , 1 , 2 , 3 , and 6[nm ] .
the slope of the ldos depends on the distance from the zigzag edge . at the edge ,
that is , at @xmath107 , since @xmath108 and @xmath109 , we see that @xmath110 , and the slope of @xmath111 is half of that of @xmath112 [ @xmath113 .
this results from the fact that the amplitudes at one of two sublattice disappears near the zigzag edge ( due to the pseudospin polarization ) , and only the half of the amplitude in the unit cell can contribute to the ldos . is due to the edge states . ] in fig .
[ fig : pspin ] , in addition to the ldos due to the standing wave , we plot the ldos ( at @xmath114[nm ] ) due to the edge states @xcite .
the edge states produce a peak at @xmath104 in the ldos .
the ldos of the edge states is calculated as follows .
the wave function of the edge state is given by @xcite @xmath115 note that the edge states appear for @xmath116 and @xmath41 ( see fig .
[ fig : sflow ] ) . since the energy eigenvalue of the edge state vanishes
, the ldos for the edge states is written as @xmath117 where @xmath118 is a phenomenological parameter representing energy uncertainty of the edge states , for which we assume @xmath119 mev . substituting @xmath120 with @xmath121 in eq .
( [ eq : rhoes ] ) , and performing the integral for @xmath31 , we obtain @xmath122 this result has been used in fig .
[ fig : pspin ] to plot the ldos of the edge states . note that @xmath123 decreases as @xmath124 , which is a slowly decreasing function compared with the exponential decaying wave function of the edge state .
it is interesting to note that @xmath125 can be derived from the fact that @xmath126 does not depend on @xmath22 for a large value of @xmath127 .
this is a sum rule for the total ldos consisting of the edge states and the standing waves , in which the density above ( below ) the fermi energy at @xmath104 must be position independent .
this argument also shows that the singular behavior of the ldos for the edge states at @xmath40 is a consequence of the wave function of the standing wave . by examining the standing wave near the graphene edge ,
we have seen that the armchair edge preserves a non - trivial berry s phase as in the bulk of graphene , on the other hand , such a non - trivial phase is absent for the standing wave near the zigzag edge . here
, we consider the indication of this result with respect to the transport in graphene nanoribbons . in the argument for the absence of the backward scattering given in the introduction ( see fig .
[ fig : abs ] ) , we find , by replacing @xmath2 with the standing wave @xmath128 , that the backscattering amplitude , @xmath129\psi_{\bf -k}$ ] , is enhanced for zigzag nanoribbons ( due to @xmath130 ) , while it still vanishes for armchair nanoribbons ( due to @xmath131 ) .
thus , we can expect that the conductance of a zigzag nanoribbon is smaller than that of an armchair nanoribbon , if we assume only scatters with long - range potential .
the simulations performed by areshkin _ et al . _
@xcite and yamamoto _ et al . _
@xcite show indeed that long - range potentials are ineffective in causing backscattering in a perfect armchair nanoribbon . for narrow metallic armchair nanoribbons ,
the linear energy dispersion originating from the dirac point provides a nearly perfect conduction , and this nearly perfect conduction might relate to the nontrivial berry s phase since the linear energy dispersion picks up the dirac singularity .
note , however , that the standing wave for the armchair edge is a short - length intervalley mode , that is , the electronic wave length of the standing wave is order of the lattice constant . as a result
, the wave function can be very sensitive to a short - range scattering potential , such as irregular edges @xcite and lattice vacancies @xcite .
an armchair nanoribbon has both advantages ( non - trivial berry s phase ) and disadvantages ( intervalley mode ) for the transport as well as that a zigzag nanoribbon has both advantages ( intravalley mode ) and disadvantages ( trivial berry s phase ) .
therefore , in a realistic situation , it is reasonable to consider that a nanoribbon does not exhibit an electronic conduction comparable to a metallic carbon nanotube .
it seems that recent tight - binding numerical studies show that the conductance of zigzag nanoribbons is more robust than that of armchair nanoribbons against edge disorders .
@xcite in the numerical studies , the transport given by the electrons which are located very close to the fermi level is concerned . for zigzag nanoribbons ,
the propagating modes in each valley contain a single one - way excess channel ( the states on the curve of @xmath80 which are close to the critical state shown in fig .
[ fig : sflow ] ) .
this feature causes a perfectly conducting channel to appear in the disordered system for the case that impurities do not give rise to intervalley scattering @xcite .
this feature also seems to cause a robust conducting channel to appear in the edge disordered system .
@xcite for our discussion of the backscattering in zigzag nanoribbons , we assume that the initial state with momentum @xmath31 accompanies the corresponding final state with @xmath132 .
this condition is not satisfied for the single one - way excess channel , for which our discussion for the berry s phase is useless .
the single one - way excess channel plays a dominant role in zigzag nanoribbons of the widths up to several nanometers for a limited energy window .
@xcite our result for the berry s phase may be useful in discussing transport of zigzag nanoribbons of the widths more than 10 nm , for which we have several number of channels having the final state with @xmath132 .
our result may have a relationship to the formation of a transport gap observed for graphene nanoribbons .
@xcite note that the standing wave near a realistic rough edge can be described as the sum of the standing waves for zigzag and armchair edges .
@xcite since the standing wave for zigzag edge is a long - length intravalley mode and that for armchair edge is a short - length intervalley one , their characters do not interfere with each other and can coexist in the standing wave near a rough edge .
this is also indicative of that a small portion of zigzag edge in a rough edge eventually governs the long - length transport behavior of a nanoribbon , as is numerically simulated by several authors .
@xcite indeed , gallagher _ et al . _
@xcite and han _ et al . _
@xcite observed that the source - drain gap has a strong dependence on ribbon s length .
it is also important to recognize that there are two distinct localized states ; the edge states and localized states which consist of the standing waves . since the localization length of the former is on the order of lattice constant , it is difficult to consider that the transport behavior of nanoribbons of the widths larger than @xmath133 nm is dominated by the edge states .
@xcite in contrast , the latter is caused by the scatters , such as a potential disorder created by charge impurities , giving rise to backward scattering . since the latter consists of the standing waves ,
there is a possibility that the localization length is the order of the width of a ribbon .
han _
et al . _
@xcite measured electron transport in lithographically fabricated nanoribbons of widths @xmath134 nm with rough edge on the order of nanometers , and confirmed that the size of a transport gap is inversely proportional to the ribbon width . in conclusion
, we have shown that the berry s phase for the standing wave near the zigzag edge is trivial .
the momentum parameter space for the standing wave near the zigzag edge does not include the dirac singularity , which is essential to the absence of a non - trivial berry s phase . as a result
, the absence of backward scattering mechanism that works well in a metallic carbon nanotube can not be used for a zigzag nanoribbon .
the absence of the dirac singularity in the momentum space of the standing wave results in the peak in the ldos due to the edge states .
an observation of the ldos peak near the zigzag edge is a direct evidence of the absence of the dirac singularity in the parameter space of the standing wave .
in contrast , a non - trivial berry s phase survives the reflection by the armchair edge . for the standing wave near the armchair edge
, we obtain the new absence of backward scattering mechanism in which a real time - reversal pair of backward scattered waves cancels with each other .
this absence of the backward scattering mechanism for the armchair edge is not robust against a short - range scattering potential since the standing wave is a short - length intervalley mode .
an armchair nanoribbon has both advantage and disadvantage for the transport as well as a zigzag nanoribbon .
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for the two dimensional kinetic ising model at finite temperature , the local mean magnetisation @xmath0 , simply related to the fraction of time spent by a given spin in the positive direction , has a limiting distribution , singular at @xmath1 , the onsager spontaneous magnetization .
the exponent of this singularity defines the persistence exponent @xmath2 .
we also study first passage exponents associated to persistent large deviations of @xmath3 , and their temperature dependence .
epsf _ submitted for publication to _ in this work we present a new approach to the study of persistence for systems undergoing phase ordering @xcite at finite temperature , which we shall illustrate on the case of the two dimensional ising model . in this approach persistence appears as a stationary property of the coarsening system , and the role of the onsager spontaneous magnetization at equilibrium @xmath4 is made apparent , thus revealing new fundamental features of phase ordering .
it departs from previous approaches to finite temperature persistence @xcite , where these features did not appear .
consider a system of ising spins @xmath5 located at sites @xmath6 , started from a random initial condition , and evolving under the heat bath dynamics at fixed temperature below the critical temperature . at each time
step a spin is picked at random , and updated with the probability @xmath7 where the sum runs over the neighbours of site @xmath8 . under this dynamics spins thermalize in their local environment .
therefore the system coarsens , i.e. domains of opposite signs grow and , in the scaling regime , the system is statistically self similar , with only one single characteristic length scale , which is the size of a typical domain @xcite . the question of persistence is to determine the fraction of space @xmath9 which remained in the same phase up to time @xmath10 @xcite ( or from time @xmath11 to time @xmath12 ) . for the two dimensional ising model at zero temperature ,
two phases coexist , corresponding to all spins equal to @xmath13 or all spins equal to @xmath14 .
hence @xmath9 is equivalently defined as the fraction of spins which did not flip up to time @xmath10 @xcite , i.e. which were not swept by an interface between domains of all spins @xmath13 or all spins @xmath14 .
numerical measurements indicate an algebraic decay @xmath15 , with @xmath16 @xcite .
the definition of persistence at finite temperature below @xmath17 is more subtle to implement because one has to make clear what is meant by ` phase ' .
by essence , in the coarsening process there is phase separation , each phase wanting to develop at the expense of the other .
it is therefore intuitive that the system , though perpetually out of equilibrium , tries to reach locally one of the two equilibrium phases , corresponding to @xmath1 , where @xmath18 is the onsager spontaneous magnetization at equilibrium @xcite .
hence in the scaling regime the average magnetization inside a domain measured on a scale of time small compared to the flipping time of the domain , should be close to the equilibrium magnetization at this temperature .
coming back to persistence , the definition of @xmath9 should reflect , in one way or the other , the fact that a given point in space remained in a phase of average magnetization equal to @xmath1 up to time @xmath10 .
this intuitive analysis is confirmed by what follows .
the central point of our approach is to consider the statistics of the local mean magnetization simply related to the fraction of time spent by a spin in the positive direction in the limit of large times .
this line of thought was already used in @xcite in the study of domain coarsening for the one dimensional ising model at zero temperature , or for the simple diffusion equation evolved from a random initial condition ( see also @xcite ) .
the idea is that since persistence probes the past history of the system , a natural quantity to consider is @xmath19 , where @xmath20 is the spin at site @xmath8 and @xmath21 ( @xmath22 ) is the length of time spent by the spin pointing upward ( downward ) , with @xmath23 .
the local mean magnetization is defined as @xmath24 for instance at zero temperature , the persistence probability @xmath9 is equal to @xmath25 since the event @xmath26 is identical to the event @xmath27 . since @xmath3 is a local quantity varying from site to site , one is naturally led to investigate the distribution of @xmath3 , @xmath28 for the one dimensional ising model at @xmath29 is was shown in @xcite by analytical arguments and numerical measurements that , when @xmath30 , @xmath31 converges to a limit distribution @xmath32 with density @xmath33 singular at @xmath34 with singularity exponent equal to @xmath35 .
it is for instance easy to show that , when @xmath30 , the limit of @xmath36 is a constant equal to @xmath37 , the laplace transform of the two - time correlation with respect to the variable @xmath38 , at argument equal to 1 @xcite .
this result therefore provides a _ stationary _ definition of persistence at zero temperature .
the same holds for the diffusion equation @xcite .
we now address the same questions at finite temperature .
we first report on numerical results .
= figure 1 depicts the histogram of the density of @xmath3 at time @xmath39 for values of @xmath40 ranging from @xmath41 to @xmath42 .
already for such a short time , and for every temperature @xmath43 , the density is maximum around @xmath1 , the equilibrium magnetization ( [ eqons ] ) . at larger times and for @xmath44 ,
the density of @xmath3 becomes peaked around zero , i.e. the mean magnetization converges toward the average magnetization per spin @xmath45 , reflecting the fact that the system reaches equilibrium . at @xmath17 the peaking of the density of @xmath3 is observed to be very slow .
finally at @xmath43 , @xmath31 converges , when @xmath30 , to a limit distribution @xmath32 with density @xmath46 given as in ( [ eqfm ] ) .
the existence of a limit law at finite temperature @xmath47 relies on the same arguments as for the zero temperature case .
for instance the convergence of @xmath48 to a constant equal to @xmath37 still holds since it only relies on the existence of a scaling regime @xcite .
= the striking fact is that now the density concentrates on @xmath49 $ ] with an exponential decay with time of @xmath50 to @xmath41 .
moreover the limit density is singular at @xmath1 , with a singularity exponent @xmath35 which defines the persistence exponent at temperature @xmath40 .
this leads to the question of the temperature dependence of persistence for @xmath43 .
the simplest assumption is that , during the coarsening process , the scales of time between a short time regime and the scaling regime decouple , yielding the following relation between the moments of the limit distributions at @xmath40 and at zero temperature , @xmath51 implying the identity of the limit distributions , if @xmath3 is rescaled by @xmath4 , and as a consequence , the temperature independence of @xmath2 . this would be in agreement with the usual view that zero temperature is an attracting fixed point for the dynamics of phase ordering @xcite .
equation ( [ eqm2 t ] ) is hard to check by numerical measurements because the convergence of the data is observed to be slow .
the difficulty is illustrated by figure 2 which depicts a plot of @xmath52 against the rescaled variable @xmath53 , at @xmath29 and @xmath54 , for @xmath55 , and at @xmath56 , for @xmath57 and @xmath58 .
though one can not be conclusive on the sole basis of numerical measurements , data collapse seems nevertheless plausible .
let us note that the limit distribution @xmath59 at @xmath29 is well approximated by a beta distribution , as was observed for the @xmath60 ising model @xcite , or for the diffusion equation @xcite .
the singularity exponent of the beta distribution is found to be around @xmath61 .
a more precise numerical determination of the exponent @xmath2 from the limit distribution of @xmath3 needs further work and will be presented elsewhere .
let us summarize at this point . for @xmath43 , the local mean magnetization @xmath3 has a limit probability density when @xmath30 , defined on the interval @xmath49 $ ] , where @xmath4 is the equilibrium magnetization , and singular at both ends .
this provides a _ stationary _ definition of persistence , which is a natural extension of the zero temperature case , where the singularity exponent defines the persistence exponent .
these are the central results of the present work .
= another new aspect of persistence introduced in @xcite is concerned with _ persistent large deviations_. the probability of persistent large deviations above the level @xmath62 ( @xmath63 ) , denoted by @xmath64 , is defined as the probability that the mean magnetization was , for all previous times , greater than @xmath62 @xcite , @xmath65 in other words one is interested in the persistence probability of the stochastic process @xmath66 @xcite . if one views the stochastic process @xmath20 as the successive steps of a fictitious random walker , then @xmath3 is the mean speed of the walker between 0 and @xmath10 , and @xmath64 is the probability that this mean speed remained larger than @xmath62 between 0 and @xmath10 .
this probability is a natural generalization of the persistence probability @xmath67 , which corresponds for the walker to always stepping to the right . for the one dimensional ising model at zero temperature ,
@xmath64 was observed to decay algebraically at large times with an exponent @xmath68 continuously varying with @xmath62 @xcite . for @xmath69 , @xmath70 , the usual persistence exponent .
figure 3 depicts a log - log plot of @xmath64 for the two dimensional ising model at zero temperature , @xmath62 varying from @xmath14 to @xmath71 , while figure 4 depicts the corresponding exponents @xmath68 , extracted from figure [ fig3 ] .
let us mention that algebraic decay of @xmath64 was also observed for the diffusion equation @xcite , and that this quantity and the corresponding exponents @xmath68 can be exactly computed for the simple model considered in @xcite .
= we now address the role of temperature for persistent large deviations . as is obvious from the first part of this work , if @xmath72 , then @xmath64 decays to zero exponentially rapidly . on the other hand , for @xmath73 one observes algebraic decay of @xmath64 , as at zero temperature .
otherwise stated , @xmath74 separates two regimes of persistent large deviations , between exponential and algebraic . as a consequence , and by analogy with the zero temperature case , one could think of extracting the persistence exponent at finite temperature from the decay at large times of @xmath64 when @xmath75 , which leads to the formal definition @xmath76 however this definition is not easy to implement in practice .
= we did not investigate the temperature dependence of @xmath68 in all generality .
we restricted our study to the case @xmath77 which corresponds to @xmath78 we find that @xmath79 with @xmath80 for @xmath81 , after a crossover , @xmath82 takes the high temperature value @xmath83 while for @xmath43 it takes the low temperature value @xmath84 .
( see figure 5 . )
the explanation of the value of @xmath82 for @xmath85 is simple .
since spins are independent , identifying as above @xmath20 , the spin at site @xmath8 , to the steps of a fictitious one dimensional symmetric random walker , @xmath86 represents the probability that the walker did not cross the origin up to time @xmath10 , which is , as is well known , decaying as @xmath87 @xcite . for decreasing temperatures ,
spins become more correlated , hence the exponent @xmath82 decreases .
note that the first passage exponent @xmath82 is defined for @xmath88 , i.e. even in absence of coarsening .
this work raises a number of questions .
for instance , what is the temperature dependence of the two time correlation in the scaling regime , for @xmath43 ? is the hypothesis ( [ eqm2 t ] ) valid ?
what is the behaviour of the distribution of @xmath3 at @xmath17 when @xmath30 ? at @xmath17 , is @xmath82 a new independent critical exponent , or is it related ( equal ? ) to the persistence exponent @xmath89 for the global magnetization @xcite ?
let us mention that for the @xmath90 ising model the quantities studied here have similar behaviour .
finally , in our view , an important point of the analysis presented here is that it may be applied to any coarsening system , since it relies mainly on scaling .
we wish to thank j - p bouchaud , i dornic and j - m luck for interesting discussions .
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this work was supported in part by the german bundesministerium fr bildung and forschung under the contract 05ht1woa3 , by the polish state committee for scientific research grant 2 p03b 060 18 for years 2000 - 2001 and by the ec contract hprn - ct-2000 - 00148 for years 2000 - 2004 .
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we discuss corrections to @xmath0 , @xmath1 and to the cp violation parameter @xmath2 in two examples of ( generalized ) minimal flavour violation models : 2hdm and mssm in the large @xmath3 limit .
we show that for @xmath4 not too heavy , @xmath5 in the mssm with heavy sparticles can be substantially smaller than in the sm due the charged higgs box contributions and in particular due to the growing like @xmath6 contribution of the double penguin diagrams involving neutral higgs boson exchanges .
hep - ph/0108226 * processes in the mssm in large limit * janusz rosiek _ physik department , technische universitt mnchen , d-85748 garching , germany _ + the determination of the ckm parameters is the hot topic in particle physics . in view of forthcoming precise results from b - factories
, it is particularly important to discuss possible effects of new physics contributions to @xmath7 processes : @xmath0 and @xmath8 mass differences and to the cp violation parameter @xmath2 . in general models of new physics
fall into the two following broad classes : + @xmath9 models in which the ckm matrix remains the unique source of flavour and cp violation - so - called minimal flavour violation ( mfv ) models @xcite and their generalizations ( gmfv models ) . in the gmfv models the non sm - like operators contribute significantly to the effective low energy hamiltonian @xcite .
+ @xmath9 models with entirely new sources of flavour and/or cp violation .
+ on the basis of the analysis @xcite we discuss here two examples of the gmfv models : the large @xmath3 limit of the 2hdm(ii ) and of the mssm , in which the ckm matrix is the only source of flavour and cp violation ( see e.g. @xcite ) .
the effective weak hamiltonian for @xmath7 transitions in the gmvf models can be written as follows [ heff ] h_eff^=2 = g_f^2m_w^216 ^ 2 _ i v^i_ckm c_i ( ) q_i . where @xmath10 are the set of 8 dimension six @xmath7 operators @xcite and @xmath11 are the appropriate ckm factors .
@xmath0,@xmath5 and @xmath2 can be expressed in terms of , respectively , three real functions @xcite f^d_tt = s_0(x_t ) 1+f_d , f^s_tt = s_0(x_t ) 1+f_s , f^_tt = s_0(x_t ) 1+f _ , @xmath12 for @xmath13 gev .
we have then @xmath14 the functions @xmath15 , @xmath16 and @xmath17 can be expressed in terms of @xmath18 as @xmath19 + { 1\over4r}c^{\rm vrr}_1(\mu_t)+ \bar p_1^{\rm lr } c^{\rm lr}_1(\mu_t ) + \bar p_2^{\rm lr } c^{\rm lr}_2(\mu_t ) \nonumber\\ & & + \bar p_1^{\rm sll}\left[c^{\rm sll}_1(\mu_t)+c^{\rm srr}_1(\mu_t)\right ] + \bar p_2^{\rm sll}\left[c^{\rm sll}_2(\mu_t)+c^{\rm srr}_2(\mu_t)\right]\end{aligned}\ ] ] where @xcite @xmath20 and the factors @xmath21 are given in @xcite .
let us consider first the 2hdm(ii ) .
at 1-loop only the charged scalar is relevant for the box diagrams contributing to and mixing . for large @xmath3 and
@xmath22 the leading terms of such contributions to the wilson coefficients @xmath18 are of the order of ( see @xcite for the complete expressions ) : @xmath23 for diagrams with @xmath24 , and @xmath25 for diagrams with @xmath26 . for large
@xmath3 the biggest contribution appears in @xmath27 and is further enhanced , compared to the sm amplitude , by the qcd renormalization effects @xcite .
it is of the opposite sign than given by the @xmath28 box diagram and can be significant only for the @xmath29-@xmath30 mixing ( so that it can compete with the sm contribution - see fig .
[ fig : bcrs9 ] ) , for which it is of the order of @xmath31 similar contributions for @xmath32-@xmath33 and transitions are suppressed by factors @xmath34 and @xmath35 , respectively , and thus very small . [ cols= " < , < " , ] unfortunately , the new cleo experimental result for the process @xmath36 set the bound @xmath37 gev @xcite .
thus , in the 2hdm(ii ) for the still allowed range of charged higgs boson mass the effects in @xmath8 can not be large . as a second realistic gmfv model we consider the mssm with the ckm as the only source of flavour / cp violation .
in the limit of heavy sparticles ( practically realized already for @xmath38 gev ) the 1-loop box diagrams involving charginos and stops are negligible .
however , for large @xmath3 even if sparticles are heavy they can still compensate the @xmath39 contribution to the @xmath40 amplitude allowing for the existence of a light , @xmath41 gev ) , charged higgs boson @xcite . from fig .
[ fig : bcrs9 ] it follows therefore that , even for @xmath42 and already at the 1-loop level the contribution of the mssm higgs sector to the @xmath43 wilson coefficient can be non - negligible . at the 2-loop level in the mssm one has to take into account the dominant 2-loop electroweak corrections , modifying the yukawa - type interactions . even for heavy sparticles these corrections can significantly modify @xcite the 2hdm(ii ) relations between the masses @xmath44 and the yukawa couplings @xmath45 , as well as induce additional , @xmath46 , terms in the charged higgs - quark couplings @xcite . for the @xmath7 processes the most important ( for non - negligible mixing of the top squarks )
is however their third effect , that is the generation of the flavour non - diagonal , @xmath3 enhanced couplings of the neutral higgs bosons to down - type quarks @xcite . additional contributions to @xmath47 , @xmath48 and @xmath49 are then generated by the double penguin diagrams involving the neutral higgs bosons exchanges .
the dominant terms obtained from such contributions are @xmath50 with @xmath51 given by ( the matrices @xmath52 and @xmath53 are defined in @xcite ) : @xmath54 where the function @xmath55 is given in @xcite and the factor @xmath56 reads as @xmath57\end{aligned}\ ] ] @xmath6 factor in eq .
( [ eqn : babucor ] ) appears because the dominant part of the effective flavour off - diagonal yukawa coupling is given @xcite by the flavour - changing chargino contribution to the @xmath58-quark self energy ( @xmath59 ) , multiplied by the tree - level yukawa coupling ( also @xmath59 ) .
two powers of the external light quark masses in ( [ eqn : babucor ] ) are canceled by the fermion propagators on internal lines . for @xmath60 , @xmath61 is close to zero @xcite , but the correction @xmath62 is proportional to @xmath63 which is not suppressed in this limit .
approximating the dimensionless factor @xmath51 by unity , it is easy to see that in the case of the mixing this correction can be for @xmath64 and @xmath65 gev as large as @xmath66 i.e. of the same order of magnitude as the sm contribution to @xmath67 .
this is illustrated in fig .
[ fig : bcrs13 ] .
an important features of the double penguin contribution are : _
i ) _ its fixed negative sign , the same as the sign of the dominant effects of the charged higgs box diagrams at large @xmath3 ; _ ii ) _ its strong dependence on the left - right mixing of the top squarks - from fig .
[ fig : bcrs13 ] it follows that large values of the stop mixing parameter @xmath68 are excluded already by the present experimental data ; _ iii ) _ it does not vanish when the mass scale of the sparticles is increased uniformly - large effects decreasing @xmath5 can be present also for the heavy sparticles provided the higgs boson masses remain low .
+ summarizing , we found that : + @xmath9 the largest effects of new contributions for large @xmath3 are seen in @xmath5 .
the corresponding contributions to @xmath0 and @xmath2 are strongly suppressed by the smallness of @xmath58-quark mass .
+ @xmath9 the dominant contributions to @xmath5 for large @xmath3 originate from double penguin diagrams involving neutral higgs particles and , to a lesser extent , in the box diagrams with charged higgs exchanges . +
* @xmath9 * the contribution of double penguins grows like @xmath6 and interferes destructively with the sm contribution , suppressing considerably @xmath5 .
it depends strongly on stop mixing , excluding large values of the mixing parameter @xmath68 .
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this research was supported by nserc - canada and by rfbr grants 98 - 02 - 17372 , 98 - 02 - 17453 and 00 - 15 - 96562 .
m.j.levine , nuovo cimento * a48 * ( 1967 ) 67 .
a.v.kuznetsov and n.v.mikheev , physics letters * b299 * ( 1993 ) 367 .
p.p.kronberg , rep .
prog . phys.*57 * ( 1994 ) 325 ; + e.asseo and h.sol , phys .
reports * 148 * ( 1987 ) 307 .
duane a. dicus , wayne w. repko , and roberto vega , hep - ph/0006264 and references therein .
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we present the standard model calculation of the optical activity of a neutrino sea .
the idea that intergalactic space is a birefrigent medium for light due to the presence of a neutrino sea has been contemplated for a long time .
some thirty years ago , royer @xcite computed in @xmath0 theory an effect of order @xmath1 . later
, stodolsky noted that due to a theorem of gell - mann @xcite , there can be no such effect with massless neutrinos and a point - like coupling , and his observation was recorded by one of the present authors in a review @xcite . in the early 1980 s data on propagation of radio waves through intergalactic space put a stringent upper bound on possible optical activity of the neutrino sea @xcite and @xcite and this led to renewed estimates for the size of such effect on the assumption of a neutrino magnetic moment , which occur for a massive neutrino @xcite .
more recently @xcite an evaluation was made for a off - mass shell photons within the standard model ( sm ) .
here we note that there is a frequency dependent effect even for real photons and we also comment on the effect for virtual photons ( both within the sm ) .
thus we shall consider three possible effects : a magnetic moment effect , an effect for virtual photons , an effect for real photons .
+ if the neutrino has a magnetic moment , this moment will interact with the magnetic field of an electromagnetic wave : @xmath2 in this equation @xmath3 is the polarization vector of a photon of momenta @xmath4 , while @xmath5 is the neutrino bispinor , and @xmath6 is the neutrino magnetic moment in bohr magnetons @xmath7 .
+ with this interaction ( 1 ) the forward scattering amplitude of a photon of momentum @xmath8 from a neutrino of momentum @xmath9 is equal to @xmath10 this amplitude differs for left - handed and right - handed photons : @xmath11 and gives rise to a nonzero optical activity for a polarizes neutrino gas . + in the standard model the neutrino has no intrinsic magnetic moment , but a magnetic moment can emerge as a one loop effect if the neutrino is massive(see figure 1 ) .
+ the order of magnitude of the magnetic moment is : @xmath12 where @xmath13 and @xmath14 is the weak mixing angle ; @xmath15 .
constant @xmath16 can be obtained by direct calculation @xcite @xmath17 thus a nonzero optical activity appears in this case as a two - loop effect @xmath18 it was understood long ago that @xmath19 scattering takes place as a one - loop process even for massless neutrino .
the first estimates were performed in the fermi theory but erroneously @xcite . according to gell - mann s theorem @xcite
the amplitude should vanish for point - like interactions .
somewhat later levine @xcite evaluated the scattering mediated by w boson exchange and found a nonzero amplitude ( see figure 2 ) . due to gauge invariance
the amplitude starts with the second power of the momentum of the photon @xmath8 .
levine computed the first non - vanishing term , which in the sm looks like : @xmath20 this amplitude is even under parity , in other words it is the same for right - handed photons and left - handed photons , so it can not contribute to optical activity .
+ very recently it has been found that by expanding to the next term of order @xmath21 one finds non - vanishing p - odd terms .
the result of this calculation is @xcite : @xmath22 this amplitude vanishes for real photons but is non - vanishing for virtual photons .
the amplitude is singular with respect to the electron mass and comes entirely from the diagram 2a . + it is interesting to ask whether one can get a parity violating amplitude for real photons .
we find a positive answer to this question . in the same order of expansion over small momenta @xmath8 ( @xmath23 )
one finds the amplitude : @xmath24 where @xmath25 this does not vanish for real photons and contributes to the optical activity of neutrino gas .
the logarithmic term in equation ( 10 ) comes from diagram 2a , while the constant receives contributions from all diagrams of figure 2 .
the calculation is straightforward .
+ before we discuss the order of magnitude of the different amplitudes we note that the authors of reference @xcite have missed one diagram which contributes to off - shell photons .
this is the diagram with @xmath26 exchange shown in figure 3 . for real photons
, @xmath27 the triangle diagrams with different fermions inside the loop cancel each other , so that the standard model is anomaly free .
but for off - shell photons each diagram gives a contribution proportional to @xmath28 , where m is the mass of the fermion running inside the loop .
the main contribution comes from the electron loop and is equal to @xmath29 of the contribution from diagram 2a ( see ref .
@xcite ) .
the sum of contributions from figure 2a and figure 3 gives @xmath30 summarizing these results , we obtain the order of magnitude estimates : @xmath31 at low frequencies the scattering due to magnetic moment dominates even though it takes place at two - loop level : @xmath32 here we assume the neutrino at rest , i.e. @xmath33 .
+ as for the relative value of @xmath34 and @xmath35 @xmath36 this is sensitive to the @xmath37 value of the virtual photon . following the authors of reference @xcite
we take for intergalactic space @xmath38 and assuming @xmath39 , we obtain : @xmath40 for visible photons the on - shell activity is smaller than for off - shell photons , but in the @xmath41-ray region the reverse is the case . + in principle the contribution @xmath35 is similar to the rotation from the intergalactic magnetic fields , but the frequency dependence is very different .
a numerical estimate shows that the contribution from the neutrino sea is many order of magnitude smaller than the magnetic ( faraday ) effect , observed with radio waves @xcite .
we estimate , for radio waves and for @xmath42 neutrino sea with fermi momenta @xmath43 an effect of order @xmath44 @xmath45 which is insignificant compared to the faraday effect . at shorter wavelengths the rotatory power due to neutrino sea would increase .
these small amplitudes are matched by very small cross sections for photon - neutrino scattering @xcite .
the scattering of laser photons with high energy intensive neutrino beam is discussed in the literature ( e.g. see @xcite ) . for this case
the considered one - loop amplitude dominates over scattering due to magnetic moment : @xmath46
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we give here explicitly the set of thirteen ordinary differential equations ( odes ) for the amplitudes of the fourier modes @xmath88 that define the truncated euler equation ( tee ) in the case @xmath89 .
note that @xmath88 are real numbers because of the free - slip boundary conditions ( see text ) .
@xmath90 \end{split}\ ] ]
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it is shown that the truncated euler equations , _ i_.e .
a finite set of ordinary differential equations for the amplitude of the large - scale modes , can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined @xmath0d navier - stokes flow with bottom friction and a spatially periodic forcing .
in particular , the random reversals of the large scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large - scale circulation velocity that are described by the related microcanonical distribution which displays transitions from gaussian to bimodal and broken ergodicity .
a minimal @xmath1-mode model reproduces these results .
the formation of large scale coherent structures is widely observed in atmospheric and oceanic flows and ascribed to the nearly bi - dimensional nature of these flows .
kraichnan showed that in two - dimensional ( 2d ) turbulence , the energy is transferred from the forcing scale to larger scales due to the conservation of both energy and enstrophy by the inviscid dynamics @xcite . in a confined flow domain and without large scale friction ,
the energy accumulates at the largest possible scale , thus generating coherent structures in the form of large scale vortices . it has been observed in laboratory experiments that the large scale circulation generated by forcing a nearly @xmath0d flow at small scale can display random reversals @xcite .
the large scale velocity has a bimodal probability density function ( pdf ) with two symmetric maxima related to the opposite signs of the large scale circulation .
this regime bifurcates from another turbulent regime with a gaussian velocity field with zero mean when the large scale friction is decreased .
when the friction is decreased further , the reversals become less and less frequent and a condensed state with most of its kinetic energy in the large scale circulation is reached @xcite .
a similar sequence of transitions is observed in numerical simulations of the @xmath0d navier - stokes equation ( nse ) with large scale friction and spatially periodic forcing @xcite .
these transitions correspond to bifurcations of a mean flow on a strongly turbulent background for which no theoretical tool exists so far .
we show in this letter that the truncated euler equation ( tee ) , _ i_.e . a finite set of ordinary differential equations ( odes ) for the amplitude of the large - scale modes without forcing and dissipation , can correctly describe these transitions . to wit , we compare the dynamical regimes observed in numerical simulations of the @xmath0d nse with the ones obtained with the tee when the characteristic scale @xmath2 of the initial conditions is changed , with @xmath3 where @xmath4 is the kinetic energy of the flow and @xmath5 is its enstrophy ( integrated squared vorticity )
. the dimensionless 2d nse reads for an incompressible velocity field @xmath6 , @xmath7 where @xmath8 is the stream function and @xmath9 is the usual poisson bracket .
the first term on the right hand side represents the frictional force in the bottom boundary layer and @xmath10 is the spatially periodic forcing , explicitly given by @xmath11 . to model flow confinement
we use free - slip boundary conditions ; therefore , the stream - function can be fourier expanded as @xmath12 the non - dimensional parameters are the reynolds number , @xmath13 and @xmath14 , which represents the ratio of the inertial term to the friction on the bottom boundary .
here , @xmath15 is a characteristic large scale velocity , @xmath16 is the length of the square container , @xmath17 is the kinematic viscosity and @xmath18 is the damping rate related to the friction .
the above equation has been made dimensionless using the length scale @xmath16 and the velocity scale @xmath15 .
( -125,182)*(a ) dns * ( -190,140)*(a1 ) @xmath19 * ( -190,97)*(a2 ) @xmath20 * ( -190,55)*(a3 ) @xmath21 * ( -190,4)*(a4 ) @xmath22 * ( -135,182)*(b ) truncated euler * ( -190,140)*(b1)@xmath23 * ( -190,97)*(b2)@xmath24 * ( -190,55)*(b3)@xmath25 * ( -190,4)*(b4)@xmath26 * + ( -160,160)*(c ) * ( -130,89 ) ( -160,160)*(d ) * we perform direct numerical simulations ( dns ) of eq . , using standard pseudospectral methods @xcite with @xmath27 collocation points and @xmath28 circular dealiasing : @xmath29 .
time stepping is performed with a second - order , exponential time differencing runge - kutta method @xcite .
dnss of the nse ( [ eq : ns_incom ] ) are carried out for @xmath30 and @xmath31 with @xmath32 .
very long time integration is needed to accumulate reliable statistics for the reversals , which become rare with increase in @xmath33 ( see below ) .
direct time recordings of the amplitude of the lowest wave number mode of the stream function , @xmath34 , are displayed in fig .
[ fig:1](a ) for @xmath31 and different values of @xmath33 . for @xmath19 , the amplitude of
the large scale circulation fluctuates around zero and its pdf is gaussian ( not shown ) .
when @xmath33 is increased , a first transition is observed within the turbulent regime and can be characterized by a change of the shape of the pdf that becomes bimodal .
@xmath34 fluctuates around two non zero most probable opposite values and displays random transitions between these two states ( see fig .
[ fig:1](a2 ) ) .
this corresponds to random reversals of the large scale circulation on a turbulent background .
the mean waiting time between successive reversals increases with @xmath33 ( see fig .
[ fig:1](a3 ) ) and finally a large scale circulation with a given sign becomes the dominant flow component ( see fig .
[ fig:1](a4 ) ) .
this is the condensed state described by kraichnan @xcite .
the regime with random reversals of the large scale circulation is therefore located in parameter space between the condensed states and the turbulent regime with gaussian velocity pdfs as observed in experiments @xcite .
we now consider the approach , introduced by lee @xcite and developed by kraichnan @xcite that relies on the 2d tee .
they showed that the euler equation , truncated between a minimum and a maximum wave number , gives a set of odes for the amplitudes of the modes that follow a liouville theorem @xcite . for 2d flows , the kinetic energy @xmath4 and the enstrophy @xmath5 ( integrated squared vorticity )
are conserved ; therefore , the boltzmann - gibbs canonical equilibrium distribution is of the form @xmath35 , where @xmath36 is the partition function and @xmath37 and @xmath38 can be seen as inverse temperatures , determined by the total energy and enstrophy . using this formalism , kraichnan @xcite derived the absolute equilibria of the energy spectrum @xmath39 and showed the existence of different regimes depending on the values of @xmath37 and @xmath38 . on the other hand ,
microcanonical distributions are defined by @xmath40 , where @xmath41 and @xmath42 are respectively the energy and enstrophy of the initial conditions , and should be used to compute the pdfs in the reversal and condensed state ( see below ) .
the tee is obtained by performing a circular galerkin truncation at wave - number @xmath43 of the incompressible , euler equation @xmath44 , which is eq . without forcing or dissipation .
the tee in spectral space reads @xmath45 with @xmath46 the kronecker delta and with fourier modes satisfying @xmath47 if @xmath48 .
note that , because of the free - slip boundary conditions eq . , the fourier modes @xmath49 are real numbers .
this truncated system exactly conserves the quadratic invariants , energy and enstrophy , given in fourier space by @xmath50 and @xmath51 . for tee
we take @xmath43 as a free parameter and the same stream function expansion as that used for the ns eq .
, thus the numerical integration method is the same as the one described above for the nse . in both cases , the minimum wavenumber is @xmath52 .
we use an initial velocity field with an energy spectra @xmath53 , where by varying @xmath37 and @xmath38 we can obtain different flow regimes in accordance with the kraichnan s absolute equilibrium predictions .
we introduce a wave - number @xmath54 given by @xmath55 , which acts as an important control parameter of the system .
we next consider the results obtained using the tee with @xmath56 ( the nse forcing wavenumber ) and initial conditions with different values of @xmath54 .
figure [ fig:1](b ) shows the transitions between different turbulent regimes when @xmath54 is decreased .
the corresponding pdfs of @xmath34 obtained for different values of @xmath54 are displayed in the inset of fig .
[ fig:1](c ) .
we observe the transition from gaussian to bimodal pdf when @xmath54 is decreased and then the transition to the condensed regime with a given sign of the large scale circulation . for nse at large @xmath57 ,
the effect of the large scale friction is to stop the inverse cascade before reaching the scale of the flow domain .
@xmath33 thus determines the largest scale of the flow that we can define using the wave number @xmath55 .
@xmath54 is displayed in fig .
[ fig:1](c ) for two values of @xmath57 .
it weakly depends on @xmath57 and monotonously decreases when @xmath33 is increased .
when @xmath33 is large ( small friction ) , the kinetic energy accumulates at the scale of the flow domain and the condensed state is obtained .
although , we do not have a quantitative agreement between the transition values for @xmath54 for the tee and @xmath33 for the nse when using the relation between @xmath54 and @xmath33 displayed in fig .
[ fig:1](c ) , the same sequence of transitions is observed in both cases .
figure [ fig:1](d ) shows that we keep this qualitative agreement when the truncation is lower , @xmath58 .
this truncation leads to only @xmath1 odes for the amplitudes of the large scale modes ( see supplemental material @xcite ) .
it is remarkable that this set of equations correctly describes the transitions observed between the different turbulent regimes observed in direct numerical simulations and experiments .
the tee model is a finite number of quadratic nonlinear odes for real variables @xmath59 ( see the remark following eq . about the amplitudes of the fourier modes noted @xmath59 hereafter to simplify the notations ) that conserve both the energy @xmath60 and the enstrophy @xmath61 ( see supplemental material @xcite ) . by making use of the identities @xmath62 and @xmath63 we can write the total microcanonical phase space volume @xmath64 as @xmath65 with @xmath66 using the steepest descent method @xcite on the integral eq .
, the expression @xmath67 furnishes an explicit parametric expression for @xmath68 at the saddle - point @xmath69 parametrize real values of @xmath70 ) ] that corresponds to values of energy and enstrophy given by the saddle conditions @xmath71 and thus to @xmath55 .
we can estimate the same way the phase space volume for a fixed value of @xmath72 by retracing the steps from eq . to , but with the product and sums going from @xmath73 to @xmath74 instead of @xmath75 to @xmath74 . by combining these parametric representations
, we obtain an explicit expression for the normalized pdf of @xmath72 that is shown in fig.[fig:1](d ) and displays a good agreement with the numerical results .
note that canonical distributions with quadratic invariants are gaussian .
when there is condensation of energy at large scale , only a few modes are present and then the canonical distribution has no reason to reproduce the microcanonical distribution results @xcite .
indeed , @xmath76 with @xmath77 and @xmath78 implies that @xmath79 .
thus the microcanonical pdf of @xmath72 has to obey @xmath80 for @xmath81 which forbids reversals of the large scale circulation .
this represents ergodicity breakdown .
our above results show that this breakdown is preceded by an ergodicity delay , in the sense that @xmath82 becomes very small @xcite .
+ between successive reversals versus @xmath33 , obtained from our dnss of nse for @xmath31 ( blue circles ) .
inset : plot of the reversal frequency @xmath18 versus @xmath54 from the dnss of tee ; it shows that the reversal frequency decreases linearly with @xmath54 with a critical @xmath83 , below which the reversals are not observed for the integration time .
, title="fig : " ] ( -95,30 ) between successive reversals versus @xmath33 , obtained from our dnss of nse for @xmath31 ( blue circles ) .
inset : plot of the reversal frequency @xmath18 versus @xmath54 from the dnss of tee ; it shows that the reversal frequency decreases linearly with @xmath54 with a critical @xmath83 , below which the reversals are not observed for the integration time .
, title="fig : " ] we now consider in more detail the regime with random reversals of the large scale circulation and its transition to the condensed regime for which the flow no longer explores the whole phase space , keeping a given sign of the large scale circulation . as shown above ( compare fig .
[ fig:1](a ) and ( b ) ) , the mean waiting time @xmath84 between successive reversals increases when @xmath33 is increased in the nse , respectively @xmath54 is decreased in the tee .
however , the divergence of @xmath84 does not follow the same law for the nse and the tee .
figure [ fig:2](a ) shows an exponential increase of @xmath84 with @xmath33 in the nse , whereas a fit of the form @xmath85 with @xmath83 is observed in the tee ( see fig . [ fig:2](b ) ) .
the later result is expected since there exists a critical value of @xmath54 below which reversals are not possible in order to fulfill the conservation of both @xmath4 and @xmath5 .
we thus expect that @xmath84 becomes infinite for a finite value of @xmath54 .
a similar trend is not observed in the nse for @xmath84 versus @xmath33 .
this can not be explained using the relation between @xmath54 and @xmath33 displayed in fig .
[ fig:1](c ) that is roughly linear close to the transition to the condensed regime .
in contrast to the tee , the nse does not involve conserved quantities that prevent reversals , even when @xmath33 is large .
in addition , all the modes above @xmath86 that are suppressed in the tee can act as an additional source of noise in the nse and trigger reversals .
+ although it can be expected that viscous dissipation is negligible for the dynamics of large scales , it is remarkable that taking into account the effect of large scale friction by selecting the value of @xmath87 in the initial conditions of the tee is enough to describe the bifurcations of the large scale flow using a small number of modes governed by the euler equation .
thus , one discards the huge number of degrees of freedom related to small scale turbulent fluctuations .
in addition , equilibrium statistical mechanics , using the microcanonical distribution related to the tee , correctly describes the pdf of the large scale velocity in the different turbulent regimes .
transitions between different mean flows are widely observed in turbulent regimes , the most famous example being the drag crisis for which the wake of a sphere becomes narrower . using the navier - stokes equation with noisy forcing @xcite
is a way to describe this type of transitions .
the tee as presented here , can provide a new method to describe the dynamics of large scales in turbulence and to model a bifurcation of the mean flow on a strongly turbulent background .
+ we thank francois ptrlis for useful discussions .
support of the indo - french centre for the promotion of advanced research ( ifcpar / cefipra ) contract 4904-a is acknowledged .
this work was granted access to the hpc ressources of mesopsl financed by the region ile de france and the project equip@meso ( reference no . anr-10-eqpx-29 - 01 ) of the programme investissements davenir supervised by the agence nationale pour la recherche . vs acknowledges supported from euhit - european high- performance infrastructure in turbulence , which is funded by the european commission framework program 7 ( grant no .
312778 ) .
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10 url # 1#1urlprefix[2][]#2 lhuillier c , sindzingre p and fouet j b 2001 _ can .
j. phys . _ * 79 * 15251535 ; arxiv : cond - mat/0009336
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spin-1/2 magnets with kagom geometry , being for years a generic object of theoretical investigations , have few real material realizations . recently , a dft - based microscopic model for two such materials , kapellasite cu@xmath0zn(oh)@xmath1cl@xmath2 and haydeeite cu@xmath0mg(oh)@xmath1cl@xmath2 , was presented @xmath3janson o , richter j and rosner h , arxiv:0806.1592@xmath4 $ ] . here ,
we focus on the intrinsic properties of real spin-1/2 kagom materials having influence on the magnetic ground state and the low - temperature excitations .
we find that the values of exchange integrals are strongly dependent on o
h distance inside the hydroxyl groups , present in most spin-1/2 kagom compounds up to date .
besides the original kagom model , considering only the nearest neighbour exchange , we emphasize the crucial role of the exchange along the diagonals of the kagom lattice .
two - dimensional ( 2d ) magnets with a kagom lattice arrangement of magnetic ions ( figure [ str ] ) are geometrically frustrated due to a triangular - like arrangement of nearest neighbours ( nn ) leading to an unusual highly degenerate classical ground state ( gs ) @xcite .
strong quantum fluctuations arising from spin-1/2 ions can lift the classical degeneracy and drive the system into a magnetically disordered quantum paramagnetic state , which might be applied for future quantum computational applications @xcite . nonetheless , in real materials couplings to further neighbours @xcite and between the kagom layers @xcite are always present and influence the gs and the thermodynamics . in a recent paper @xcite we have performed dft calculations for two new natural isostructural materials with spin-1/2 kagom lattice kapellasite cu@xmath0zn(oh)@xmath1cl@xmath2 @xcite and haydeeite cu@xmath0mg(oh)@xmath1cl@xmath2 @xcite .
the local density approximation ( lda ) yields a metallic solution ( left panel in figure [ dos ] ) in contrast to the experimentally observed insulating behaviour due to underestimation of strong on - site correlations for the cu@xmath5 sites . to account for this deficit
, we carried out an effective one - band tight - binding ( tb ) fit of the lda bands ( figure [ dos ] , right panel ) and subsequently mapped the resulting transfer integrals @xmath6 to an extended hubbard model and to a heisenberg model with @xmath7 .
this mapping provides an estimate for the antiferromagnetic ( afm ) part of the exchange . to obtain the values for the total exchange , consisting of afm and ferromagnetic contributions
, we have performed supercell calculations for various spin arrangements .
the correlations were treated in the lsda+@xmath8 approximation . we have found that besides the standard kagom model based on nn interactions ( @xmath9 ) , an additional coupling along diagonals of kagom hexagons ( @xmath10 ) is relevant ( figure [ str ] ) .
since the absolute values of @xmath9 and @xmath10 change only the overall temperature scale , their ratio @xmath11 is crucial for the gs and low - energy excitations .
kagom layers in the crystal structure of kapellasite ( haydeeite ) .
cuo@xmath12 plaquettes are shown in yellow , zno@xmath1 ( mgo@xmath1 ) octahedra are shown in blue .
the relevant exchange paths are highlighted.,width=283 ] left panel : total and atom - resolved densities of states of kapellasite cu@xmath0zn(oh)@xmath1cl@xmath2 ( left ) and the corresponding band structure ( right ) .
right panel : band structure of antibonding @xmath13 bands crossing the fermi level and the effective one - band tight - binding fit for different o
h distances ( experimental : 0.78 , lda bands dashed lines , tb fit green squares ; optimized : 1 , lda bands solid lines , tb fit red circles).,width=604 ] finite kagom lattice of @xmath14 sites .
the solid lines represent the nn bonds @xmath9 and the dashed lines the diagonal bonds @xmath10 .
the coloured circles indicate the spin orientations of the twelve - sublattice classical gs relevant for @xmath15 . along the chains formed by diagonal bonds
there is an antiparallel ( nel ) spin alignment ( e.g. the spins at the light and dark green circles ) . on each triangle formed by nn bonds
@xmath9 ( e.g. spins on sites 21 , 22 , 23 ) there is a @xmath16 spin arrangement .
in addition , two antiparallel ( nel ) spin - sublattices are perpendicular to one other group of two nel - like sublattices , i.e. ; ; .,width=604 ] gs spin - spin correlation @xmath17 in dependence on @xmath18 for the finite lattice of @xmath14 sites shown in figure [ 36 ] . the site indices @xmath19 correspond to those of figure [ 36 ] .
@xmath20 @xmath20 @xmath20 @xmath20,width=362 ] one focus of this paper is to emphasize the influence of side groups ( which are commonly neglected in model physics ) on different exchange paths .
a key issue of theoretical investigations of real materials is the construction of a relevant model , which describes experimental data and has predictive power . for cuprates , that are insulators with a @xmath21 cu@xmath5 configuration ,
the extension from the initial heisenberg idea to consider only spin degrees of freedom ( heisenberg model ) to a multiband hubbard model with a separate treatment of cu @xmath22 and o @xmath23 orbitals became computationally feasible only recently .
on the other hand , the famous goodenough - kanamori - anderson ( gka ) rules formulated quite early on an empirical basis provide a simple intuitive picture for the exchange interactions based on geometrical quantities . due to the simplicity of the gka rules
, they are violated in many cases . a natural way to search for a possible origin of these phenomena
is to improve the model by including ligand fields effect . in general ,
such models are very difficult to evaluate . even in case of simple systems like cugeo@xmath0 the models are quite complicated and contain many parameters @xcite . here
, we present the results of a dft - based modeling for cu@xmath0zn@xmath3mg@xmath4$](oh)@xmath1cl@xmath2 . in these compounds ,
the leading exchange interactions are strongly dependent on the position of h atoms the simplest possible side groups . for haydeeite , the experimentally defined h position yields an unusually short o
h distance ( @xmath24 ) of 0.78 ( for kapellasite the h position has not been reported ) .
therefore , in lda calculations we relaxed the h position which resulted in an optimized @xmath25 for both systems . in order to elucidate the influence of @xmath24 on the exchange integrals
, we have studied the range @xmath26 which covers the experimental errors ( that are large due to weak x - ray scattering by h atoms ) .
the details of our calculations are described in @xcite .
we find that @xmath24 has a strong influence on the nn exchange @xmath9 , while @xmath10 is almost unaffected .
obviously , this modifies their ratio @xmath27 and consequently the physical properties .
the underlying reason is the shift of oxygen states down in energy caused by the shortening of an o h bond . in consequence ,
cu o hybridization decreases .
therefore , for a fixed cu o geometry , a decreased @xmath24 reduces the afm contribution to the total exchange .
this is supported by a strongly reduced bandwidth ( which is related via @xmath28 to the afm exchange ) for a reduced @xmath24 ( figure [ dos ] , right panel ) and fits well the results of supercell calculations .
there , we find a quasi - linear @xmath29 dependence with a positive slope for both compounds .
@xmath9 is 2.5 times larger for @xmath30 than for @xmath31 in kapellasite ( 2.5 mev versus 1 mev ) , while it changes the sign in haydeeite ( @xmath32 mev versus @xmath33 mev ) .
the next step towards a physically relevant picture of real materials is to evaluate the influence of @xmath27 onto the gs and the low - lying excitations . here
, we use a @xmath9-@xmath10 spin-1/2 heisenberg model on the kagom lattice ( @xmath9-@xmath10 model ) .
the classical gs of the pure kagom system ( @xmath34 ) is known to be highly degenerate .
the additional diagonal bond @xmath10 reduces this degeneracy drastically and selects non - coplanar classical gs s with twelve magnetic sublattices ( see figure [ 36 ] ) among the huge number of classical gs s existing for @xmath34 .
for the quantum spin-1/2 model we have calculated the gs by exact diagonalization for the finite lattice shown in figure [ 36 ] . obviously , the quantum gs spin - spin correlations are drastically affected when including @xmath10 .
while for _ _
j__@xmath35 all spin correlations except the nn correlation function @xmath36 are close to zero we find a well pronounced short - range order for @xmath37 , that corresponds to the classical magnetic structure . for @xmath38
the strongest spin correlations are @xmath39 , @xmath40 , @xmath41 , and they belong to the chains formed by @xmath10 bonds .
the nn correlation function @xmath42 monotonously decreases with @xmath10 .
the correlation function @xmath43 belongs to two sites located on sublattices being perpendicular in the classical gs , and it is almost zero for @xmath38 .
this leads to the conclusion that even in the quantum model the gs might have a non - coplanar magnetic structure giving rise to enhanced chiral correlations .
to summarize , we have shown the relevance of the ligand field for the low - energy physics by example of two spin-1/2 cuprates with a kagom lattice geometry .
we found that there are two relevant exchange integrals in both materials .
the ratio @xmath18 is strongly dependent on the o
h distance and has a drastic impact on the physical properties : the quantum gs of a corresponding spin-1/2 heisenberg @xmath9-@xmath10 antiferromagnet on the kagom lattice exhibits strong magnetic correlations along @xmath10 bonds for @xmath44 , whereas the other correlations remain weak .
therefore , the low - energy excitations might be @xmath45 spinons causing an effectively one - dimensional low - temperature physics as has been discussed previously for other 2d models such as the crossed - chain model @xcite as well as for the anisotropic triangular lattice @xcite . as the physical properties are related to @xmath27 which is strongly dependent on the o
h distance , the h position should be reinvestigated experimentally with better accuracy .
generally , a precise determination of the position of side group atoms is crucial for an accurate model derivation .
we suggest that the dependence of exchange integrals on side group positions can be used to tune a magnetic ground state under changed conditions like external pressure .
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this work is supported in part by fundao de amparo pesquisa do estado de so paulo ( fapesp ) and the conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) .
m.s.h is the martin gutzwiller fellow , 2007/2008 .
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we derive closed expressions for the nuclear breakup cross sections in the adiabatic limit using the austern - blair theory .
these expressions are appropriate for the breakup of weakly bound nuclei .
the concept of an exit doorway that mediates the coupling between the entrance channel and the breakup continuum is used .
we prove the validity of the scaling law that dictates that the nuclear breakup cross section scales linearly with the radius of the target .
we also compare our results for the nuclear breakup cross section of @xmath0be , @xmath1b on several targets with recent cdcc calculation .
the breakup of nuclei is a common occurance when the bombarding energy is high enough and/or the binding energies are sufficiently low . in the case of weakly bound nuclei the threshold for breakup is small and more so for bound unstable nuclei .
the mechanism of breakup is assumed to consist of elongating the projectile , through the action of the interaction , which eventually leads to the production of two or more fragments .
this interaction is composed of a short range , nuclear piece and a longer ranged electromagnetic one .
a debate has been going on in the literature concerning the way the nuclear part of the breakup cross section depends on the mass of the target nucleus which supplies the interaction . in most references @xcite , it is assumed that the dependence goes as the cubic root of the mass number . in reference
@xcite , however , it is claimed that this dependence is more like linear ! in a recent paper @xcite , through a careful continuum discretized coupled channels ( cdcc ) calculation , the former dependence ( a@xmath2 ) has been established , which corroborates the contention that the nuclear breakup cross section should follow the prediction of the serber model @xcite .
it is interesting to compare the numerical cdcc calculation alluded to above with those of simpler analytical models .
specifically , the ausern - blair adiabatic theory for inelastic scattering comes to mind .
if one assumes that the breakup proceeds through a so - called exit doorway @xcite , then the process can be treated as an inelastic excitation .
the idea of exit doorway has been used in the case of the influence of breakup on fusion @xcite and in the excitation of giant resonances @xcite with success . in a very recent paper
@xcite a comparison of a preliminary cdcc calculation for the breakup cross section of the system @xmath3he@xmath4al at low bombarding energies with a simple formula derived by us]using the austern - blair model , showed that such an idea is quite reasonable and encouraged us to pursue the matter further . we do this in the present paper , where we fully develop the austern - blair model for the nuclear elastic breakup reaction cross section assumed to proceed through the excitation of an exit doorway @xcite .
the exit doorway concept has been used in the devolopment of reaction theories invloving the the excitation of a `` doorway '' in the final state , in contrast to the conventional cases where such resonances are populated in the entrance channel @xcite . in the breakup reactions of halo nuclei
one may envisage that the process proceeds through the breakup doorway dipole , quadrupole etc . ) into the continuum .
as such , the detailed description of the exclusive reaction , where the final channels are especified , will necessarily contain the full information about the exit doorway ( its energy , width etc . ) .
this is the case that was encountered in the theory of the excitation of multiple giant resonances @xcite and of the influence of the pygmy resonance on the fusion of halo nuclei @xcite . in the current paper
we will be content with the inclusive quantity of the integrated breakup cross section and the only reference to the exit doorway is made implicitly as a final state that has to be populated for breakup to occur .
the full hamiltonian which describes the colliding ions can be written as @xmath5 where @xmath6 is diagonal in open channel space , @xmath7 is the intrinsic part that describes the structure of the projectile and the target nuclei , @xmath8 is the kinetic energy operator and @xmath9 is the optical potential which contains the complex nuclear plus the coulomb parts .
the operator @xmath10 describes the coupling among the open channels .
the intrinsic hamiltonian @xmath7 , which for simplicity is taken here to represent the excitable projectile nucleus with the target considered structureless , is now written as : @xmath11 + \nonumber \\
\sum_{ij}[|i>\omega_{ij}<j| + cc ] \end{aligned}\ ] ] the first three terms on the rhs above refer to the ground , exit doorway and discretized continuum states , respectively . the fourth term couples the doorway to the discretized continuum states , and the last term represents the continuum - continuum coupling .
if we remove the doorway from the above we get the continuum disctretized coupled channels ( cdcc ) intrinsic hamiltonian @xmath12 + \nonumber \\
\sum_{ij}[|i>\omega_{ij}<j| + cc ] \end{aligned}\ ] ] the exit doorway modulated cdcc hamiltonian , eq . ( 2 ) , is our subject of study here . a full development of this new cdcc will be left for a future work .
here we concentrate our effort on understanding the consequence of reaching the breakup continuum from the entrance channel only through the exit doorway @xmath13 . for this purpose
we ignore the last term in eq .
( 2 ) and remind ourselves that , whereas @xmath14 and @xmath15 are eigenstates of @xmath7 , @xmath13 is not . the full doorway - modulated cdcc equations can be obtained as follows .
the full schrodinger equation of the colliding system is , @xmath16|\psi>=0\ ] ] which when projecting onto the different channels gives : @xmath17 @xmath18 we now invoke the `` exit - doorway '' hypothesis , @xmath19 the overlaps @xmath20 and @xmath21 and can be easily obtained from eq.(2)(without the last term)@xcite , @xmath22 \label{alfa}\ ] ] where @xmath23 the exit - doorway spreading width describing its average coupling to the continuum states of the projectile , is related to the @xmath24 factors through , @xmath25 where @xmath26 is an average value and @xmath27 is the average density of discretized continuum sates in the vicinity of d. clearly the need to the continuum - continuum coupling terms would be very important if exclusive cross sections are to be calculated , since through them ( and through the doorway ) the elastic channels coupling to the breakup channel continuum can be fully acounted for . including the c - c coupling term ,
would result in a more complicated expression for @xmath28 than that of eq.[alfa ] .
equations ( 5 ) and ( 6 ) can be recast into the following , after setting @xmath29 and @xmath30 , @xmath31 @xmath32 the breakup cross section , within the exit doorway model then becomes , @xmath33 where the sum over @xmath34 has been performed by appropriate contour integration over @xmath35 .
note that the q - value in @xmath36 is complex owing to the non - zero width of the exit dooway whose energy is @xmath37 .
a simple way to see how the complex q - value arises is to eliminate @xmath38 in eq.(5 ) in favor of @xmath39 by employing eq.(6 ) , which gives @xmath40 . with this eq.(5 ) becomes @xmath41 . with the exit doorway hypothesis , the polarization potential contribution , @xmath42 becomes @xmath43 this suggests defining the exit - dorway scattering wave function by setting @xmath44 such that eqs.(10 ) and ( 11 ) become : @xmath45 @xmath46 the `` inelastic '' cross - section is thus given by eq.(13 ) above with the aforementioned proviso that the q - value of the excited state is complex .
the width of this q - value is a measure of the continuum contribution to the coupling . at this point
we comment on the inclusion of the continuum - continuum coupling , namely the last term in eq.(3 ) .
in this situation the amplitudes @xmath47 are obtained by matrix diagonalization and , among other things , the resulting overlap probability @xmath28 , deviates from the breit - wigner form of eq.(8 ) . a possible form which may incorporate
some of the c - c effects is a lorentzian : @xmath48 the above form results in an equation for @xmath49 with a modified form factor which depends on the position and width of the exit doorway @xmath50 where @xmath51 is generally complex .
accordingly the cross - section would be : @xmath52 . in the limiting case of @xmath53 , the factor @xmath51
is approximately given by @xmath54 resulting a cross - section given by : @xmath55 .
in the case of coupling to the breakup continuum considered here , the other limit , @xmath56 is more appropriate , as @xmath57 is roughly given by the q - value of the breakup ( @xmath58mev ) while @xmath59 measures the extent in continuum excitation the discretization is performed ( @xmath60 mev ) .
the function @xmath51 can be calculated in such a situation , but we leave this for a future investigation . the important point we are making here , is that a dwba calculation with complex excitation energy in the final state , and with a form factor of the type @xmath61 , should be an adequate candidate to treat the elastic breakup process . in the following
we take the exit doorway to be excited sates of different multipolarities and use the austern - blair sudden / adiabatic theory @xcite .
we employ the distorted wave born approximation for @xmath39 and @xmath36 . the elastic breakup cross section and its dependence on the target masss can be analysed within the distorted wave born approximation ( dwba ) .
if we treat the breakup as an inelastic multipole process , the amplitude @xmath62 would look like : @xmath63 the unpolarized cross section of the dipole transition is then obtained from the expression @xmath64 the radial integrals @xmath65 for pure nuclear excitation are given by , @xmath66 where the form factors @xmath67 are given by the following expressions for the monopole,@xmath68 , dipole , @xmath69 and quadrupole excitations @xmath70 @xcite , @xmath71 & & \\ f_1(r)=-\delta_1^{(n)}(\frac{3}{2 } ) ( \frac{\delta r_p}{r_p})[\frac{dv(r)}{dr}+(\frac{r}{3})\frac{d^2v(r)}{dr^2 } ] & & \\ f_2(r)=-\delta_2^{(n)}\frac{dv(r)}{dr } & & \end{aligned}\ ] ] with @xmath72 being the difference between the rms radii of the neutron and proton distributions of the projectile , and @xmath73 is the elastic scattering channel optical potential .
the quantities @xmath74 and @xmath75 can be extracted from the analysis of refs.@xcite . in ref.@xcite
a power expansion in @xmath76 was employed in the analysis of @xmath77-inelastic scattering from neutron skin nuclei . in the adiabatic limit , @xmath78 , and for large orbital angular momenta , @xmath79 , the radial integral
can then be evaluated in closed form following the procedure of austern and blair @xcite . for the dipole and quadrupole cases we have @xmath80 where @xmath81 is
the nuclear deformation length given by @xmath82 with @xmath83 being the nuclear deformation parameter and @xmath84 is the radius of the excited projectile .
the above expression for the radial integrals can be associated with the nuclear elastic breakup radial integral .
thus we can obtain analytical expression for the integrated nuclear breakup cross section by simply integrating the cross section formula , eq.(2 ) . in performing this calculation the angular momentum
coupling coefficients are evaluated exactly and the sum over @xmath85 can be performed by putting the coulomb phase shifts both as functions of @xmath86 .
the amplitude of eq .
( 13 ) is given now by @xmath87 with the condition that @xmath88 if @xmath89 is odd .
the integrated pure nuclear breakup cross section containing dipole and quadrupole contributions then becomes the following @xmath90 \sum_{l=0}^\infty(2l+1)|\frac{ds^{(n)}(k)}{dl}|^2\ ] ] where terms proportional to the second derivative of @xmath91 have been dropped .
a simple estimate of the above formula can be made by approximating the sum in @xmath92 by an integral in @xmath93 : @xmath94 assuming a real nuclear s - matrix which depends on @xmath95 through @xmath96^{-1}$ ] then the derivative of @xmath97 would peak around the grazing angular momentum @xmath98 with a width given by @xmath99 .
the integral ( [ i ] ) is then obtained as : @xmath100 for @xmath101 . using @xmath102 @xmath103 , with @xmath104 and @xmath105 being the diffuseness of the optical potential we find the simple formula for @xmath106 .
@xmath107 \frac{r}{3a } \label{ab}\ ] ] where @xmath108 is a constant normalization factor which depends among other things on the exit doorway nature of the excited state exemplified by the factor @xmath109 .
it is clear that @xmath106 depends linearly on the radius of the target and , more importantly on the square of the nuclear dipole and quadrupole deformation lengths .
thus , the @xmath110dependence is established . in the calculation to follow we use the cluster model to calculate de deformation lengths for the different multipolarities @xcite this model assumes that the projectile is composed of two clusters , a core of mass and charge @xmath111 and @xmath112 and a `` valence '' particle with @xmath113 and @xmath114 . the separation energy is denoted by @xmath115 , the q - value of the breakup .
calling the spectroscopic factor of finding the cluster configuration in the ground state of the projectile , @xmath97 one obtains the following expression for the distribution of @xmath116 in the excitation energy @xmath117 @xcite .
@xmath118 where @xmath119 is normalization factor which takes into account the finite range , @xmath120 of the @xmath121 potential .
the latter is assumed to be such as to give a yukawa type wave function at large distances , @xmath122 with @xmath123 and @xmath124 .
it is easy to obtain @xmath125 by simply integrating of eq.([b ] ) and employing the expression : @xmath126 } \prod_{k=1}^{2\lambda+1}(\lambda+\frac{3}{2}-k)\ ] ] we get for the cluster - model deformation lenghts @xmath127 and @xmath128 the following : @xmath129 @xmath130 where @xmath131 refers to the projectile . for our three nuclei discussed here
, we have @xmath0be@xmath132be@xmath133 , @xmath134b@xmath135be@xmath136 and @xmath137be@xmath138he@xmath139he , which define their cluster character , with the corresponding breakup q - values , @xmath140 mev , @xmath141 mev and @xmath142 mev .
the factor @xmath143 could be related to the asymptotic normalization coefficient ( anc ) of the bound state wave function and is taken as a parameter to be adjusted so as to account for the experimentally known @xmath125 .
cdcc calculations for the nuclear breakup ( dots ) compared to the results of eq .
( [ ab ] ) .
see text for details.,scaledwidth=50.0% ] simple estimate of @xmath144 can be obtained from @xcite who gave @xmath145 . in table 1
we present the results of the deformation lengths obtained using the cluster model of formulas [ d1 ] and [ d2 ] .
for @xmath0be , we used @xmath146 e@xmath147fm@xmath147 @xcite and we get @xmath149 fm@xmath147 .
further,@xmath150 fm@xmath147 from the same reference . in figure 1
we compare our results , eq.([ab ] ) using @xmath151 fm with the cdcc calculation of ref.@xcite at @xmath152 mev.a .
clearly we underestimate the cdcc calculation .
the reason resides in the neglect , in our model , of the higher - order channel coupling terms alluded to above .
we also show in figure 1 the comparison with the cdcc calculation for @xmath1b and @xmath153be using the formula([ab ] ) .
clearly the scaling law is better obeyed in the `` normal '' nucleus @xmath153be as has already been discussed in @xcite .
the value of the normalization @xmath108 is close to unity for the `` normal '' nuclei @xmath153be for the @xmath1b the normalization is very close the average value of the asymptotic normalization coefficient @xmath154 measured in ref.@xcite . for the @xmath0be a higher normalization
is obtained probably due to higher order effects which are not accounted by our dwba description .
.deformation lengths for the @xmath153be , @xmath1b and @xmath0be projectiles .
the deformation lengths for @xmath153be and @xmath134b have been calculated using formulas ( [ d1 ] ) and ( [ d2 ] ) using @xmath155 for the @xmath0be the @xmath156 and @xmath157 are the values from ref.@xcite [ cols="^,^,^,^,^",options="header " , ] in conclusion , we derived an expression for the nuclear breakup cross - section using the austern - blair theory .
the obtained cross - section exhibits the scaling law and should serve to supply a simple mean for an estimate of the nuclear breakup contribution .
the expression found should be contrasted with the purely geometric serber - like expression @xmath158@xcite .
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i am indebted to l. m. simons who , inspired by the colella - overhauser - werner experiments , put forward the idea of measuring the gravitational interaction of antimatter with a mach - zehnder interferometer at a conference in the 1980s .
he has also repeatedly argued that muonium would be almost completely antimatter and thus be a reasonable target for an experimental test of its gravitational interaction - suffering only from insufficient source strength .
i am also thankful to d. taqqu for discussions and explanations concerning the production of a suitable muonium beam .
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recently a new technique for the production of muon ( @xmath0 ) and muonium ( @xmath1 ) beams of unprecedented brightness has been proposed .
as one consequence and using a highly stable mach - zehnder type interferometer , a measurement of the gravitational acceleration @xmath2 of muonium atoms at the few percent level of precision appears feasible within 100days of running time .
the inertial mass of muonium is dominated by the mass of the positively charged - antimatter - muon .
the measurement of @xmath2 would be the first test of the gravitational interaction of antimatter , of a purely leptonic system , and of particles of the second generation .
the gravitational acceleration of antimatter has not been measured so far .
an experiment with antiprotons ( see @xcite and references therein ) did not succeed because of the extreme difficulty to sufficiently shield the interaction region from electromagnetic fields . for a similar reason , results of measurements with electrons @xcite are discussed very controversial and the plan to eventually compare with positrons was never realized . not affected by these problems are neutral systems like antihydrogen and , consequently , considerable effort today is devoted to the preparation of suitable samples of antihydrogen ( compare @xcite ) . a possibility to measure
the effect of gravitation on neutral particles is via a phase acquired in the gravitational potential in a suitably built interferometer , demonstrated in the classic colella
overhauser werner ( cow ) experiment @xcite . in case of limited source performance ,
when one has to deal with extended sources , comparatively large beam divergence and poor energy definition , mach - zehnder type interferometers have striking advantages @xcite
. their performance has been demonstrated , among others , with neutrons @xcite and atoms @xcite .
the idea to apply interferometry to the measurement of an antimatter system was inspired by the cow - experiments and dates back , as far as i know , to the 1980s @xcite but was put into print , with explicit mentioning of antihydrogen , positronium and antineutrons , first in 1997 @xcite .
common problems of the species are the quality of the particle beams and the availability of suitable , sufficiently large gratings .
the case of positronium was further elaborated suggesting the use of standing light waves as diffraction gratings @xcite but the realization of an experiment appears still very challenging . in the meantime also other experimental approaches to measure the gravitational interaction have been proposed for antihydrogen and positronium , see @xcite for an overview .
no discussion about a gravity experiment using muonium atoms ( m = @xmath1 ) appeared in the literature yet and the original idea of using m atoms for testing antimatter gravity is again by simons @xcite .
the suitability of m atoms comes from the fact that the inertial mass of the muon is some 207 times larger than the one of the electron , thus , muonium is almost completey , to 99.5% , antimatter .
an interesting feature is that m atoms are almost exclusively produced at thermal energies by stopping @xmath0 in matter which they often leave again as thermalized , hydrogen - like , m atom .
however , up to until recently , a gravitational experiment with muonium would have been science fiction .
the reason for this publication is , that there is now the real chance to perform such an experiment within the next few years .
an experiment with m atoms would constitute the first test of the gravitational interaction of antimatter with matter .
it would also be the first and probably unique test of particles of the second generation . while it would also be the first test in a purely leptonic system one should note that tests of the equivalence principle proving at a high level of precision that the gravitational interaction is independent of composition of test masses also in principle prove ( to still impressive precision ) that electrons fall in the same way as the rest of the material . for a recent review on tests of the equivalence principle see @xcite .
as a first measurement , the determination of the sign of interaction could be already interesting ( for a discussion of antigravity see @xcite , but also , e.g , @xcite ) , however , a reasonable first goal for such an experiment would be to determine @xmath2 to better than 10% .
one should add here , that it is not at all obvious that there could be a discrepancy between the gravitational interaction of matter and antimatter , see @xcite .
but the universality of @xcite has been disputed and possible scenarios have been sketched in @xcite .
anyhow , an experimentalist will probably favor the direct measurement ( and this , again , not only with respect to antimatter but also to a lepton of the second generation ) over the discussion of models .
the following quote from @xcite for _ antiprotons _ holds equally well for _ muonium atoms _ : `` it would be the first test of gravity , i.e. general relativity , in the realm of antimatter .
even if the experiment finds exactly what one expects , namely that antimatter falls toward the earth just as matter does , it would be , a classic , one for the text books. .... of course , if a new effect were found in the _ antiproton _ gravity experiment , then there would be no telling what exciting physics could follow . ''
the muonium experiment appears feasible now because of two recent inventions : ( i ) a new technique to stop , extract and compress a high intensity beam of positive muons , to reaccelerate the muons to 10kev and focus them into a beam spot of 100@xmath3 m diameter or even less @xcite ; and ( ii ) a new technique to efficiently convert the muons to m atoms in superfluid helium at or below 0.5k in which they thermalize and from which they get boosted by 270k perpendicular to the surface when they leave into vacuum @xcite .
assuming an existing surface muon beam of highest intensity as input , see e.g. @xcite , it should be possible to obtain an almost monochromatic beam of m atoms ( @xmath4 ) with a velocity of about 6300 m/s ( corresponding to 270k or a wavelength @xmath5 ) and a 1-dimensional divergence of @xmath6mrad at a rate of about @xmath7s@xmath8 m atoms @xcite .
this is a many orders of magnitude brighter beam than available up to now . following the approach of @xcite a mach - zehnder type interferometer
should be used in the muonium experiment . the principle with the source , the three grating interferometer and the detection region
is sketched in fig .
[ setup.eps ] .
we assume here three identical gratings and use the first two for setting up the interference pattern which is scanned by moving the third grating .
the setup is rather short , because the decay length of the m atoms is about 1.4 cm only ( @xmath9s ) .
the whole system from source to detection may be 4 decay lengths long , and without further collimation the source illuminates a cross section of less than 5 mm over the length of the interferometer .
the three free - standing gratings can be made sufficiently large with existing , proven technology with a period of 100 nm @xcite resulting in a diffraction angle @xmath10mrad .
the optimum distance @xmath11 between two gratings is slightly larger than one decay length ; however , for simplicity here @xmath12 cm . assuming another length @xmath11 each , for distances of the source and the detector to the nearest interferometer grating , results in 4 decay lenghts .
decay and transmission loss by the three 50% open ratio gratings reduces the initial m rate by a factor @xmath13 , yielding @xmath14s@xmath8 detected m. because only the indicated first order diffraction carries the desired information but essentially all transmitted m are detected , the interference pattern has a reduced contrast of somewhat below 4/9 . assuming a contrast of @xmath15 and using eqn .
( 3 ) of @xcite yields the statistical sensitivity of the experiment : @xmath16 which means that the sign of @xmath2 is fixed after one day and 3% accuracy can be achieved after 100days of running .
scheme of the experimental setup : the m beam comes from the cryogenic @xmath0 beam target on the left hand side , enters and partially traverses the interferometer and reaches the detection region on the right hand side .
the dimensions are not to scale and the diffraction angle @xmath17 is in reality smaller than the divergence . ] with the quite satisfactory statistics , the next important issues are the alignment and stability of the interferometer .
the gravitational phase shift to be observed is ( using the notation of @xcite ) @xmath18 this is rather small but still an order of magnitude larger than the phase shift due to the acceleration induced by the rotation of the earth ( sagnac effect : @xmath19 ) .
other accelerations of the system as a whole , e.g. from environmental noise , mainly affect the contrast and must therefore be suppressed .
the same is true for misalignments of the gratings and their drifts .
the effects must be kept below the phase shift , for example , for an unwanted translation @xmath20 of the third ( scanning ) grating perpendicular to the m beam and the lines of the grating one requires @xmath21 and consequently @xmath22 rotational misalignment of the gratings around the m beam must be much less than the period over beam height ratio , 100nm/5 mm , or 20@xmath3rad and corresponding drifts must not exceed 20nrad . in a similar way , limits for all other static or dynamic deviations from the perfect alignment of the three identical , equidistant , parallel gratings can be obtained .
the relatively small size of the interferometer is a major advantage for the stabilization . as in previous matter interferometry experiments
@xcite the muonium experiment must use ( multiple ) laser interferometry for alignment , monitoring and feedback position stabilization .
the gratings for the laser interferometry are ideally integrated in the m atom gratings as perfect alignment is required .
state of the art piezo systems can be used for positioning the gratings and for scanning of the third grating with a step precision of 50pm or better . because it will not be possible to perform a comparative experiment with the ( @xmath23 ) system
, the gravitational effect on the m atoms will be calibrated with a laser interferometer : in case of a vertical separation of the beam paths in the inteferometer , as in fig .
[ setup.eps ] , the m fringes will be shifted by gravity with respect to the laser interference pattern . by a @xmath24 rotation of the setup around the beam axis
the gravity shift can be removed from the m interference while the laser stays unaffected ( the gravitational redshift effects can be safely neglected at the level of precision required here ) .
of course , the sagnac effect has to be taken into account , which does not affect the interference in the vertical but will do so in the horizontal position .
given the possible sensitivity of the experiment it might be just at the edge , but in principle can be measured by horizontal measurements at @xmath24 and @xmath25 rotations of the setup , respectively .
also a setup with a vertical m beam would allow studying the sagnac effect and the stability of the interferometer .
this could also be a first stage of the experiment because for practical reasons the m beam development would probably first produce a vertical beam @xcite .
the muonium detection itself is straight forward : the @xmath26 from the @xmath0 decay must be detected in a way that the position of the m atom can be determined to be behind the interferometer .
a coincidence signal from the detection system is desirable in order to suppress background coming from the 3 orders of magnitude more muon decays upstream of the third grating .
one way to achieve this is to detect both , the decay @xmath26 and the remaining atomic @xmath27 from the m atom decay in flight .
the @xmath26 ( energies up to 53mev ) could be detected in a scintillation detector which is segmented along the beam axis and surrounds the m beam .
the @xmath27 ( energies of 10ev ) will be electrostatically accelerated out of the decay region onto , e.g. , a microchannel plate . in conclusion
, a muonium gravity experiment appears feasible today .
the development of the required m beam could be done within a few years and the interferometer set up in parallel and tested with its lasers interferometry offline .
it appears possible to have the gravitational acceleration of m atoms measured to better than 10% within the next 5 years .
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the author is grateful to v.o .
manturov for idea of @xmath0-polynomial for long virtual knots and fruitful consultations .
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we construct new invariant polynomial for long virtual knots .
it is a generalization of alexander polynomial . we designate it by @xmath0 meaning an analogy with @xmath0-polynomial for virtual links .
a degree of @xmath0-polynomial estimates a virtual crossing number .
we describe some application of @xmath0-polynomial for the study of minimal long virtual diagrams with respect number of virtual crossings .
virtual knot theory was invented by kauffman around 1996 @xcite .
long virtual knot theory was invented in @xcite by m. goussarov , m. polyak , and o. viro .
@xmath0-polynomial for virtual link was introduced independently by several authors ( see @xcite,@xcite,@xcite,@xcite ) , for the proof of their coincidence , see @xcite .
the idea of two types of classical crossings in a long diagram , which were called @xmath1 ( circle ) and @xmath2 ( star ) , was invented by v.o .
manturov ( see @xcite,@xcite ) . in present paper we called @xmath1 and @xmath2 crossings by _ early overcrossing _ and _ early undercrossing _ respectively . to consider early overcrossings and early undercrossings is the basis idea for a construction of @xmath0-polynomial in the case of long virtual knots . by a _ long virtual knot diagram _ we mean a smooth immersion @xmath3 such that : \1 ) outside some big circle
, we have @xmath4 ; \2 ) each intersection point is double and transverse ; \3 ) each intersection point is endowed with classical ( with a choice for underpass and overpass specified ) or virtual crossing structure .
a _ long virtual knot _ is an equivalence class of long virtual knot diagrams modulo generalized reidemeister moves . by an _
arc _ of a long virtual knot diagram we mean a connected component of the set , obtained from the diagram by deleting all virtual crossings ( at classical crossing the undercrossing pair of edges of the diagram is thought to be disjoint as it is usually illustrated ) .
we say that two arcs @xmath5 belong to the same _ long arc _ if there exists a sequence of arcs @xmath6 and virtual crossings @xmath7 such that for @xmath8 the arcs @xmath9 are incident to @xmath10 from opposite sides . throughout the paper , we mean that initial and final long arcs , @xmath11 and @xmath12 , form united long arc @xmath13 .
let @xmath14 be a long virtual diagram with @xmath15 classical crossings .
hence , there is a natural pairing of all classical crossings and all long arcs : classical crossing @xmath16 and long arc @xmath17 , which emanates from @xmath16 , are paired .
we say that classical crossing @xmath16 is _ early overcrossing _
( _ early undercrossing _ ) if we have an arc passing over ( under ) @xmath16 at first , in the natural order on long virtual diagram ( see also @xcite , p. 139 ) .
an _ incidence coefficient _
t=\mathbb{z}[p , p^{-1},q , q^{-1}]/((p-1)(p - q),(q-1)(p - q))$ ] of classical crossing @xmath16 and arc @xmath19 is defined as a sum of some of three polynomials : @xmath18={\varepsilon}_1 1+{\varepsilon}_2 ( t^{sgn\,v}-1)+{\varepsilon}_3(-t^{sgn\,v})$ ] , where @xmath20 ; @xmath21 if @xmath16 is early overcrossing , @xmath22 if @xmath16 is early undercrossing ; @xmath23 denotes _ local writhe number _ of @xmath16 .
we set @xmath24 arc @xmath19 is emanating from @xmath16 ; @xmath25 @xmath19 is passing over @xmath16 ; @xmath26 @xmath19 is coming into @xmath16 . if @xmath16 and @xmath19 are not incident we set @xmath18=0 $ ] .
let us enumerate all classical crossings of @xmath14 by numbers @xmath27 in arbitrary way and associate with each classical crossings the emanating long arc .
our generalization of alexander polynomial for long virtual knots is defined as determinant of @xmath28-matrix @xmath29 with elements @xmath30s^{deg\,a}\in t[s , s^{-1}]\ ] ] the function @xmath31arcs of d@xmath32 is defined according to the rules : \(1 ) if arc @xmath19 is a first at a long arc , @xmath33 ; \(2 ) if arcs @xmath19 and @xmath34 are neighbour on a long arc , @xmath19 precedes @xmath34 , then @xmath35 , if we pass from the left to the right with respect to the transversal arc , and @xmath36 otherwise .
in the first case we called such virtual crossing _ increasing _
, in the second case
_ decreasing_. it easy to see that polynomial @xmath37 does not depend on a numeration of classical crossings . by analogy with @xcite
we formulate following three theorems .
[ theorem0 ] if virtual diagrams @xmath38 are equivalent then @xmath39 for some integer @xmath40 .
the invariance of @xmath0 for reidemeister moves @xmath41 is evident .
the checking of invariance for @xmath42 and @xmath43 is similar to the case of @xmath0-polynomial for virtual link ( see @xcite,@xcite ) .
there are two types of the first reidemeister move @xmath44 : @xmath45 , if we have early overcrossing , and @xmath46 , if we have early undercrossing .
it easy to calculate that @xmath47 , @xmath48 .
it is convenient to use the laplace theorem ( about determinants ) to check that @xmath49 .
we check equality for @xmath50 pair of @xmath51-minors of matrices @xmath52 and @xmath53 .
two of these pairs give equalities only if we set @xmath54 .
[ theorem1 ] let @xmath55 be the number of virtual crossings on a long virtual diagram @xmath14 .
then @xmath56 . from theorems [ theorem0 ] and [ theorem1 ]
we easily conclude [ cor1 ] if @xmath57 then @xmath14 has minimal virtual crossing number .
for checking of minimality by using corollary [ cor1 ] it is convenient to use [ theorem3 ] the @xmath58-th coefficient of @xmath59 is equal to @xmath60 , where @xmath61 $ ] if @xmath62 s.t .
@xmath63#of increasing virtual crossings on @xmath64 , and @xmath65 otherwise , @xmath66 . example . in * figure
* we draw long virtual diagram @xmath67 which closure is unknot .
arcs @xmath68 , @xmath69 , are marked by thick lines . by theorem [ theorem3 ]
the @xmath58-th coefficient of @xmath59 is equal to @xmath70|_{i , j=1, ... ,n}=$ ] @xmath71 in the ring @xmath72 .
consequently , @xmath67 is minimal by corollary [ cor1 ] . by using our @xmath0-polynomial
we can proof following conjecture in a particular case . here
symbol @xmath73 denotes usual product of long knots .
conjecture .
_ if @xmath14 is a minimal long virtual diagram with respect number of virtual crossings , k is a long classical knot diagram , then @xmath74 is also minimal . _
[ theorem2](the particular case of conjecture ) + if @xmath14 is a minimal long virtual diagram s.t .
@xmath75 is equal to virtual crossing number of @xmath14 , @xmath76 is a long classical knot diagram , then @xmath74 is minimal . for a proof of theorem [ theorem2 ] we use following lemmas .
let @xmath77 be a number of long arc @xmath78 , where @xmath11 and @xmath12 are initial and final long arcs respectively
. then @xmath79s^{deg\,a}=$ ] @xmath80s^{deg\,a}+\sum_{a\subset { \gamma}_{+ } } \,[v_i : a]s^{deg\,a}$ ] .
consequently , @xmath81 , where @xmath82s^{deg\,a}$ ] , @xmath83 for @xmath84 .
thus , we have the natural decomposition of @xmath0-polynomial : @xmath85 , where @xmath86 . [ lem1 ] @xmath87 ; @xmath88 .
[ lem2 ] @xmath89/((p-1)(p - q),(q-1)(p - q))$ ] is zero divisor @xmath90 @xmath91 . by lemma [ lem1 ] @xmath92 @xmath93 @xmath94 , because @xmath95 .
consequently , @xmath96 if @xmath97 is not zero divisor .
it easy to check that @xmath98 , where @xmath99 denotes alexander polynomial .
it is known that @xmath100 .
hence , by lemma [ lem2 ] @xmath101 is not zero divisor , because @xmath102 .
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10 k. aamodt et al .
( alice collaboration ) , j. instrum . * 3 * , s08002 ( 2008 ) k. nakamura et al .
( particle data group ) , j. phys .
* g37 * , 075021 ( 2010 ) k. oyama et al .
( alice collaboration ) , proc . of the workshop
`` lhc lumi days '' , to be published .
lhcb collaboration , cern - lhcb - conf-2010 - 014 a. pulvirenti et al .
( alice collaboration ) , these proceedings .
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low mass vector meson ( @xmath7 ) production provides key information on the hot and dense state of strongly interacting matter produced in high - energy heavy ion collisions . among them , strangeness enhancement can be accessed through the measurement of @xmath0 meson production , while the measurement of the @xmath8 spectral function can be used to reveal in - medium modifications of hadron properties close to the qcd phase boundary .
vector meson production in pp collisions provides a reference for these studies .
moreover , it is interesting by itself , since it can be used to tune particle production models in the unexplored lhc energy range .
the alice experiment at the lhc can access vector mesons produced at forward rapidity through their decays in muon pairs , and at central rapidity in the di - electron decay channel .
the detector is fully described in @xcite . in this paper ,
results from the analysis of the data collected during the 2010 pp run at @xmath2 tev are reported .
the measurement in the dimuon channel was performed using the forward muon spectrometer , that consists of an absorber acting as muon filter , a set of cathod pad chambers ( five stations , each one composed of two chambers ) for the track reconstruction in a dipole field , two stations of two resistive plate chambers for the muon trigger , two absorbers and an iron wall acting as a muon filter . the data sample used for the analysis in the dimuon channel amounts to an integrated luminosity of approximately 85 nb@xmath9 .
since only a fraction of the data contained the full information relevant for the extraction of the integrated luminosity , a subsample corresponding to @xmath10 was used for the measurement of the @xmath0 cross section , while the full sample was used to extract the @xmath11 distribution .
muon pairs were selected asking that each muon track reconstructed in the tracking chambers matches the corresponding tracklet in the trigger stations in the position in the ( x - y ) plane and in the slope in the ( r - z ) plane .
a cut on the muon rapidity @xmath12 was applied in order to remove the tracks close to the acceptance borders .
about 291,000 opposite sign @xmath13 , 197,000 like - sign @xmath14 muon pairs survived these selections .
the combinatorial background was evaluated using the event mixing technique , and normalized to @xmath15 , where @xmath16 , and @xmath17 is the acceptance for a ( @xmath18 ) pair .
the event mixing was checked by comparing the results obtained for like - sign mixed pairs with the real ones .
the shapes of the background calculated with the two methods are identical , while the amount of like - sign pairs estimated with the event mixing differs from the one in the real data by @xmath19 .
we take this value as the uncertainty on the background normalization .
the signal - to - background ratio for @xmath4 gev/@xmath20 is about 1 at the @xmath0 and @xmath1 masses .
alternatively , the combinatorial background contribution to the opposite sign mass spectrum for a given @xmath21 mass bin can be evaluated from the like sign mass spectra using the formula : @xmath22 the two techniques are in good agreement for @xmath23 gev/@xmath20 , while for lower pair transverse momenta both methods fail in describing the background .
the analysis is thus limited to @xmath4 gev@xmath5 . after subtracting the combinatorial background from the opposite sign mass spectrum ,
we obtain the signal mass spectrum shown in fig .
[ fig : phi_all ] ( left ) .
the invariant mass spectrum is fitted with the contributions given by the light meson decays into muons and open charm / beauty contributions .
the free parameters of the fit are the normalizations of the @xmath24 , @xmath25 , @xmath26 and open charm signals .
the other processes ( @xmath27 , @xmath28 , @xmath29 , @xmath30 and open beauty ) are fixed according to the relative branching ratios or cross sections .
the main sources of systematic uncertainty are due to the uncertainty in the background normalization and on the relative normalization of the sources , mainly due to the error on the branching ratios for the @xmath1 and @xmath31 dalitz decays .
the raw number of @xmath0 and @xmath32 resonances obtained from the fit is @xmath33 and @xmath34 .
the @xmath0 production cross section was evaluated in the range @xmath6 , @xmath35 gev/@xmath20 through the formula @xmath36 , where @xmath37 is the measured number of @xmath0 mesons corrected for the efficiency and the acceptance , @xmath38 is the branching ratio in lepton pairs , obtained as a weighted average of the branching ratios in @xmath39 and @xmath40 pairs @xcite , @xmath41 is the number of minimum bias collisions , @xmath42 is the alice minimum bias cross section in pp collisions at @xmath2 tev and @xmath43 is the ratio between the number of single muons in the region @xmath44 , @xmath45 gev/@xmath20 collected with the minimum bias trigger and with the muon trigger .
the minimum bias cross section was measured in a van der meer scan @xcite .
its value is @xmath46 .
the number of minimum bias collisions was corrected , run by run , for the probability of having multiple interactions in a single bunch crossing .
the ratio @xmath43 strongly depends on the data taking conditions and was evaluated run by run .
we obtain @xmath47 .
the systematic error comes from the uncertainty on the background subtraction ( @xmath48 ) , the muon trigger efficiency ( @xmath49 ) , the tracking efficiency ( @xmath50 ) the uncertainty on the @xmath0 branching ratio into dileptons ( @xmath51 ) , on the minimum bias cross section ( @xmath52 ) and on the ratio @xmath43 ( @xmath50 ) . the @xmath3-differential cross section @xmath53 is shown in fig .
[ fig : phi_all ] ( triangles ) . point to point uncorrelated systematic uncertainties are indicated as red boxes .
the fully correlated systematic uncertainty , represented by a blue box on the left side of the plot , amounts to @xmath54 .
a fit to the differential cross section with a power law function , @xmath55^n$ ] , gives @xmath56 gev/@xmath20 and @xmath57 .
the measurements in kaon pairs performed by lhcb in a similar rapidity range ( @xmath58 , open circles ) @xcite and by alice at midrapidity ( @xmath59 , full circles ) @xcite are also plotted , showing that the shapes are similar .
the rescaling of the lhcb cross section to @xmath4 gev@xmath5 and to @xmath60 leads to @xmath61 mb .
there is a @xmath62 difference between the alice and lhcb measurements .
considering the alice statistical error and the part of the systematic uncertainty which are certainly not correlated among the two experiments , the two measurements are in agreement .
the ratio @xmath63 was measured as a function of the transverse momentum , showing a flat trend with an average value of @xmath64 . in order to extract the @xmath1 cross section ,
the @xmath8 and @xmath1 contributions must be disentangled , leaving the @xmath8 normalization as an additional free parameter in the fit to the dimuon mass spectrum .
the result of the fit gives @xmath65 .
the systematic uncertainty was evaluated changing the normalizations of the @xmath66 and @xmath67 according to the uncertainties in their branching ratios , and the background level by @xmath68 , twice the uncertainty in the normalization . from these results ,
it was possible to extract the ratio @xmath69 . this ratio is plotted as a function of @xmath3 in fig .
[ fig : dsigmaomegadydpt ] ( left ) .
the @xmath1 production cross section , calculated from this ratio , is @xmath70 . in fig .
[ fig : dsigmaomegadydpt ] ( right ) the @xmath1 differential cross section is shown .
data are fitted with the power law function , obtaining @xmath71 gev/@xmath20 and @xmath72 . in conclusion ,
the @xmath0 and @xmath1 @xmath3 differential cross sections were measured in pp collisions at @xmath2 tev .
the ratio between the @xmath0 and the @xmath1 cross sections is flat as a function of @xmath3 . in pb - pb collisions work
is in progress to measure @xmath0 production as a function of centrality .
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9 g. nakamoto _ et al . _ , j. phys .
soc . jpn . * 64 * , 4834 ( 1995 ) .
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previously measured thermopower data of cenisn exhibit a significant sample dependence and non - monotonous behavior in magnetic fields . in this paper
we demonstrate that the measured thermopower @xmath0 may contain a contribution from the huge nernst coefficient of the compound , even in moderate fields of 2 t. a correction for this effect allows to determine the intrinsic field dependence of @xmath0 .
the observed thermopower behavior can be understood from zeeman splitting of a v - shaped pseudogap in magnetic fields . *
introduction : * the orthorhombic system cenisn has been classified as a kondo semimetal , in which an anisotropic pseudogap opens in the density of states ( dos ) below approximately 10 k @xcite .
various experimental probes confirm the presence of a finite quasiparticle dos at the fermi level , such as the metal - like resistivities of samples with high purity @xcite and the linear - in-@xmath1 dependence of the thermal conductivity below 0.3 k @xcite .
magnetic fields of the order of 10 t along the easy magnetic @xmath2 axis suppress the gap formation significantly , while fields along @xmath3 and @xmath4 are less effective @xcite .
the thermopower @xmath5 of cenisn is highly anisotropic and exhibits a significant sample dependence @xcite .
below 10 k , the absolute values of @xmath5 are enhanced , which has been attributed to the gap formation in this temperature range @xcite .
consequently , application of a magnetic field of 8 t along the easy @xmath2 axis has been found to induce a significant lowering of the thermopower along @xmath2 , @xmath6 , at low @xmath1 @xcite .
however , the experimental data for @xmath7 and @xmath8 with @xmath9 are inconsistent .
first measurements at 4.2 k and 1.3 k showed a lowering of @xmath8 in magnetic fields @xcite , while investigations on samples of higher purity revealed an increasing @xmath8 upon increasing field @xcite .
likewise , for @xmath7 at 1.3 k either a monotonous decrease up to 10 t @xcite or an increase in 4 t with a subsequent decrease in 8 t @xcite was found . for correlated semimetals as cenisn
a large nernst coefficient has been predicted @xcite .
the corresponding transverse thermal voltage can become comparable in magnitude to the longitudinal one even in moderate magnetic fields .
in such a case , small deviations from the ideal contact geometry may give rise to a non - negligible nernst contribution to the measured thermopower , an effect which has not been considered previously @xcite .
it is expected to be most relevant for @xmath7 , which exhibits significantly lower absolute values at low temperatures than @xmath6 and @xmath8 . in this work
we discuss the intrinsic behavior of @xmath10 for @xmath9 and @xmath4 determined from measurements in positive and negative fields up to 7 t. + + * experiments : * the investigated samples originate from the single crystal # 5 , which was grown by the czochralski technique and subsequently purified by the solid - state electrotransport ( sse ) technique as described in ref .
@xcite .
thermopower and nernst coefficient have been measured with a steady - state method .
the temperature gradient along the samples was determined using a chromel - aufe@xmath11 thermocouple . in order to correct for spurious contributions due to a non - ideal contact geometry
all measurements were performed in @xmath12 .
two configurations are presented : the heat current @xmath13 was applied along @xmath3 with @xmath9 ( sample 1 ) and @xmath14 ( sample 2 ) . + + * results : * and @xmath9 .
exemplary error bars indicate the uncertainty of the data at low @xmath1 and high @xmath15 . above 4 k
it is typically less than 1 @xmath16v / k .
the inset compares the zero - field data obtained on two samples of cenisn .
( b ) the in - field antisymmetric contribution of the measured thermopower @xmath17 in comparison to the nernst signal @xmath18 in 2 t. @xmath19 can be scaled to @xmath18 by a factor of 8 .
[ cns - svst],scaledwidth=96.0% ] the zero - field thermopower of cenisn measured along @xmath3 , @xmath10 , is shown in the inset of fig .
[ cns - svst ] .
the data sets obtained on two different samples agree relatively well in the whole investigated temperature range . above 10 k , the thermopower exhibits a similar @xmath1 dependence as reported for a high - purity single crystal without sse treatment @xcite : @xmath10 is positive with maxima at 20 k and 100 k. the two maxima are attributed to kondo scattering from the ground - state doublet and thermally populated cef levels . a contribution from paramagnon drag
was also suggested as a possible origin for the maximum at 20 k @xcite .
toward lower @xmath1 , @xmath20 changes sign at 8 k and goes through a large negative minimum of @xmath21 @xmath16v / k at around 3.5 k. this value represents the largest negative @xmath5 ever observed for a cenisn sample .
below 1.7 k the thermopower assumes again positive values .
a similar temperature dependence has been observed in samples grown by a czochralski technique without sse treatment @xcite .
these samples exhibit a negative @xmath7 between 2.5 and 7 k , however , with significantly smaller absolute values of at most -6 @xmath16v / k .
by contrast , investigations on single crystals of similar quality as those presented here yielded a thermopower @xmath10 with opposite sign and a maximum value of 8 @xmath16v / k at 3.5 k @xcite . ,
@xmath22 ( a ) , and the thermopower value at @xmath22 , @xmath23 ( b ) , for different orientations of @xmath15 .
the dashed lines are meant guides to the eye .
the solid line in the left plot is a linear fit to the data for @xmath24 .
the error bars given for @xmath23 represent the scattering of the data around @xmath22 .
[ cns - min_vsb],scaledwidth=96.0% ] the effect of a magnetic field along @xmath2 on the low-@xmath1 thermopower @xmath7 is shown in the main plot of fig .
[ cns - svst ] . with increasing @xmath15 the minimum at @xmath25 k
shifts to lower @xmath1 and the absolute values at the minimum @xmath26 are enhanced for @xmath27 t. in 7 t a weak lowering of @xmath26 is observed , which , however , is of the order of the uncertainty in the data .
application of a magnetic field along @xmath4 gives rise to a similar evolution of @xmath0 ( not shown ) .
compared to the configuration @xmath9 , the shift of the minimum is less pronounced and no lowering of @xmath26 is observed around 7 t , ( cf .
fig .
[ cns - min_vsb ] ) . fig .
[ cns - min_vsb]a shows the field dependence of @xmath22 for @xmath9 and @xmath14 .
it clearly reveals that the shift of the minimum is stronger for @xmath9 . a linear extrapolation of @xmath28 to zero temperature yields a critical field of 14 t.
this value is comparable to the energy - gap quenching field along @xmath2 of 18 t determined from resistivity measurements @xcite .
therefore , it is supposed that the shift of the minimum is related to the closing of the gap in field .
fig .
[ cns - min_vsb]b shows the evolution of the thermopower value at the minimum , @xmath29 for @xmath9 and @xmath30 .
while @xmath26 first increases with increasing field , a saturation and subsequent reduction is observed at higher @xmath15 .
again , the effect of a field along @xmath2 is more pronounced than that of @xmath31 .
it is suspected that @xmath26 for both orientations is further reduced in higher @xmath15 as the minimum shifts to lower @xmath1 .
+ + * discussion : * the strong sample dependence of @xmath0 of cenisn reported in literature has been related to the differing purity of the investigated crystals @xcite . however , two other effects have to be taken into account .
firstly , in view of the sensitive direction dependence of @xmath0 it can not be excluded that tiny misorientations are responsible for at least part of the reported variations .
the large negative @xmath7 observed around 3 k in the present investigation could be easily diminished by a small contribution from the huge positive @xmath6 and @xmath8 of up to 70 @xmath16v / k expected in the same temperature range @xcite . in this context , the small discrepancy of about 20 % between @xmath32 of sample 1 and 2 might be attributed to this effect . secondly , the large nernst signal expected for cenisn can contribute to the thermopower voltage for a non - ideal contact geometry . in the present investigation ,
this effect was already appreciable in 2 t as demonstrated in fig .
[ cns - svst]b . below 8 k
the measured thermopower curves @xmath33 for positive and negative fields differ significantly .
the in - field antisymmetric contribution @xmath34 $ ] exhibits a temperature dependence similar to that of the nernst signal @xmath35 in 2 t and can be scaled to it by a factor of 8 .
this corresponds to a misorientation of the contacts by only @xmath36 . for the presented data
this effect has been corrected for by averaging between @xmath37 and @xmath38 .
however , it has generally not been accounted for in previous investigations @xcite .
in particular , the drastic and non - monotonous change in @xmath7 around 2 k reported for crystals of similar quality as those investigated here @xcite might be to some extent influenced by the huge nernst signal in this @xmath1 range .
therefore , the current study represents the first investigation with the intrinsic temperature dependence of @xmath7 in fields up to 7 t. for different @xmath15 .
( c ) field dependence of the logarithmic derivative of the dos at @xmath39 .
[ cns - fig4x1],scaledwidth=96.0% ] application of magnetic fields @xmath40 is found to induce a systematic shift of the thermopower minimum at 3.5 k to lower temperatures ( cf .
fig .
[ cns - min_vsb]a ) .
the estimated critical field along @xmath2 of 14 t as well as the stronger effect for @xmath9 compared to @xmath41 confirms that the minimum is related to the gap formation in cenisn . in this context , the enhancement of @xmath42 in parallel with the closing of the gap in field ( fig .
[ cns - min_vsb]b ) is surprising .
apparently , application of a magnetic field does not only suppress the gap but also influences the residual dos inside the gap .
a similar effect has been found from investigations of the specific - heat @xmath43 of cenisn @xcite , which revealed an enhancement of @xmath43 at low @xmath1 in applied magnetic fields @xmath9 .
this observation had been interpreted within a simple model assuming zeeman splitting of a modified v - shaped dos @xcite .
an illustration , how the same mechanism can give rise to an enhancement of @xmath44 at low @xmath15 is depicted in fig .
[ cns - fig4x1 ] : application of a magnetic field induces a shift of the sub - bands for the spin - up and spin - down states to different energetic directions , as shown for an arbitrary field in fig . [ cns - fig4x1]a .
thus , the resulting total dos around the gap structure depends sensitively on the field magnitude ( fig .
[ cns - fig4x1]b ) .
if the fermi level is situated slightly off the symmetry line of the gap structure , the slope of the dos at @xmath45 becomes field dependent ( fig .
[ cns - fig4x1]c ) . within this simplified picture , the absolute values of the thermopower @xmath46
@xcite increase in small magnetic fields and decreases for higher @xmath15 .
it is admitted , that the sketched behavior is much too simple to explain in detail the behavior of @xmath47 observed for cenisn .
nevertheless , the presented picture demonstrates that the increase in @xmath44 is not fundamentally contradictory to the suppression of the gap in magnetic fields .
zeeman splitting appears a possible and simple mechanism to understand the observed behavior .
it seems likely that a realistic band structure and the allowance for thermal broadening enables a more sophisticated description of the data . in conclusion ,
the minimum in @xmath10 of cenisn at 3 k is attributed to the gap formation and the observed field dependence is related to the suppression of the gap in magnetic fields .
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for @xmath166 , @xmath167 , and @xmath168 the functions @xmath48 and @xmath47 are given by @xmath169 \
, , \end{aligned}\ ] ] @xmath170 \ , , \end{aligned}\ ] ] @xmath171 \ , , \end{aligned}\ ] ] @xmath172 \ , , \end{aligned}\ ] ] and @xmath173 and @xmath174 .
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we study the spin polarization and its associated spin - hall current due to electric - dipole - induced spin resonance in disordered two - dimensional electron systems .
we show that the disorder induced damping of the resonant spin polarization can be strongly reduced by an optimal field configuration that exploits the interference between rashba and dresselhaus spin - orbit interaction .
this leads to a striking enhancement of the spin susceptibility while the spin - hall current vanishes at the same time .
we give an interpretation of the spin current in geometrical terms which are associated with the trajectories the polarization describes in spin space .
the ability to coherently control the spin of charge carriers in semiconductor nanostructures is the main focus of spintronics@xcite .
band - structure and confinement effects in these systems lead to a strong spin - orbit interaction ( soi ) offering the possibility to efficiently access the charge carrier spin via the control of its orbital motion@xcite .
a versatile and efficient scheme of spin control is electric dipole induced spin resonance ( edsr)@xcite where electric radio frequency ( rf ) fields give rise to internal fields coupling to the spin .
choosing an adequate configuration of the electric rf fields and a static magnetic field defining a quantization axis for the spin , arbitrary spin rotations can be realized .
this is analogous to standard paramagnetic spin resonance techniques , has the advantage , however , that it can be integrated in gated nanostructures thereby avoiding magnetic rf coils . in a two - dimensional electron gas ( 2deg ) with pure rashba
soi the amount of spin polarization which can be achieved by edsr is severely limited by disorder@xcite .
similar limitations are found for pure dresselhaus soi .
however , if both dresselhaus and rashba soi are present interference between the two soi mechanisms can occur and qualitatively new behavior emerges , such as anisotropy in spin relaxation@xcite and transport@xcite . for spin relaxation
this anisotropy is most pronounced if both sois have equal strength . in this case , the spin along the @xmath0 $ ] direction [ see fig .
[ fig : vectorfields ] ] is conserved@xcite , and the associated spin relaxation rates vanish , whereas they become maximal along the perpendicular direction @xmath1 $ ] . for the driven system considered here
we show that similar interference effects occur and that not only the internal rf field but also the edsr linewidth becomes dependent on the direction of the magnetic field . in a microscopic approach we show then that due to this dependence an optimal configuration exists where the linewidth and the internal field simultaneously become minimal and maximal , resp . , and that , as a remarkable consequence , the spin susceptibility gets dramatically enhanced . in other words , this optimal configuration allows one to obtain a high spin polarization with relatively small electric fields and thus making the power consumption for spin polarization minimal . due to spin - orbit interaction angular momentum can be transferred between spin and orbital degrees of freedom .
this fact leads , in particular , to a dynamical coupling between spin and spin current described by the heisenberg equation of motion@xcite . exploiting this coupling
we show that the spin current can be interpreted in geometrical terms : the spin dynamics generated by the rf fields describes an elliptical trajectory .
the spin - hall conductivity can then be expressed entirely in terms of the semi - minor and semi - major axis and the tilt angle of this ellipse .
since the spin dynamics ( trajectories ) is experimentally accessible , for instance with optical methods@xcite , this opens up the possibility for a direct measurement of the spin - hall current .
finally , we find that for the optimal configuration the spin current vanishes , in stark contrast to the spin polarization which , as mentioned , becomes maximal .
we consider a non - interacting 2deg consisting of electrons with mass @xmath2 and charge @xmath3 which are subject to a random impurity potential @xmath4 .
we take into account linear soi @xmath5 of the rashba - and dresselhaus type where @xmath6 are the pauli matrices and @xmath7 is the canonical momentum . taking the coordinate axes along the @xmath8 , [ 010 ] , [ 001]$ ] crystallographic directions , the internal magnetic field @xmath9 is then given by ( cf .
fig.[fig : vectorfields ] ) @xmath10 where @xmath11 and @xmath12 is the strength of the rashba and dresselhaus soi , respectively .
additionally , the external static magnetic field is given by @xmath13 with @xmath14 , and the external electric rf field by @xmath15 , where @xmath16 are the angles enclosed with the @xmath17 $ ] direction .
the system is described by the hamiltonian @xmath18 where @xmath19 is the vector potential associated with @xmath20 and @xmath21 with @xmath22 the electron g - factor and @xmath23 the bohr magneton .
_ spin polarization .
_ we turn now to the calculation of the spin polarization ( magnetization@xmath24 ) per unit area , @xmath25 , evaluated in linear response to an applied electric field @xmath26/2 $ ] and in the presence of both rashba and dresselhaus soi . due to the interference between these two soi mechanisms
we need to carefully extend earlier calculations@xcite , which were restricted to rashba ( dresselhaus ) soi only , to this new situation . we will then be able to identify a configuration that allows one to obtain a maximum degree of spin polarization due to this interference .
working in the linear response regime , @xmath27 is obtained from a kubo formula averaged over the random distribution of impurities in the 2deg .
we evaluate this average with standard diagrammatic techniques assuming the impurities to be short - ranged , isotropic and uniformly distributed . in this case , the impurity average @xmath28 is @xmath29 - correlated and proportional to the momentum relaxation time @xmath30 .
we further take the fermi energy @xmath31 to be the largest energy scale in the problem . then , to leading order in @xmath32 with @xmath33 the mean free path , the averaged spin polarization is given by the diffuson diagram , giving rise @xcite to a correction @xmath34 of the spin vertex ( cf . ref .
@xcite ) in the kubo formula .
thus , the spin susceptibility defined by @xmath35 is given by @xmath36 \omega_{kj } \ , , \end{aligned}\ ] ] where @xmath37 is the two - dimensional density of states and @xmath38 .
we evaluate the vertex correction @xmath39 of eq.([eq : spin - pol - rd ] ) for the case of a magnetic field @xmath40 that is large compared to the internal fields induced by soi .
this regime is characterized by @xmath41 and @xmath42 .
the components @xmath43 with @xmath44 of the vertex correction are then found to be given by @xmath45 where @xmath46 . here , the functions @xmath47 and @xmath48 are second order in @xmath49 and @xmath50 , and depend on the frequency @xmath51 , the larmor frequency @xmath52 , and the angle @xmath53 and @xmath48 are too lengthy to be written down here . in the following , only @xmath54 , @xmath55 and the linear combination @xmath56 are relevant for the linewidth and the spin - hall conductivity , resp . , which are explicitly given below in eqs.([eq : damping2 ] ) and ( [ eq : spin - current - explicit ] ) . ] . in the edsr system ,
pauli paramagnetism gives rise to a constant equilibrium polarization @xmath57 along @xmath58 which is independent of the electric field .
the polarization dynamically generated by @xmath59 , however , depends on the amplitude of the oscillating internal field perpendicular to @xmath58 .
it is thus instructive to consider the longitudinal ( along @xmath60 ) and the transverse ( along @xmath61 and @xmath62 ) polarization components given by @xmath63 and @xmath64 , @xmath65 , resp . as a result
, we find the polarization @xmath66 in terms of the transformed susceptibility @xmath67 .
to lowest order in @xmath68 only the transverse components ( @xmath69 ) are finite .
they are given by @xmath70 \notag \\ & \times w_{i } \left ( \frac{1}{\omega_l - \omega + \delta \omega -i\gamma } + \frac{1}{\omega_l + \omega -\delta \omega + i\gamma } \right ) , \end{aligned}\ ] ] where @xmath71 for the in - plane ( @xmath72 ) and @xmath73 for the out - of - plane component ( @xmath74 ) , and @xmath75 is proportional to the drude conductivity ) we can identify the component of the internal rf field @xmath76 ( which effectively drives the spin dynamics ) in terms of the electrically induced momentum drift @xmath77 .
we find that @xmath78 is given by the projection of the internal rf field ( induced by @xmath79 ) on the transverse direction @xmath80 .
note that due to disorder scattering the fourier transform @xmath81 and @xmath82 are phase - lagged with respect to @xmath83 . ] .
close to resonance the scattering from disorder leads to a renormalization of the magnetic field dependence .
the resonance is shifted by a term @xmath84 \frac { \omega_l \tau}{1 + ( \omega_{l } \tau)^2 } , \notag\end{aligned}\ ] ] corresponding to an effective g - factor which depends both on the amplitude and the orientation of the magnetic field . the linewidth @xmath85 of the resonance peak is given by @xmath86}\right ] .\notag\end{aligned}\ ] ] note that in @xmath85 the rashba and dresselhaus soi do not simply add up but can interfere with each other , enabling a strong enhancement of the susceptibility as we will see next . in fig .
[ fig : vectorfields ] we plot the spin susceptibility at resonance , @xmath87 /\gamma$ ] for the case @xmath88 measured in @xcite .
the angle @xmath89 has been tuned to maximize @xmath90 which displays a pronounced dependence on the magnetic field direction . in eq .
( [ eq : damping2 ] ) we note that @xmath85 scales with the mean square fluctuations of the internal magnetic fields @xmath91 and @xmath92 , where @xmath93 denotes a uniform average over all ( in - plane ) directions @xmath94 .
comparison with a simple model@xcite of spin relaxation ( bloch equation ) shows that the first term in eq .
( [ eq : damping2 ] ) comes from pure dephasing , i.e. from disorder induced fluctuations of the internal fields along @xmath58 , while the second term is due to fluctuations along @xmath80 . choosing a configuration with @xmath95 and tuning the soi strengths to @xmath96 the first term vanishes while the second is subject to narrowing due to the magnetic field .
the width becomes @xmath97 $ ] where @xmath98 is the dyakonov - perel spin relaxation rate for rashba soi .
increasing the frequency such that ( at resonance ) @xmath99 will lead to an increase of the inverse width @xmath100 and , hence , of the susceptibility at resonance , given by @xmath101 for comparison , we find the ratio to the resonance susceptibility @xmath102 in the pure rashba case as @xmath103 $ ] growing quadratically with @xmath104 .
thus , the spin polarization can be substantially enhanced by tuning the sois to equal strengths and by increasing the magnetic field .
finally , the range of validity for the linear response regime can be estimated as follows . assuming full polarization ( @xmath105 ) and parameters for a gaas 2deg @xcite with spin - orbit splitting @xmath106 , fermi wavelength @xmath107 and @xmath108
, we find from eq .
( [ eq : pol - res ] ) that the linear response is valid for electric fields with amplitudes up to @xmath109 . _ polarization and spin current .
_ we consider the spin current defined by @xmath110 .
using the heisenberg equation of motion the spin current components @xmath111 and @xmath112 along @xmath80 and @xmath113 can be expressed in terms of the polarization at frequency @xmath51 as @xmath114 ( i \omega s'^1 + \omega_l s'^3 ) - i \omega \beta \cos ( 2 \theta ) s'^2\\ ( \alpha + \beta \sin ( 2 \theta ) ) i \omega s'^2 - \beta \cos ( 2 \theta ) ( i \omega s'^1 + \omega_l s'^3 ) \notag \end{array } \right ) \ , .\end{aligned}\ ] ] we consider the configuration @xmath115 such that the soi induced internal rf field is perpendicular to @xmath58 and the longitudinal component @xmath116 is not altered in linear response in @xmath117 . note that in this case eq.([eq : spin - current - rd ] ) simplifies such that @xmath118 .
this relation differs from the naive model of an average spin - orbit field equating the internal field @xmath119 with its average @xmath120 .
contrary to eq.([eq : spin - current - rd ] ) , we then find @xmath121 where @xmath122 is a phenomenological transverse relaxation rate .
discrepancies to the model of an averaged spin - orbit field occur similarly for other effects such as the generation of an out - of plane polarization@xcite and zitterbewegung@xcite .
we proceed by evaluating the spin - hall current @xmath111 in terms of the vertex correction eq.([eq : vertex - correction ] ) which was obtained in the diagrammatic approach and is valid up to second order in @xmath68 .
the linear combination @xmath123 cancels in lowest order ( cf .
eq.([eq : sus3trans ] ) ) such that @xmath111 is given by the second order terms @xmath124 . from eq.([eq : spin - pol - rd ] ) and eq.([eq : spin - current - rd ] ) we find the spin - hall conductivity , defined as @xmath125 , to be given by @xmath126 remarkably , for high frequencies @xmath127 and @xmath128 eq.([eq : spin - current - explicit ] ) reaches the universal limit @xmath129 ( independent of the disorder details ) .
this limit depends only on the ratio @xmath130 of the strengths of the sois , but not on their absolute values , and agrees with the clean limit found in@xcite for @xmath131 . indeed , for the condition @xmath132 ( @xmath133 )
, many cycles of the electric rf field pass through between subsequent scattering events such that the system effectively behaves as ballistic .
this regime can be exploited to achieve high spin polarizations as described above .
moreover , the singularity in eq.([eq : spin - current - rd ] ) for @xmath96 is removed in eq.([eq : spin - current - explicit ] ) up to the accuracy @xmath134 considered here and we find that @xmath135 vanishes in the configuration where @xmath136 is maximal , i.e. for @xmath96 and @xmath137 .
we turn now to a geometrical interpretation of the spin hall current relating it to the trajectories @xmath138 followed by the tip of the polarization vector . for an applied electric field
@xmath139/2 $ ] with frequency @xmath140 this trajectory is given by the polarization ( as a function of time ) @xmath141 with the matrix @xmath142 containing the fourier components @xmath143 of the susceptibility evaluated at @xmath144 .
eq .
( [ eq : spin - polarization ] ) constitutes a quadratic form for the trajectory given by @xmath145 with real , positive eigenvalues @xmath146 ( say @xmath147 ) of the defining matrix @xmath148 .
thus , @xmath149 is of elliptic shape with semi - major and semi - minor axis @xmath150 and @xmath151 , resp .
we can further determine the angle @xmath29 enclosed by the semi - major axis of @xmath149 and the @xmath152 direction since the matrix @xmath153 is diagonalized by a rotation @xmath29 around @xmath154 .
the polarization of eq .
( [ eq : spin - polarization ] ) can thus be written as @xmath155{cc } \cos \delta & -\sin \delta \\
\sin \delta & \cos \delta \\ \end{array } \right ) \left(\begin{array}[c]{c } a \cos(\omega_0 t + \varphi ) \\
b \sin(\omega_0
t + \varphi ) \end{array } \right)\,.\end{aligned}\ ] ] here , @xmath156 is a phase shift between the electric field and the polarization . from eqs .
( [ eq : spin - polarization ] ) and ( [ eq : ellipse ] ) , we can relate the real and imaginary part of the susceptibilities @xmath157 and @xmath158 to the parameters @xmath159 , @xmath156 , and @xmath29 .
in particular , we obtain the spin hall current ( eq.([eq : spin - current - rd ] ) ) at resonance ( @xmath160 ) as @xmath161 eq . ( [ eq : spin - current - ab - res ] ) provides a remarkable interpretation of the spin hall current in terms of the geometric properties of the orbit @xmath149 .
the component @xmath111 is given by a complex phase depending on the rotation angle @xmath29 and the difference between the semi - minor and semi - major axis @xmath162 . in the linear response regime , the spin hall current characterizes the deviation from a circular orbit with @xmath163 to an elliptic shape ( with @xmath164 ) .
therefore , @xmath111 becomes accessible in terms of simple geometric properties of @xmath165 in experiments capable of resolving individual polarization components . in conclusion , we predict a substantially enhanced spin polarization due to interference effects of rashba and dresselhaus soi .
the spin hall current associated with this polarization can be interpreted in terms of the trajectory in spin space and vanishes if the polarization is maximal .
we thank o. chalaev , d. bulaev , j. lehmann , and h .- a .
engel for helpful discussions .
we acknowledge financial support from the swiss nf , nccr nanoscience basel , and the onr .
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this work was supported by research funds of chonbuk national university ( 2004 ) and the korea research foundation grant ( moehrd ) ( r14 - 2002 - 059 - 01000 - 0 ) ( hh ) , and by the research grants council of the hksar under project 2017/03p and hong kong baptist university under project frg/01 - 02/ii-65 ( lht ) .
k. binder , in _ finite - size scaling and numerical simulation of statistical systems _ , edited by v. privman ( world scientific , singapore , 1990 ) , p. 173 ; k. binder and d. w. heermann , _ monte carlo simulation in statistical physics .
an introduction _ , 3rd ed .
( springer , berlin , 1997 ) .
y. kuramoto , in _ proceedings of the international symposium on mathematical problems in theoretical physics _ , edited by h. araki ( springer - verlag , new york , 1975 ) ; y. kuramoto , _ chemical oscillations , waves , and turbulence _
( springer - verlag , berlin , 1984 ) ; y. kuramoto and i. nishikawa , j. stat . phys . * 49 * , 569 ( 1987 ) .
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the binder cumulant ( bc ) has been widely used for locating the phase transition point accurately in systems with thermal noise . in systems with quenched disorder
, the bc may show subtle finite - size effects due to large sample - to - sample fluctuations .
we study the globally coupled kuramoto model of interacting limit - cycle oscillators with random natural frequencies and find an anomalous dip in the bc near the transition .
we show that the dip is related to non - self - averageness of the order parameter at the transition .
alternative definitions of the bc , which do not show any anomalous behavior regardless of the existence of non - self - averageness , are proposed .
the characterization of phase transitions relies mainly on the singularity structure of physical quantities at the transition , which can be quantified by critical exponent values . in numerical efforts ,
the accuracy of the estimated exponents heavily depends on the precision of locating the phase transition point . in the case of most thermal systems , the binder cumulant ( bc )
is widely believed to provide one of the most accurate tools for estimating the transition point @xcite . the critical bc value at the transition is also believed to be universal , even though there is still controversy over its universality @xcite . in some complex systems @xcite
, the bc shows an anomalous negative dip in finite systems , which represents a rugged landscape ( multi - peak structure ) in the probability distribution function ( pdf ) of the order parameter .
great care is required in analyzing numerical data to see whether the dip will vanish in the thermodynamic limit .
if it does , the negative dips in the finite systems can be attributed to long - living metastable states .
otherwise , a nonvanishing negative dip usually implies that the transition is not continuous , but is of the first order . in systems with quenched disorder
, the disorder fluctuation may also generate an anomalous negative dip in the conventional bc , which is defined as the ratio of the disorder - averaged moments of the order parameter . in this case
, the negative dip may be related to the non - self - averageness ( nsa ) of the order parameter , which usually implies an extended and/or multi - peak structure in the disorder - averaged pdf @xcite .
we consider a typical nonequilibrium dynamical system with quenched disorder , such as the kuramoto model of interacting limit - cycle oscillators with random natural frequencies @xcite .
the dynamic synchronization transition is dominated by space - time fluctuations of the order parameter .
the quenched disorder is , by definition , perfectly correlated in the time direction , so it may generate strong disorder fluctuations similar to quantum systems with random defects @xcite .
in fact , we recently showed that the disorder fluctuation was anomalously strong near the synchronization transition @xcite .
we take the globally coupled kuramoto model , which can be solved analytically to some extent .
the model is defined by the set of equations of motion @xmath0 where @xmath1 represents the phase of the @xmath2th limit - cycle oscillator @xmath3 .
the first term @xmath4 on the right - hand side denotes the natural frequency of the @xmath2th oscillator , where @xmath4 is assumed to be randomly distributed according to the gaussian distribution function @xmath5 characterized by the correlation @xmath6 and zero mean @xmath7 .
we note that the natural frequency @xmath4 plays the role of _ quenched disorder"_. the second term of eq .
( [ eq : model ] ) represents global ( all - to - all ) coupling with equal coupling strength @xmath8 .
the sine coupling form is the most general representation of the coupling in the lowest order of the complex ginzburg - landau ( cgl ) description@xcite , and its periodic nature is generic in limit - cycle oscillator systems .
we consider the ferromagnetic coupling ( @xmath9 ) , so the neighboring oscillators favor their phase difference being minimized .
the scattered natural frequencies and the coupling of the oscillators compete with each other .
when the coupling becomes strong enough to overcome the dispersion of natural frequencies , macroscopic regions in which the oscillators are synchronized by sharing a coupling - modified common frequency @xmath10 may emerge .
collective phase synchronization is conveniently described by the complex order parameter defined by @xmath11 where the amplitude @xmath12 measures the phase synchronization and @xmath13 indicates the average phase .
when the coupling is weak , each oscillator tends to evolve with its own natural frequency , resulting in the fully random desynchronized phase ( @xmath14 ) . as the coupling increases
, some oscillators with @xmath15 become synchronized , and their phases @xmath16 start to show some ordering ( @xmath17 ) .
equation ( [ eq : model ] ) can be simplified to @xmath18 decoupled equations @xmath19 where @xmath12 and @xmath13 are to be determined by imposing self - consistency . in the steady state ( @xmath20 ) , the self - consistency equation reads @xmath21 with @xmath22 and @xmath23 @xcite .
this equation has a nontrivial solution only when @xmath24 : @xmath25 with @xmath26 .
we note that the exponent @xmath26 corresponds to the mean field ( mf ) value for systems of locally coupled oscillators @xcite .
now , we perform numerical integrations of eq .
( [ eq : model ] ) by using heun s method @xcite for various system sizes of @xmath27 to @xmath28 . for a given distribution of disorder @xmath29
, we average over time in the steady state after some transient time . after the time average
, we also average over disorder .
typically , we take the time step @xmath30 , the maximum number of time steps @xmath31 , and the number of samples @xmath32 . for convenience ,
we set @xmath33 ( unit variance ) ; then , the corresponding critical parameter value is @xmath34 .
figure [ fig : gl_m ] shows the behavior of the phase synchronization order parameter @xmath12 against the coupling strength @xmath35 for various system sizes @xmath18 . in the weak coupling region ( @xmath36 )
, we find that the order parameter approaches zero as @xmath37 , which is a characteristic of the fully random phase . in the strong coupling region @xmath38 ,
@xmath12 saturates to a finite value , indicating a phase transition at @xmath39 in the thermodynamic limit @xmath40 , which is consistent with the analytic result . to pin down the transition point @xmath41
precisely , we use the binder cumulant method @xcite .
the fourth - order cumulant of the order parameter , the binder cumulant ( bc ) , is defined in thermal systems as @xmath42}{3[\langle \delta^2 \rangle]^2 } b_{\delta } = 1-\frac{\langle \delta^4 \rangle } { 3\langle \delta^2 \rangle^2 } , \label{eq : gl_bm}\ ] ] where @xmath43 represents the thermal ( time ) average . in systems with quenched disorder ,
on the other hand , we should consider the disorder average besides the thermal one .
we may first consider the bc as the disorder - averaged moment ratio @xcite @xmath44}{3[\langle \delta^2 \rangle]^2 } , \label{eq : gl_bm_old}\ ] ] where @xmath45 $ ] denotes the disorder average , i.e. , the average over different realizations of @xmath29 .
figure [ fig : gl_bm_old ] displays @xmath46 as a function of the coupling strength @xmath35 for various system sizes @xmath18 . in the region of weak coupling @xmath47 , we expect the random nature of the oscillator phases @xmath48 to yield an asymmetric poisson - like probability distribution function ( pdf ) characterized by @xmath49 with a constant @xmath50 , which leads to @xmath51 . on the other hand , in the strong - coupling region ,
the pdf becomes a @xmath52-like function with a very narrow variance , which leads to @xmath53 .
the numerical data in fig .
2 are consistent with our predictions
. however , near the transition , the @xmath46 shows a big anomalous _ dip " _ on the desynchronized side . as the system size increases , the dip develops initially with a broad width and then becomes sharper and also deeper .
the dip s position moves toward the transition point .
the crossing points seem to nicely converge to the critical point @xmath54 . however , as the system size increases , the presence of the dip starts to hinder us in locating the critical point accurately . in this letter , we explain why the dip develops in this system and propose alternative definitions of the binder cumulant that do not show any dip in the same system .
we measure the disorder ( sample - to - sample ) fluctuations defined as @xmath55}{[\langle{\cal o}\rangle]^2}-1 , \label{eq : a}\ ] ] where @xmath56 is any observable , such as @xmath12 and @xmath57 , in a system .
this quantity is positive definite and is supposed to vanish in the thermodynamic limit in _ self - averaging _ systems and to remain finite in non - self - averaging systems @xcite . as one can see in fig .
[ fig : gl_a ] , the disorder fluctuation @xmath58 is quite sizable in the range of @xmath35 where the dip appears ( @xmath59 shows a similar behavior ) .
in other words , the @xmath46 shows a dip where the system is not well self - averaged .
a careful finite - size analysis on @xmath58 reveals that it vanishes as @xmath60 away from criticality , but saturates to a finite value at criticality .
the non - self - averageness at criticality is not surprising because the quenched randomness in natural frequencies should be relevant at this transition .
strong disorder fluctuations may cause non - negligible spreading of the _ effective _ coupling constants over different realizations of disorder @xcite .
figure [ fig : pdf ] shows for 20 independent samples , the pdf of @xmath12 just below the transition and obtained from the time series of @xmath12 after the system had reached the steady state .
indeed , a large part of the sample - to - sample variations can be interpreted as a shift in the @xmath41 of individual samples .
the two quantities @xmath61}$ ] and @xmath62 $ ] in eq .
( [ eq : gl_bm_old ] ) can be considered as the second and the fourth moments of the disorder - averaged pdf , which is much broader than the individual pdfs near the transition .
one can easily see that broadening yields a larger value for the ratio @xmath62/[\langle\delta^2\rangle]^2 $ ] and , hence , a smaller bc .
the effect is particularly pronounced on the small @xmath35 side of the transition , where @xmath12 itself is small , in which case a shift in @xmath41 has a stronger influence on the moments .
an alternative definition for the binder cumulant for systems with quenched disorder ( especially non - diminishing disorder fluctuations ) is@xcite @xmath63 .
\label{eq : gl_bm_new}\ ] ] we note that the disorder average is performed over the ratio of the time - averaged moments .
the moment ratio is calculated for each sample first and , is then averaged over disorder .
it is clear that this definition of the binder cumulant should eliminate the most dominant contribution from the disorder fluctuations , i.e. , the anomaly caused by the spreading of the effective coupling constants .
this definition has been adopted mostly in quantum disorder systems , where strong disorder fluctuations are anticipated @xcite .
figure [ fig : gl_bm_new ] displays @xmath64 versus @xmath35 .
we note that the dip shown in fig .
[ fig : gl_bm_old ] disappears and that the crossing points nicely converge to @xmath41 , implying that @xmath64 should serve better for locating the transition point than the conventional one , which is confirmed numerically ( not shown here ) .
yet another definition of the binder cumulant is @xmath65}{3[\langle \delta^2 \rangle^2]}. \label{eq : gl_bm_newer}\ ] ] we expect that @xmath66 may also behave smoothly near the transition because it does not involve disorder fluctuation terms such as @xmath45 ^ 2 $ ] included in @xmath46 .
figure [ fig : gl_bm_newer ] displays @xmath67 versus @xmath35 .
as expected , we find no anomalous behavior in @xmath66 .
we can directly relate @xmath46 and @xmath66 through the disorder fluctuation @xmath58 .
simple algebra leads to @xmath68 as the disorder fluctuation @xmath58 becomes larger , @xmath69 shows a bigger dip .
this explains quantitatively the size and the location of the dip in @xmath69 .
the critical value of @xmath70 ( @xmath71 ) provides additional information on the temporal variations of @xmath12 .
one can show that @xmath72/[\langle\delta^2\rangle^2]$ ] , where @xmath73 .
our numerical result indicates that the relative temporal fluctuations are almost negligible even at criticality . in this case
, @xmath70 is not practically useful in locating the transition point accurately . in summary , we studied binder cumulants in the quenched disorder system . for the kuramoto model
, we found that the conventionally defined bc shows a big anomalous dip near the transition .
this dip is shown to be directly related to the disorder fluctuation ( non - self - averageness ) .
alternative definitions of the bc , which did not show any anomalous behavior were proposed and may be useful in locating the transition point accurately in general systems with quenched disorder .
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this work is supported in part by the national natural science foundation of china .
see , e.g. , t. fukuyama and h. nishiura , hep - ph/9702253 ; r. n. mohapatra and s. nussinov , _ phys .
d _ * 60 * , 013002 ( 1999 ) ; z. z. xing , _ phys .
d _ * 61 * , 057301 ( 2000 ) ; _ phys .
d _ * 64 * , 093013 ( 2001 ) ; _ phys .
d _ * 74 * , 013010 ( 2006 ) .
s. luo , j. w. mei and z. z. xing , _ phys .
d _ * 72 * , 053014 ( 2005 ) ; s. luo and z. z. xing , _ phys .
b _ * 632 * , 341 ( 2006 ) ; _ phys . lett .
b _ * 637 * , 279 ( 2006 ) ; z. z. xing and h. zhang , _ commun .
* 48 * , 525 ( 2007 ) .
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assuming the majorana nature of massive neutrinos , we generalize the friedberg - lee neutrino mass model to include cp violation in the neutrino mass matrix @xmath0 . the most general case with all the free parameters of @xmath0 being
complex is discussed .
we show that a favorable neutrino mixing pattern ( with @xmath1 , @xmath2 , @xmath3 and @xmath4 ) can naturally be derived from @xmath0 , if it has an approximate or softly - broken @xmath5-@xmath6 symmetry .
we also point out a different way to obtain the nearly tri - bimaximal neutrino mixing pattern with @xmath7 and non - vanishing majorana phases . recently
, a novel neutrino mass model has been proposed by friedberg and lee ( fl).@xcite the neutrino mass operator in the fl model is simply given by @xmath8 where the parameters @xmath9 , @xmath10 , @xmath11 and @xmath12 are all assumed to be _ real _ , and the charged - lepton mass matrix is taken to be diagonal .
a salient feature of @xmath13 is its partial gauge - like symmetry ; i.e. , its @xmath9 , @xmath10 and @xmath11 terms are invariant under the transformation @xmath14 ( for @xmath15 ) with @xmath16 being a space - time independent constant element of the grassmann algebra.@xcite from eq .
( 1 ) , one can directly write down the neutrino mass matrix : @xmath17 two interesting features can be inferred from the diagonalization of @xmath0 .
first , the neutrino mass matrix takes a magic form,@xcite in which the sums of rows and columns are all equal to @xmath12 .
the unitary matrix used to diagonalize @xmath0 must have one eigenvector with three equal components @xmath18 .
second , when @xmath19 holds , it is very easy to check that the neutrino mass operator @xmath13 has the exact @xmath5-@xmath6 symmetry ( i.e. , @xmath13 is invariant under the exchange of @xmath5 and @xmath6 indices).@xcite in addition , one may consider to remove one degree of freedom from @xmath13 or @xmath0 ( for instance , by setting @xmath20).@xcite to include cp or t violation into the fl model , one may insert the phase factors @xmath21 into eq .
( 1 ) by replacing the term @xmath22 with the term @xmath23.@xcite the resultant neutrino mass matrix is no longer symmetric , hence it describes dirac neutrinos instead of majorana neutrinos .
however , in most of the realistic models , the majorana nature is preferable to the dirac nature of neutrinos .
hence , in this work , we aim to generalize the fl model to include cp and t violation for massive majorana neutrinos .
let us start from the generic analysis with all the parameters of @xmath0 in eq .
( 2 ) being complex . for majorana neutrinos
, @xmath0 is symmetric and can be diagonalized by the transformation @xmath24 , in which @xmath25 ( for @xmath26 ) stand for the neutrino masses .
after a straightforward calculation , the neutrino mixing matrix @xmath27 turns out to be @xmath28 where the explicit expressions of @xmath29 and @xmath30 are @xmath31 and the definitions of @xmath32 , @xmath33 and @xmath34 can be found in ref . .
furthermore , three mass eigenvalues of @xmath0 and two majorana phases of @xmath27 are found to be @xmath35 and @xmath36 again , the expressions of @xmath37 have also been listed in ref . .
we proceed to consider two special but interesting scenarios of the generalized fl model and explore their respective consequences on three neutrino mixing angles and three cp - violating phases .
scenario ( a ) : @xmath9 and @xmath12 are real , and @xmath38 are complex .
note that the @xmath5-@xmath6 symmetry of @xmath0 is softly broken in this case , because @xmath39 holds . by using the generic results given in eqs .
( 3)-(6 ) , one can easily arrive at @xmath40 $ ] , @xmath41 , @xmath42 ^ 2 + 3 \left [ { \rm i m } \left ( b \right ) \right ] ^2 } ~ - a + { \rm re } \left ( b \right ) \ ; , \ \ \ \ \
m^{}_2 = m^{}_0 \ ; , \nonumber \\ m^{}_3 & = & \sqrt{\left [ m^{}_0 + a + 2{\rm re } \left ( b \right ) \right ] ^2 + 3 \left [ { \rm i m } \left ( b \right ) \right ] ^2 } ~ + a - { \rm re } \left ( b \right ) \ ; , % ( 7)\end{aligned}\ ] ] together with @xmath43 . comparing our results with the well - known standard
parametrization,@xcite we immediately obtain @xmath44 , @xmath45 , and @xmath46 $ ] .
the leptonic jarlskog parameter @xmath47 , which is a rephasing - invariant measure of cp violation in neutrino oscillations,@xcite reads @xmath48 .
if @xmath49 holds , the tri - bimaximal neutrino mixing pattern ( with @xmath50 or @xmath51 , @xmath2 and @xmath52)@xcite will be reproduced .
one can see that the soft breaking of @xmath5-@xmath6 symmetry leads to both @xmath53 and @xmath54 , but it does not affect the favorable result @xmath2 given by the tri - bimaximal mixing pattern . on the other hand , @xmath55 is an excellent approximation , since @xmath29 must be small to maintain the smallness of @xmath56 . in view of @xmath57,@xcite
we obtain @xmath58 and @xmath59 .
it is possible to measure @xmath60 in the future long - baseline neutrino oscillation experiments .
the neutrino masses in scenario ( a ) rely on four real model parameters @xmath12 , @xmath9 , @xmath61 and @xmath62 .
thus it is easy to fit the neutrino mass - squared differences @xmath63 and @xmath64.@xcite such a fit should not involve any fine - tuning , because ( a ) the number of free parameters is larger than the number of constraint conditions and ( b ) three neutrino masses have very weak correlation with three mixing angles .
a detailed numerical analysis can be found in ref . , and a remarkable feature is that only the normal neutrino mass hierarchy ( @xmath65 ) is allowed in this scenario .
scenario ( b ) : @xmath9 , @xmath10 and @xmath11 are all real , but @xmath12 is complex . by using eqs .
( 3)-(6 ) , we obtain @xmath66 , and @xmath67 ^ 2 + \left [ { \rm i m } \left
( m^{}_0 \right ) \right ] ^2 } \;\ ; , \ \ \ \ \ m^{}_2 = \left
|m^{}_0 \right | \ ; , \nonumber \\ m^{}_3 & = & \sqrt{\left [ m^{}_{+ } + { \rm re } \left ( m^{}_0 \right ) \right ] ^2 + \left [ { \rm i m } \left ( m^{}_0 \right ) \right ] ^2 } \;\ ; , % ( 8)\end{aligned}\ ] ] where @xmath68 .
two majorana phases @xmath69 and @xmath70 are given by @xmath71 although @xmath69 and @xmath70 have nothing to do with the behaviors of neutrino oscillations , they may significantly affect the neutrinoless double - beta decay.@xcite comparing our formulae with the standard parametrization , we arrive at @xmath72 , @xmath73 , @xmath74 together with @xmath7 for the dirac phase of cp violation .
the results for @xmath75 and @xmath56 in this scenario are the same as those obtained in scenario ( a ) , but the jarlskog parameter @xmath47 is now vanishing .
because of the @xmath5-@xmath6 symmetry breaking , @xmath76 may somehow deviate from the favorable value @xmath77 .
given @xmath78 corresponding to @xmath57 , @xmath76 is allowed to vary in the range @xmath79 .
the neutrino masses depend on five real model parameters @xmath9 , @xmath10 , @xmath11 , @xmath80 and @xmath81 .
hence there is sufficient freedom to fit two observed neutrino mass - squared differences @xmath82 and @xmath83 . our careful numerical analysis , which has been done in ref .
, shows that both normal ( @xmath65 ) and inverted ( @xmath84 ) neutrino mass hierarchies are allowed , and the majorana phases ( @xmath69 , @xmath70 ) are less restricted in scenario ( b ) .
although our discussions about the generalized fl model are restricted to low - energy scales , it can certainly be extended to a superhigh - energy scale ( e.g. , the gut scale or the seesaw scale ) . in this case
, one should take into account the radiative corrections to both neutrino masses and flavor mixing parameters when they run from the high scale to the electroweak scale.@xcite we conclude that the @xmath5-@xmath6 symmetry and its slight breaking are useful and suggestive for model building .
we expect that a stringent test of the generalized fl model , in particular its two simple and instructive scenarios , can be achieved in the near future from the neutrino oscillation and neutrinoless double - beta decay experiments .
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the financial support provided by the deutsche forschungsgemeinschaft is gratefully acknowledged .
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we present an example revealing that the sign of the `` momentum '' @xmath0 of the wigner `` distribution '' function @xmath1 is not necessarily associated with the direction of motion in the real world .
this aspect , which is not related to the well known limitation of the wigner function that traces back to the heisenberg s uncertainty principle , is particularly relevant in transport studies , wherein it is helpful to distinguish between electrons flowing from electrodes into devices and vice versa .
recently , we critically analyzed @xcite the manner in which the wigner `` distribution '' function was used in studies on molecular transport relying upon finite isolated clusters @xcite .
these studies made us aware of a limitation of using the wigner function as a true momentum distribution function for transport , which we could not find in the literature and want to present here .
the wigner function @xmath1 is employed in many physical studies , including transport s , in spite of its physical limitations .
the limitation known from textbooks @xcite traces back to the heisenberg s uncertainty principle
. the wigner `` distribution '' function can be negative and should be not interpreted as a probability distribution , but rather as `` one step in the calculation never the last step , since '' is not measurable but `` is used to calculate other quantities that can be measured the particle density and current '' and ``
no problems are encountered as long as one avoids interpreting @xmath2 as a probability density '' ( quotations from ch .
3.7 , p. 203 of ref . ) . in transport , it is helpful to distinguish between incoming and outgoing electrons , i. e. , flowing from electrodes into devices and from devices into electrodes @xcite . to this aim
, it is necessary to use a physical property whose sign enables to indubitably assess that electrons are , say , left- or right - moving .
the averages of the particle ( probability ) current operator @xmath3 , \ ] ] or of the _ physical _ momentum @xmath4 do represent such properties .
above , @xmath5 and @xmath6 are annihilation and creation field operators for ( spinless ) electrons moving in one dimension , and @xmath7 stands for electron s mass . to see whether the momentum `` variable '' @xmath0 of the wigner function @xmath8 justifies to speak of left- or right - moving electrons depending on the sign of @xmath0 , let us consider the ground state @xmath9 of @xmath10 noninteracting electrons confined within a one - dimensional square well of width @xmath11 and infinite height .
electrons occupy energy levels @xmath12 , whose single - electron wave functions @xmath13 with @xmath14 , @xmath15 . in the ground state @xmath9 , the lowest @xmath10 levels are occupied up to the fermi `` momentum '' @xmath16 .
computing the wigner function of this system is straightforward @xmath17 see , e. g. , ref . ,
ch .
3.7 , pp .
202 - 203 .
( color online ) wigner function for two sizes @xmath11 computed at two points @xmath18 indicated in the legend .
gold s fermi wave vector @xmath19nm@xmath20 is used.,scaledwidth=45.0% ] one might think that one could use the wigner function as if it were a distribution function in cases where its shape resembles a fermi distribution . let us inspect the curves for @xmath1 computed as indicated above and presented in fig .
[ fig : wf ] .
in fact , at smaller sizes ( close to the linear size of the au@xmath21-clusters used in molecular wigner transport studies @xcite ) the wigner function does not bear much resemblance to a fermi function ( the lower curves of fig . [
fig : wf ] ) , at larger sizes ( much larger than those one could hope to tackle within ab initio calculations to correlated molecules , for which such a wigner - transport approach @xcite was conceived ) the curves ( the upper part of fig .
[ fig : wf ] ) become more similar to a step function , and one may think that this is encouraging . in reality , the contrary is true : as visible in fig .
[ fig : wf ] , _ mathematically _ one can calculate the wigner function for positive and negative `` momentum '' variables @xmath0 _
separately_. however , this mathematical separation does not reflect a physical reality : for any single - particle eigenstate @xmath22 the electron momentum vanishes @xmath23 left- and right - traveling waves are entangled with equal weight , and one can not speak of single - particle eigenstates representing left- or right - moving electrons only because wigner functions with positive or negative @xmath0-arguments can be computed .
this represents a further limitation of the usefulness of the wigner function , not related to the heisenberg s principle , which is particularly relevant for transport .
the current has a direction , and if one wants to unambiguously specify this direction , the wigner function @xmath1 is inappropriate ; a wigner function with negative ( positive ) `` momentum '' , @xmath24 ( @xmath25 ) , does not imply that left- and right - moving particles exist in the real physical world .
so , using @xmath26 and @xmath27 as if they were true momentum distributions of incoming electrons , as done in wigner approaches of molecular transport based on finite isolated clusters @xcite is not justified in quantum mechanics .
the above example demonstrates that , indeed , the textbook s warning mentioned in the beginning of this section is pertinent .
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this work has been supported by rfbr grants 14 - 02 - 00914 , 14 - 02 - 31816 .
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neutrino flavour oscillations in a nonuniformly moving matter are considered . the neutrino oscillation resonance condition in presence of matter , in the most general case when matter is moving with acceleration ,
is derived for the first time .
we predict that the effect of matter acceleration can have significant influence on neutrino oscillations pattern in different astrophysical environments .
in this short note we generalize our previous studies @xcite on neutrino flavour oscillations in an uniformly moving matter to the case when matter moves with acceleration .
consider two flavour ( electron and muon ) neutrino oscillations in nonuniformly moving matter .
the effective lagrangian of neutrino interactions with background matter can be expressed in the form @xmath0 where @xmath1 is the fermi constant , @xmath2 is the speed of matter , @xmath3 and @xmath4 and @xmath5 for the electron and muon neutrino correspondingly . here
@xmath6 is the electron number density in the reference frame for which the total speed of matter is zero .
note that we consider the matter composed of neutrons , protons and electrons .
obviously , neutrons and protons do not influence the oscillations . for the effective hamiltonian of neutrino flavour states evolution
we obtain @xmath7 where @xmath8 , @xmath9 is the vacuum mixing angle , @xmath10 is the neutrino energy and @xmath11 is the angle between the speed of matter and the direction of neutrino propagation ( it is supposed that the neutrino is propagating along @xmath12 direction ) . in the adiabatic approximation
the oscillation probability has the usual form @xmath13 , where the effective mixing angle @xmath14 and oscillation length @xmath15 are determined by the elements @xmath16 of the evolution hamiltonian ( [ h ] ) .
the straightforward calculations yields the neutrino flavour oscillations resonance condition @xcite @xmath17 in the considered case the neutrino oscillations probability gets its maximum value in a set of points @xmath18 that are the solutions of eq .
( [ res_cond ] ) . note that in general case eq .
( [ res_cond ] ) is not linear in respect to @xmath12 . obviously , that in case of monotonic @xmath12 dependence of the density @xmath19 there can be only one resonance point
.
consider matter motion with a constant acceleration @xmath20 .
then the matter velocity is given by @xmath21 where @xmath22 is the initial matter speed .
in this case one can obtain the effective electron matter density in the form @xmath23 in fig .
[ fig01 ] the function @xmath24 is plotted .
it follows that @xmath19 significantly increases in case of matter motion against neutrino propagation ( @xmath25 ) and almost vanishes in the opposite case ( @xmath26 ) . in the nonrelativistic and ultrarelativistic limits of matter motion eq .
( [ ne_v ] ) can be simplified as follows @xmath27 in the corresponding evaluation it is supposed that @xmath28 . when nonrelativistic matter is moving with a constant acceleration the corresponding shift of the electron number density
is given by @xmath29 , where @xmath30 and @xmath31 .
for instance , during a supernova core - collapse @xmath32 is up to 0,2 that yields an order of @xmath33 shift to the effective resonance number density @xmath19
. it would be interesting to consider possibilities of observing this effect . and the angle of propagation @xmath11 .
the red arrows defines the direction of a neutrino flux .
the left and right plots corresponds to @xmath28 and @xmath34 accordingly.,title="fig : " ] and the angle of propagation @xmath11 .
the red arrows defines the direction of a neutrino flux .
the left and right plots corresponds to @xmath28 and @xmath34 accordingly.,title="fig : " ]
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we have calculated the cross section of radiative capture process @xmath10 .
we applied pionless eft to find numerical results for the m1 contributions for this capture process for incident neutron energies relevant for bbn , @xmath12 mev . at these energy our calculation is dominated by s - wave state and magnetic transition m1 contribution .
the error estimate in the cross section in comparison with evaluated nuclear data file endf @xcite is shown in fig .
[ error ] and table 1 .
errors estimate are 20 - 30 percent at leading order , below 10 percent up to nlo and by insertion of three - body force at n@xmath2lo , this error is reduced to below 1@xmath13 percent .
specially , our calculation shows minimum error for energy 60 - 70 kev up to at n@xmath2lo .
comparison of the lo with the nlo and n@xmath2lo results demonstrate convergence of the effective field theory .
finally , three - body forces will enter at higher orders of the eft approach and reduce the theoretical uncertainty .
the authors would like to thanks u. van kolck for helpful discussions .
we would like to thanks p.f .
bedaque and harald w. griehammer for useful comments and valuable mathematica code .
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the cross section of neutron - deuteron radiative capture @xmath0 is calculated at energies relevant to big - bang nucleosynthesis ( @xmath1 kev ) with pionless effective field theory . at these energies ,
magnetic transition m1 gives the dominant contribution .
the m1 amplitude is calculated up to next - to - next - to leading order(n@xmath2lo ) with insertion of three - body force .
results are in good agreement within few percent theoretical uncertainty in comparison with available calculated data below e=200 kev . * h. sadeghi * and * s. bayegan * + _ department of physics , university of tehran , p.o.box 14395 - 547 , tehran , iran . _ + 1.0 cm 26.35.+c , 21.30.fe , 25.40.lw , 11.80.jy , 27.10.+h [ cols= " > , < " , ] at lo and nlo , this is the only three - body force entering , but at n@xmath2lo , where we saw that @xmath3 is required , it is determined by the triton binding energy @xmath4 .
we solve integral equation by expansion in order of @xmath5 and properly iterating the kernel .
then , the resulted @xmath6 will be folded to electromagnetic interaction order by order and properly integrated on the involving momentum . in kev .
the short dashed , long dashed and solid line correspond to the error up to lo , nlo and n@xmath2lo , respectively . ] for our calculation source of error due to low cutoffs and low momentums for very low energy calculation is neglected .
the cutoff variation decreases steadily as we increase the order of the calculation and is of the order of @xmath7 , where @xmath8 is the order of the calculation .
we used @xmath9 mev , for the smallest cutoff .
the cross section calculation for neutron radiative capture by deuteron as function of the center - of - mass energy at lo , nlo and n@xmath2lo is shown in fig . [ crosssection ] .
table 1 shows numerical results for the eft @xmath10 cross section for various nucleon center of mass energies e up to n@xmath2lo and errors estimate at every order in comparison with the last column .
the corresponding values for the cross section from the online evaluated nuclear data file endf / b - vi @xcite are shown in the last column .
the eft results for this cross section are presented up to only two significant digits(the third digit is shown for better comparison with endf data ) . in fig
. [ error ] , we show the error due to available evaluated data endf @xcite in percentage versus nucleon center - of - mass kinetic energy @xmath11 in kev .
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the authors would like to thank the w.m .
keck foundation for the financial support , akos vertes who provided the laser for our experiments , infrared fiber systems , silver spring , md , who provided the optical fibers for this study , william rutkowsky for helping with the instrumentation , andrew gomella and craig s pelissier for helping with the detector calibration process , jyoti jaiswal , mary ann stepp and gauri tadvalkar for helping with the cell culture , and alexander jeremic for providing us the facility to culture cell samples .
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we report the near - field ablation of material from cellulose acetate coverslips in water and myoblast cell samples in growth media , with a spot size as small as 1.5 @xmath0 m under 3 @xmath0 m wavelength radiation .
the power dependence of the ablation process has been studied and comparisons have been made to models of photomechanical and plasma - induced ablation .
the ablation mechanism is mainly dependent on the acoustic relaxation time and optical properties of the materials .
we find that for all near - field experiments , the ablation thresholds are very high , pointing to plasma - induced ablation as the dominant mechanism .
near - field scanning optical microscopy ( nsom)@xcite is a promising technique that overcomes the diffraction limit of conventional optical microscopy @xcite and by doing so has created a number of potential applications in biological imaging .
the combination of nsom for ablation with mass spectrometry is of particular interest to obtain detailed molecular information with spatial resolution better than that of the conventional optical spectrometry .
a step in this direction , ultraviolet - nsom - based mass spectrometry with a lateral resolution of 170 nm in ambient conditions , has demonstrated soft ablation capabilities @xcite .
an improvement would be ablating in the infrared rather than the uv regime so that the native water in a cell plays the role of the ablation matrix due to its strong absorption at 2940 nm @xcite . by this approach
, cells can be probed _ in - vivo _ or _ in - vitro _ , but in the far - field , the spatial resolution is limited by diffraction effects and by the quality of available optics to a spot size of about 50@xmath1 .
there are in the literature a number of reports of ablation of conventional solids@xcite and organic molecules @xcite .
there are fewer in which the energy delivered to the sample is characterized well enough to measure ablation thresholds @xcite . in these ,
the ablation thresholds are many orders of magnitude larger than would be expected for far - field approaches .
schematic diagram of nsom setup , modified for ir ablation ( a ) ir fiber tip , attached to the tuning fork , scale bar:1 mm .
( b ) sem image of an ir fiber tip etched using modified tube etching technique @xcite , scale bar:200 @xmath0m.,width=268 ] we report the first integration of ir with near - field techniques applied to the ablation of live cells .
we obtain ablation features as small as 1.5 @xmath1 under 3 @xmath1 wavelength radiation , in hard and soft materials , and in air and water environments .
the ablation threshold and fluence dependence of these processes are discussed here .
of particular interest is the successful ablation of cellular samples in growth media , which we describe in this paper .
the experiments described here are performed with an nsom apparatus , modified for operation in the mid - ir region @xcite . in all experiments ,
the mid - ir output of a nd : yag laser , coupled to a pumped - optical - parametric oscillator ( opo ) ( set to 2940 nm , 100 hz , 5 ns pulse width ) is used @xcite . a schematic diagram of the experimental setup is shown in fig.[scan1 ] .
a schwarzschild objective is used to focus the laser beam and couple it to an nsom tip , which is fabricated by tube etching @xcite from a short stub of germanium - based glass fiber @xcite .
afm images of the sample , using normal force feedback , are obtained using the same tip . to calibrate the fluence through the tip
, laser power measurements are made using a pbse photoconductive detector @xcite positioned beneath the fiber tip in place of the sample . to set a baseline for understanding the ablation process , material from a cellulose acetate plastic coverslip is ablated in air in the far - field configuration .
the ablation threshold for the sample is determined by conducting a series of single - spot ablations . in these
, the laser beam is focused at normal incidence on the surface of the sample with a repetition rate of 100hz .
the sample is simultaneously observed under the microscope , and the ablation threshold is defined to be the fluence below which we do not observe visible permanent modification of the surface of the sample . in later experiments , a cellulose acetate plastic coverslip is ablated in air and in water by near - field techniques .
laser ablation in the water medium is achieved by immersing the cellulose acetate cover slip in water , 2 mm - deep , and then coupling the laser beam into the tip , which is partially immersed in the water .
the near - field ablation threshold is determined by comparing the afm images for different laser powers .
afm images of the cellulose acetate coverslip before and after ablation in water are shown in fig .
[ scan ] .
3d topographic image of a plastic cover slip in water ( a ) before and ( b ) after near - field ablation .
( c ) profile of the crater , whose size is 1.25 @xmath0 m , as measured by the full - width at half maximum ( fwhm ) of the profile .
scalebar : 2 @xmath0m , width=336 ] myoblast cells are studied in specially prepared sample holders formed from a pdms ring molded directly on a glass cover slip to hold the sample and surrounding media during the process of scanning and ablating .
cells from the c2c12 cell line are cultured directly in these molded wells .
plated cells attach to the glass surface of the sample holders during the incubation period .
topographic images of myoblast cells in media before and after ablation are shown in fig .
[ myo2 ] . under the near - field laser illumination conditions of our experiment
, it is clear from the afm images that the damage is localized to a well defined region , approximately 2.5 @xmath0 m in size . from the profile images of the craters
, we can see that there is some redeposition of the ablated material in the vicinity of the ablation crater , as is the case with the cellulose acetate material .
topographic image of myoblast cells in media ( a ) before ablation .
( b ) after ablation .
( c ) profile of the crater , whose size is 2.5 @xmath0 m .
scalebar : 10 @xmath0 m .
, width=336 ] in our study , the ablation thresholds for a single material , cellulose acetate , varied greatly , from 0.22 j/@xmath2 to 301 j/@xmath2 as seen in table i. with such a big disparity between the near - field and the far - field ablation thresholds , the question naturally arises about whether the mechanism has differs . in the far - field regime ,
the low thresholds are characteristic of a photomechanical ablation mechanism that starts with thermoelastic stresses arising from local expansion of the sample upon high - intensity laser heating@xcite .
the thermo - accoustic process is favored when the pulse width and spot size are suitable to locally contain the thermal and accoustic energy deposited by the beam @xcite .
for example , the region of the sample ablated by each pulse is set by the thermal diffusion length @xmath3 @xcite , @xmath4 where @xmath5 is the laser pulse width , and @xmath6 is the thermal diffusivity of the material . for the cellulose acetate and myoblast samples , the thermal diffusion length , @xmath3 , is approximately 25 nm , small enough to minimize lateral thermal damage .
heating leads to local thermal expansion , and ablation occurs when conditions for stress or inertial confinement are achieved .
that is , for laser pulse durations shorter than a characteristic acoustic relaxation time@xcite , @xmath7 here @xmath8 is the speed of sound , and @xmath9 is the optical absorption coefficient of the material
. a theoretical model for the photomechanical mechanism has been developed by a number of research groups @xcite , which indicates the possibility for thermo - accoustic ablation in our samples . for cellulose acetate , the speed of sound in the sample is @xmath10 m / s and the absorption coefficient is @xmath11 m@xmath12 , to yield an acoustic relaxation time @xmath13 of about @xmath14 ns .
this is much longer than the laser pulse width and satisfies the inertial confinement condition @xmath15 . for cellular samples in growth media
, we expect the properties of water to determine the length and time scales for the ablation process .
the absorption coefficient of water is of the order of @xmath16 m@xmath17 and the velocity of the sound is 1497 m / s , for which the acoustic relaxation time is equal to 0.5 ns .
comparing to our laser pulse width of 5 ns , the condition for the stress confinement , @xmath18 , is not satisfied , and photomechanical stress is not likely to be a factor , or at least will be reduced significantly .
.results of ablation threshold studies on cellulose acetate and myoblast cells in far - field ( ff ) and near - field ( nf ) regime .
each of these has been reproduced at least five times .
the laser wavelength is 2940 nm , and the pulse width is 5 ns [ cols="^,^,^,^ " , ] the measured thresholds for the near - field ablation for all cases reported here and for a number reported in the literature @xcite are much higher than expected for photomechanical mechanism .
this is not surprising for the ablation of cells but is so for the cellulose acetate . in both cases
the measured ablation threshold points to a second , higher energy mechanism , plasma induced ablation @xcite . here , the intense laser beam ionizes molecules in the sample , and subsequent collisions within the ablated plume and with ambient gas molecules lead to the formation of a hot dense plasma above the sample surface .
the vapor plume then expands perpendicularly from the surface and is further ionized by incoming radiation .
consequently , the plasma absorbs more energy from the trailing part of the laser pulse by photoionization or inverse bremsstrahlung processes @xcite .
near - field ablation alters the conditions for plasma - induced ablation .
the laser beam s path through the plasma is a narrow gap of only about 10 nm before reaching the sample , and hence the interaction with the plasma is weaker .
furthermore , the presence of a sharp probe in the vicinity of the sample leads to localization of the ionizing field and the probe itself physically blocks the expansion of the plume in the direction normal to the sample , leading to the movement of the plume laterally away from the ablated spot @xcite . in water ,
the conditions for plasma - induced ablation are enhanced and we observe a decrease in the threshold fluence .
this change is primarily due to the different optical properties of the water , namely the refractive index , which leads to an overall reflectivity decrease from @xmath19 to @xmath20 upon immersion of the sample in water .
( the reflectivity is calculated in the usual way , r = @xmath21 , here @xmath22 = 1.48 , the refractive index of cellulose acetate . )
overall , the absorptivity of water - sample system is larger than that of the air - sample system @xcite .
also , in water , the plasma is confined and there is a delay in its expansion .
hence , the induced pressure created by the laser sample interaction is much greater in water than in air and the plasma is compressed @xcite , an effect that is further enhanced by near - field confinement of the light . as has been observed by a number of groups @xcite high fluences
are required to achieve near - field ablation in organic @xcite and metallic samples @xcite .
likewise , higher fluences lead to plasma induced ablation conditions for the myoblast cells . due to the presence of water above and below the cell membrane , the induced plasma pressure in the myoblast cell samples
is enhanced compared to that of the cellulose acetate sample .
this effect is further amplified by the near - field confinement of the light .
the reflectivity of myoblast cell samples , in growth media , is found to be @xmath23 , assuming @xmath24,@xcite which helps to explain the smaller threshold fluence of myoblast cells compared to cellulose acetate . in conclusion , as a first step towards _ in - vitro _ mass spectral analysis of biological samples , we have successfully demonstrated the ablation of hard and soft materials in liquids , and , in particular , of biological samples in nutrient media . to understand the ablation mechanism , we have measured the ablation threshold of different types of samples .
the mechanisms that are involved in the ablation process are mainly photomechanical and plasma induced ablation for samples in air . since the acoustic relaxation time of water
rich samples is of the order of the laser pulse width , the ablation mechanism in cellular samples is easily seen to be plasma induced rather than photomechanical . for cellulose acetate ,
the necessity for this large increase in the near - field ablation threshold in air is not as clear .
however , for cells in media , their thermal and acoustic properties lead to conditions that favor the plasma - induced mechanism over the photomechanical one .
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prior knowledge about the shapes to be segmented is required for segmentation of images involving limited and low quality data . in many applications ,
object shapes come from multiple classes ( i.e. , the prior shape density is multimodal " ) and the algorithm does not know the class of the object in the scene .
for example , in the problem of segmenting objects in a natural scene ( e.g. , cars , planes , trees , etc . )
, a segmentation algorithm should contain a training set of objects from different classes .
another example of a multimodal density is the shape density of multiple handwritten digits , e.g. , in an optical character segmentation and recognition problem . in this paper , we consider segmentation problems that involve limited and challenging image data together with complex and potentially multimodal shape prior densities . [ cols= " < , < , < , < , < " , ] + & & & & & & + & & & & & & + & & & & & &
we have presented a mcmc shape sampling approach for image segmentation that exploits prior information about the shape to be segmented .
unlike existing mcmc sampling methods for image segmentation , our approach can segment objects with occlusion and suffering from severe noise , using nonparametric shape priors .
we also provide an extension of our method for segmenting shapes of objects with parts that can go through independent shape variations by using local shape priors on object parts .
empirical results on various data sets demonstrate the potential of our approach in mcmc shape sampling .
the implementation of the proposed method is available at spis.sabanciuniv.edu/data_code .
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segmenting images of low quality or with missing data is a challenging problem .
integrating statistical prior information about the shapes to be segmented can improve the segmentation results significantly .
most shape - based segmentation algorithms optimize an energy functional and find a point estimate for the object to be segmented .
this does not provide a measure of the degree of confidence in that result , neither does it provide a picture of other probable solutions based on the data and the priors . with a statistical view , addressing these issues would involve the problem of characterizing the posterior densities of the shapes of the objects to be segmented . for such characterization , we propose a markov chain monte carlo ( mcmc ) sampling - based image segmentation algorithm that uses statistical shape priors .
in addition to better characterization of the statistical structure of the problem , such an approach would also have the potential to address issues with getting stuck at local optima , suffered by existing shape - based segmentation methods .
our approach is able to characterize the posterior probability density in the space of shapes through its samples , and to return multiple solutions , potentially from different modes of a multimodal probability density , which would be encountered , e.g. , in segmenting objects from multiple shape classes .
we present promising results on a variety of data sets .
we also provide an extension for segmenting shapes of objects with parts that can go through independent shape variations .
this extension involves the use of local shape priors on object parts and provides robustness to limitations in shape training data size .
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we gratefully acknowledge project management support by p. leisching .
the work was funded by the german ministry of economics and technology ( zim projects kf2303709 and kf2806204 ) .
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we investigate the tuning behavior of a novel type of single - frequency optical synthesizers by phase comparison of the output signals of two identical devices .
we achieve phase - stable and cycle slip free frequency tuning over 28.1 ghz with a maximum zero - to - peak phase deviation of 62 mrad .
in contrast to previous implementations of single - frequency optical synthesizers , no comb line order switching is needed when tuned over more than one comb line spacing range of the employed frequency comb .
a single - frequency optical synthesizer ( sos ) is the analog of an electrical synthesizer in the optical domain .
it provides a single - frequency optical field whose frequency and phase can be arbitrarily adjusted within a certain spectral range and resolution while it is related to a reference signal in a phase - coherent fashion . in an ideal case
it combines the best of two worlds in a tunable way , i.e. the spectral resolution of narrow linewidth frequency stabilized lasers with the broad spectral coverage of frequency combs .
+ such an sos has applications in basic research e.g. in precision spectroscopy @xcite , frequency metrology and quantum optics but could also enable the development or improvement of a number of useful practical devices in optical metrology like high resolution optical spectrum or vector analyzers , phase stable cw thz - synthesizers @xcite , high resolution , high dynamic range optical frequency domain reflectometers based on frequency scanning , optical coherence tomography @xcite or traceable range finders and laser trackers .
+ the invention of frequency combs @xcite , which consist of a broadband fixed frequency grid with well - defined frequency and phase relations between the individual comb lines , has enabled the most sophisticated sos approaches up to date @xcite .
these implementations are based on phase - locking of a single - frequency `` clean - up '' laser to an individual comb line of a frequency comb and subsequent tuning of the repetition rate @xcite or tuning of the offset frequency between `` clean - up '' laser and comb line @xcite .
the tuning speed and range is hampered in the first approach due to the required macroscopic resonator length variation .
+ the approaches based on offset frequency tuning suffer from an ambiguity at `` critical '' optical frequencies , where the beat frequency between a specific comb line and the `` clean - up '' laser becomes zero or coincides with the beat frequency from an adjacent comb line .
this leads to forbidden frequency gaps and limits the agility of the sos .
moreover , these ambiguities need to be resolved in a complex way which imposes additional technical overhead . if forbidden frequency gaps at the critical frequencies are to be avoided , an offset frequency tuning based sos needs to take a smart action when approaching these frequencies . in @xcite for example , a second , frequency shifted beat note is generated and used as alternative error signal when critical frequencies are approached . in that case , the sos will have problems to preserve phase stability when switching between error signals .
+ the aforementioned complications can be circumvented by combining a novel method for frequency shifting the carrier frequency of frequency combs that we have reported on in an earlier paper @xcite , with a fixed offset phase lock of a `` clean - up '' laser to a single comb line . in this case , frequency tuning of the comb spectrum avoids any problematic ambiguity since all rf - beat frequencies are constant in time and conjugate beat signals never coincide .
such a universal sos consists of three elements , i.e. a mode - locked laser acting as frequency comb generator ( ofc ) , the mentioned frequency shifter for frequency combs @xcite and a single line selector @xcite which ensures single frequency operation .
the latter is implemented with a phase lock of a `` clean - up '' laser to a single comb line .
+ in this paper we present the first characterization of the tuning properties of an sos based on such a frequency shifter . in order to characterize the carrier phase fidelity of the sos during tuning action we duplicate the system and use the second sos as reference enabling the treatment in the rotating frame .
we show that we can achieve phase - stable tuning of our sos with respect to the reference over the full tuning range of the `` clean - up '' laser . + the principle of the frequency shifter relies on serrodyne frequency shifting @xcite the carrier frequency of an ofc .
it exploits the fact that the spectrum of the ofc corresponds to a periodic pulse train in the time domain and shifts the optical carrier phase between subsequent pulses of the ofc . to explain the principle , the definition of the instantaneous frequency of the monochromatic signal of the comb line with order number @xmath0 can be used .
@xmath1 the frequency evolution @xmath2 of the signal is given by a constant frequency @xmath3 plus a change of the carrier phase @xmath4 per unit time @xmath5 . in the frequency domain the frequency comb can be treated as a multitude of monochromatic fields , each experiencing the frequency evolution according to eq.([equ1 ] ) . in the time domain
the monochromatic fields interfere destructively at the dark intervals of the pulse train .
the natural time interval for an incremental phase change is thus the temporal pulse - to - pulse spacing , i.e. @xmath6 . the pulse - to - pulse change @xmath4 of the carrier phase is implemented by adding consecutively changing phase command values by means of an electro - optic phase modulator ( eom ) .
the sequence of phase command values required for a target temporal evolution of the frequency shift is determined by temporal integration of the second term in eq.([equ1 ] ) .
a simple constant carrier frequency shift thus corresponds to a linear stepwise increase of the carrier phase between subsequent pulses .
the cubic phase evolution of fig.[setup]a accordingly leads to a parabolic frequency evolution of the comb .
since the electromagnetic field is @xmath7 periodic , the phase command values can be applied modulo @xmath7 , avoiding their divergence and thus of the voltages applied to the eom .
+ the incremental phase changes @xmath4 between pulses and the @xmath7 flyback events in fig.[setup]a occur at the dark intervals of the pulse train . due to the pulsed nature of the comb electromagnetic field , only the effective phase values which are present during the optical pulse length contribute to the frequency shifting effect .
thus , possible imperfections of the phase steps , e.g. transients or ringing , are concealed and the generation of spurious frequencies is suppressed .
consequently , the phase evolution between the pulses can be designed in an arbitrary fashion . + a detailed description of the frequency shifter for frequency combs and experimental proof that frequency tuning of a comb over multiple @xmath8 is readily achieved can be found in @xcite .
+ in our experiment we use an integrated eom ( eospace ) to set the carrier phase of subsequent pulses of a frequency comb . ,
arrows : @xmath7 flybacks , solid line : possible phase evolution at eom .
b : setup of the experiment consisting of two channels cha and chb .
fpga : field programmable gate array , eom : electro - optical modulator , bpf : band pass filter ( 50 ghz ) , dl1/2 : external cavity diode laser 1 and 2 , pd : photo diode , pll : phase locked loop , ool : out of loop measurement . ]
the setup consists of two identical , but independent channels , a and b , as shown in fig.[setup]b .
a fiber splitter divides the output power of a frequency comb generator ( toptica femtoferb sync , @xmath9=1550 nm , @xmath8= 56 mhz ) in two equal parts of 30 mw average power and sends them to the two identical eoms .
the phase command values for both channels are generated with the help of two independent numerically controlled oscillators ( nco , details in @xcite ) that are implemented in one fpga ( stratix iii ) .
the light fields in both channels are thus manipulated completely autonomously by digital - to - analog converting the phase command values and sending them to the eom by means of an rf - signal .
the fpga is clocked with a harmonic of @xmath8 , hence the changes @xmath4 and the @xmath7 flybacks of the phase command value are synchronized with the comb s pulse train .
the signal gain of the phase command values is adjusted such that the @xmath7 flybacks correspond to a carrier phase shift of @xmath7 to the eom . for the sake of simplicity
we describe only channel a in the following .
the transmitted , frequency shifted light is filtered by a 50 ghz band - pass filter to improve the signal to noise ratio of the beat between a selected comb line @xmath0 and an external cavity diode laser ( dl1 , toptica dl pro design , @xmath9=1550 nm ) .
the light fields from dl1 and the filtered comb are superimposed on a photodiode ( pd1 ) using polarization maintaining fiber splitters .
the electric beat signal of the pd is used to phase - lock dl1 to the selected comb line @xmath0 .
the feedback loop is closed using a fast analog controller ( toptica falc 110 ) which acts on the piezo and current of dl1 .
additionally , the fpga presets the desired frequency evolution of dl1 by means of a baseband feed - forward signal to the piezo of the laser .
thus , the phase locked loop ( pll ) has to compensate only for imperfections of the feed - forward control , e.g. due to piezo hysteresis .
+ the remaining light from dl1 is split in two parts by a polarization maintaining fiber splitter .
one part is sent to an out - of - loop measurement ( ool ) of the laser frequency implemented with a delayed self - heterodyne interferometer of known length .
the other part is sent to a photodiode ( pd3 ) where it is superimposed with the corresponding light field from channel b , i.e. the light field from dl2 phase - locked to the same comb line @xmath0 .
the offset frequencies of the phase locks of laser dl1 and dl2 to the selected comb line @xmath0 differ by 1 mhz .
the resulting 1 mhz beat signal is detected at pd3 , digitized with a sampling rate of 7 mhz by a 16 bit daq card and stored on a computer . in a post - processing step ,
the differential phase of the two light fields during a desired frequency scan is extracted .
this phase deviation is deduced from the raw data using a software phase - retrieval algorithm @xcite .
effectively , it compares the phase of the 1 mhz beat with the differential phase between the pll offset signals .
since both channels share the same frequency comb generator , frequency and phase fluctuations of this `` master oscillator '' are ruled out via common - mode rejection and solely the phase deviation introduced by the frequency shifter is measured . + the setup in fig.[setup]b represents a two - beam interferometer with two separate paths from the eoms to pd3 .
both signal path lengths of about 11 m have to be matched to a degree of 300 @xmath10 in order to avoid moving spectral fringes during tuning .
length matching is accomplished with the help of a free space delay line in channel a. + to characterize the tuning behavior of the sos both lasers are swept simultaneously over 500 comb lines , or 28.1 ghz following a sinusoidal frequency evolution . of the beat note between dl1 and dl2 during the same sweep ( blue ) .
sidebands at 50 and 100 hz are caused by power line noise .
fft - filter function used to calculate @xmath11 ( red ) .
bottom : result of the out - of - loop frequency measurement of dl1 during the sweep .
the coarse frequency measurement has an uncertainty of @xmath12 mhz . ]
fig.[fig3 ] shows the phase deviation between dl1 and dl2 ( top ) and the frequency of dl1 ( bottom ) for this experiment . during this 2 s long sweep
the relative phase is stable within @xmath13 rad .
high frequency technical noise contributions ( @xmath14 = 55 mrad , 4 hz @xmath15 3.5 mhz ) are found together with a slow drift on the timescale of seconds .
this slow drift is caused by fluctuations of the interferometer path length difference which is not controlled .
although the frequency shifting method is in principle capable of controlling the absolute phase value of the optical output field in a deterministic way , this has not yet been implemented here .
unwanted interferometric phase drifts could be reduced , e.g. by reducing the length of the fiber pigtails of the components , temperature stabilizing the interferometer setup or ultimately by active fiber length stabilization .
+ the inset in fig.[fig3 ] shows in blue the power spectral density @xmath11 ( rf - spectrum ) of the beat note between both dls during their common , 28.1 ghz wide frequency sweep .
@xmath11 was calculated from the raw data of fig.[fig3 ] ( top ) using fft .
its 3db - width of 0.7 hz is slightly broader than the fft - filter function ( in red , 0.5 hz ) used to calculate the spectrum and the peak is 30 db suppressed with respect to the filter function .
this reflects the onset of a carrier collapse of the filtered rf - signal and is in accordance with the interferometric phase drifts found in fig.[fig3 ] ( top ) .
the spectrum represents a beat - note in the `` rotating - frame '' and is characteristic for the two - sos - scheme employed here .
+ the tuning speed of the sos reaches 45 ghz / s at the steepest slope of the 28.1 ghz sinusoidal modulation or up to 120 ghz / s for frequency modulations on the order of 200 mhz .
tuning speed and range are solely limited by the `` clean - up '' technique , i.e. by the bandwidth of the locking electronics and the mode - hop free tuning range of the diode laser .
the frequency shifting method itself , disregarding the clean - up laser , is capable of tuning speeds of up to tens of thz per second combined with a tuning range over the whole bandwidth of the frequency comb .
+ to illustrate that the sos is running cycle slip free , we intentionally provoke a cycle slip in the pll of dl1 by a modification of the locking parameters .
the cycle slip manifests itself as a @xmath7 jump in the differential phase between dl1 and dl2 as shown in fig.[fig4]a . from the absence of this signature in measurements with optimized locking parameters
, we deduce that the sos is operating cycle slip free .
+ in the following an experiment to characterize the phase deviations at the critical frequencies mentioned in the introduction is described . for our sos
the most relevant critical frequencies correspond to the frequencies of the initially unshifted comb lines .
this can be explained by small deviations of the eom driving signal from the ideal shape , e.g. non perfect-@xmath7-flybacks , which are sampled by the light pulses and lead to spurious sidebands at the optical output spectrum of the frequency shifter .
the spurs occur at the frequencies of the initial unshifted comb lines and are typically suppressed by 30 db with respect to the shifted comb lines .
these spurs in the frequency domain induce pll error signal glitches upon tuning the `` clean - up '' laser across the spur and show up as systematic phase deviations with amplitudes comparable to the technical noise of fig .
[ fig3 ] in our measurements . to measure the systematic phase deviations of an individual sos
, any common - mode suppression of the glitch during tuning transits across the spur has to be avoided .
dl1 and dl2 are thus linearly swept over the spur with a frequency shifter induced detuning of 5 mhz .
then , phase - tracking their beat note while tuning reveals the individual phase deviations at the glitch positions since the individual transits of both sos take place at different time instants .
thus , the true phase deviation of dl1 at the critical frequency is measured , while dl2 acts as an unperturbed local oscillator .
fig.[fig4]b shows the phase deviation produced when dl1 is slowly swept over such an unshifted comb line .
the trace shows the average of 100 consecutive sweeps with interferometric phase offsets subtracted .
the averaging reduces the technical noise and reveals a systematic phase glitch with a zero - to - peak deviation of 62 mrad .
the interference between the swept signal and the spur leads to an oscillatory shape of the glitch .
the deviation from a pure oscillatory shape results from the pll response .
the observed phase glitch is consistent with the expected phase error caused by spurs which are 30 db below the carrier . to the best of our knowledge
this is the first time phase stable tuning of an sos across critical frequencies is achieved and characterized . + the frequency resolution of the frequency shifter is given by the product of @xmath8= 56 mhz and the length of the nco tuning word .
the algorithm implemented in the fpga firmware uses a phase resolution of @xmath16 bits , leading to a frequency resolution of the soss center frequency of 200 nhz .
it should be noted that the smaller amplitude resolution ( 12 bit ) of the digital to analog converter used to generate the analog phase command values does not influence the frequency resolution , but determines the so called spur - free dynamic range and thus the maximum phase error .
the frequency resolution could even be improved by using more bits in the fpga implementation .
+ the short - term linewidth of dl1 locked to the free running unshifted comb corresponds to the linewidth of dl1 in the non - rotating frame and was measured to be well below 10 khz at 1550 nm using a delayed self - heterodyne interferometer .
the minimum absolute frequency uncertainty achievable with this sos depends on the choice of frequency reference for the frequency comb and can approach the values of the best optical / microwave clocks .
+ in conclusion , we have demonstrated phase - predictable tuning of two independent , common mode suppressed soss .
in contrast to previous implementations of soss , our approach allows for arbitrary , set - point - switching free tuning over the full spectrum of the used comb oscillator , limited only by the applied `` clean - up '' technique ( 28.1 ghz ) .
the maximum zero - to - peak deviation of the phase found at critical frequencies was shown to be 62 mrad and the maximum tuning speed realized was 120 ghz / s . in combination with well studied methods for self - referencing of frequency combs and widely tunable lasers
, this technique could stimulate the development of a compact sos , capable of delivering a tunable absolute optical frequency of high precision .
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the components of @xmath22 associated with ( [ higgsn ] ) are ( @xmath67 ) w. nahm , phys . lett .
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fermion zero modes for abelian bps monopoles are considered . in the spherically symmetric case
the normalisable zero modes are determined for arbitrary monopole charge @xmath0 .
if @xmath1 the zero modes are zero along @xmath2 half - lines emanating from the monopole .
keywords : abelian gauge theory , bps monopoles , weyl equation , fermion zero modes .
fermion zero modes for bps monopoles can be constructed via the same nahm transform used to obtain the monopoles @xcite .
the construction is , however , cumbersome for magnetic charge @xmath3 . in this letter
we obtain zero modes for abelian bps monopoles .
our approach is to directly integrate the weyl equations in three - dimensional space rather than use nahm s method ( which has been adapted to abelian monopoles in @xcite ) .
the abelian bps equations read @xmath4 where @xmath5 is a real higgs field and @xmath6 is a magnetic field derived from a vector potential @xmath7 .
the maxwell equation @xmath8 implies that the higgs field @xmath5 obeys the laplace equation . the higgs field
@xmath9 with @xmath10 and @xmath11 constant , is harmonic away from @xmath0 singularities @xmath12 ( @xmath13 ) .
physically , the system comprises @xmath0 dirac monopoles each with magnetic charge @xmath11 interacting with a higgs field . here
@xmath10 fixes the asymptotic value of the higgs field .
consider the weyl operators @xmath14 where @xmath15 is the electric charge of the fermion and @xmath16 .
these weyl operators assume a real yukawa coupling in minkowski space .
however , identifying @xmath5 as @xmath17 they are also weyl operators for self - dual monopoles defined in euclidean space .
the dirac quantisation condition requires @xmath18 with @xmath19 integer . if @xmath20 and @xmath21 , @xmath22 has @xmath0 normalisable zero modes . in the @xmath23 case
we have @xmath24 taking the origin as the location of the monopole and @xmath25 denote spherical polar coordinates . here
the dirac string lies on the positive @xmath26-axis .
one can verify that @xmath27 is a zero mode of @xmath22 ( the components of @xmath22 in spherical polar coordinates are given in the appendix ) .
@xmath28 has no zero modes .
this result can be obtained by taking the large @xmath29 limit of the jackiw - rebbi zero mode @xcite for the basic @xmath30 bps monopole after performing a gauge transformation which diagonalises the higgs field .
the fermion density @xmath31 is spherically symmetric and has an integrable singularity at the monopole centre .
the general @xmath0 case is more complicated .
however , in the spherically symmetric case where the positions of the @xmath0 monopoles coincide the higgs field is @xmath32 and @xmath7 is the @xmath23 potential multiplied by @xmath0 . here
@xmath22 has @xmath0 normalisable zero modes : @xmath33 these solutions resemble ( in particular with respect to the @xmath34 and @xmath35 dependence ) known solutions of the dirac equation in the background of abelian monopoles @xcite .
however , our solutions are written directly in terms of trigonometric functions rather than spherical harmonics is even the zero modes can be expressed in terms of standard spherical harmonics . if @xmath0 is odd spin - weighted or monopole harmonics @xcite are required . ] .
our solutions are normalisable with @xmath36 norm @xmath37 . as the zero modes @xmath38 are annihilated by the hamiltonian it is not clear to us whether the presence of the higgs field cures the self - adjointness problem associated with monopole hamiltonians . to address this issue one needs to study the scattering states @xcite .
note that the densities @xmath39 are not spherically symmetric for @xmath1 .
for @xmath40 the first zero mode is zero along the positive @xmath26-axis while the second mode is zero on the negative @xmath26-axis . by taking a suitable linear combination of @xmath41 and
@xmath42 one can obtain a zero mode with a zero along any half - line emanating from the monopole ; the zero mode is axially symmetric about the axis on which the half - line lies .
the @xmath1 zero modes are zero along @xmath2 half - lines .
our @xmath41 and @xmath43 have zeros of strength @xmath2 on the positive and negative @xmath26-axes , respectively ; the remaining @xmath44 modes have lower strength zeros on both the positive and negative @xmath26 axes .
again one can adjust the directions of the @xmath2 half lines by taking different linear combinations of the @xmath0 zero modes .
for a discussion of zeros of fermion zero modes in a different context see @xcite . in general , the @xmath3 zero modes are not axially symmetric even though the @xmath38 are all axially symmetric about the @xmath26-axis . if @xmath45 is a linear combination of the @xmath38 , the density @xmath46 has the form @xmath47 where @xmath48 is a function of @xmath34 and @xmath35 . for @xmath3
one can see that the zero modes vanish at the position of the monopole and the zero modes peak somewhere on the sphere @xmath49 .
the function @xmath50 has up to @xmath2 zeros ; if there are fewer than @xmath2 zeros these are repeated zeros associated with coincident half - lines .
examination of @xmath50 for several zero modes indicates that @xmath50 has a single peak .
for example , the @xmath51 zero mode @xmath52 yields an @xmath50 function with two zeros and a maximum on the equator @xmath53 ( these three points are equally spaced on the equator ) .
the maximum of @xmath50 is a point on the unit sphere except for some axially symmetric solutions where @xmath50 peaks on a circle . in figure 1
spherical plots of @xmath50 are given for an @xmath40 and an @xmath51 zero mode .
for an @xmath40 and @xmath51 zero mode ( the scale depends on the normalisation and hence @xmath10 ) .
the left plot shows @xmath50 for an @xmath40 zero mode ( @xmath54 ) which has one zero and is axially symmetric .
two zeros are visible for the @xmath51 zero mode ( @xmath55 ) on the right .
both zero modes have a single peak . ]
van baal @xcite has given an ansatz which provides a solution of the weyl equation for any solution of the abelian bps equation : @xmath56 satisfies @xmath57 where @xmath58 and @xmath59 satisfies the laplace equation .
this works as the bps equation implies @xmath60 is a scalar and ( [ vanbaal2 ] ) gives @xmath61 .
taking @xmath62 yields our @xmath5 and @xmath7 .
however , @xmath63 is not normalisable ) does provide one normalisable zero mode for a different class of higgs fields ; here @xmath5 has @xmath64 singularities representing @xmath0 positively charged and @xmath0 negatively charged monopoles @xcite .
] though for @xmath65 it agrees with our zero mode @xmath41 .
indeed , @xmath41 is the @xmath65 van baal solution multiplied by @xmath66 . as the van baal construction does not rely on spherical symmetry
this approach may provide information about the @xmath65 limit of the general case where the @xmath0 monopoles are separated .
we have considered fermion zero modes for bps monopoles and have obtained solutions for arbitrary magnetic charge @xmath0 .
the spherically symmetric abelian case we have solved may provide a model for non - abelian magnetic bags ; although higher charge @xmath30 bps monopoles are never spherically symmetric , solutions with approximate spherical symmetry may emerge for large @xmath0 @xcite
. it would be interesting to investigate the extent to which our zero modes approximate the fermion zero modes of magnetic bags .
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this work has been partially supported by the brazilian research agencies cnpq and fapesp .
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nucl.phys.b 422 ( 1994 ) 449 _ c. ahn and r.i .
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in this note we study la baxter @xcite the possible integrable manifolds of the asymmetric eight - vertex model .
as expected they occur when the boltzmann weights are either symmetric or satisfy the free - fermion condition but our analysis clarify the reason both manifolds need to share a universal invariant .
we also show that the free - fermion condition implies three distinct classes of integrable models .
= -0.33 in = -0.3 in ufscar - th-02 + + _ departamento de fsica , universidade federal de so carlos _ _ caixa postal 676 , 13565 - 905 , so carlos , brazil _
july 2002 exactly solved vertex models play a fundamental role in classical statistical mechanics .
the most important of these is the so - called eight - vertex model which contains as special cases most systems on a plane square lattice @xcite .
the general asymmetric eight - vertex model possesses six different boltzmann weights @xmath0 and @xmath1 whose transfer matrix can be written as @xmath2\ ] ] where the trace is over the ordered product of local operators @xmath3 which are given by the following @xmath4 matrix @xmath5 and @xmath6 are pauli matrices acting on the sites @xmath7 of an one - dimensional lattice .
the asymmetric eight - vertex model is known to be solvable in the manifolds @xmath8 @xmath9 @xmath10 where @xmath11 and @xmath12 are arbitrary constants .
the manifold ( 3 ) is the so - called free - fermion model whose free - energy was first calculated by fan and wu @xcite and later re - derived by felderhof @xcite who devised a method to diagonalize the corresponding transfer matrix .
the integrability of the free - fermion manifold is usually assumed from the fact that its transfer matrix commutes with the @xmath13 hamiltonian as shown by krinsky @xcite who used a procedure first developed by sutherland @xcite .
later on barouch @xcite and kasteleyn @xcite have revisited the problem of commuting asymmetric eight - vertex transfer matrices and generalized heisenberg hamiltonians which leaded kasteleyn @xcite to point out the existence of the manifolds ( 4 ) . as stressed by this author , however , such manifolds are trivial because they can be seen as set of independent one - dimensional models and therefore they should be disregarded . on the other hand ,
the solution of the symmetric manifold was found by baxter through quite general approach denominated commuting transfer matrix method which culminated in the famous `` star - triangle '' relations @xcite .
the fact that the symmetric eight - vertex transfer matrix commutes with a related @xmath14 hamiltonian @xcite has then been made more precise because the latter is essentially a logarithmic derivative of the former @xcite .
it would be quite desirable to extend the baxter s method to the asymmetric eight - vertex model and to rederive the manifolds ( 3 ) and ( 5 ) from a unified point of view . since this approach does not assume a priori the existence of a specific local form for the corresponding hamiltonian it can lead us to new integrable manifolds not covered by the analysis of barouch @xcite and kasteleyn @xcite .
we recall that much of the work on this problem , see e.g. refs .
@xcite , has been concentrated to analyze the yang - baxter equations directly in terms of spectral parameters . though this is a valid approach it often hides the general integrable manifolds in terms of specific parameterizations which need to be found by a posteriori guess - work . a more direct way would be first to determine the solvable manifolds by an algebraic study la baxter of the corresponding `` star - triangle '' equations and afterwards to parameterize them by using the theory of uniformization of biquadratic polynomials @xcite .
it appears that kasteleyn @xcite was the first to make an effort toward such analysis but the best he could do was to guess the manifold ( 3 ) from known results by felderhof besides clarifying the origin of the pseudo one - dimensional manifold ( 4 ) as the linearization of the yang - baxter equation around a non - identity @xmath15 @xmath16-matrix . since the later possibility leads us to trivial manifolds we will disregard it , as did kasteleyn , from our forthcoming analysis .
the probable reason that such generalization has not yet been carried out seen technical since in the asymmetric model we have to deal with the double number of equations as compared to the symmetric eight - vertex model . at first sight
this appears to be a cumbersome task , but here we show that it is possible to simplify this problem , without recoursing to computer manipulations , to a number of simple equations that will clarify the common origin of the above two integrable manifolds . besides that , this approach allows to show that manifold ( 3 ) is one between three possible different integrable branches satisfying the free - fermion condition .
the `` star - triangle '' relations are sufficient conditions @xcite for commuting transfer matrices and for the asymmetric eight - vertex model they are given by @xmath17 @xmath18 @xmath19 @xmath20 @xmath21 @xmath22 note that each of these equations possesses two possibilities and we shall denote them by eqs.(@xmath23,@xmath24,@xmath25 ) .
altogether we have twelve linear homogeneous equations and only six weights , say @xmath26 and @xmath27 , are at our disposal to be eliminated in terms of remaining set of weights @xmath28 and @xmath29 .
therefore we have to choose the appropriate equations to start with and our solution goes as follows .
we first eliminate the weights @xmath30 with the help of the pair of equations ( @xmath31,@xmath32 ) and by substituting the result in eqs.(@xmath23 ) we find the following relations @xmath33\end{aligned}\ ] ] next we apply similar procedure in the case of eqs.(@xmath34,@xmath35 ) and the corresponding relations between the weights @xmath36 , @xmath37 and @xmath38 are @xmath39\end{aligned}\ ] ] from eqs.(@xmath40 ) it is not difficult to eliminate the weights @xmath36 , leading us to constraints between @xmath37 and @xmath38 @xmath41= d^{"}\left[a_{\mp}b_{\mp}f(a_{\pm}^{'},b_{\pm}^{'},c^{'},d^{'})-a_{\pm}^ { ' } b_{\pm}^{'}f(a_{\pm},b_{\pm},c , d ) \right]\ ] ] at this point it is tempting to use such equations and the previous results for @xmath30 and @xmath36 to eliminate five weights ratios and to substitute them in the remaining equations , namely eqs.(@xmath25 ) and either eqs.(@xmath42 ) or eqs.(@xmath43 ) .
this is , however , not so illuminating because it leads us to carry out simplifications in complicated expressions .
we find that it is more profitable to repeat the procedure described above but now we first eliminate the weights @xmath44 and in the end we use eqs.(@xmath25 ) instead of eqs.(@xmath23 ) .
this leads us to a different constraint between @xmath37 and @xmath38 given by , @xmath45= c^{"}\left[a_{\mp}b_{\mp}f(a_{\pm}^{'},b_{\pm}^{'},c^{'},d^{'})-a_{\mp}^ { ' } b_{\mp}^{'}f(a_{\pm},b_{\pm},c , d ) \right]\ ] ] now we reached a point that enables us to make conclusions on the way the set of weights @xmath46 and @xmath47 should be related to each other .
in fact , from eqs.(@xmath48,@xmath49 ) we find that the necessary conditions for the weights @xmath37 and @xmath38 not to be all zero are @xmath50 and either @xmath51 or @xmath52 we are already in the position to conclude that the asymmetric eight - vertex model has indeed only two possible integrable manifolds , one is singled out by the free - fermion condition ( 17 ) while the other ( 18 ) turns out to be a mixed type of conditions that relate the set of weights both alone and between each other .
one important point of our analysis is that it makes clear that both manifolds need to share a common invariant given by eq.(16 ) to close our analysis it remains to check the consistency between eqs.(@xmath23 ) and eqs.(@xmath25 ) which can in principle be a source of further constraints . from such equations
one can easily calculate the ratios @xmath53 and @xmath54 , namely @xmath55 @xmath56 which in principle can be compared with our previous results for the same ratios . before proceeding with that ,
however , there exists one property that we have not yet explored . instead of starting our analysis by eliminating the weights @xmath57,@xmath36 , @xmath37 and @xmath38 we could choose to begin with the other two sets of weights as well .
because the `` star - triangle '' equations are not symmetric by exchanging a given two sets of weights we expect that each possibility will leads us to different kind of constraints .
this means that we can use the asymmetry of the weights in our favour which may help us in further simplifications .
for example , the relations ( @xmath23-@xmath25 ) are invariant under the exchange of weights @xmath58 and @xmath59 only after the transformation @xmath60 is performed for all set of weights .
this means that if we had started our procedure by eliminating the weights @xmath61 and @xmath1 the same analysis we have carried out so far will lead us to the following constraints @xmath62 besides that either @xmath63 or @xmath64 by the same token if we had started by eliminating @xmath65 and @xmath66 we will find @xmath67 and that either @xmath68 or @xmath69 let us now analyze the consequences of this observation for each possible integrable manifold and here we begin with the second manifold .
it is not difficult to see that the consistency of the equations ( 18),(23 ) and ( 26 ) , to what concern relations within the same set of weights , impose severe restrictions on the second type of the manifold , namely @xmath70 and similar conditions for the other sets @xmath71 and @xmath72 .
it turns out , however , that the only possibility compatible with the `` universal '' constraints ( 16 ) , ( 21 ) and ( 24 ) is the totally symmetric case @xmath73 and @xmath74 leading us therefore to the baxter s model ( 5 ) .
note that in this situation the compatibility between eq(@xmath23 ) and eq.(@xmath25 ) is trivial because both equations ( 19 ) and ( 20 ) are automatically satisfied .
we now turn our attention to the free - fermion manifold . in this case
we have much less restrictive constraints since we are only left with relations between different weights , namely eqs.(16,21,24 ) .
altogether these equations provide us the following relation @xmath75 whose compatibility with eqs.(@xmath23,@xmath25 ) can be implemented by evaluating the left - hand side of eq.(28 ) with the help of eqs.(19,20 ) .
after few manipulations , in which the free - fermion condition is explicitly used , we end up with a `` separable '' equation @xmath76 for the weights @xmath59 and @xmath77 and the polynomial @xmath78 is given by @xmath79 \nonumber\\ & & \times \left [ a_{-}b_{+}a_{-}^{'}b_{+}^{'}-a_{+}b_{-}a_{+}^{'}b_{-}^ { ' } \right ] \nonumber\\ & & \times \left [ ( a_{+}^{2}+b_{-}^2-a_{-}^2-b_{+}^2)(a_{+}^{'}b_{-}^ { ' } + a_{-}^{'}b_{+}^ { ' } ) - ( a_{+}b_{- } + a_{-}b_{+ } ) ( { a_{+}^{'}}^{2}+{b_{-}^{'}}^2-{a_{-}^{'}}^2-{b_{+}^{'}}^2 ) \right ] \nonumber\\\end{aligned}\ ] ] from this equation we conclude that we have three possible free - fermion integrable manifolds given by either @xmath80 or @xmath81 or still @xmath82 besides of course the free - fermion conditions for both @xmath59 and @xmath83 together with the `` universal '' relation ( 16 ) .
note that the free - fermion case ( 31 ) can not be related to the manifold ( 4 ) beginning by the fact that in the former model the weight @xmath84 can be different of the weight @xmath1 . in figure 1
we have summarized all the results obtained so far .
let us now compare our results with previous work in the literature .
contrary to what happened to the symmetric manifold ( 5 ) we recall that eqs.(30,31 ) do not imply that the ratios @xmath85 and @xmath86 are necessarily constants but only that they are invariants for two distinct set of weights .. ] this is the reason why general solutions of the yang - baxter equation satisfying the free - fermion condition are expected to be non - additive @xcite .
in fact , in the appendix @xmath87 we show that the additional assumption of additivity provides us extra restriction to the weights . in this sense ,
the manifold ( 30 ) turns out to be a generalization of the original result ( 3 ) by krinsky @xcite .
next the manifold ( 31 ) has been only partially obtained in the literature , more precisely in the special case @xmath73 and @xmath88 @xcite and @xmath74 is contained in the baxter s solution . ] .
finally , to the best of our knowledge the last branch ( 32 ) is new in the literature .
the probable reason that such general manifolds have been missed in previous work , see for example refs.@xcite , is related to analysis of the yang - baxter equation in terms of spectral parameters .
there it was required that at certain value of the spectral parameter ( initial condition ) the weights should be regular , i.e that the corresponding @xmath89 operator be proportional to the four dimensional permutator .
note that the @xmath89 operator of manifold ( 32 ) can not be made regular and therefore does not have a local associated hamiltonian .
this is also the reason barouch @xcite and kasteleyn @xcite missed such manifold since they used the assumption of local forms of hamiltonians .
we recall that though the property of regularity guarantees that the logarithmic derivative of the transfer matrix is @xmath90 this is by no means a necessary condition for integrability . in summary
, we have analyzed according to baxter the integrable branches of the asymmetric eight - vertex model . besides recovering the baxter s model we shown that the free - fermion condition produces three different set of integrable manifolds . a natural question to be asked
is whether or not the new manifolds ( 31 ) and ( 32 ) can be solved by the method devised by felderhof originally proposed to diagonalize the transfer matrix of the krinsky s manifold ( 3 ) .
this is of interest since these systems maybe the corner stone of highly non trivial models as recently have been discussed in refs.@xcite .
in fact , we have evidences that the manifold ( 31 ) is related to a staggered @xmath13 model . because both the baxter symmetric model and the free - fermion manifolds ( 30 - 32 ) share a common algebraic structure , the yang - baxter algebra , it is plausible to think that baxter s generalized bethe ansatz can be adapted to include the solution of the free - fermion models too .
this problem has eluded us so far though some progress has been made in the case of the simplest free - fermion branch ( 31 ) .
* appendix a * the purpose here is to demonstrate that the hypothesis of additivity of the weights leads us to much more restrictive conditions for the free - fermion manifold as compared with the results ( 30 - 32 ) of the main text . in order to see that lets us consider as usual that the weights @xmath91 are parameterized by the variables @xmath92 and @xmath93 and similarly that @xmath94 and @xmath95 are parameterized by @xmath96 and @xmath97 , respectively .
the consistency between the `` universal '' relations ( 16 ) , ( 21 ) and ( 24 ) implies a remarkable separability condition for the ratio @xmath98 where @xmath99 is an arbitrary function . the additional assumption that the weights are additive means that this function is necessarily a constant which ultimately leads us to the relation @xmath100 as a consequence of that the possible manifolds satisfying the the free - fermion condition are either @xmath73 or @xmath74 .
now by imposing the consistency between eqs(@xmath23 ) and eqs.(@xmath25 ) it turns out that these two possibilities becomes either @xmath101 or @xmath102 where @xmath103 is a constant
. clearly , these are special cases of the manifolds ( 31 ) and ( 30 ) , respectively .
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quarks and leptons are believed to be the fundamental particles of matter yet their nature is still far from being understood .
one of the exciting puzzles is a formula involving the masses of the three leptons , discovered by yoshio koide @xcite : @xmath0 see table 1 for the corresponding numerical values ( from @xcite ) .
[ tbl:1 ] .lepton masses .
[ cols= " < , < , < , < , < " , ] * 4 . coda . * whether this intriguing connection with geometry will contribute to an understanding of the masses of leptons remains an interesting question and requires further investigation .
the analogy described above may turn out to be merely superficial , but given the current state of understanding about the matter , any interesting structural parallels are worthy of our consideration in the effort of reconstructing deeper patterns .
i am grateful to philip feinsilver for his encouraging interest and priceless comments .
j. kocik , a matrix theorem on circle configuration ( _ arxiv:0706.0372v2)_. y. koide , fermion - boson two body model of quarks and leptons and cabibbo mixing , _ lett .
nuovo cimento _ * 34 * ( 1982 ) 201 .
y. koide , _ mod .
_ a5 ( 1990 ) 2319
. y. koide , challenge to the mystery of the charged lepton mass formula ( 2005 ) , hep - ph/0506247 . _ _
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a remarkable formal similarity between koide s lepton mass formula and a generalized descartes circle formula is reported . +
* keywords : * lepton mass , koide formula , descartes circle theorem , geometry .
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we have used the hartree - fock random phase approximation ( hf - rpa ) to study the interacting electron gas in a quantum wire .
the spectra of intersubband spin - flip excitations reveal a considerable red shift with respect to single - particle hf energies . that signals on appearance of collective intersubband spind - density excitations due to the exchange interaction .
the long wavelength dispersions of the intrasubband collective spin - density excitations are linear , but the sound velocities are renormalised due to the exchange interaction and screening
. the in - phase intrasubband charge - density excitation has the long wavelength form @xmath0^{1/2}$ ] .
we found good qualitative agreement of our results with experimental observations .
a semiconductor quantum wire can be fabricated by applying a voltage with a microstructured gate to a 2d electron gas .
the single - particle energy ( spe ) spectrum typically consists of subbands separated by several @xmath1 . at a 1d electron density about @xmath2@xmath3
more than one subband can be occupied . in recent years progress has been made in spectroscopic study of such systems . in angular
resolved raman spectra of gaas quantum wires@xcite , collective spin - density excitations ( sde ) and charge - density excitations ( cde , or plasmons ) were observed .
the measured spectra cover from low to high frequency , and for low - frequency intrasubband excitations , the wave - vector dependence of the spin - wave energy was found to be linear .
the correct interpretation of these experiments allows us not only to understand the interesting physical processes , but also to access important physical parameters .
the exchange interaction is crucial to the collective intrasubband and intersubband sde .
similar to the direct long range coulomb interaction which leads to the depolarisation shift of single - particle excitations and to the appearance of collective plasma modes , exchange interaction gives rise to the red - shift of spe . if the red - shift is sufficiently large , the collective sde with a sizable oscillator strength splits off the continuum of spe . for semiconductor quantum wells
, such split - off appears in the hartree - fock random phase approximation ( hf - rpa)@xcite .
thus , it is important to perform a hf - rpa analysis on collective electron excitations in a realistic gaas quantum wire with full coulomb interaction , and compare the results with measured spectra@xcite .
the selfconsistent and conserving@xcite hf - rpa is suitable for this task , because we will calculate the two - particle spectra but not single - particle properties . as in using any approximation
, the hf - rpa calculation also contains error . however , our hf - rpa results agree very well with experimental measurements . in this letter we will first outline the hf - rpa method to express the spin or charge correlation functions in terms of the corresponding spin- or charge - density induced matrix , which satisfies an eigenvalue matrix equation . in a realistic quantum wire , for high energy intrasubband excitations and for excitation spectra when more than one subband is occupied , the self - consistent equations have to be solved numerically .
however , for the case that only one subband is occupied , the analytical expressions of sde and cde are derived . with additional approximation applied to the hf - rpa analytical solutions
, they reduce to known results .
this may be coincidental , but is outside the central theme of this letter . to define the quantum wire , let us start from an electron gas in a narrow quantum well with interfaces parallel to the @xmath4-@xmath5 plane , and the well width shorter than any other relevant length scales .
we consider the realistic experimental situation that only the lowest subband in the quantum well is occupied , so the motion of electrons in the @xmath6 direction can be ignored . by applying a properly designed gate potential ,
a quantum wire along the @xmath4 axis is fabricated with a longitudinal constant electrostatic potential , but a transverse parabolic one @xmath7/2 along the @xmath5 axis , where @xmath8 is the electronic effective mass .
the quantum wire has a finite length @xmath9 , and we assume periodic boundary conditions . in the absence of electron - electron interaction , the single particle eigenfunctions are simply @xmath10=@xmath11 , with the corresponding eigenenergies @xmath12=@xmath13+@xmath14+@xmath15 , where @xmath16 is the subband index , @xmath17=@xmath18 , and @xmath19 the harmonic oscillator wave function .
the transverse confinement length of the electrons is @xmath20=@xmath21^{1/2}$ ] .
when the electron - electron interaction is turned on , we will use the hf approximation @xmath22 to derive a complete orthonormal basis @xmath23 of quasi - particle hartree - fock states , where @xmath24 is the hartree - fock energy , @xmath25 is the fermi occupation factor and @xmath26 the dielectric constant of the surrounding medium . in terms of this basis set , we will use the corresponding time dependent hf ( hf - rpa ) to calculate the charge - density and the spin - density correlation functions , which describe the self - consistent linear response of the electron gas to an external perturbation . in this letter
, we restrict ourselves to the nonmagnetic ground state . for inelastic light scattering not close to the band gap resonance , the raman intensities in polarised and depolarised scattering geometries are proportional , respectively , to the imaginary parts of the charge - charge correlation function and spin - spin correlation function@xcite .
if we define @xmath27 as the fourier transform of the charge - density operator and @xmath28 the fourier transform of the spin - density operator along the spin quantisation axis , then , these correlation functions are @xmath29 | > $ ] with @xmath30=@xmath31 , where @xmath32 denotes a thermodynamic average . in hf - rpa , they can be expressed as @xmath33 in terms of the @xmath30-density induced matrix @xmath34 .
the charge - density induced matrix satisfies @xmath35 \ , , \label{indcharge}\end{aligned}\ ] ] where @xmath36=@xmath37 is the coulomb matrix element in the basis of the hf single - particle states .
when @xmath30=@xmath38 , there is no direct coulomb interaction ( hartree term ) between the excited spins .
if the exchange term ( fock term ) is neglected , the spin - density excitation spectra are simply given by the quasi - particle hartree - fock energies . in hf - rpa ,
the spin - density excitations are shifted from the quasi - particle hartree - fock energies due to the exchange terms . as a result
, collective spin - density excitation may appear in the spectra .
this shift is a measure of the effective strength of the exchange term in the system .
we will use the above hf - rpa analysis to investigate collective excitations in a multisubband system .
this has to be done numerically , and the first step is to derive the quasiparticle basis set @xmath23 by solving ( [ hfground ] ) self - consistently via iteration
. the functional basis must be sufficient large in order to ensure the required accuracy for the quasiparticle energy spectra and the spatial electron density of the ground state . when using this self - consistent quasiparticle basis set to calculate collective excitations , the required accuracy for the peak position of an excitation energy is the experimental linewidth @xmath390.1 @xmath1 .
the intrinsic materials parameters of a gaas quantum wire required in our calculation are @xmath8=@xmath40 and @xmath26=12.4 , where @xmath41 is the electron mass .
the extrinsic sample parameters are the total electron density @xmath16 , the transverse potential @xmath42 , and the length of the quantum wire @xmath9 .
our calculations are based on a sample with @xmath16=10.4@xmath43@xmath44 @xmath45 , which has been investigated experimentally in ref .
@xcite .
the transverse potential @xmath42 is determined by the experimental conditions . since the experimental sample does not have a parabolic potential as can be seen from the intersubband cde
, we will adjust @xmath46 to fit the subband electron occupations .
when we set @xmath46=7.9 @xmath1 ( @xmath20=120 ) , the derived self - consistent hartree - fock subband spacing between the two lowest subbands is 5.4 @xmath1 at the zone center and 5.6 @xmath1 at the fermi level .
the electron densities in the two occupied subbands are @xmath47=6.4@xmath43@xmath44 @xmath45 for the lowest subband ( labeled by @xmath48=0 ) , and @xmath49=4.0@xmath43@xmath44 @xmath45 for the second lowest subband ( labeled by @xmath48=1 ) .
the physical properties of the quantum wire are insensitive to its length , provided that the wire is sufficiently long .
therefore , we set @xmath9=1.0 @xmath50 m .
the experiment@xcite was done at a temperature 1.7 k , which is the temperature used in our calculation .
let @xmath51 and @xmath52 be , respectively , the fermi wave - vector and the fermi velocity of the @xmath48th subband . in the limit of long wavelength @xmath53@xmath540 , the transverse excitation is forbidden ( or allowed ) if the excitation energy is low ( or high ) , and so the corresponding physical processes are dominated by intrasubband ( or intersubband ) transitions .
when @xmath53 increases to @xmath55-@xmath56 , the low energy intersubband transitions are activated .
however , collective excitations with large energy and/or large @xmath53 are strongly damped . to illustrate the main features of long wavelength excitations ,
let us consider a very simply case by neglecting the electron self - energies and the vertex correction .
in this case we obtain analytical results for both the sde and cde energies .
the sde dispersion is linear in @xmath53 , with sound velocities @xmath57 and @xmath58 for intrasubband excitations in the two lowest subbands
. with two subbands occupied , the cde dispersion has two branches .
the in - phase mode is @xmath59 where @xmath60 is the fourier transform of the coulomb potential @xmath61-@xmath62+@xmath63-@xmath64^{-1/2}$ ] , and the out - of - phase mode is @xmath65 such two - band sde and cde dispersions are exactly the same results as schulz@xcite obtained for a two - subband tomonaga - luttinger model ( tlm)@xcite .
when we turn on the intrasubband and intersubband exchange interaction , as well as the screening , the sde and cde energies will be modified from the above expressions , and have to be derived numerically . in the polarised spectrum , the higher energy region
is dominated by strong intersubband cde . at @xmath53=0 ,
the calculated cde spectrum consists of a single peak which is characteristic to transverse parabolic confinement potentials@xcite . in the region of finite @xmath53
the intersubband hf spe energies show up at 5.5 @xmath1 . in the spin - flip
depolarised raman spectrum , we obtained the red shift of intersubband sde with respect to hf spe .
the sde has a dominant weight of the spectrum and appears at a resonance energy of 2.1 @xmath1 , which is shifted from the hf spe , indicating the importance of vertex correction .
these overall features of our numerical results have been observed experimentally@xcite . in fig .
[ cdesde ] we show the calculated intrasubband sde and cde in the region of energy and frequency for which accurate experimental data are available .
the mode with strong intensity is labeled as sde0 , and the next mode with weaker intensity is labeled as sde1 .
the inset gives their dispersions , which are linear for small @xmath53 , with the corresponding sound velocities @xmath66=0.9@xmath57 and @xmath67=0.7@xmath58 .
these sound velocities are smaller than the respective fermi velocities because of the exchange screening of sde induced by the intersubband virtual transitions , and the renormalisation due to the intra- and intersubband exchange interaction .
the experiments@xcite seem to have detected only @xmath66 , which corresponds to the high energy sde near the fermi wave vector @xmath55 of the lowest subband .
it is probably the low electron density in the second lowest suband that causes the velocity @xmath67 of the sde around @xmath56 to be too small to be observed experimentally .
the spin velocity @xmath66 agrees quantitatively with the experiment@xcite . for sufficiently large @xmath53 , because of the finite length of our quantum wire , the landau damping shows up in fig .
[ cdesde ] in the form of enhanced intensities of satellite single - particle peaks around the sde , instead of a broad band if the quantum wire is infinitely long .
the mode sde1 is not damped , since it does not enter the region of spes .
the dispersions of the cde and sde are also plotted in the inset in fig .
[ cdesde ] . in the long wavelength limit
, the in - phase cde+ can be well fitted with @xmath0^{-1/2}$ ] according to ( [ 2charge+ ] ) , as expected for 1d electron gas@xcite .
the out - of - phase cde- mode , corresponding to ( [ 2charge- ] ) with an expected linear dispersion@xcite , appears in our calculation as a weak band with a sound velocity @xmath68 .
this happens to be the same as the sound velocity of the sde0 , an accidental result for this specific sample .
the cde- mode has a much lower intensity compared to the cde+ mode .
checking against the experiment@xcite , we believe the the lower energy band observed in the low frequency polarised raman spectrum is our calculated cde- mode . the decay of the plasmon ( cde+ ) and the cde- at higher @xmath53 due to the landau damping is stronger than the decay of sde0 and sde1 , as shown in fig .
[ cdesde ] . after the above complete study of a realistic quantum wire , perhaps it is worthwhile to mention some surprising findings for the special case that only the lowest subband occupied at zero temperature , namely , the pure 1d system including spin degrees of freedom . in this case we can drop the band index . for sde , the eigenvalue equation has the form @xmath69 where @xmath70=@xmath71 for the quasi-1d coulomb potential @xmath61-@xmath62+@xmath72^{-1/2}$ ] , which has a logarithmic singularity @xmath73 as @xmath53@xmath540 . since the hartree self - energy is canceled by the potential energy due to the positive background charges , the difference in exchange self - energy at the fermi energy , @xmath74=@xmath75+@xmath76-@xmath75-@xmath76 is @xmath77 $ ] .
we define @xmath78 by @xmath79@xmath80@xmath81-@xmath82 , and at @xmath17=@xmath55 rewrite ( [ sdeone ] ) as @xmath83 \nonumber \\ & & = 2\pi\left [ \hbar\omega -\hbar qv_0 - \delta_x(q)\right ] y(k_0 ) \ , .
\label{add1}\end{aligned}\ ] ] in the limit of long wavelength , by substituting @xmath84@xmath39@xmath85 and @xmath86@xmath39@xmath87 into the above equation , we see that the contributions of coulomb potential in the exchange energy and the vertex corrections , both behave as @xmath88 , exactly cancelling each other .
hence , the @xmath88 behaviour of low - energy single - particle excitation due to the exchange self - energy is removed from the sde and the plasmon energies , and so only the exchange terms at momentum @xmath89 are important . an equation similar to ( [ add1 ] ) for @xmath90 at @xmath91 can also be derived . in the long wavelength limit
, we find the sde energy @xmath92 where @xmath93=@xmath94 .
the sound velocity is reduced with respect to the bare fermi velocity .
we have also solved for the cde and found the plasmon dispersion @xmath95 for a long - range coulomb potential @xmath60@xmath96@xmath97 , the plasmon energy is not affected by the exchange interaction within the hf - rpa .
these results ( [ 1spin ] ) and ( [ 1charge ] ) , derived with full coulomb interaction , differ from those obtained by schulz@xcite for the tlm including a nonsingular backscattering matrix element at @xmath89 , which mixes the right- and the left - traveling modes . nevertheless , in ( [ sdeone ] ) , if we replace the hartree - fock energy by the bare single - particle energy , and omit the singular part around v(q=0 ) in the vertex correction , then our results exactly reduce to schulz s results for the sde and the plasmon energies .
whether this finding , as well as ( [ 2charge+ ] ) and ( [ 2charge- ] ) , is simply a coincidence remains to be clarified . to close this letter , we should mention that according to our analysis , the frequency of the lowest energy sde decreases as @xmath53 increases towards @xmath56 , and then vanishes at a value of @xmath53 close to @xmath98 .
this behaviour suggests a low temperature intrinsic instability of the electron gas against the formation of spin - density waves ( sdw ) , namely , the peierls instability . when this instability emerges from our mean - field analysis
, we must take into account the sdw long - range order in the ground state .
on the other hand , the results of the tlm with backscattering predict the absence of sdw long - range order , but instead the appearance of a slowly decaying wigner crystal at @xmath99@xcite .
we guess that the tlm with backscattering describes the real ground state better than hf - rpa .
hence , we believe that our analysis based on a ground state without sdw long range order is a better approximation than that including sdw in the ground state .
a. b. would like to thank a. sudb and l. j. sham for stimulating discussions .
this research was supported in part by a norfa grant .
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the authors thank professors i. a. sanda and a. soni for useful discussions .
m. a. acknowledges support from the science and technology agency of japan .
e. k. acknowledges support from the japanese society for the promotion of science .
3.0 cm * figure 1 * : non - spectator contribution to @xmath99 decays .
+ 0.5 cm * figure 2 * : the current - current ( tree ) contribution to @xmath42 decay .
+ 0.5 cm * figure 3 * : the triangle quark loop diagram for gluon - gluon-@xmath4 vertex .
+ 0.5 cm * figure 4 * : branching ratio versus cp asymmetry as @xmath63 varies from @xmath100 to @xmath101 for three sets of @xmath102 values .
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we calculate the branching ratio and direct cp asymmetry in nonleptonic two body @xmath0 decays @xmath1 . it is shown that the tree diagram and gluon fusion mechanism via penguin diagram have comparable contributions to these decays which , as a result , could provide an interesting venue for investigating cp violation .
our estimate shows that the direct cp asymmetry in the above decays could be as large as @xmath2 which along with a branching ratio @xmath3 should be accessible to experiment in the near future .
the discovery by the cleo collaboration of a larger than expected branching ratio for fast @xmath4 production in hadronic @xmath0 decays@xcite has led to extensive theoretical work on investigating the underlying mechanism .
one explanation which is based on the gluon - gluon-@xmath4 anomalous coupling , has been proposed by atwood and soni@xcite . in this mechanism , @xmath4 is produced by the fragmentation of the virtual gluon of the qcd @xmath5 penguin .
on the other hand , in order to formulate the inclusive @xmath6 and exclusive @xmath7 decays under the same mechanism , we proposed a nonspectator gluon fusion process using anomaly driven @xmath8 vertex@xcite .
the exclusive branching ratio @xmath9 obtained this way is in good agreement with experimantal data . in this paper , we focus on a different hadronic decay mode of @xmath0 mesons which may receive a significant contribution from the above mechanism .
indeed , if the anomalous @xmath8 vertex in conjunction with the qcd penguin is the underlying process for the fast @xmath4 production in @xmath0 meson decays , one could expect that the same mechanism also be an important part of the two body @xmath1 decay modes . the difference between these decay modes and @xmath7
decay is in that , unlike the latter one , the former decays receive a comparable contribution from the tree diagrams . as a result
, one could expect that the above decay modes provide a suitable avenue to investigate the direct cp asymmetry in charged @xmath0 meson decays . in this work
, we estimate the branching ratio and direct cp asymmetry in @xmath1 decays and point out a possible large cp asymmetry which should be accessible to experiment in the near future .
we start by repeating the derivation of the nonspectator contribution to @xmath7 matrix element from reference @xcite . using the dominant chromo - electric component of the qcd penguin , we obtain the following effective hamiltonian corresponding to figure 1 .
@xmath10 where @xmath11\;\ ; .\ ] ] the coefficient function @xmath12 is defined as @xmath13 where @xmath14 with @xmath15 being the internal quark mass@xcite .
@xmath16 is the form factor parametrizing the @xmath8 vertex @xmath17 using the decay mode @xmath18 , @xmath19 is estimated to be approximately 1.8 gev@xmath20@xcite . a re - arrangement of eq . ( 1 ) via fierz transformation @xmath21\;\ ; , \end{aligned}\ ] ] and using the definition of the decay constants for the @xmath0 and @xmath22 mesons in the context of factorization
, we can express the matrix element for @xmath7 decay as the following : @xmath23 leading to the exclusive decay rate @xmath24 where @xmath25 and @xmath26 are the three momentum of the k meson @xmath27}^{\frac{1}{2}}\;\ ; , \ ] ] and the energy transfer by the gluon emitted from the light quark in the b meson rest frame , respectively . inserting @xmath28 gev@xmath29 , @xmath30 gev and @xmath31 gev in eq .
( 7 ) results in the following exclusive branching ratio : @xmath32 which is in good agreement with the experimental data@xcite . on the other hand
, we observe that if @xmath8 anomalous coupling along with qcd @xmath5 penguin ( fig .
1 ) is indeed the dominant underlying mechanism for @xmath33 decay , then one could expect that a similar process , with @xmath34 quark replacing the strange quark , to contribute significantly to @xmath35 decay mode .
however , in this case , unlike @xmath7 decay , the tree and penguin amplitudes receive a ckm suppression factor of the same order and therefore there is a good possibility for existence of a large direct cp asymmetry in @xmath1 decay modes . the nonspectator contribution to @xmath36
can be obtained from eq .
( 6 ) by replacing @xmath37 with @xmath38 and @xmath39 with @xmath40 , where @xmath41+v_{ub}v_{ud}^*[e(x_u)-e(x_c)]\right ) \;\ ; .\ ] ] thus , including the expression for the tree amplitude ( fig .
2 ) which are derived by using the factorization assumption , the total matrix element for @xmath42 can be written as : @xmath43 @xmath44 and @xmath45 are the combinations of the wilson coefficients of the current - current operators and @xmath46 and @xmath47 , are defined as follows : @xmath48 @xmath49f_1^{b\to p}(q^2 ) \\ & + & \frac{m_b^2-m_p^2}{q^2}q^\mu f_0^{b\to p}(q^2 ) \;\ ; , \end{aligned}\ ] ] where @xmath50 .
we estimate @xmath51 gev by using the two - angle formalism for the @xmath52 mixing@xcite . as for the form factors , we use the numerical results @xmath53 and @xmath54 obtained in the bsw model@xcite . using the wolfenstein parametrization@xcite and the unitarity triangle convention for the phases of the quark mixing matrix elements@xcite @xmath55 where @xmath56 and @xmath57 are well determined@xcite .
the parameters @xmath58 and @xmath59 are correlated and their range have been determined from ckm unitarity fits@xcite . in order to show the dependence of the branching ratio and cp asymmetry on these parameters , we present our results for central values of @xmath60 and @xmath61 and three different sets of @xmath62 values from the constrained range . we note that different weak ( cp odd ) phases enter nonspectator ( penguin ) and tree amplitudes . also , an overall strong ( cp even ) phase @xmath63 for the nonspectator amplitude
is assumed which in combination with the weak phases leads to cp asymmerty in @xmath1 decays .
in fact , one possible source of this strong phase could be the form factor which parametrizes the @xmath8 vertex .
for example , in the sigma model formulation of @xmath64 where this form factor is derived from a quark triangle loop ( fig .
3 ) , an absorptive part is generated for @xmath65 ( @xmath66 is the mass of the quarks participating in the triangle loop i.e. , @xmath67 in the @xmath4 case)@xcite .
for @xmath68gev@xmath29 the magnitude of this strong phase can be around @xmath69 to @xmath70 depending on the input value for @xmath66 .
inserting @xmath71 ( @xmath72gev@xmath20 as mentioned before ) in eq . ( 11 ) and using the same input values for various parameters as in @xmath7 case , we obtain the following branching ratio for @xmath42 @xmath73\;\ ; .\end{aligned}\ ] ] the first term in eq .
( 15 ) is due to current - current operators ( tree ) while the second term is the contribution of the nonspectator gluon fusion process ( penguin ) and the last entry is the cross term .
the branching ratio for the cp conjugate process @xmath74 is obtained from the eq .
( 15 ) by changing the sign of the weak angles @xmath75 and @xmath76 .
consequently , the cp asymmetry which is defined as @xmath77 is proportional to @xmath78 and @xmath79 . in figure 4
, we illustrate the variation of @xmath80 and @xmath81 when @xmath63 takes on values in the range @xmath82 for three different sets of @xmath58 and @xmath59 .
it is quite interesting that a large asymmetry is possible due to the comparable nonspectator and tree contributions to this process .
for example , for @xmath83 and @xmath84 ( i.e. @xmath85 ) , @xmath86 could be as large as @xmath2 and @xmath3 if @xmath87 .
a larger branching ratio @xmath88 is possible for @xmath89 and @xmath90 ( i.e. @xmath91 ) however , the asymmetry is somewhat smaller around @xmath92 . on the other hand , for the preferred values @xmath93 and @xmath94 ( i.e. @xmath95 ) , the resulting branching ratio and asymmetry are @xmath96 for @xmath97 , which is the case if we assume that the absorptive part of the gluon - gluon-@xmath4 form factor @xmath16 is responsible for the strong phase . in conclusion
, we emphasize that a comparison between our results and other theoretical estimates@xcite indicates that @xmath98 and @xmath81 could receive a significant enhancement due to the inclusion of the nonspectator gluon fusion mechanism .
these decay modes could be accessible to experiment in the near future and therefore , provide a clean testing ground for cp violation mechanism in the standard model .
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40 j. bielcikova ( star collaboration ) , j. phys . *
g74 * , s929 ( 2007 ) ; l. molnar ( star collaboration ) , j. phys . *
g74 * , s592 ( 2007 ) ; b. abelev ( star collaboration ) , arxiv:0705.3371 ; c. nattrass ( star collaboration ) , proceedings of quark matter conference , jaipur , india , feb . 4 - 10 , 2008 .
r. c. hwa , arxiv:0708.1508 ; p. romatschke , phys .
rev . c*75 * 014901 ( 2007 ) ; s. a. voloshin , nucl .
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c*76 * , 047901 ( 2007 ) ; v. s. pantuev , arxiv:0710.1882 ; a. dumitru , y. nara , b. schenke , and m. strickland , arxiv:0710.1223 ; a. dumitru , f. gelis , l. mclerran , and r. venugopalen , arxiv:0804.3858 ; s. gavin , l. mclerran , and g. moschelli , arxiv:0806.4718 .
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in a central nucleus - nucleus collision at high - energies , medium partons kicked by a near - side jet acquire a momentum along the jet direction and subsequently materialize as the observed ridge particles .
they carry direct information on the early parton momentum distribution which can be extracted by using the ridge data for central auau collisions at @xmath0 gev .
the extracted parton momentum distribution has a thermal - like transverse momentum distribution but a non - gaussian , relatively flat rapidity distribution at mid - rapidity with sharp kinematic boundaries at large rapidities that depend on the transverse momentum . in central high - energy heavy - ion collisions ,
the state of the parton medium during the early stage of a nucleus - nucleus collision is an important physical quantity .
it furnishes information for the investigation of the mechanism of parton production in the early stages of the collision of two heavy nuclei .
it also provides the initial data for the evolution of the system toward the state of quark - gluon plasma .
not much is know about the early state of the partons from direct experimental measurements .
recently , the star collaboration observed a @xmath1-@xmath2 correlation of particles associated with a near - side jet in central auau collisions at @xmath0 gev at rhic , where @xmath1 and @xmath3 are the azimuthal angle and pseudorapidity differences relative to a high-@xmath4 trigger particle @xcite .
the near - side correlations can be decomposed into a `` jet '' component with fragmentation and radiation products at @xmath5@xmath6(0,0 ) , and a `` ridge '' component at @xmath7@xmath80 with a ridge structure in @xmath3 .
while many theoretical models have been proposed @xcite , a momentum kick model was presented to describe the ridge phenomenon @xcite .
the model assumes that a near - side jet occurs near the surface and it kicks medium partons , loses energy along its way , and fragments into the trigger particle and other fragmentation products in the `` jet '' component .
the kicked medium partons , each of which acquires a momentum kick @xmath9 from the near - side jet , materialize by parton - hadron duality as ridge particles .
the ridge particle momentum distribution is related to the initial momentum distribution by a momentum shift .
we can therefore extract the initial parton momentum distribution from the ridge data for central auau collisions at @xmath0 gev obtained by the star collaboration @xcite .
we parametrize the normalized initial parton momentum distribution as @xmath10 where @xmath11=@xmath12 , @xmath13 is a normalization constant defined by @xmath14 , @xmath15 is the rapidity fall - off parameter , @xmath16 is beam parton rapidity , @xmath17 is the beam parton mass , @xmath18 = @xmath19 , and @xmath20 is a mass parameter introduced to give a better description of low-@xmath4 ridge data . for lack of a definitive determination ,
we set @xmath16 equal to the nucleon rapidity @xmath21 and @xmath17 equal to @xmath18 , pending future definitive measurements of the ridge boundary .
the observed ridge distribution is the final parton momentum distribution after jet - parton collisions , scaled by the average number of charged kicked partons per trigger , @xmath22 , that may be modified by a ridge attenuation factor @xmath23 .
[ h ] [ h ] = -3 = -9.0 cm * fig . 1 * transverse momentum distribution of associated particles in @xmath24 and central auau collisions .
= -3 = 7.6 cm * fig . 2 * azimuthal angular and pseudorapidity distributions in @xmath24 and central auau collisions . to infer the associated ridge particle yield from experimental data , we need to know the jet component in central auau collisions .
the scaling relation between the fragments in the jet component and the trigger particle @xcite allows us to treat the auau jet component per trigger as a @xmath24 near - side jet distribution , attenuated by a semi - empirical attenuation factor @xmath25 @xcite . the experimental @xmath24 near - side jet data , shown as open circles in figs . 1 and 2 , can be described by the dash - dot curves in these figures obtained from @xmath26}/{2\sigma_\phi^2 } } \ } } { 2\pi\sigma_\phi^2},\end{aligned}\ ] ] where @xmath27=@xmath28 , @xmath29=0.5 , @xmath30=1.1 gev , @xmath31=0.75 , and @xmath32=0.55 gev .
theoretical evaluation of both the jet component and the ridge component for central auau collisions leads to the total yield of associated particles . a self - consistent comparison of the momentum kick model results with experimental data in figs . 1 , 2 , and 3 then allows us to search for the initial parton momentum distribution .
we find that the totality of the star associated particle data @xcite from @xmath33 gev to 4 gev and @xmath34 from 0 up to 3.9 in central auau collisions at @xmath0 gev , can be described by the average momentum kick per jet - parton collision @xmath35 gev , the average effective numbers of kicked partons per jet @xmath36 , and the normalized initial parton momentum distribution eq .
( [ dis2 ] ) with @xmath37 fig .
1 shows good agreement between theoretical @xmath38 results with experimental charged hadron yields .
note that the theoretical ridge @xmath39 ( the dashed curve ) has a peak at @xmath40 gev , as a result of the momentum kick .
in fig . 2 , comparison of theoretical and experimental associated particle data indicates general agreement over azimuthal angles [ figs .
2(a ) and 2(b ) ] and over pseudorapidities [ figs . 2(c ) and 2(d ) ] , for both @xmath41 gev [ figs .
2(a ) and 2(c ) ] and @xmath42 gev [ figs .
2(b ) and 2(d ) ] .
the forward rapidity data in fig .
3 have large uncertainties and the value of @xmath43 gives reasonable agreement with experiment .
[ h ] [ h ] = -3 = -9.0 cm * fig .
3 * azimuthal distributions of associated particles in forward rapidities for central auau collisions .
= -3 = 7.4 cm * fig .
4 * theoretical predictions of associated particle pseudorapidity distribution within the acceptance region of the phobos detector . using the parameters we have extracted from the star ridge data for central auau collisions at @xmath44 gev
, we can predict the pseudorapidity distribution of associated particles for the phobos experimental acceptance defined by @xmath451 , [email protected] , [email protected] gev .
the total associated particle yield is shown as the solid curve in fig .
4 .
the @xmath24 jet yield and the associated ridge particle yield are shown as the dash - dot and the dashed curves respectively .
the result has been corrected for @xmath3 acceptance .
the present prediction up to large @xmath48 was found to agree well with experimental measurements obtained by the phobos collaboration @xcite .
= 9 = 3.2 in the distribution ( [ dis2 ] ) with parameters of eq .
( [ bestset ] ) gives the normalized initial parton momentum distribution at the moment of jet - parton collision .
we show this distribution in figs .
5 .
it can not be separated as the product of two independent distributions of @xmath4 and @xmath49 . in fig .
5(a ) , the momentum distribution for @xmath50 and high @xmath4 has a slope parameter @xmath51 that is intermediate between that of the jet and the inclusive particles .
this indicates that partons at the moment of jet - parton collision are at an intermediate stage of dynamical equilibration .
the distribution as a function of @xmath4 does not change much as @xmath49 increases from 0 to 2 . for @xmath52 ,
the maximum value of @xmath4 is 1.54 gev and the distribution changes significantly as the kinematic boundary is approached . for @xmath53 ,
the boundary of @xmath4 is located at 0.55 gev .
[ h ] = -3 = -8.cm * fig . 5 * initial parton momentum distribution at the moment of jet - parton collisions .
= -11 = 3.2 in in fig .
5(b ) , the momentum distribution as a function of @xmath49 for a fixed @xmath4 is essentially flat at mid - rapidity and it extends to a maximum value of @xmath54 that depends on @xmath4 .
the distribution decreases rapidly as it approaches the kinematic limit and covers a smaller allowed region of @xmath49 as @xmath4 increases .
the flat rapidity distribution at mid - rapidity for a fixed @xmath4 gives rise to the ridge structure that is observed experimentally .
= -2 = 3.2 in in conclusion , the application of the momentum kick model for ridge particles associated with a near - side jet in central auau collisions at @xmath0 gev allows us to extract unique and valuable information on the medium parton momentum distribution at the moment of jet - parton collision . in the process
, we also find the average momentum kick per jet - parton collision @xmath55 , and the average effective numbers of kicked medium partons per near - side jet @xmath56 , in these collisions .
they provide important empirical data for future investigations on the dynamics of parton production , parton evolution , and jet momentum loss .
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the authors acknowledge some financial support from the spanish project dgicyt - fis2009 - 13364-c02 - 01 .
also thanks to the consejera de economa , comercio e innovacin of the junta de extremadura ( spain ) for financial support , project ref . gru09011 .
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in this work , a one - dimensional model of crystalline solids based on the dirac comb limit of the krnig - penney model is considered . from the wave functions of the valence electrons , we calculate a statistical measure of complexity and the fisher - shannon information for the lower energy electronic bands appearing in the system .
all these magnitudes present an extremal value for the case of solids having half - filled bands , a configuration where in general a high conductivity is attained in real solids , such as it happens with the monovalent metals .
and crystalline solid models ; electronic band structure ; statistical indicators the application of information theory measures to quantum systems is a subject of interest in the last years @xcite
. some relevant properties of the hierarchical organization of atoms @xcite and nuclei @xcite are revealed when these indicators are calculated on these many - body systems . on the one hand
, they display an increasing trend with the number of particles , electrons or nucleons . on the other hand
, they take extremal values on the closure of shells . moreover , in the case of nuclei , the trace of magic numbers is displayed by these entropic products such as the fisher - shannon information @xcite and a statistical complexity measure @xcite .
also these statistical quantifiers have revealed a connection with physical measures , such as the ionization potential and the static dipole polarizability @xcite in atomic physics .
all of them , theoretical and physical magnitudes , are capable of unveiling the shell structure of atoms , specifically the closure of shells in the noble gases .
a strategy to calculate these indicators is to quantify the discrete hierarchical organization of these multiparticle systems through the fractional occupation probabilities .
these probabilities capture the filling of the shell structure of these systems . from them ,
the different statistical magnitudes are derived .
the metallic clusters is another system that has also been studied with this method @xcite . as in the case of atoms and nuclei ,
the shell structure of the valence electrons is well displayed by the spiky behavior of the statistical complexity and the magic numbers are unveiled by relevant peaks of the fisher - shannon information . a different strategy to compute these entropic magnitudes is to use the probability density of the quantum system as the basic ingredient .
this can be analytically obtained in some cases such as the h - atom @xcite or numerically derived in other cases from a hartree - fock scheme @xcite or a density functional - theory for atoms and molecules @xcite . in this work
, we address the problem to calculate these statistical indicators in a solid by this last strategy . for this purpose ,
the band structure of the solid has to be determined .
the krnig - penney ( kp ) model @xcite is a one - dimensional model of crystalline solids that presents a band structure sharing many properties with band structures of more sophisticated models .
moreover , it also has the advantage that allows to analytically find such electronic band structure .
the kp model considers that electrons move in an infinite one - dimensional crystal where the positive ions are located at positions @xmath0 with @xmath1 , generating a periodic potential of period @xmath2 .
a simplified version of the kp model is obtained when this periodic potential is taken with the form of the dirac comb @xcite : @xmath3 where @xmath4 is the planck constant , @xmath5 is the electronic mass and @xmath6 is the intensity of the potential . in this case , the spatial part @xmath7 of the electronic wave function is determined from the time independent schrdinger equation : @xmath8\psi(x ) = e\psi(x ) , \label{eqschro}\ ] ] where the eigenvalue @xmath9 is the energy of the eigenstate @xmath7 . for a periodic potential ,
the bloch s theorem @xcite establishes the form of the general solution of the eq .
( [ eqschro ] ) .
this is a plane wave , with wave number @xmath10 , modulated by a periodic function , @xmath11 : @xmath12 where @xmath11 has the periodicity of the crystal lattice : @xmath13 .
it implies the following translation property , @xmath14 let us observe that we have the case of free electrons when @xmath15 and the solutions of type ( [ eq - u_k ] ) recover the plane wave form with a total wave number @xmath16 , with @xmath17 .
it suggests that the solution of eq .
( [ eqschro ] ) in the region @xmath18 , where @xmath15 , can be associated in some way with a wave number @xmath19 and then written in the general form : @xmath20 and , by the translation property ( [ eq - trans ] ) , this solution in the region @xmath21 is @xmath22 , \label{eq - psi2}\ ] ] with @xmath23 , @xmath24 complex constants and @xmath25 .
two boundary conditions must be fulfilled by @xmath26 at the point @xmath27 : on the one hand , the continuity of the wave function and , on the other hand , the jump in the derivative provoked by the delta function ( [ eq - dirac ] ) .
this gives the relations : @xmath28 from these boundary conditions applied to the wave functions ( [ eq - psi1]-[eq - psi2 ] ) , the following homogeneous linear system is obtained for the unknonws @xmath23 and @xmath24 : @xmath29 to have a non - trivial solution , the determinant of this @xmath30 matrix has to be zero .
then , the following quantization relation for @xmath19 is obtained @xcite : @xmath31 the electronic band structure of the one - dimensional crystal is contained in this equation . when @xmath10 varies its value in the different brillouin zones , given by @xmath32 , with @xmath33 for the first brillouin zone , @xmath34 for the second brillouin zone , etc .
, only certain intervals of @xmath19 are allowed .
these intervals for @xmath19 are the energy bands of the electronic system .
the positive and negative parts of these intervals correspond with the positive and negative parts of the brillouin zones , respectively . in the limit @xmath35 ,
the free electron problem is recovered , then the solutions are the plane waves with @xmath16 . in the limit @xmath36
, we have the square well problem , then the wave number of the eigenstates verify @xmath37 . for an intermediate case , @xmath38 , eq . ( [ eq - band ] )
has to be solved .
concretely , for the particular value @xmath39 , that has also been used in ref .
@xcite , the lower energy bands obtained in this system for @xmath40 are : @xmath41 the bands are symmetrically found for @xmath42 .
observe that , to finally get the wave function of the electronic states , we additionally need the normalization condition to completely determine @xmath7 , except a global phase factor . for our calculations , by taking @xmath39 , we will perform this normalization in the unit cell @xmath43 $ ] .
the basic ingredient to calculate the statistical entropic measures in which we are interested is the probability density of the electronic states .
this is given by @xmath44 . from this density , we proceed to compute the statistical complexity and the fisher - shannon information . notice that the wave function @xmath7 for a given @xmath19 is transformed in @xmath45 for @xmath46 , therefore all the magnitudes depending on the density are the same in both cases , and then we reduce our study to the positive part ( @xmath40 ) of the electronic bands .
let us recall the definition of the statistical complexity @xmath47 @xcite , the so - called @xmath48 complexity , that is defined as @xmath49 where @xmath50 represents the information content of the system and @xmath51 gives an idea of how much sharp is its spatial distribution . as a quantifier of @xmath50
, we take a version used in ref .
@xcite .
this is the simple exponential shannon entropy @xcite , that has the form , @xmath52 where @xmath53 is the shannon information entropy @xcite , @xmath54 for the disequilibrium we take the form originally introduced in refs .
@xcite , that is , @xmath55 , vs. the adimensional wave number , @xmath56 , for @xmath40 .
only the four lower electronic bands given in expression ( [ eq - bandas ] ) are shown.,width=453 ] the entropy , @xmath53 , and the statistical complexity , @xmath47 , for the lower electronic bands of the present one - dimensional crystalline solid are given in fig .
[ fig1 ] and fig .
[ fig2 ] , respectively .
when this hypothetical solid is in a situation of high conductivity , i.e. when it contains a band that is partially filled and partially empty , it can be observed in the figures that the more energetic electrons attain the highest entropy and the lowest complexity in the vicinity of the half - filled band .
this is the point where in general the highest conductivity is also attained .
take , for instance , the real case of the monovalent metals , that include the alkali metals ( li , na , k , rb , cs ) and the noble metals ( cu , ag , au ) .
these metals present all the bands completely filled or empty , except an only half - filled conduction band @xcite .
compared with other solids , these metals display a very high conductivity , that in the cases of ag and cu it is the highest in nature .
then , it is remarkable this coincidence at the point of half - filled band where , on the one hand , the entropy and the statistical complexity are extrema for this model of solids and , on the other hand , the conductivity reaches its upper values for the real cases of monovalent metals . , vs. the adimensional wave number , @xmath56 , for @xmath40 .
only the four lower electronic bands given in expression ( [ eq - bandas ] ) are shown.,width=453 ] now , we check that other statistical entropic measures also display this behavior when the solid has half - filled bands .
let us take , for instance , the fisher - shannon information , @xmath57 , that has been applied in different contexts @xcite for atomic systems .
this quantity is given by @xmath58 where the first factor @xmath59 is a version of the exponential shannon entropy @xcite , and the second factor @xmath60 ^ 2\over \rho(x)}\ ; dx\;,\ ] ] is the so - called fisher information measure @xcite , that quantifies the stiffness of the probability density .
observe in fig .
[ fig3 ] the confirmation of the previous results obtained in figs .
[ fig1 ] and [ fig2 ] for @xmath53 and @xmath47 , in the sense that the extremal values of @xmath57 for this model of solids are also reached at the half - filling band points .
, vs. the adimensional wave number , @xmath56 , for @xmath40 .
only the four lower electronic bands given in expression ( [ eq - bandas ] ) are shown.,width=453 ] the former calculations are done orbital by orbital , i.e. thinking that the solid is a set of individual and independent orbitals , each one identified by its own wave number @xmath19 .
we can change the point of view of the problem and to think that the solid stands in some kind of collective state whose probability density @xmath61 is the normalized sum of all the allowed electronic densities obtained from the orbitals with wave numbers in the interval @xmath62 $ ] ; @xmath63 will be the minimal electronic wave number of the solid , i.e. the lowest @xmath19 obtained in the first band , and @xmath64 will be the upper @xmath19 corresponding to the most energetic electron of the solid .
the expression for @xmath61 is @xmath65 observe that @xmath61 is normalized in the interval @xmath43 $ ] , @xmath66 , and that in the present model of solid @xmath67 as given in formulas ( [ eq - bandas ] ) .
the calculation of the statistical complexity @xmath68 for this @xmath61 is presented in fig .
[ fig4 ] . in this case , the minimal values of @xmath68 are also located in the vicinity of the half - filled electronic bands such as the behavior of @xmath47 for the individual orbitals shown in fig .
[ fig2 ] . in the hypothetical limit case of a solid where @xmath69 ,
let us remark that the density @xmath61 will tend to the uniform density and then the lowest value of complexity , @xmath70 , can be reached , as it can be seen in fig .
[ fig4 ] . , vs. the adimensional wave number , @xmath56 , for @xmath40 .
only the four lower electronic bands given in expression ( [ eq - bandas ] ) are shown.,width=453 ] in summary , this work puts in evidence that certain conformational properties of many - body systems are reflected by the behavior of the statistical complexity @xmath47 and the fisher - shannon information @xmath57 . in the present study , the electronic band structure of a model of solids has been unfolded and the measurement of these magnitudes for such a model has been achieved .
it is remarkable the fact that the extremal values of @xmath47 and @xmath57 are attained on the configurations with half - filled bands , which is also the electronic band configuration displayed by the solids with the highest conductivity , let us say the monovalent metals .
therefore , the calculation of these statistical indicators for a model of solids has unveiled certain physical properties of these systems , in the same way that these entropic measures also reveal some conformational aspects of other quantum many - body systems .
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* characteristic oscillations in the photon number distribution of dynamically displaced number states for @xmath68 , @xmath108 and @xmath109 . in ( a ) and ( b ) the quantum limit , eq . ( [ 8 ] ) , is displayed for different values of @xmath57 . in ( c ) and
( d ) the semi - classical limit , eq .
( [ 9 ] ) , is shown for different values of @xmath110 .
the full line shows the poisson distribution with @xmath70 . *
coherent destruction of tunneling monitored in the cavity field .
displayed is eq .
( [ 9 ] ) for @xmath111 , @xmath108 as a function of @xmath72 : upper figure @xmath112 , lower figure @xmath113 . the first root of @xmath91 occurs at @xmath114 .
the effect is exhibited by a decrease of the amplitude modulation in the cavity mode oscillations .
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* abstract . * the mechanism of coherent destruction of tunneling found by grossmann _
et al . _ [ phys .
rev . lett . * 67 * , 516 ( 1991 ) ] is studied from the viewpoint of quantum optics by considering the photon statistics of a single mode cavity field which is strongly coupled to a two - level tunneling system ( ts ) . as a function of the interaction time between ts and cavity the photon statistics displays the tunneling dynamics . in the semi - classical limit of high photon occupation number @xmath0 , coherent destruction of tunneling
is exhibited in a slowing down of an amplitude modulation for certain parameter ratios of the field .
the phenomenon is explained as arising from interference between displaced number states in phase space which survives the large @xmath0 limit due to identical @xmath1 scaling between orbit width and displacement .
pacs number(s ) : 42.50.ct , 42.50.lc , 05.30.-d 2 in recent years the idea of modulation and therefore control of tunneling by a monochromatic electromagnetic field has been subject of considerable interest .
the typical hamiltonian describes a particle in an isolated double - well potential ( dwp ) which is periodically driven by an external force @xmath2 here @xmath3 is the hamiltonian of the dwp , @xmath4 is the amplitude and @xmath5 the driving frequency .
the attention has mainly focused on a possible enhancement or suppression of coherent tunneling : lin and ballentine@xcite have demonstrated that the tunneling probability is highly enhanced due to periodic modulation for high field strengths and driving frequencies close to the classical oscillation frequency at the bottom of each well . in the opposite limit grossmann , hnggi and coworkers@xcite found complete suppression of tunneling such that a particle initially localized in one of the two wells will never escape to the other well .
they termed this effect `` coherent destruction of tunneling '' .
it has since been of continuing interest.@xcite the most surprising feature is the _ periodicity _ of the destruction of tunneling for certain parameter ratios of @xmath4 and @xmath5 . so far there is no clear understanding of this phenomenon .
by using the floquet formalism it has been shown that many characteristic features of the tunneling suppression can already be described in a two - level approximation of the dwp.@xcite many different aspects of the effect have been illuminated in this framework : makarov@xcite , plata and gomez llorente@xcite quantized the electromagnetic field and recovered the effect in the limit of large number of photons in the field .
wang and shao@xcite mapped the driven two - level dynamics to a classical one of a charged particle moving in a harmonic potential plus a magnetic field in a plane .
kayanuma@xcite explained the suppression of tunneling as an effect arising from interference at periodic level crossings . in the present rapid communication ,
an alternative explanation is proposed which uses the concept of _ phase space interference _ known from squeezed states@xcite and displaced number states@xcite in quantum optics .
let us start with a description of the physical situation we have in mind . at @xmath6 a single two - level tunneling system ( ts ) prepared in a state localized in
say the left well @xmath7 is injected in a cavity and starts to tunnel between its left and right state @xmath7 and @xmath8 .
the cavity contains a single mode which has been prepared in a number state @xmath9 .
we consider an ideal cavity , i.e. , we neglect any kind of dissipation . after an interaction time
@xmath10 the ts leaves the cavity and the photon number distribution is measured irrespective of the state of the ts .
thus our aim will be to calculate the transition probability from the product state @xmath11 at @xmath6 to another product state @xmath12 for any @xmath13 at time @xmath14 , @xmath15 where @xmath16 . for fixed @xmath10
this is the photon number distribution with @xmath17 . for fixed @xmath18
, @xmath19 describes the dynamics of the cavity mode interacting with a ts .
an important aspect of the present problem is the strong coupling between the cavity mode and the ts , and the separation of timescales between the _ slow _ tunneling motion and the _ fast _ field oscillations .
if we denote the coupling energy by @xmath20 and the tunneling frequency by @xmath21 , this means that we are interested in the limit @xmath22 and @xmath23 for low , and @xmath24 for high number of photons in the field . in this regime
the field strongly dresses the ts and both ts + field have to be treated as a single unit .
in contrast to the situation where @xmath25 , we are confronted in the present case with a situation in which the rotating - wave approximation is not applicable .
thus instead of using the jaynes - cummings hamiltonian@xcite , we must include counter - rotating terms so that our hamiltonian reads @xmath26 where @xmath27 , @xmath28 are the bosonic creation and annihilation operators of the cavity mode
. we have identified @xmath29 and @xmath30 and @xmath31 , the usual formulation of the hamiltionian in quantum optics is recovered .
we also have added a constant energy shift for later convenience .
it has to be noted that suppression of tunneling does not occur in the jaynes - cummings model.@xcite hence the situation here is fundamentially different from the one usually encountered in quantum optics .
owing to the separation of timescales between the tunneling and the oscillation dynamics , @xmath19 can be calculated in perturbation theory in @xmath32 by introducing dressed states @xmath33 and @xmath34\ ] ] where @xmath35 .
equivalently , one may perform the polaron transformation @xmath36 and continue to use the product state instead of the dressed state basis . in the dressed state basis
the hamiltonian can be written as @xmath37 with @xmath38 and @xmath39 where @xmath40 $ ] is the shift operator and @xmath41 an associate laguerre polynomial ( @xmath42 ) .
the diagonal part @xmath43 builds a ladder of tunneling doublets with intra - doublet spacing @xmath44 and inter - doublet spacing @xmath45 . because @xmath46 , @xmath47 induces only mixing between dressed states belonging to _ different _ doublets
hence , corrections to the dressed states are @xmath48 . neglecting @xmath47 for this reason we find for the transition probability if @xmath42 @xmath49 and @xmath50 if @xmath51 .
_ we conclude that the tunneling dynamics can be seen in the spectrum of the transition probability of the cavity field . _ in addition to harmonics of @xmath5 its power spectrum also contains resonances at @xmath52 arising from the tunneling motion .
this behavior strongly depends on the initial preparation of the ts .
injecting it in its ground state @xmath53 , for instance , yields only resonances at @xmath54 where @xmath55 depending on whether @xmath56 is even or odd , respectively , and hence only rapid oscillations .
the effect of a strong coupling between the cavity mode and the ts is to mix a coherent amplitude @xmath57 with the intial number state of the mode .
this happens by displacing the oscillator wave function @xmath58 like @xmath59 in dimensionless coordinates @xmath60 .
thus by injecting a ts which strongly couples to the cavity field , it is possible to realize displaced number states@xcite .
the statistical properties of displaced number states have been discussed in ref . . the photon number distribution is simply given by @xmath61 because @xmath62 is the probability amplititude of finding @xmath18 photons in a displaced number state @xmath63 .
the photon distribution ( [ 8 ] ) for number states which are _ dynamically _ displaced by the tunneling process resembles @xmath64 .
the displaced number states can either be shifted into the same well ( first term in ( [ 8 ] ) ) or opposite wells ( second term in ( [ 8 ] ) ) . as expected for @xmath65 , @xmath66 independent of the interaction time . in fig .
1(a),(b ) we have plotted @xmath19 for @xmath67 , @xmath68 and @xmath69 0.25 and 0.5 .
whereas a coherent state with @xmath70 obeys the familiar poisson distribution@xcite shown by the line , the dynamically displaced number states exhibit oscillations shown by the histogram . increasing @xmath71 results in further oscillations
this effect is independent of the specific value of @xmath72 for @xmath73 , though the absolute value of @xmath19 depends on @xmath72 .
these modulations are clearly exhibited in the asymptotic expansion of ( [ 8 ] ) in the semi - classical limit ( bohr s correspondence principle ) : @xmath74 , @xmath75 with @xmath76 finite .
if we scale the coupling constant between the field and the ts as @xmath77 and note that associate laguerre polynomials asymptotically approach bessel functions@xcite one finds in the semi - classical limit @xmath78 here @xmath79 is the renormalized tunneling frequency , and @xmath80 is the rabi frequency . in fig .
1(c),(d ) , eq .
( [ 9 ] ) is plotted for @xmath81 1 and 3.3 .
it shows the same oscillations as the exact expression ( [ 8 ] ) .
the expression ( [ 9 ] ) is easily understood .
the cosine and the sine factors represent the probability for the tunneling particle to stay in the well where it has been prepared initially , or to escape to the other well , respectively.@xcite the bessel functions represent the probability for the corresponding displaced number state to contain @xmath18 photons if it had @xmath0 initially ( note that @xmath82 in the semi - classical limit ) . hence the bessel functions represent the photon statistics of displaced number states with an effective displacement @xmath83 or @xmath84 . from ( [ 10 ] ) and the cosine and sine factors in ( [ 9 ] ) we further see that for specific parameter values of the driving field where @xmath85 hits the roots of the zero order bessel function
the tunneling process is completely suppressed .
note that localization can not occur for small photon number in the field because there is no value of @xmath71 which is simultaneously a root of all laguerre polynomials .
we conclude that in the large @xmath0 limit _ the tunneling dynamics is displayed in the photon statistics by an amplitude modulation of the cavity mode oscillations .
_ _ coherent destruction of tunneling manifests itself in the slowing down of this amplitude modulation _ as depicted in fig .
2 .
we now show how the present picture can give a quantitative understanding of this effect .
first recall that the oscillation in the photon number statistics in fig .
1 originate from phase space interference:@xcite the @xmath0th number state can be associated with a circular band of width @xmath86 around its orbit with radius @xmath87 and centered around the origion .
analogously so can the @xmath18th displaced number state which is shifted by @xmath88 . according to the area - of - overlap concept@xcite ,
the transition amplitude between two states is governed by the sum of all possible overlap areas weighted with appropriate phases .
this results in the interference between contributions from different overlaps which can be constructive or destructive , thus , giving rise to oscillations in the photon number distribution .
another way of understanding the oscillations is that they arise from the possibility of a constructive and destructive overlap between displaced harmonic oscillator wave functions by noting that @xmath89 is a polynomial in @xmath18 of order min@xmath90 , i.e. , has min@xmath90 roots . from this consideration
it becomes apparent that varying the relative radius of the two bands by changing @xmath18 vs. @xmath0 , or varying the relative displacement by changing @xmath71 for fixed @xmath18 and @xmath0 should result in similar effects .
if one notices further that the factor @xmath91 in ( [ 10 ] ) is the probability amplitude of finding in a number state displaced by @xmath92 again exactly the same number of photons , one expects that the periodic suppression of tunneling arises from destructive and constructive interference of displaced harmonic oscillator wave functions in phase space . to verify this argument
, the @xmath93 limit has to be considered with care .
first take the number of photons in the cavity mode to be finite . the factor @xmath94 which scales the doublet splitting ( [ 6 ] ) is the overlap between a number state @xmath9 shifted into the right well with the same number state shifted into the left well .
if we note that @xmath94 is a polynomial in @xmath95 of degree @xmath0 , and consequently has @xmath0 zeros , we expect @xmath96 oscillations of @xmath94 as a function of @xmath71 .
the @xmath0 zeros between the maxima result from the @xmath0 possible ways of destructive overlap between displaced harmonic oscillator wave functions .
this explains fig . 2 in ref .
however , contrary to the claim there , this picture remains also valid if the oscillator energy is larger than the reorganization energy @xmath97 , and even in the limit @xmath93 .
the simple reason for this is that in the semi - classical limit @xmath93 with @xmath98 fixed , both the distance between the nodes of the harmonic oscillator wave function ( as a function of @xmath99 ) _ and _ the displacement @xmath92 scale as @xmath1 .
based on this argument , we expect oscillation to become noticable as soon as the displacement exceeds the bandwidth , i.e. , @xmath100 .
finally , we note that with the scaling chosen above both displaced orbits will always intersect no matter how large @xmath101 is .
this results in the infinite number of oscillations seen in the zero order bessel function ( [ 10 ] ) .
summarizing coherent destruction of tunneling is explained as a _ quantum effect arising from the destructive interference of displaced harmonic oscillator wave functions in phase space_. the effect is strictly speaking only observable in the semi - classical limit for the reasons mentioned above , and survives the large @xmath0 limit since both _ the bandwidth of the orbit and the displacement scale in the same way to zero as _ @xmath102 . in closing , we note that experimental study of the effects described here is presently still out of range . in a microcavity
the driving frequency @xmath5 is @xmath103(ghz ) whereas typical values of the one - photon rabi frequency @xmath104 are @xmath103(khz).@xcite hence a number state with large @xmath0 is needed which is difficult to realize experimentally . possibly a trap is more suited because of its lower driving frequency @xmath105(mhz).@xcite damping of the cavity mode which is too strong will also render an observation impossible .
furthermore the tunneling dynamics must still be coherent .
finally the superposition state @xmath106 is difficult to prepare because the particle experiences strong electromagnetic fields when it enters and leaves the cavity@xcite ( for a discussion of this point if one uses quantum wells , see ref . ) .
@xmath107 @xmath107 this research has been supported in part by the nsf and by the alexander von humboldt foundation .
we also thank prof .
peter hnggi and prof .
herbert walther , as well as dr .
min cho and dr .
jochen rau for discussions .
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we thank all members of the flavianet kaon working group ( http://www.lnf.infn.it/wg/vus ) for comments , discussion , and suggestions .
this work is supported in part by eu contract mtrn - ct-2006 - 035482 ( flavianet ) .
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we update the numerical results for the s - wave @xmath0 scattering phase - shift difference @xmath1 at @xmath2 from a previous study of isospin breaking in @xmath3 amplitudes in chiral perturbation theory .
we include recent data for the @xmath4 and @xmath5 decay widths and include experimental correlations .
la - ur-07 - 8412 the authors of refs . and have shown the importance of taking into account radiative corrections in @xmath6 decays both in experimental measurements and theoretical predictions .
in particular , these corrections are enhanced in the extraction of the phase shifts @xmath1 by the @xmath7 rule by a factor of @xmath822 . on the experimental side
, kloe has measured with high accuracy the ratio of branching ratios ( brs ) for the decay @xmath9 to the decay @xmath10 @xcite .
this measurement is fully inclusive of radiation for the @xmath11 channel , allowing an unambiguous comparison with theoretical predictions . for the extraction of the phase shift @xmath1 from @xmath6
we follow ref . .
in the presence of electromagnetic interactions , the usual isospin decomposition of amplitudes becomes : @xmath12 where @xmath13 are the infrared - finite amplitudes for the decays @xmath14 and @xmath15 , respectively , and @xmath16 is the amplitude for the decay @xmath17 .
these amplitudes are related to the decay rate by : @xmath18 where @xmath19 .
the factor @xmath20 is associated with the effect of real and virtual photons @xcite .
the following notation is introduced in ref .
: @xmath21 where in the absence of electromagnetic interactions the @xmath22 are the standard isospin amplitudes and the phases @xmath23 are identified with the s - wave @xmath0-scattering phase shifts @xmath24 .
otherwise , we have , by eqs .
( 7.20 ) and ( 7.33 ) of ref .
: @xmath25 and : @xmath26 where @xmath27 are the coefficients of the leading ( @xmath28 ) octet and 27-plet weak non - leptonic chiral operators ( as defined in eqs .
( 2.6 ) and ( 2.7 ) of ref . ) .
note that in order to arrive at eqs .
( [ sviluppo ] ) , a number of higher - order chiral effective couplings are needed .
here we adopt the estimates given in section 5 of ref . .
the coefficients @xmath29 , @xmath30 , and @xmath31 depend on the chiral renormalization scale @xmath32 ( from the matching uncertainty in the low - energy constants ) and the tree - level @xmath33 mixing angle @xmath34 given by : @xmath35 we have obtained the values of these coefficients from the authors of ref . .
eqs .
( [ finiteampl ] ) , ( [ expampl ] ) , and ( [ isospinampl ] ) can be combined to obtain @xmath36 where @xmath37 . using the expansion of eq .
( [ sviluppo ] ) , this system of equations can be written in terms of the three unknowns , @xmath38 , @xmath39 , and @xmath40 .
we solve this system using a numerical minimization procedure .
the experimental inputs used to obtain the amplitudes by eq .
( [ expampl ] ) are listed in table [ tab : exp ] .
as noted above , the brs for @xmath41 and @xmath42 are obtained from the kloe measurement of their ratio , which accounts for the large value of the correlation coefficient .
the br for @xmath5 is weakly correlated with the value of the @xmath43 lifetime by the fit performed in ref . .
these correlations are taken into account in the minimization procedure . while the kloe measurement of the ratio of @xmath4 brs is fully inclusive of radiation in the @xmath11 channel , the inclusiveness of the value for @xmath44 from the fit in ref .
is less well defined .
however , this is of lesser concern because of the dominance of the @xmath45 amplitudes .
.experimental inputs used to obtain @xmath46 , @xmath40 , and @xmath47 .
[ cols="<,^,^,^",options="header " , ] with @xmath48 gev and @xmath49 we obtain : @xmath50 the error matrix is : @xmath51 we compute the systematic error by varying the chiral renormalization scale parameter @xmath32 from 0.5 gev to 1 gev , and the tree - level @xmath33 mixing angle @xmath34 from @xmath52 to @xmath53 , and taking half of the total variation as the uncertainty .
the systematic error matrix is : @xmath54 including the systematic errors and using eq .
( [ deltacorr ] ) we have : @xmath55 the results in eqs .
( [ phaseresult ] ) have been obtained using the central values for the higher - order chiral couplings estimated in ref .
within the large-@xmath56 expansion .
the quoted uncertainty reflects only the uncertainty in the matching scale . as already pointed out in ref .
( section 7.3 ) , the extraction of @xmath46 and @xmath40 is rather sensitive to the input on the chiral couplings .
for instance , even a simple variant of the large-@xmath56 procedure leads to changes in @xmath46 and @xmath40 at the @xmath57 level @xcite .
this implies that the values for @xmath27 quoted in eq .
( [ phaseresult ] ) are affected by an unknown systematic offset of at least @xmath57 .
( see also the analysis of ref . . ) on the other hand , the extraction of the phase difference @xmath47 is quite insensitive to the input on the effective chiral couplings .
the result for the phase difference @xmath1 instead depends quite sensitively on the estimate of the isospin - breaking correction of eq .
( [ deltacorr ] ) .
it is interesting to compare the present result for @xmath1 with predictions from phenomenological evaluations .
the roy equations @xcite determine the @xmath0 scattering amplitude in terms of its imaginary part at intermediate energies , up to two subtraction constants : the s - wave scattering lengths @xmath58 and @xmath59 .
colangelo , gasser , and leutwyler @xcite obtain values for @xmath58 and @xmath59 by matching a representation of the @xmath0 scattering amplitude from @xmath60 calculations in chiral perturbation theory with a phenomenological representation based on the roy equations .
they obtain @xmath61 , which differs from our result by @xmath62 .
kamiski , pelez , and yndurin @xcite fit experimental @xmath0 scattering amplitudes at both low and high energies with parameterizations that satisfy analyticity at low energy , constrained to satisfy the forward dispersion relations and the roy equations .
they obtain @xmath63 , which differs from our result by @xmath64 .
it is important to note that their input data includes low - energy s - wave phase shift determinations from @xmath65 and @xmath3 decays , including a preliminary result for @xmath66 from this update , differing very little from the value presented here .
the significant discrepancies between our result , based on @xmath67 and @xmath43 br measurements , and the results of phenomenological analyses of @xmath0 scattering ( with or without constraints from chiral symmetry ) are puzzling and deserve further investigation .
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imaging atmospheric cherenkov telescopes use large mirror areas to reflect the cherenkov photons from cosmic - ray and gamma - ray air showers onto a photo - detector camera , usually comprised of photo - multiplier tubes ( pmts ) .
the number of photoelectrons generated at each pmt is directly proportional to the charge under the pmt output pulse , and this is easily measured using adcs with fixed - length integration gates .
an alternative method is to use `` flash '' adcs to rapidly sample the output pulse and record the complete pulse shape .
this allows to maximize the signal - to - noise ratio for individual pulses at the analysis stage , lowering the effective energy threshold of the telescope @xcite .
a number of authors have suggested that the timing and pulse shape information could also be used to improve sensitivity through improved gamma - hadron separation or better reconstruction of shower parameters ( core location , primary energy , etc . )
@xcite . [ cols="^,^ " , ]
a preliminary examination of the timing information provided by the veritas fadcs verifies that the effective charge integration gate can be reduced to @xmath0 .
while the muon background currently limits the analysis threshold of the first veritas telescope , the reduced charge integration gate should result in a correspondingly reduced energy threshold for analysis when stereo data becomes available .
this preliminary study does not indicate a significant improvement in gamma - hadron separation using the timing information ; however , this situation may also change when the muon background is removed .
refinements to the simulations , the application of advanced digital signal processing methods and an increased dataset of gamma - ray source observations should all help future studies in this area .
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the 499 pixel photomultiplier cameras of the veritas gamma ray telescopes are instrumented with 500mhz sampling flash adcs .
this paper describes a preliminary investigation of the best methods by which to exploit this information so as to optimize the signal - to - noise ratio for the detection of cherenkov light pulses .
the fadcs also provide unprecedented resolution for the study of the timing characteristics of cherenkov images of cosmic - ray and gamma - ray air showers .
this capability is discussed , together with the implications for gamma - hadron separation .
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the authors wish to thank tom bullock for comments on an earlier version of this paper . t.h . and j.s .
acknowledge financial support from the academy of finland ( grant no .
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we show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have .
we define the joint measurability region of a probabilistic theory which describes the amount of added noise needed to make any pair of observables in the theory jointly measurable .
we then show that the joint measurability region of quantum theory is contained in the joint measurability region of any other probabilistic theory .
quantum theory has a number of important features not known in classical physics , ranging from the superposition and indeterminacy principles formulated by the pioneers to the more recently discovered no - cloning and no - broadcasting theorems .
it is an old problem to identify operationally significant properties of quantum theory that distinguish it from other probabilistic theories . in recent years
many features have been under intensive investigation from this perspective , including information processing @xcite , optimal state discrimination @xcite , entropy @xcite , purification @xcite and discord @xcite .
it has been found that some properties are quite generally valid in any non - classical ( no - signaling ) probabilistic theories while others are specifically quantum .
the existence of pairs of incompatible observables marks one of the most striking distinctions between quantum theory and classical physical theories .
there are many manifestations of incompatibility , perhaps the most famous being the heisenberg uncertainty principle @xcite . in this letter
we demonstrate that quantum theory contains observables that are as incompatible ( in a sense to be defined ) as observables in any probabilistic theory can be .
our aim is thus to compare quantum theory to other possible probabilistic physical theories .
we first need to set some minimal constraints .
a probabilistic theory is a framework that provides a description of physical systems in terms of states and observables with the following general properties : + + ( i ) the states of a system are represented by the elements of a convex subset of a real vector space . + ( ii ) an observable is represented as an affine mapping from the set of states into the set of probability distributions on some outcome space . for simplicity , we restrict ourselves here to observables with a finite or countable number of outcomes . + ( iii ) any affine mapping from the set of states into the set of probability distributions is a valid observable . + we consider a particular probabilistic theory ( pt ) as given by a family of convex sets of states with associated sets of observables that share some properties specific to that pt
. one may think of each pair consisting of a set of states with associated set of observables as an _ instance _ of a pt representing a particular type of physical system .
given a pt , we denote by @xmath0 the probability of obtaining a measurement outcome @xmath1 when an observable @xmath2 is measured in a state @xmath3 .
hence , @xmath4 and @xmath5 .
we will typically label the measurement outcomes by integers .
in quantum theory the states are described by density operators and observables correspond to povms @xcite .
their duality is given by the trace formula ( with @xmath3 a density operator and @xmath6 a povm ) @xmath7 } \ , .\ ] ] another example of a probabilistic theory is a classical theory , where the states are probability measures on a phase space @xmath8 and observables are traditionally represented as functions @xmath9 ; the associated affine maps from states @xmath3 to probability distributions are then given by the formula @xmath10 continuing our discussion on general probabilistic theories , we note that it follows from the required properties ( i)-(iii ) that the set of observables is a convex set ; a mixture of two observables is an observable .
physically mixing corresponds to an experiment where we switch between two measurement apparatuses with a random probability .
we can directly write a mixture of two observables with the same set of measurement outcomes . if the sets of measurement outcomes differ , we can still write a mixture by first adding enough outcomes and then embedding both sets into @xmath11 . another consequence of the basic requirements is that every constant mapping @xmath12 , where @xmath13 is a fixed probability distribution , is an observable and we call it a _ trivial observable_. a trivial observable @xmath14 corresponds to a dice rolling experiment , where we randomly pick the measurement outcome according to a given fixed probability distribution , without manipulating the state at all . in quantum theory , trivial observables are described by povms @xmath15 such that each operator @xmath16 is a multiple of the identity operator , i.e. , @xmath17 for some @xmath18 with @xmath19 .
the concept of _ joint measurement _ can be defined in any probabilistic theory .
two observables @xmath20 and @xmath21 are _ jointly measurable _ if there exists an observable @xmath2 such that @xmath22 in this case @xmath2 is called a _ joint observable _ of @xmath20 and @xmath21 . if @xmath20 and @xmath21 are not jointly measurable , then we say that they are _
incompatible_.
any probabilistic theory contains jointly measurable pairs of observables .
namely , a trivial observable @xmath12 is jointly measurable with any other observable ; we can write a joint observable @xmath23 for the trivial observable and any other observable @xmath20 . this simply corresponds to an experiment where we measure @xmath20 and simultaneously roll a dice .
it is a well known fact that , in quantum theory , an observable which is jointly measurable with all other observables is necessarily a trivial observable .
indeed , any povm element of such an observable commutes with all projections and must therefore be a scalar multiple of the identity ( e.g. ( * ? ? ?
* theorem iv.1.3.1 ) ) .
the following simple observation is a key ingredient for our discussion .
[ prop : sum=1 ] let @xmath20 and @xmath21 be two observables and @xmath24 .
then @xmath25 and @xmath26 are jointly measurable for any choice of trivial observables @xmath27 and @xmath28
. this proposition can be proved with the following construction .
first , let @xmath29 and @xmath30 be the probability distributions related to @xmath27 and @xmath28 .
we define an observable @xmath2 by formula @xmath31 for a fixed @xmath3 , the right hand side is clearly a probability distribution .
moreover , the right hand side is an affine mapping on @xmath3 ; therefore @xmath2 is an observable .
the marginal observables are @xmath32 this proves prop .
[ prop : sum=1 ] . the physical idea behind this construction is the following .
in each measurement run we flip a coin and , depending on the result , we measure either @xmath20 or @xmath21 in the input state @xmath3 . in this way we get a measurement outcome for either @xmath20 or @xmath21 .
in addition to this , we roll a dice and pretend that this is a measurement outcome for the other observable . in this way we get an outcome for both observables simultaneously .
the overall observable is the one given in formula . for two observables @xmath20 and @xmath21 ,
we denote by @xmath33 the set of all points @xmath34\times[0,1]$ ] for which there exist trivial observables @xmath35 such that @xmath25 and @xmath36 are jointly measurable , and we call @xmath33 the _ joint measurability region _ of @xmath20 and @xmath21 .
the joint measurability region thus characterizes how much noise ( in terms of trivial observables ) we need to add to be able to have a joint realization of @xmath20 and @xmath21
. clearly , @xmath20 and @xmath21 are jointly measurable if and only if @xmath37 .
the joint measurability region @xmath33 is a convex region which can be plotted in the plane .
this follows as a simple application of prop . 2 of @xcite ; with the help of this lemma it also follows from prop . [
prop : sum=1 ] that @xmath38\times[0,1 ] : \lambda+\mu \leq 1 \ } \subseteq j({\mathfrak{m}}_1,{\mathfrak{m}}_2 ) \ , .\ ] ] for two orthogonal spin-@xmath39 measurements is a quadrant of the unit disk .
the region @xmath40 ( light ) is a subset of @xmath33 for any pair @xmath41 , while the surplus region ( dark ) depends on the specific pair under consideration.,width=94 ] as an example , suppose that we are within quantum theory and @xmath20 and @xmath21 correspond to spin-@xmath39 measurements in two orthogonal directions , say @xmath42 and @xmath43 -axes .
we then describe them with two povms @xmath44 and @xmath45 , where @xmath46 and @xmath47 are the usual pauli matrices in @xmath48 .
it has been shown in @xcite that for the uniformly distributed trivial observable @xmath49 ( hence describing an unbiased coin ) , the two observables @xmath50 and @xmath51 are jointly measurable if and only if @xmath52 . by prop .
3 of @xcite this inequality is a necessary condition for the joint measurability of any pair @xmath53 and @xmath54 , where @xmath55 are arbitrary trivial observables .
therefore , we conclude that @xmath56\times[0,1 ] : \lambda^2+\mu^2\leq 1\ } \ , .\ ] ] this region is depicted in fig .
[ fig : qubit ] .
we are now ready to define the main concept of this letter .
a global joint measurability feature of a probabilistic theory pt is characterized by the intersection of all the sets @xmath33 across all instances of pt , and we denote @xmath57\times[0,1 ] : ( \lambda,\mu)\in j({\mathfrak{m}}_1,{\mathfrak{m}}_2 ) \\ & \textrm{for all pairs of observables $ { \mathfrak{m}}_1 $ and $ { \mathfrak{m}}_2$}\\ & \textrm{in all instances of pt } \}.\end{aligned}\ ] ] we call @xmath58 the _ joint measurability region _ of @xmath59 .
we always have @xmath60 , but @xmath58 can be larger than @xmath40 .
the larger the surplus region is , the more jointly measurable the theory is globally ; see fig . [
fig : regiong ] .
if @xmath61 , this means that there is a pair of observables @xmath20 and @xmath21 such that the mixtures @xmath25 and @xmath36 are incompatible with any choice of trivial observables @xmath27 and @xmath28 . since @xmath58 can be defined in any probabilistic theory , we can compare the joint measurability regions of different probabilistic theories .
we obviously have @xmath62\times [ 0,1]$ ] in any probabilistic theory where all measurements are jointly measurable , such as the classical probability theory .
in the most incompatible case we would have @xmath63 .
we will next show that quantum theory is , globally , as incompatible as a probabilistic theory can be .
[ th : main ] in quantum theory @xmath64 . in particular , @xmath65 for any probabilistic theory pt . in quantum theory
every observable @xmath2 corresponds to a unique povm @xmath6 by equation .
we will prove that for any pair @xmath66 , there are quantum observables @xmath67 and @xmath68 such that the mixtures @xmath69 and @xmath70 are incompatible with any choice of trivial observables @xmath55 . our proof is based on a recent result @xcite on the joint measurability region for two complementary observables , which is a generalization of the result illustrated in fig .
[ fig : qubit ] .
we have earlier seen that @xmath60 , so we need to show that @xmath71 .
let @xmath66 , i.e. , @xmath72 .
fix @xmath73 such that @xmath74 .
we then choose @xmath75 to be a positive integer satisfying @xmath76 ( this can be done since the left hand side @xmath77 when @xmath78 . )
we will consider a quantum system that is described by a @xmath75-dimensional hilbert space @xmath79 .
let @xmath80 be an orthonormal basis for @xmath79 .
we define another orthonormal basis @xmath81 for @xmath79 by @xmath82 the orthonormal bases @xmath80 and @xmath81 are mutually unbiased , i.e. , @xmath83 .
we define two povms @xmath67 and @xmath68 by @xmath84 we thus obtain a pair of @xmath75-outcome observables on @xmath79 . since @xmath67 and @xmath68 consist of projections and @xmath85 , it follows that they are incompatible . as proved in @xcite , the observables @xmath86 and @xmath87 are incompatible for any choice of trivial observables @xmath55 whenever @xmath88 since @xmath89 we conclude that @xmath90 . in the proof of theorem [ th : main ]
we have used quantum observables with arbitrarily many ( but a finite number of ) outcomes .
it is natural to ask if this is a necessary trick in order to reach the conclusion @xmath64 . to present a partial answer to this question ,
let us investigate the joint measurability region in the case of pairs of binary quantum observables .
our aim is to show that @xmath91\times[0,1 ] : \lambda^2+\mu^2\leq 1\ } \subseteq j ( { \mathsf{m}}_1 , { \mathsf{m}}_2)\ ] ] for any binary observables @xmath67 and @xmath68 , regardless of the dimension of the hilbert space .
in other words , we will show that two orthogonal spin observables are as incompatible as any binary observables can be . to this end , let us note that two binary quantum observables are incompatible if and only if they enable a violation of the bell - chsh inequality @xcite .
we must therefore look at the bell expression @xmath92 let us denote @xmath93 and @xmath94 . using , e.g. , ( * ? ? ? * theorem 1 ) one verifies that by varying the observables @xmath95 and @xmath96 and the bipartite state the numbers @xmath97 and @xmath98 obtain all of the values from the disc @xmath99 .
if we now mix the observables @xmath95 with the trivial observable @xmath100 with some weights @xmath101 and @xmath102 we see that the pair @xmath103 turns into @xmath104 , thus changing the bell expression from @xmath105 to @xmath106 .
we must therefore determine those @xmath107 for which @xmath108 for all @xmath103 satisfying @xmath99 ( see fig .
[ fig : bell ] ) .
but the boundary curve for this region is obtained when the equations @xmath109 and @xmath110 have at most one common solution . by inserting @xmath111 into the first equation the problem reduces to determining when the discriminant is negative or zero , and one readily verifies that this is the case exactly when @xmath112 . in conclusion ,
given any pair of binary observables @xmath67 and @xmath68 , and weights @xmath101 and @xmath102 with @xmath113 , the mixtures @xmath114 and @xmath115 can not be used to violate the bell - cshs inequality and must therefore be jointly measurable .
we note that in the case @xmath116 the same result using a different technique has been obtained by banik _
et al .
_ @xcite .
we note that in the proof of theorem [ th : main ] we have used quantum observables acting on an arbitrarily high ( but finite ) dimensional ( but finite ) hilbert space
. we can again ask if this is a necessary trick in order to reach the conclusion @xmath64 . since
two mutually unbiased bases are expected to be among the most incompatible observables in a fixed dimension @xmath75 , our construction suggests that the conclusion @xmath117 can be reached only if one considers arbitrarily high dimensions .
a proof of this fact is , however , lacking .
we show finally that the conclusion @xmath117 can be reached by using a _ single pair _ of incompatible observables if we consider an infinite dimensional system and observables with infinite number of outcomes .
let @xmath118 be an infinite dimensional hilbert space and write it as a direct sum of finite @xmath75-dimensional hilbert spaces @xmath79 , @xmath119 . in each @xmath79
consider a pair of mutually unbiased orthonormal bases @xmath120 and @xmath121 , where the latter is obtained from the first one by the formula .
we define two povms @xmath122 and @xmath123 via @xmath124 these observables act in the infinite dimensional hilbert space @xmath118 and @xmath75 in is an index labeling the different outcomes .
the outcome space of @xmath125 and @xmath126 is @xmath127 .
[ th : totalmub ] the observables @xmath125 and @xmath126 defined in satisfy @xmath128 .
let @xmath29 and @xmath30 be two probability distributions defined on @xmath129 .
assume that @xmath130 and define two observables @xmath131 via @xmath132 we need to show that @xmath133 and @xmath134 are incompatible . to prove this
, we make a counter assumption that @xmath131 are jointly measurable .
this implies that for any projection @xmath135 on @xmath118 , the projected observables @xmath136 and @xmath137 acting on a subspace @xmath138 are jointly measurable .
( if @xmath139 is a joint observable of two observables @xmath140 , then @xmath141 is a joint observable of @xmath142 in @xmath138 . )
especially , the projections of @xmath133 and @xmath134 to any subspace @xmath79 should be jointly measurable .
but from the result cited in the proof of theorem [ th : main ] we know that for @xmath75 large enough , the projections to @xmath79 are incompatible .
hence , @xmath133 and @xmath134 are incompatible .
we note that the observables @xmath125 and @xmath126 defined in are not the only pair that satisfy @xmath128 .
namely , we can modify @xmath125 and @xmath126 in any chosen subspace @xmath79 but the conclusion @xmath128 is still true since it depends on the fact that @xmath125 and @xmath126 contain mutually unbiased bases in arbitrarily high dimension .
an interesting question within quantum theory is to find a characterization of all pairs of quantum observables @xmath140 that satisfy @xmath143 .
do they exist in finite dimensions ?
can they have finite numbers of outcomes ?
|
for metal element @xmath122@xcite , @xmath123 is the state density which have some spin directions on the fermi surface .
@xmath124 according to the assumption@xmath125 formalism([renormal ] ) can be written as@xmath126 so from bcs theory , the cooper pair lies in the attraction area , ie @xmath127 . on substitution of @xmath128ev
@xcite we can estimate @xmath129 , so @xmath108ev .
wu , m. s. byrd , and d. a. lidar , _ phys rev lett . _
* 89 * , 057904 ( 2002 ) ; j. dukelsky , j. m. romn , and g. sierra , _ phys rev lett . _ * 90 * , 249803 ( 2003 ) ; l .- a .
wu , m. s. byrd , and d. a. lidar , _ phys rev lett . _ * 90 * , 249804 ( 2003 )
|
we propose a new simulation computational method to solve the reduced bcs hamiltonian based on spin analogy and submatrix diagonalization .
then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by bogoliubov transformation method .
the exponential problem of @xmath0-dimension matrix is reduced to the polynomial problem of @xmath1-dimension matrix .
it is essential to validate this method on a real quantum computer and is helpful to understand the many - body quantum theory .
bcs theory@xcite and its subsequent extension is a well established theory to explain the mechanism of superconducting property . with two gross simplifications : the free electron approximation and the effective interaction approximation@xcite@xmath2 @xcite , a simplified bcs model is obtained and described by the reduced bcs hamiltonian . there has been much work on solving this hamiltonian .
the mean field method is exact in the limit of large number of electrons where fluctuation can be neglected but disabled in the case of small number of electrons . since richardson s work@xcite in the 60 s to now , the exactly solvable bcs hamiltonian attracts much attention in connection with the problems in different areas of physics such as superconductivity , nuclear physics , physics of ultrasmall metallic grains . recently in _
l .-
a .
wu et al . _
s paper@xcite an nmr experiment scheme performing a polynomial - time simulation of pairing model was reported .
based on this work we propose an explicit theory method to diagonalize the reduced bcs hamiltonian through the spin analogy and submatrix diagonalization . compared with the conventional method it is more useful in solving practical problem .
the problem is solved in the spin space , which is convenient related to the qubit system .
it gives a senseful alive method , quantum simulation , instead of the numerical diagonalization calculation .
and it shows the potential to solve many - body problem by quantum computer .
in fact more and more people concentrate on the research of simulating other physics systems by quantum computer@xcite .
the experimental quantum simulations about quantum harmonic oscillator@xcite , three - spin artifical hamiltonian@xcite and migration of excitation in a one - dimensional chain@xcite _ et al .
_ have been realized .
recently a relative experiment is performed to get the eigenvalues of the bcs hamiltonian through selecting a proper initial state and realizing hamiltonian evolution.@xcite@xcite the exact solvable model , _
i.e. _ the reduced bcs hamiltonian considered in this paper is@xcite@xmath2@xcite@xmath2@xcite : @xmath3 where @xmath4 are the electron number operators , @xmath5 is the fermionic creation ( annihilation ) operator .
the coupling coefficient is simplified as a constant @xmath6@xcite@xmath2@xcite .
note that the summation indexes @xmath7 represent all of relevant quantum numbers , and the electron pairs are labelled by the the quantum number @xmath8 and @xmath9 , according to the cooper pair situation where the paired electrons have equal energies but opposite momenta and spins : @xmath10 and @xmath11 .
introduce the pair creation operator @xmath12 and the pair annihilation operator @xmath13 .
so one can write the hamiltonian ( [ bcsh ] ) as@xcite : @xmath14 where @xmath15 is the free electron kinetic energy from fermi surface ( @xmath16 is the fermi energy ) . there are two possible cases for every pair state @xmath8 : occupation `` and empty '' , which are denoted respectively by : @xmath17 where the spin up state @xmath18 indicates occupation `` and the spin down state @xmath19 indicates empty '' .
obviously , @xmath20 and @xmath21 , then we can get the so - called spin - analogy corresponding of the pair annihilation operator @xmath22 as @xmath23 in the same way , the spin - analogy corresponding of the pair creation operator @xmath24 becomes @xmath25 from the pair number operator @xmath26 has the eigenvalue 2 ( which represents the electron number in every cooper pair ) when operating on @xmath18 , and 0 when operating on @xmath19 , it follows that @xmath27 in fact , the fermionic pair operators satisfy the commutation algebra : @xmath28 , _ i.e. _ @xmath29 . from formulas ( [ bx])-([nk ] )
one can express @xmath30 in terms of the spin operators : @xmath31 where @xmath32 .
in fact the spin analogy of the bcs hamiltonian is well known and exact diagonalization of the pairing model in the spin space has been carried out in several previous works@xcite . in this paper
we propose a computational simulation method which is potential to realize in future with the development of quantum computer .
the primary advantage of our method lies in the practical realization in experiments .
especially we can solve superconducting energy gap by this simulation method conveniently as following paragraphs .
it is more practicably than other solution of energy gap , because it can simplify a @xmath0-dimension problem to an @xmath1-dimension problem .
we know eigenvalues may not be solvable for high dimension matrix in principle .
now a @xmath0-dimension problem , exponential problem(ep ) can be simplified an @xmath1-dimension problem , polynomial problem(pp ) . in the following part
we will describe how to transform ep to pp in detail .
firstly the total hamiltonian @xmath33 will be expressed as the direct - sum of a set of submatrices@xcite .
@xmath34 the system states with the same spin - up state number form an absolute subspace .
the subscripts @xmath35 representing the number of the spin - up state in the corresponding subspace are respectively @xmath36 .
secondly we will prove that the eigenvalues of @xmath37 in sub1 submatrix justly are the eigenvalues of @xmath38 note the @xmath39 operator as@xmath40 whose @xmath41-th diagonal element is noted as @xmath42 $ ] @xmath43 .
it is easy to see the non - diagonal elements of @xmath44 are zero . for the hamiltonian as @xmath45 , the @xmath46-th element satisfies@xmath47=\delta_{im}\text { \ \ \ \ } ( i , m=1,2,\cdots , n )
\label{delta im}%\ ] ] we will prove this lemma by mathematic induction . when @xmath48 , it is easy to get three matrices , @xmath8 value is 1 , 2 , 3 respectively . @xmath49 here the non - diagonal elements are all 0 .
these diagonal elements with @xmath50 @xmath50 are
@xmath51 $ ] .
it is easy to validate @xmath51=\delta_{im}$ ] and the last diagonal element @xmath52 $ ] is zero .
if when @xmath53 , @xmath54=\delta_{im}\text { \ } ( i , m=1,2,\cdots , l ) \label{l(1)}%\ ] ] is right and @xmath55=0 \label{l(2)}%\ ] ] then we should examine whether @xmath56=\delta_{im}$ ] @xmath57 is right when @xmath58 . we will discuss it in two cases : @xmath59 and @xmath60 .
\(1 ) @xmath59 : @xmath61 from eq.([h(l+1,m ) ] ) it is easy to see@xmath62=h(l+1,m)[2x-1]=h(l+1,m)[x ] \label{h(2x)}%\ ] ] so@xmath63=h(l , m)[2^{l}-2^{l - i}]=\delta_{im}\text { \ } ( i=1,2,\cdots , l)\ ] ] next we should also know the value of @xmath64 $ ] . from eq.([l(2 ) ] ) and eq.([h(2x ) ] ) there is @xmath65=h(l , m)[2^{l}]=0 $ ] .
from above discussion for @xmath66 , the equality @xmath56=\delta_{im}$ ] is valid .
\(2 ) @xmath60:@xmath67 from the above expressions the @xmath68-th diagonal element @xmath69=1 $ ] , that is to say@xmath70=1\text { } ( i = l+1)\ ] ] and then for @xmath71 , @xmath70=0\ ] ] because the even - th diagonal elements are all zero obviously .
so for @xmath60 there is also @xmath72=\delta_{im}$ ] . from the discussion ( 1 ) and ( 2 )
we have proved that when @xmath58 , @xmath56=\delta_{im}$ ] @xmath57 is valid .
so for @xmath73 ( @xmath1 is the natural number ) , there is the equality@xmath47=\delta_{im}\text { \ \ \ \ } ( i , m=1,2,\cdots , n)\ ] ] after the preparation we will prove that the eigenvalues of @xmath37 in sub1 subspace justly are the energy spectrum of quasiparticle excitation of @xmath30 .
firstly set a diagonal hamiltonian as @xmath74 , @xmath75 and @xmath76 are the quasiparticle number operators .
according to the previous analogy rule of number operators eq.([nk ] ) the spin - analogy form of @xmath77 is@xmath78 @xmath79 s submatrix in sub1 subspace is denoted as @xmath80 and the @xmath41-th diagonal element of @xmath81 as @xmath82 $ ] . we can find the @xmath41-th diagonal element of @xmath80 is the @xmath46-th diagonal element of the total hamiltonian @xmath79 . according to eq.([delta i m ] ) @xmath83 & = h_{\mathrm{spin}}^{\mathrm{diag}}% [ 2^{n}-2^{n - i}]\\ & = \sum_{m=1}^{n}e_{m}h(n , m)[2^{n}-2^{n - i}]\\ & = \sum_{m=1}^{n}e_{m}\delta_{im}\\ & = e_{i}%\end{aligned}\ ] ] so the spin - analogy hamiltonian of a diagonal bcs hamiltonian @xmath77 in sub1 subspace has the same eigenvalues as the diagonal bcs hamiltonian s energy spectrum .
we can deduce further that the eigenvalues of @xmath37 are justly the eigenvalues of @xmath30 whether which is diagonal or not , because @xmath30 can be written as the diagonal form like @xmath77 generally , i.e. @xmath84 , while we need not care about how to obtain the diagonal form hamiltonian .
consequently it implies that if we have diagonalized @xmath37 we can get the energy spectrum of quasiparticle excitation of @xmath30 and get energy gap further avoiding complex computation .
one of the classical methods to diagonalize @xmath30 , bogoliubov transformation method is available under the mean - field approximation , which is not exact especially in the case of limited @xmath1 .
another famous method about the exact solution of bcs hamiltonian has been proposed in the 60 s by richardson@xcite .
he considered the system with @xmath85 pairs electrons and constructed a set of operators @xmath86 @xmath87 commuted with @xmath30 , finally gave the expression of the eigenvalues @xmath88 of @xmath86 through solving the @xmath89 coupled algebraic equations .
@xmath88 is not yet the eigenvalue of @xmath30
. we can more directly and simply give the energy spectrum of @xmath30 .
thus it can be seen the idea of diagonalizing @xmath37 instead of solving the eigenvalues of @xmath30 directly is better than those classical ones .
now a key problem that how to get the submatrix @xmath37 is placed to the front .
the correlative work@xcite in our group has proved that the general form of @xmath37:@xmath90=\epsilon_{i},\text { \ \ \ } h_{sub1}[i , j]=-v\text { \ } % ( i , j=1,2,\cdots , n;\text { } i\neq j ) \label{wang}%\ ] ] @xmath91 $ ] is the matrix element of @xmath37 .
finally it is necessary to check the methods in numerical computation . here
we will compare our solution with the result of the mean - field approximation by the value of superconducting energy gap @xmath92 @xmath93 . according to the physics meaning of @xmath92 ,
the energy required to excite at least a quasiparticle from the fermi surface , the energy of the element excitation is written as @xmath94.@xcite in fact the element excitation energy is also the eigenvalue of @xmath30 . after getting the eigenvalue of @xmath37 , _
i.e. _ the eigenvalue of @xmath30 , we can get the value of @xmath92 by solving equation @xmath94 .
but in order to get rid of the effect of energy zero the equation@xmath95 is used to solve @xmath92 in practice , because the difference of eigenvalues dosent depend on the energy zero and that we find the energy difference between the ground and the first excited state @xmath96 is by far larger than @xmath97 in the course of the numerical computation . here
@xmath98 and @xmath99 .
the another kind of solution used in comparing is the following energy gap equation@xcite:@xmath100 here we consider the reduced bcs model whose energies are given for simplicity by @xmath101@xcite@xmath2@xcite@xmath2@xcite , here @xmath102 is the average level spacing which is inversely proportional to the size of the grains . in the strong coupling regime , corresponding to large grains or strong coupling constants , @xmath103 . in the weak coupling region , corresponding to small grains or small coupling constants , @xmath104 @xcite . from much research about ultrasmall superconducting grains@xcite
, the mean - field theory is not suitable in the weak coupling region .
it has been proved that the corrections to the mean - field results are small in large grains become important in the opposite limit@xcite .
so in this paper we carry out the numerical computation in the first case , @xmath103 .
we also take the coupling constants @xmath105 in order to discuss conveniently . in order to give the numerical pictures ,
we suppose the value of @xmath6 by the rough estimate . from bcs theory
, the cooper pair lies in the attraction area , _
i.e. _ @xmath106@xcite . for metal debye energy @xmath107ev
, we can set @xmath108ev by rough estimate .
the estimate process is put to the later appendix .
another two variables @xmath109 and @xmath1 are taken as the independent variables of the energy gap .
we list our results in diagrams . in fig.[gap - m ] setting @xmath110 , the energy gap is plotted as the function of energy level number , which is the mono - increasing function of the energy level number @xmath1 . as a function of the number of energy level , @xmath1 .
we give the comparison between two results by the different methods .
the solid line is the solution of the energy - gap equation ; the dashed line is our result by spin analogy and diagonalizing submatrix .
here we choose @xmath108ev , @xmath110 .,scaledwidth=50.0% ] the relative error between our result and that of the eq.([gapeqold ] ) is not more than 5% in the range from @xmath111 to @xmath112 and fixed @xmath113 .
it shows that the result from our method is well consistent with the solution of the energy gap equation . in order to check the universality of this new method
, we also give the dependence relation between the energy gap and the level spacing , see fig.[gap - namda ] .
, energy gap @xmath92 is plotted as the function of @xmath109 .
we give the comparison between two results : the solid line is the solution of the energy - gap equation ; the dashed line is our result by spin analogy and diagonalizing submatrix . here
we choose @xmath108ev.,scaledwidth=50.0% ] it is clear to see @xmath92 changes gently with @xmath109 when @xmath109 is large enough .
that is to say , @xmath92 is almost independent of @xmath102 when @xmath102 is small enough .
it shows the rationality of @xmath114 on the inverse hand . obviously , in fig.[gap - namda ]
, when @xmath109 is larger than 80 , the relative error is less than 1.1% .
we also consider a small departure from the fermi surface , that is @xmath115 .
@xmath116 is a small value , @xmath117 @xmath102 . in the following discussion we note @xmath118
, @xmath119 is the natural number .
the small departure from the fermi surface reduces the energy gap and energy gap is not a real root when the departure reaches a critical value , see fig.[gap - b ] .
ev , @xmath110 , @xmath120 the energy gap @xmath92 is ploted as the function of @xmath119 through the quantum simulation . when @xmath121 , there is not the real root for @xmath92 .
the result is obtained by the quantum simulation .
similar result will be obtained by the energy gap equation eq.([gapeqold]).,scaledwidth=50.0% ] according to above comparison we know two results are consistent well , while our method to solve the energy spectrum does nt include approximation , which indicates that our result includes that obtained by mean field theory and is superior to it . in summary
, we have proposed an exact numerical simulation method to calculate the energy spectrum of the reduced bcs hamiltonian by spin analogy and diagonalizing submatrix .
a numerical computation to verify the validity of our computational method is given .
we make a comparison between our method and energy gap equation eq.([gapeqold ] ) , and two results are well consistent in numerical computation . by examining the change of the energy gap value under the change of the parameter
, we include the excellent consistency between the two results by the different methods is independent on the particular parameter .
it implies that one can implement this quantum simulation on a quantum computer and the result will be believable . currently a new experiment about 2-qubit simulation of the pairing hamiltonian on an nmr quantum computer has been realized and get the energy spectrum of the pairing hamiltonian successfully@xcite . with the development of quantum computer , especially the manipulation and control of multi - qubit system
, this new simulation computation method has the great potential in practical application .
we are grateful xiaosan ma , wanqing niu , zhao ningbo , zhu rengui and su xiao - qiang for helpful discussion .
this work was founded by the national fundamental research program of china with no .
2001cb309310 , partially supported by the national natural science foundation of china under grant no .
60173047 and the natural science foundation of anhui province .
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fig . 1 . the external periodic magnetic field as modeled by ( [ eq : field ] ) .
+ fig . 2 . the band structure for a spinless problem compressed in the ( longitudinal ) @xmath12-direction .
@xmath22 t , @xmath38 , @xmath24 , @xmath25 .
the states in the broad " bands ( upper quarter ) have a finite longitudinal mobility , while the longitudinal mobility of the states in the narrow " bands ( left and right quarters ) is infinitesimal .
the band structure on a bigger scale . only every fourth of the narrow " bands is shown .
the box in the middle is magnified for the spinless problem ( fig .
2 ) , and the problem with spin ( fig . 4 ) .
same as in fig . 2 but for the problem with spin .
fermi surface for @xmath39 a. u. only a part of the first brilloin zone is shown .
shaded areas are populated by electrons .
the curved lines of the fermi surface ( in the middle ) correspond to the states in the broad " bands , while the vertical lines on the extreme right and left represent the states in the narrow " bands .
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we analyze the energy band structure of a two - dimensional electron gas in a periodic magnetic field of a longitudinal antiferromagnet by considering a simple exactly solvable model .
two types of states appear : with a finite and infinitesimal longitudinal mobility .
both types of states are present at a generic fermi surface .
the system exhibits a transition to an insulating regime with respect to the longitudinal current , if the electron density is sufficiently low .
+ pacs number(s ) : 75.50.rr , 75.70.ak = 8.75 in = -0.25 in = -0.75 in * electronic band structure in a periodic * * magnetic field * andrey krakovsky _ department of physics , new york university , new york , new york 10003 _ august 18 1995 the interest in the magnetoconductance properties of the two - dimensional electron gas in spatially
periodic lateral magnetic fields has been further stimulated by the recent experimental availability of such systems @xcite . in the work of carmona
et al .
@xcite spatial modulation of a magnetic field was produced by means of equidistantly located superconducting stripes where magnetic vortices were trapped by impurities resulting in periodic inhomogeneity of the external magnetic field , while in the work of of ye et al .
@xcite it was produced by deposition of ferromagnetic microstructures on top of the high - mobility two - dimensional ( 2d ) electron gas .
vast theoretical efforts on 2d electron gas in an inhomogeneous external magnetic field range from the theory of momentum - dependent tunneling through a magnetic barrier @xcite to properties of electronic states and transoprt in a weakly spatially modulated magnetic field [ 47 ] . in this short paper
we will be concerned with the one - electron energy band structure of the 2d electron gas under a periodic lateral magnetic field of an antiferromagnet , which is a limiting case of a strong periodic modulation .
we will show that two types of states appear : with a finite and infinitesimal longitudinal mobility .
both types of states are present at a generic fermi surface .
the system exhibits a transition to an insulating regime with respect to the longitudinal current if the electron density is sufficiently low .
the effect of a uniform magnetic field on energy bands produced by the periodic ( electric ) potential is well known @xcite .
the impact of the slightly inhomogeneous magnetic field on the landau levels of a free electron was considered by mller @xcite .
he showed that the energy bands exhibit a pronounced asymmetry in the lateral direction .
for a spatially modulated magnetic field a common theoretical model @xcite employs magnetic field perpendicular to the plane of the two - dimensional electron gas which has a carrier " field @xmath0 with a periodic modulation on top of it : @xmath1 in the work of peeters and vasilopoulos @xcite the effect of a periodic electric and weakly modulated magnetic field ( @xmath2 ) was considered .
they showed that the broadening of the landau levels is roughly proportional to the modulation amplitude @xmath3 .
the hofstadter - like " spectrum was obtained by wu and ulloa @xcite , and collective excitations were analyzed in @xcite by the same authors . in this work
we will deal with an electron gas confined to a plane in a perpendicular periodic magnetic field without a carrier " field .
in other words , in ( [ eq : period ] ) we take @xmath4 .
this corresponds to the extreme case of the other limit , @xmath5 .
such a periodic field will create an energy band structure of its own .
we see this type of arrangement experimentally realizable by bringing a two - dimensional electron gas in close contact with a mesoscopic longitudinal antiferromagnetic ( sandwich ) structure , without an external magnetic field .
when dealing with the one - electron spectrum it is useful to have some exactly solvable models ( potentials ) , as they elucidate the whole structure of the energy bands @xcite .
below we show that the energy bands can be obtained exactly in a simple way for a reasonably idealized periodic magnetic field .
we present full band picture for both electron with spin and spinless , and discuss the topology of the fermi surface .
the hamiltonian for a free spinless electron in a magnetic field is : @xmath6 in our case the electron is confined to a plane , and a periodic magnetic field of lateral antiferromagnet is superimposed .
the magnetic field can be modeled as ( see fig .
1 ) @xmath7 { \bf \hat{z}}\ , , \label{eq : field}\ ] ] for such a magnetic field the vector potential takes a form @xmath8 where @xmath9 we proceed with solution of eqs .
( 2)(6 ) in a standard way .
we look for the solution in a form @xmath10 thus , in the gauge ( [ eq : gauge ] ) the solution is a plane wave in the @xmath11-direction . for the @xmath12-dependent part of the wave function @xmath13 we arrive at @xmath14 \chi ( y ) = \left(e - \frac{k_x^2}{2 } \right)\chi ( y)\ , .
\label{eq : kronig}\ ] ] here atomic units are adopted ; @xmath15 is the energy up to an unimportant constant ; @xmath16 , the dimensionless magnetic length " is given by @xmath17 , @xmath18 is the bohr radius .
the eq .
( [ eq : kronig ] ) is precisely the schrdinger equation for the kronig - penney model , and can be easily solved exactly .
the resulting dispersion relation is given by @xmath19 where @xmath20 @xmath21 is the quasimomentum in the longitudinal direction .
the band structure for @xmath22 t , @xmath23 m , @xmath24 , @xmath25 is presented on figs . 2 and 3 .
it is compressed in the ( longitudinal ) @xmath12-direction . as was pointed out in @xcite ,
the pronounced asymmetry along the @xmath11-direction is the signature of the energy spectrum in the inhomogeneuos magnetic field . in the upper quarter of fig .
2 is the region where broad " bands are formed .
these bands have a finite width in the @xmath12-direction , and the particles occupying these states will have a finite mobility in the longitudinal direction .
the other set of narrow " bands occupies the left and right quarters of fig . 2 .
from the point of view of the kronig - penney model ( [ eq : kronig ] ) they correspond to the valence bands of the periodic potential .
these bands are infinitesimally narrow in the @xmath12-direction , and electrons populating them would have a vanishingly small longitudinal mobility . of course , in the transverse direction states in both types of bands would have some finite mobility . fig .
3 represents the same band structure on a bigger scale . the part of the spectrum shown on the fig .
2 corresponds to the area inside the box of fig .
3 .
dark areas of fig .
3 represent regions of broad " bands , while the parabolas represent narrow " bands . in order not to overcomplicate the picture
we show only every fourth of the latter .
as we will see below , the peculiarity of the energy spectrum in a periodic magnetic field will appear in the fact that at the fermi surface both types of states will appear . in this model
the problem of electrons with spin is equally easy to treat .
this amounts to simply adding the spin - dependent term to the left - hand side of eq .
( [ eq : kronig ] ) : @xmath26 \chi ( y ) = \left(e - \frac{k_x^2}{2 } \right)\chi ( y)\ , ; \label{eq : kronig.sp}\ ] ] which results in a slightly modified dispersion relation : @xmath27 with the same @xmath28 and @xmath29 as in eqs .
( [ eq : alpha ] ) , ( [ eq : beta ] ) . the detailed band structure for the same magnetic field as before
is shown on fig .
4 .
the main structure of the whole spectrum is still represented by the fig .
3 .
as compared to the spinless problem , the broad " bands are characterized by wider gaps in the density of states , while narrow " bands only slightly change their locations .
as we fill the spin up " and down " states up to the fermi level , the ground state exhibiting transverse oscillatory spin oscillations in the spirit of the ones discussed by chudnovsky @xcite may result .
possible fermi surface corresponding to cutting the energy manifold at @xmath30 a.u .
is presented on fig . 5 .
we show only a part of the first brilloin zone .
shaded areas are populated by electrons .
curved lines of the fermi surface ( in the middle ) correspond to states in the broad " bands . as discussed above
, these states have a finite mobility in both directions and will always contribute to the conductivity of the sample .
the vertical lines on the extreme right and left correspond to the sections of narrow " bands which are flat in the longitudinal direction . thus , these states will not contribute to the longitudinal current , while always contributing to the transverse one .
if the electron gas is dilute enough so that the fermi level drops below the @xmath31 level ( see fig .
3 ) , the sample will not conduct in the longitudinal direction at all as all the states at the fermi surface will have a vanishing mobility in the @xmath12-direction .
it is easy to estimate the electron density for transition to an insulating regime by counting the states with @xmath32 . in our range of parameters
we can totally neglect the width of the narrow " bands .
the transition density is given by @xmath33 ( all quantities are in atomic units ) .
the summations are over all @xmath34 for which the radicals remain positive . for our values of @xmath35 , @xmath36 , and @xmath3 , @xmath37 .
if any of the summations turns out to be restricted to the first narrow " band , one has to account for the bandwidth in the @xmath12-direction as well . in a realistic experimental situation
many - body effects will be present .
the simplest of them is screening .
screening will smear " the effective single - particle potential , which may result in suppressing smaller gaps predicted in the calculation .
however , these effects do not change the overall structure of the spectrum . in conclusion ,
we have presented a simple exactly solvable model of the electronic band structure in a spatially periodic magnetic field .
all the conclusions derived from the model are not restricted to this particular model , but illustrate the general structure of the energy bands of the two - dimensional electron gas in a periodic lateral magnetic field .
this type of system can be realized by imposing magnetic field of a longitudinal antiferromagnet on a high - mobility two - dimensional electron gas .
the band structure exhibits a pronounced asymmetry in the lateral direction and consists of the two types of bands .
the states in broad " bands will have a finite mobility in the longitudinal direction while the longitudinal mobility of the electrons occupying states in narrow " bands is infinitesimally small .
a generic fermi surface will contain both types of states .
if the electron density is sufficiently low , only the narrow " bands will be occupied resulting in vanishing of the longitudinal conductivity . the electron density for transition to the insulating regime has been estimated .
an interesting extension of this work is to account for screening in such a system .
the author would like to thank professors a. d. kent , p. m. levy , j. k. percus , and especially j. l. birman for useful discussions .
partial support of the physics department at new york university is gratefully acknowledged .
9 h. a. carmona , a. k. geim , a. nogaret , p. c. main , t. j. foster , m. henini , s. p. beaumont , and m. g. blamire , _ phys .
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p. d. ye , d. weiss , r. r. gerhardts , m. seeger , k. von klitzing , k. eberl , and h. nickel , _ phys .
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lett .
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a. matulis , f. m. peeters , and p. vasilopoulos , _ phys .
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j. e. mller , _ phys .
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lett . * 68 * _ , 385 ( 1992 ) . f. m. peeters and p. vasilopoulos , _ phys .
rev.b , * 47 * _ , 1466 ( 1993 ) .
x. wu and s. e. ulloa , _ phys .
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b , * 47 * _ , 10028 ( 1993 ) .
x. wu and s. e. ulloa , _ phys .
rev .
b , * 47 * _ , 7182 ( 1993 ) .
d. r. hofstadter , _ phys .
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for the recent work see o. khn , v. fessatidis , h. l. cui , r. e. selbmann , and n. j. horing , _ phys .
rev .
b , * 47 * _ , 13019 ( 1993 ) ;
p. w. wiegmann and a. v. zabrovin _ phys .
rev .
lett . * 72 * _ , 1890 ( 1994 ) ; y. hatsugai , m. kohmoto , and y .- s .
wu _
phys .
rev .
lett . * 73 * _ , 1134 ( 1994 ) .
j. c. slater , _ phys .
rev .
* 87 * _ , 807 ( 1952 ) .
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this work was supported in part by the eu under grant ist - topqip , `` topological quantum information processing '' ( contract ist-2001 - 39215 ) .
a.c . acknowledges support from marie curie rtn project conquest . v.v . acknowledges also support from epsrc and the british council in austria .
g. m. palma , k .- a .
suominen , and a. k. ekert .
lond . , a * 452 * 567 , ( 1996 ) ; l .- m . duan and g .- c .
lett . , * 79 * 1953 , ( 1997 ) ; p. zanardi and m. rasetti .
* 79 * 3306 , ( 1997 ) ; d. a. lidar , i. l. chuang , and k. b. whaley .
* 81 * 2594 , ( 1998 ) ; a. barenco et al . , * 26 * 1541 , ( 1997 ) .
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we propose a new way to generate an observable geometric phase by means of a completely incoherent phenomenon .
we show how to imprint a geometric phase to a system by `` adiabatically '' manipulating the environment with which it interacts . as a specific scheme we analyse a multilevel atom interacting with a broad - band squeezed vacuum bosonic bath .
as the squeezing parameters are smoothly changed in time along a closed loop , the ground state of the system acquires a geometric phase .
we propose also a scheme to measure such geometric phase by means of a suitable polarization detection .
whenever a pure quantum state undergoes a parallel transport along a closed path , it gathers information on the geometric structure of the hilbert space in which it lies . in this letter
we will show that a a possible way to generate such a parallel transport is by way of an irreversible quantum evolution .
in several models of interaction with the environment there are some `` protected '' subspaces , like the decoherence free subspaces ( dfs ) , which are left unaffected @xcite .
states lying in these subspaces are _ stationary _ , i.e. they do not evolve in time .
a typical example is the ground state of an atomic system , which , trivially , remain unaffected by the coupling with the electromagnetic field .
however , there are situations in which the interaction between a system and an engineered environment can generate non - trivial ground states @xcite . for instance , when a group of atoms collectively interacts with a broad band squeezed vacuum , the highly non - classical correlations which are present in the field are transferred to the atomic system , which relaxes in a complex pure equilibrium state . in such a scenario ,
the control over the engineered reservoir allows an indirect control on the state of the system to which it is coupled @xcite .
of particular interest is the possibility to change in time the reservoir parameters in such a way that the `` protected '' system subspace evolves in a controlled fashion .
here we show that if this change in time is made slowly enough , a state lying in such a subspace evolves coherently and acquires information about the geometry of the space explored . as an explicit example , we consider a suitable multilevel atomic system interacting with a broad band squeezed vacuum . to be more specific let us consider first a three level atom whose interaction with an electromagnetic field in the rotating wave approximation is described by the following hamiltonian : @xmath0 d\omega \text{,}\nonumber\ ] ] where @xmath1 is the free atomic hamiltonian , @xmath2 is the atomic operator describing the absorption of an excitation and @xmath3 is the annihilation operator of the mode with frequency @xmath4 ( @xmath5 . the field , which we treat as a reservoir , is assumed to be in a broad band squeezed vacuum state . in mathematical terms , this is obtained from the ordinary field vacuum state by means of the unitary operator @xmath6 @xmath7 where @xmath8 d\omega\right\}\ ] ] is a multimode squeezing transformation @xcite , which correlates symmetrical pairs of modes around the carrier frequency @xmath9 and @xmath10 is the squeezing parameter , whose polar coordinates @xmath11 and @xmath12 are called phase and amplitude of the squeezing , respectively .
the use of the born markov approximation , justified by the broadband nature of the field , leads to the following master equation for the atomic degrees of freedom @xcite : @xmath13 where @xmath14 , and @xmath15 from ( [ rmatrix ] ) follows that the state @xmath16 with @xmath17 and @xmath18 , satisfies @xmath19 .
in other words , this state is unaffected by the environment , i.e. it is decoherence free .
moreover @xmath20 represents the new ground state , as all the other states of the atomic system relax to it .
as anticipated , the key idea is to smoothly change the squeezing parameter of the field in order to `` adiabatically '' drag a state initially prepared in @xmath21 into @xmath22 , where @xmath23 is the time dependent squeezing parameter .
we will show the existence of an `` adiabatic '' limit such that the transition probability of @xmath24 to the orthogonal subspace vanishes as the rate of change of @xmath25 becomes sufficiently small .
furthermore , we will show that after a cyclic evolution of @xmath25 , the state @xmath26 acquires a geometric phase .
it is worth stressing that this procedure , although reminiscent of the usual adiabatic evolution , is a different physical phenomenon .
the usual adiabatic approximation refers to a coherent evolution , obtained by tuning the parameters of the system hamiltonian , while the `` steering process '' discussed here is achieved manipulating the environment .
the essential difference is that in the latter case the system state can be adiabatically controlled entirely by means of an incoherent phenomenon and no hamiltonian term contributes to its time evolution . to show how this incoherent adiabatic steering process can take place ,
consider the time dependent version of equation ( [ mastereq ] ) where @xmath27 is explicitly dependent on time through @xmath23 .
it is useful to express the equation of motion in the reference frame where @xmath28 is time independent . to this end , consider the following unitary transformation @xmath29 from the basis @xmath30 , @xmath31 , @xmath32 to the time dependent basis @xmath33 , @xmath34 , @xmath35 , where @xmath26 coincides with @xmath35 under this change of frame , the equation of motion becomes @xmath36\text{,}\ ] ] where , in this new frame , @xmath37 , @xmath38 , @xmath39 and @xmath40 is a hamiltonian term arising from the change of picture . moreover , in this frame the lindbladian term , @xmath41 , assumes a simple diagonal form : @xmath42 the main advantage of this transformation is that it allows to formulate clearly the adiabatic condition , since the rate of change of the environment parameters are contained in the operator @xmath43 .
the limit that we are interested in is the one in which the dominant contribution in equation ( [ eq : newmastereq ] ) comes from the incoherent terms , i.e. @xmath44 . an interesting case is the one in which the squeezing amplitude is kept constant while its phase is slowly changed from @xmath45 to @xmath46 .
this adiabatic evolution can be easily achieved by tuning , for example , the carrier frequency , @xmath47 , of the squeezed state slightly off resonance from the two photon transition @xmath48 . by introducing this detuning , @xmath49 , ( assuming @xmath50 )
, the master equation obtained has the form of eq .
( [ mastereq ] ) and ( [ rmatrix ] ) where @xmath51 is replaced by @xmath52 .
hence , a sufficiently small value of @xmath49 determines the required adiabatic evolution . under this condition
the operator @xmath43 assumes the form @xmath53 where @xmath54 and @xmath55 .
we show that , when @xmath56 is small enough , the state @xmath57 is adiabatically decoupled from its orthogonal subspace and a cyclic evolution in @xmath51 results in a geometric phase acquired by @xmath26 depending only , in this case , on the amount of squeezing @xmath58 . note however that , since the steering process is essentially incoherent , any phase information acquired by a superposition of @xmath28 and a state belonging to the orthogonal subspace is inevitably lost , as the latter is subject to decoherence .
the only way to retain such information is to consider an auxiliary level @xmath59 , unaffected by the noise , playing the role of a reference state for an interferometric measurement . for simplicity
assume that @xmath59 is unaffected by the environment during the whole evolution , and , hence , is time independent . as a consequence ,
the action of the unitary transformation @xmath60 on @xmath59 is trivial , and equation ( [ eq : newmastereq ] ) remains essentially unchanged .
the whole information about the geometrical phase and the coherence retained by the system during its evolution is then recorded in the phase and amplitude of the density matrix term @xmath61 , whose evolution is described by the following set of coupled differential equations @xmath62 where @xmath63 .
assume that initially the excited states @xmath30 and @xmath31 of the system are not populated , hence @xmath64 and the coherence @xmath65 evolves as @xmath66 \text{,}\end{aligned}\ ] ] where @xmath67 in the limit @xmath68 we obtain for the coherence @xmath65 @xmath69 where @xmath70 .
we are interested in a cyclic evolution , corresponding to @xmath71 . by retaining only the leading terms in @xmath72 the total evolution at time @xmath73
is given by @xmath74 and @xmath31 and between @xmath31 and @xmath30 is @xmath9 .
the transitions between these level are coupled to the modes @xmath3 of the reservoir .
the reference state @xmath59 is decoupled from the reservoir .
+ ( b ) five - level system , transitions @xmath75 and @xmath76 are coupled to modes @xmath77 and @xmath78 and @xmath79 are coupled to modes @xmath80 of the reservoir . ] where we have substituted @xmath81 with @xmath82 .
finally , going back to the original frame by means of @xmath83 ) , the corresponding coherence @xmath84 is given by : @xmath85 for example a state initially prepared in @xmath86 , after closing the loop , evolves into @xmath87 where @xmath88 .
it is clear from this expression , that in the limit @xmath89 the dominant contribution to the time evolution is just a phase factor @xmath90 , with @xmath91 .
this proves that in the `` adiabatic '' approximation , the system preserves its coherence .
in fact , according to equation ( [ eq : cohe ] ) , the amplitude damping of @xmath92 occurs only when we take into account the first order contribution in @xmath93 , which shows an exponential decay rate of the order of @xmath94 .
this proves that for small @xmath56 the system admits an adiabatic limit , in which the subspace @xmath95 spanned by @xmath96 and @xmath59 is adiabatically decoupled from its orthogonal subspace @xmath97 .
for this reason , @xmath95 is decoupled from the effects of the decoherence , which only affect states lying in its orthogonal subspace . within this approximation , then
, a state prepared in the space @xmath98 is adiabatically transported rigidly inside the evolving subspace @xmath95 .
as a result of this adiabatic steering , when the system is brought back to its initial configuration , the coherence @xmath92 acquires a phase @xmath91 .
this phase can be interpreted as the geometric phase accumulated by the state @xmath96 . by using the canonical formula for the berry phase , it easy to see that the geometric phase of @xmath96 is given by @xmath99
as expected the value of @xmath100 depends only on the squeezing , and vanishes as the squeezing tends to zero .
moreover , notice that the phase @xmath100 is purely geometrical , i.e. there is no dynamical contribution arising from an existing hamiltonian , since , in absence of any steering process , the states inside @xmath101 have a trivial dynamics .
this makes the measurement of this phase a relatively easy task .
usual procedures to measure geometric phases make use of suitably designed techniques to eliminate dynamical phase contributions , such as spin - echo @xcite or parallel transport conditions @xcite . in this setup ,
the geometric phase is the only contribution to the phase accumulated by @xmath26 , and hence , it is straightforward to measure by a suitable interferometric setup .
a simple scheme to measure the geometric phase obtained by such a steering process can be realized with a simple variation of our system .
let us consider the five - level atomic system shown in picture [ fig : sys4l](b ) .
it essentially consists of two replicas of the three - level system discussed above , with the level @xmath31 in common .
the important ingredient is that transitions @xmath102 and @xmath103 are coupled with modes of the reservoir which are different from those coupled to the transitions @xmath104 and @xmath105 .
a simple way to achieve this , is to choose , for example , polarisation selective transitions , say , left - circularly polarised modes for the former transitions and right - circularly polarised for the latter ones .
the complete hamiltonian of such system is : @xmath106 d\omega \text{,}\nonumber\end{aligned}\ ] ] where @xmath107 , and @xmath108 and @xmath109 , and @xmath110 is the annihilation operator of the mode with the energy @xmath4 and polarization @xmath111 .
assume broadband squeezed vacuum states for the set of modes @xmath77 and modes @xmath80 with different squeezing parameters @xmath112 and @xmath113 : @xmath114 where @xmath115 are the analogous of the operator ( [ eq : squeezer ] ) acting on the set of modes @xmath116 .
under the same assumptions which lead to equation ( [ mastereq ] ) we obtain the master equation : @xmath117 where @xmath118 , and @xmath119 .
this system admits a two - dimensional decoherent - free subspace , spanned by states @xmath120 and @xmath121 whose definition is the analogous of state @xmath26 of equation ( [ eq : dfstate ] ) .
we assume again time dependent squeezing parameters @xmath122 , and again we examine the time dependence of the system in a rotating frame , i.e. a frame where the state @xmath123 appear stationary .
this leads to the following master equation for the five - level system in the rotating frame : @xmath124\text{,}\ ] ] where @xmath125 , @xmath126 being the unitary transformation producing the change of frame .
assume again , for simplicity , that the parameters @xmath127 and @xmath128 are kept constant and that @xmath129 . under this assumption , the master equation can be exactly solved .
the solution is analogous to the one obtained for system previously analyzed .
suppose that the system is initially prepared in a coherent superposition of state @xmath130 and @xmath131 , for example : @xmath132 .
at a later time one has @xmath133 with @xmath134 and @xmath135 .
when the parameter @xmath51 closes a loop , at @xmath136 , the coherence has gained a phase @xmath137 which is the difference between the geometric phases @xmath138 acquired by the states @xmath139 , respectively . as in the previous scheme ,
the visibility is reduced by a factor which is linear in the `` adiabatic parameters '' @xmath140 , which guarantees the existence of the adiabatic limit .
the advantage of this modified scheme is that the value of the geometric phases can be readily measured from the polarisation of the light emitted when the system relaxes .
infact , if the value of the squeezing parameters @xmath141 is suddenly switched to zero , the states @xmath139 are no longer decoherece free , and decay to a superposition of the ground states @xmath32 and @xmath142 .
this dissipation process is accompanied by two photon emissions into the reservoir . due to the structure of the interaction ( [ hamiltonian2 ] ) with the reservoir , the photon emitted due to the transitions @xmath143 and @xmath144 , is polarised according to the geometric phase accumulated between @xmath120 and @xmath121 .
for example , if @xmath77 and @xmath80 are right and left circularly polarised modes , respectively , the first dissipation process produces the linearly polarised photon : @xmath145 the detection of the polarisation of the emitted photon makes possible a direct measurement of of the geometric phase .
we have presented a scheme to generate a geometric phase via a completely incoherent control procedure .
this scheme is conceptually different from the usual coherent adiabatic control .
the latter is realized through a smooth evolution of suitable hamiltonians , whereas here , the adiabatic steering is the effect of an externally controlled environment .
the phase generated is purely geometrical , and , therefore , experimentally detectable without resorting to techniques for the elimination of dynamical contributions . due to its very nature , this scheme is immune from unwanted environmental effects .
moreover , like any geometric effects , it presents an inherent degree of robustness against uncertainties in the control parameters .
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i.n . and m.l.s .
contributed equally to this work .
the authors gratefully acknowledge w. venstra for useful discussion , a. di michele and p. sassi for raman analysis .
the authors acknowledge financial support from the european commission ( fpvii , grant agreement no : 318287 , landauer ) .
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the engineering through strain of material properties is very interesting for a wide list of applications , specially for atomically thin membranes made of mos@xmath0 because of its high young modulus and fracture strength .
the modification of the electronic structure and electronic transport under tensile and compressive strain of mos@xmath0 and other transition metal dichalcogenides has been predicted by _
ab initio _ calculations in several works .
a transition from semiconductor to metallic transport has been predicted for single and few layer mos@xmath0 . in this article
we present the observation of this effect on a few layer mos@xmath0 ribbon under a maximum uniaxial tensile strain of 0.14 .
experimental data are accompanied with _
ab initio _ calculations showing that uniaxial strain on few layer mos@xmath0 provokes the closing of the energy band - gap .
molybdenum disulphide , mos@xmath0 , is a layered crystalline solid with an hexagonal structure similar to graphene .
single layer mos@xmath0 is formed by a plane of mo atoms sandwiched and covalently bonded to two planes of s atoms .
few layer and bulk mos@xmath0 is formed by successive stacking of this hexagonal structure through vdw forces . in a bulk form , mos@xmath0 is an indirect semiconductor with energy gap eg= while its single layer counterpart shows a direct transition at the k point with eg=. this material among other transition metal dichalcogenides ( tmds ) has been widely studied in the last years because of their outstanding optical , mechanical and electrical properties along very strong electro - mechanical and opto - mechanical coupling@xcite . in particular
, mos@xmath0 has been suggested to show a transition from semiconductor to metal under mechanical strain@xcite and not only for monolayer mos@xmath0 but also for few - layer mos@xmath0 and even bulk@xcite .
although several studies based on _ ab initio _ calculations have predicted this particular effect no experimental validation has been published until now . in this letter
we report the evidence of the semiconductor - metal transition of mos@xmath0 under tensile strain .
in particular , changes in the direct - current electronic transport properties of mos@xmath0 are characterized for a length mos@xmath0 suspended ribbon under uniaxial tensile strain up to @xmath1=0.14 .
the transition , that occurs at @xmath2 0.12 , is seen as a drastic increment in the current flowing through the device which is related to the closure of the gap that , typically , semiconductors show in their two terminal i - v curve .
first principles band structure calculations of the material are then performed to corroborate the experimental data .
the results shown in this letter paves the way for the development of new micro and nanodevices exploiting the strong electro - mechanical coupling of mos@xmath0 such as strain sensors , straintronic devices , among others .
mos@xmath0 flakes are obtained by mechanical exfoliation from a single crystal @xcite with the `` scotch tape method '' and then transferred to a polydimethylsiloxane ( pdms ) holder as in ref .
@xcite .
an optical microscope is used to locate few layer mos@xmath0 candidates for the analysis . in particular
we focused our attention on the flake reported in fig .
[ f : layers](a ) . to determine the number of layers composing the structure we have performed an optical analysis considering the light transmittance of single - layer mos@xmath0 to be 94.5% @xcite . increasing the number of layer the transmittance decrease linearly proportional to the number of layers , @xmath3 @xcite .
the analysis for the selected flake is reported in fig .
[ f : layers](b ) , where the transmittance profile is relative to the section highlighted in fig .
[ f : layers](a ) .
each horizontal gray line is relative to the expected transmitted light for a specific number of layers starting from the top line corresponding to single layer , @xmath3=1 .
the red curve corresponds to the transmitted light of a 7-layers mos@xmath0 , matching the data obtained for the lighter part of the flake reported in fig .
[ f : layers](a ) .
we also performed raman analysis to further characterize the sample .
mos@xmath0 shows two peaks in the raman spectrum , @xmath4 and @xmath5 , the position and intensity of which have been demonstrated to be indicators of the number of layers @xcite .
fig .
[ f : layers](c ) shows the measured spectrum for the present device ( points ) excited by line and a double gaussian fitted curve ( line ) centered at and which is in good agreement with ref . for @xmath3=7
the relative intensity is @xmath6 , also supporting @xmath3=7 . in order to characterize the semiconductor - to - metal transition of a seven layer mos@xmath0 ribbon ,
we have carried out first principles calculations within density functional theory ( dft ) to investigate the electronic band structure of the material .
we used the siesta package @xcite , norm - conserving pseudopotentials and the perdew - burke - eznerhof parameterization ( pbe ) of the generalized gradient approximation ( gga ) @xcite to obtain the electronic structure of infinite seven layer mos@xmath0 .
an optimized double-@xmath7 polarized basis set to expand the one - electron wave - function is considered . a @xmath8 grid of k - points to sample the brillouin zone is used and the structure is relaxed until all forces are lower than . the unit - cell of stacked mos@xmath0 is shown in fig .
[ f : simulation](a ) .
the values for the energy gap ( eg= ) and the lattice constant ( a= ) are in good agreement with those in the literature@xcite .
strain is introduced to the system by deforming the unit cell along the zig - zag ( x - axis ) and arm - chair ( y - axis ) direction , so @xmath9 .
the evolution of the energy band - gap under tensile strain is shown in fig .
[ f : simulation](c ) decreases as the strain increases .
however , one should note that , while predictions of the pressure coefficients based on dft calculations are very reliable , the band - gaps are notoriously underestimated .
therefore , data shown in fig . [
f : simulation ] ( b ) represent just a trend of @xmath10 . .
a ) atomic structure of stacked mos@xmath0 in the x - y plane , showing bonding between mo and s atoms .
b ) energy gap of seven layer mos@xmath0 as function of applied uniaxial strain along @xmath11 and @xmath12 directions . ] to test the semiconductor to metal transition in our experiment the selected flake has been transferred over a au / cr coated ( ) sio@xmath0 substrate .
the hospital substrate has been mounted on a high precision nanopositioning stage clamping one end of the substrate to the reference frame and the other to the movable part .
the positioning system is capable of a maximum extension of with a pull force of .
thanks to a incision made on the bottom of the substrate the force applied by the stage during elongation is capable of producing a clean crack and split it into two movable parts separated by a gap @xmath13 .
once the substrate is split in two parts the flake is transferred over the trench .
the initial gap was set to @xmath14 determining the length of the metal - semiconductor - metal ( msm ) device to @xmath15 .
after the mos@xmath0 ribbon is transferred , two blobs of epoxy resin mechanically fix both clamps of the ribbon preventing slipping during strain application .
the complete procedure of substrate preparation and transfer of the flake is depicted in detail in fig .
[ f : scheme ] .
flake .
] strain is applied to the msm device by increasing the gap , @xmath13 , being @xmath16 .
the effective value of the gap is directly measured through the integrated positioning sensor of the linear actuator .
a possible way to directly observe semicondutor to metal transition under tensile strain is monitoring the current flowing through the material varying the polarization @xmath17 in a two terminal configuration , _
i.e. _ , the i - v curve
. the typical semiconductor behavior is observed in the suspended unstrained ( @xmath1=0.00 ) mos@xmath0 flake , the i - v curve in this condition is represented by the green curve in fig .
[ f : iv](a ) . in this scenario
the current remains low for @xmath18 . under tensile strain , the current flowing through the flake increases even for @xmath19 , increasing substantially the slope of the curve from to and reaching a linear relation , which corresponds to a metallic behavior ( red curve of fig .
[ f : iv](a ) ) .
the lighter curves presented in fig .
[ f : iv](a ) are relative to intermediate strains between those relative to green and red curves . after increasing the strain beyond @xmath1=0.014
the 7 layer ribbon broke and only the bulk part remained .
the contribution from the bulk mos@xmath0 part of the flake has been then measured and is represented by the dashed line in fig .
[ f : iv](a ) . in this case
no change on the i - v curve was measured as @xmath13 increased .
the i - v curves for unstrained and strained device subtracting the static contribution of the bulk part of the ribbon are presented in fig .
[ f : iv](b ) .
fig .
[ f : iv](c ) show a color map of the current flowing through the device as function of applied strain and polarization voltage , showing an increase in current as function of the voltage for higher applied tensile strains . in fig .
[ f : model](a ) the equivalent circuit of the measured device is presented .
@xmath20 represents the contact resistance , @xmath21 the background impedance that accounts for the bulk mos@xmath0 and parasitic currents . finally , @xmath22 is the strain dependent impedance of the seven layer mos@xmath0 . according to this model
the total conductance of the device can be expressed as : @xmath23 where the conductance @xmath24 is the inverse of the impedance @xmath25 .
we now focus on the conductance of the seven layer mos@xmath0 .
according to ref .
@xcite the conductance of a semiconductor under strain can be expressed as @xmath26 \label{eq:2}\ ] ] where @xmath27 stands for the conductance of the unstrained mos@xmath0 flake , @xmath28 is the boltzmann constant and @xmath29 is the temperature .
the measured conductance of the device is represented by black dots in fig .
[ f : model](b ) .
the fit of eq .
[ eq:1 ] and eq .
[ eq:2 ] , obtained with experimental data is represented by red solid line in fig .
[ f : model](b ) showing a good agreement between experiment and model .
the fitting parameters are reported in table [ t : parameters ] , which are in very good agreement with literature@xcite .
.fitted parameters from the conductance model .
[ cols="^,^",options="header " , ] [ t : parameters ] we have also calculated the gauge factor for the present msm device at @xmath17= ( @xmath30 / \delta \varepsilon $ ] ) to be @xmath31110 in the range 5 - 10% of strain .
the obtained value exceeds the values of conventional and other 2d based strain sensor , like graphene based devices@xcite . = . ] in conclusion , we have measured the semiconductor - metal transition under monoaxial tensile strain of a 7 layer mos@xmath0 ribbon .
an increase of more than an order of magnitude of the current flowing through the device , and a linear i - v relation are obtained at @xmath1=0.014 . a sound model of the conductance as function of the applied strain
is given showing a good agreement between experimental data and theory .
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1 . solar neutrino spectrum in the ssm [ 1 ] .
2 . typical oscillation factor .
3 . typical lwvo solutions [ 8 ] .
eigenvalues of propagation matrix versus density @xmath37 .
5 . the msw bathtub ; suppression factor @xmath95 versus @xmath96 . 6 .
msw solutions [ 16 ] .
typical exotic neutral current effects on msw solutions [ 19 ] .
modulation factors for @xmath7 neutrino spectrum .
electron spectra due to @xmath7 neutrinos [ 16 ] .
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this is an invited review talk , presented at the international conference on non - accelerator particle physics ( icnapp-94 ) , bangalore , india , 2 - 9 january 1994 .
6.5 in 8.5 in .25 in # 1#20=1=to0@xmath00=1=to0#2#1 - 01
@=11 tempcntc citex[#1]#2@fileswauxout tempcnta@tempcntb@neciteaciteforciteb:=#2citeo#1 citeotempcnta > tempcntbciteacitea , tempcnta = tempcntbtempcnta
@=12 = cmssbx10 scaled 2 to * solar neutrino oscillations * + r.j.n .
phillips + _ rutherford appleton laboratory , chilton , didcot , oxon ox11 0qx , england _
* 1 .
introduction*. standard solar models ( ssm ) predict the @xmath1 flux of fig.1 with some uncertainties@xcite . measurements by capture in @xmath2 @xcite , @xmath3 scattering@xcite and capture in @xmath4 @xcite , with differing @xmath5 thresholds , find three different deficits : @xmath6 & observation / ssm[2 ] \\
\nu -e & 7.5 mev & 0.51\pm .07\pm .07 & 0.66\pm .09\pm .16 \\ ^{37}cl & 0.81 mev & 0.29\pm .03\pm .04 & 0.36\pm .04\pm .08 \\
^{71}ga & 0.24 mev & 0.62\pm .10\pm .03 & 0.67\pm .11\pm .04 \\ \end{array}\ ] ] where the first error is experimental , the second is from the ssm .
these numbers suggest a differential suppression , with the top and bottom of the accessible range less suppressed than the middle .
they pose the solar neutrino problem 1994 .
re - tuning the solar model gives no easy solution@xcite .
a lower central temperature would suppress @xmath7 production and the @xmath3 rate , but to explain @xmath2 rates the @xmath8 line must then be obliterated - a bit unlikely given that @xmath7 is made from @xmath8 .
neutrino oscillations offer several possible explanations , that i briefly compare . *
2 . long wavelength vacuum oscillations ( lwvo)*. suppose the weak eigenstate @xmath1 , emitted by @xmath9-decays in the sun , is actually a superposition of two mass eigenstates : @xmath10 with @xmath11 .
the mass eigenstates propagate independently with time t , each picking up a different phase factor @xmath12 , so that after a distance @xmath13 the projection back onto @xmath1 becomes @xmath14 $ ] .
the probability that this evolved state can interact like @xmath1 is then @xmath15 where @xmath16 .
figure 2 illustrates this oscillatory probability .
for @xmath17 there is negligible effect ; for values @xmath18 there are resolvable oscillations ; for values @xmath19 the oscillations are averaged in practice , either by source / detector size or by energy resolution . averaged 2-neutrino oscillations
suppress by at most 1/2 , but n - neutrino mixing can give 1/n ; however this suppression is flat and therefore unsuited to the 1994 solar problem . on the other hand , resolved oscillations provide a strongly varying suppression .
try overlaying figs.1 and 2 ( they have the same horizontal log scale ) .
if we tune @xmath20 such that the first minimum falls around a few mev , the kamiokande @xmath21 rate will be somewhat suppressed ; if we fine - tune to put the @xmath22 line in a minimum , the @xmath2 rate will be somewhat more suppressed ; meanwhile the @xmath4 rate suffers less , since its dominant pp neutrinos encounter only average suppression .
we clearly have the makings of one or more solutions here , with these just - so " oscillations@xcite .
figure 3 shows typical recent fits @xcite in the @xmath23 ) parameter plane ; the disconnected regions put the @xmath8 line in different minima of p. note that for @xmath24 mixing , @xmath25 scattering contributes a bit to the kamiokande signal and helps to explain why it is less suppressed than @xmath2 ; @xmath26 sterile flavour mixing lacks this help and is harder to fit .
two special features arise from lwvo resolved oscillation patterns @xcite .
+ i ) there is an oscillatory modulation on the shape of the high- energy @xmath7 spectrum contribution .
the shape ( though not the magnitude ) of this ssm component is model - independent ; the modulation would be detectable at sno@xcite and super - kamiokande .
see figs .
8,9 at the end .
+ ii ) the @xmath8 line with fixed @xmath5 has oscillatory strength , because the earth - sun distance @xmath27 has small seasonal variations .
this line strength could be measured directly by borexino@xcite ; the effect is diluted in @xmath2 and @xmath4 signals . + ( present @xmath28 , @xmath29 and @xmath30 data constrain ( i ) and ( ii ) rather weakly ) .
but mixing @xmath1 with @xmath31 or @xmath32 affects @xmath1 spectra from supernovae ; sn1987a data may disfavour large @xmath33 @xcite , including solutions like fig.3a . *
3 . oscillations in matter*. coherent forward scattering in matter generates a refractive index and affects propagation@xcite .
z - exchange processes are the same for @xmath1 , @xmath31 , @xmath32 , generating a common phase that can be ignored , but w - exchange contributes only to @xmath34 scattering and significantly changes the propagation equation : @xmath35 where @xmath36 is the vacuum mixing angle , @xmath37 is the electron number density and @xmath38 or @xmath32 ( for sterile @xmath39 see later ) . diagonalizing the propagation matrix above
, we find that the mixing angle in matter @xmath40 depends on @xmath41 : @xmath42 if @xmath43 the mixing is enhanced ; it becomes maximal ( @xmath44 ) where the denominator vanishes - sometimes called a resonance .
as neutrinos travel out from the solar core , the mixing angle @xmath40 and the matter - propagation eigenstates @xmath45 change continuously .
this gives a possibility for efficient @xmath46 conversion via adiabatic level crossing ( the msw effect@xcite ) .
suppose that @xmath41 is far above the resonance value at the point of @xmath1 creation in the solar core ( i.e. @xmath47 ) ; then @xmath48 here .
if subsequent propagation is adiabatic , the local eigenstate components are essentially preserved : @xmath49 .
emerging from the sun , the dominant @xmath50 component becomes the vacuum mass eigenstate @xmath51 if the vacuum mixing angle @xmath36 is small ; thus initial @xmath1 ends up as mostly @xmath31 .
fig.4 shows how the two eigenvalues @xmath52 of the propagation matrix behave versus @xmath37 ; solid lines give the case of no mixing , @xmath53 ; dashed lines show how the eigenvalues cross over when mixing is present .
if @xmath37 changes slowly enough for the mixing to act ( adiabatically ) , the physical state follows the full eigenstates ( dashed lines ) ; but if @xmath54 changes too suddenly , the physical state follows the unmixed eigenstates ( solid lines ) .
there are 2 conditions for adiabatic level crossing .
+ ( i ) central density is above the resonance value : @xmath55 \simeq 10 ^ 5,\ ] ] ( ii ) density changes slowly enough near the resonance for adiabaticity : from the landau - zener approximation we obtain@xcite @xmath56cos2\theta \simeq 3 \times 10 ^ 8 sin^22\theta.\ ] ] the detailed consequences require big calculations , but we need no computer to see the main features .
the msw effect gives a bathtub - shaped suppression factor ; see fig.5 .
the steep left - hand end is determined by the resonance - crossing condition ( i ) ; the sloping right - hand end covers the range where adiabaticity breaks down , determined by condition ( ii ) .
we can choose almost any bathtub we please , versus energy @xmath5 , by selecting @xmath57 to get the left - hand end and @xmath58 to get the length ( and also the depth ) of the bathtub .
however , condition ( ii ) excludes msw effects in the lwvo region .
notice also that efficient @xmath59 conversion does not require big vacuum mixing @xmath36 ; on the contrary , the best conversion is with small @xmath36 .
the msw bathtub offers an immediate explanation of the apparent differential suppression of the @xmath1 spectrum : let the sloping end lie across the @xmath3 scattering range @xmath60 ( moderate @xmath3 suppression ) ; let the flat bottom lie across the rest of the @xmath2 capture range @xmath61 ( more @xmath2 suppression ) ; let the steep end fall near @xmath62 at the top of the @xmath63 spectrum contribution ( less @xmath4 suppression ) .
this simple prescription leads to the best msw solution : @xmath64 with @xmath65 , fig.6a shows a typical recent fit@xcite .
there is also a large-@xmath36 region , not really a good solution but a local @xmath66 minimum , where the bathtub is much shallower and wider .
matter effects can also arise in the earth .
they are not msw ( no chance of adiabatic level crossing ) , just amplified vacuum oscillations through which @xmath31 can convert back to @xmath1 when the sun is below the horizon , giving day / night and summer / winter asymmetries in counting rates .
there are 2 conditions for big earth effects . + ( i ) near - resonant amplification in the earth ( @xmath67 large ) : @xmath68 assuming rock density @xmath69 .
+ ( ii ) matter oscillation wavelength ( @xmath70 ) less than earth diameter ( @xmath71 ) . at resonance
the matter - eigenvalue difference is @xmath72 , giving @xmath73 both these conditions can be approached or satisfied in a small region of @xmath74 for given @xmath75 , but @xmath36 can not be very small . for the kamiokande @xmath3 range , @xmath76 mev , a region near the msw large-@xmath36 solution is sensitive to earth effects ; the absence of a day / night asymmetry@xcite excludes this region ( labelled `` excluded 90% c.l . '' in fig.6 ) .
future experiments will enlarge this region of sensitivity .
similar things can happen for @xmath1 mixing with sterile @xmath39 , but now z - exchange no longer drops out ; coherent @xmath34 and @xmath77 z - exchanges cancel and the net effect is to replace @xmath37 above by @xmath78 where @xmath79 is the neutron number density@xcite .
the critical parameters change a bit and the large-@xmath36 solution vanishes ( fig.6b ) .
three- or four - flavour neutrino mixing offers more complicated possibilities , with more free parameters , that we do not need yet and shall not discuss today .
* 4 . exotic neutral current effects . *
if there are new neutral - current interactions , such as @xmath80 flavour - conserving or flavour - flipping scattering via r - parity - violating b - squark exchanges@xcite , new terms will appear in the matter - propagation matrix .
in the most general case with diagonal and off - diagonal contributions from scattering on @xmath81 distributions in matter , this matrix can be put in the form @xmath82 in the approximation @xmath83 , with just two constant parameters @xmath84 and @xmath85 describing the new physics in units of the standard matter effect . if @xmath86 , we have mixing and oscillations even in the absence of vacuum mixing ( @xmath87)@xcite .
these new terms modify the previous msw solutions .
fig.7 compares @xmath88 solutions in the cases @xmath89 , @xmath90 , @xmath91 ( with @xmath92)@xcite . adding this small exotic mixing
scarcely affects the large-@xmath36 solution but distorts or even splits the small-@xmath36 solution . *
5 . outlook .
* the different two - flavour - mixing scenarios can be distinguished ( or rejected ) by future measurements of the @xmath7 spectrum modulation ( sno , super - kamiokande , see figs.8,9 ) , the @xmath8 0.86 mev line contribution ( borexino ) , and possible day / night effects ( sno , super - kamiokande , icarus ) : @xmath93 furthermore , the charged - current / neutral - current event ratio [ sno , borex , icarus ] will distinguish whether the neutrino flavour mixed with @xmath1 is active ( @xmath94 ) or sterile ( @xmath39 ) .
the problem will become much more clearly defined .
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10 url # 1#1urlprefix[2][]#2 aad g _
et al . _ ( atlas collaboration ) 2012 _ phys.lett . _ * b716 * 129 ( _ preprint _ )
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in this talk the methods and computer tools which were used in our recent calculation of the three - loop standard model renormalization group coefficients are discussed .
a brief review of the techniques based on special features of dimensional regularization and minimal subtraction schemes is given .
our treatment of @xmath0 is presented in some details .
in addition , for a reasonable set of initial parameters the numerical estimates of the obtained three - loop contributions are presented .
the renormalization group ( rg ) proves to be a useful and powerful tool in studying high - energy behavior of the standard model . before the discovery of the higgs boson rg equations ( rge ) were used , among other things , to bound the value of the higgs self - coupling .
however , the bounds significantly depends on the scale at which one expects the appearance of new physics .
the observation of the higgs boson in 2012 @xcite in some sense finalizes the sm since the information about the values of all the sm couplings become available from experiments . due to this fact
, the interest to rg studies of the standard model arises again , but at a new level of precision .
one- and two - loop results for sm beta - functions have been known for quite a long time @xcite and are summarized in @xcite . the first paper with full three - loop calculation of gauge coupling beta - functions within the sm was published in ref .
@xcite .
the next step was carried out by another group from karlsruhe @xcite , which considered , in the specific limit , the three - loop beta - functions for the top quark yukawa coupling , the higgs self - coupling and mass parameter . at this state of things
our group entered the game .
we not only confirmed the results of refs .
@xcite but also provided the three - loop expressions for beta - functions of all the yukawa couplings @xcite corresponding to the fermions of third generation .
contrary to chetyrkin and zoller , we include the dependence on electroweak couplings . in the beginning of 2013
our group started the calculation of missing three - loop terms in beta - functions of the higgs potential parameters .
it turns out that the same problem was considered by the authors from karlsruhe .
unfortunately to us , they managed to obtain and make their results @xcite public a week earlier than our group @xcite .
nevertheless , in such a complicated calculation it is important to have a confirmation from an independent source . in what follows , we are going to discuss the peculiarities of the procedure used to obtain the results published by our group in a series of papers @xcite .
first of all , let us mention that the calculation of the three - loop sm beta - functions requires evaluation of millions of feynman diagrams .
this task definitely requires automatization by means of a computer .
fortunately , all the necessary tools were available on the market , so we only needed to combine them in a proper way .
it is due to nice features of dimensional regularization and minimal subtraction scheme so one can significantly simplify the calculation . since in @xmath1-scheme all the renormalization constants can be extracted from the ultraviolet ( uv ) divergent parts of corresponding green functions , one can modify infra - red ( ir ) structure of the model to simplify the calculation of counter - terms .
this is the essence of the so - called infrared - rearrangement ( irr ) trick , which was originally proposed by vladimirov a.a @xcite .
this kind of modifications can lead to a spurious ir divergences which should be removed consistently by the so - called @xmath2 @xcite .
however , in many practical cases one can avoid this kind of complications .
in our series of paper we made use of two variants of ( naive ) irr procedure .
for the calculation of gauge and yukawa coupling beta - functions it is possible to convert all the required two- and three- point green functions to the massless propagator - type feynman integrals .
it is done via neglecting all internal masses and setting the higgs boson external momenta entering yukawa vertex to zero .
the evaluation of massless three - loop propagators is performed via a form @xcite package ` mincer ` @xcite .
this kind of manipulations does not introduce spurious ir divergences .
moreover , since we are only interesting in uv counter - terms it is possible to work from the very beginning within the unbroken phase of the sm , in which all the fields are massless and the higgs doublet @xmath3 does not have a vacuum expectation value .
the second approach to irr , which was used in calculation of higgs potential parameter beta - functions , is the introduction of an auxiliary mass parameter @xmath4 in every propagator via iterative application of the following formula @xcite @xmath5 where @xmath6 and @xmath7 are linear combinations of internal and external momenta correspondingly .
it is clear that if one applies this kind of decomposition a sufficient number of times the last term can be neglected in the calculation of uv divergences ( after subtraction of subdivergences ) .
it turns out that for the scalar four - point green functions considered only the first term in eq .
is necessary .
consequently , we are left with massive vacuum integrals , which can be calculated by either public ` matad ` package @xcite or private ` babma ` code written by velizhanin . in such an approach
no spurious ir divergences appear so it can be used in the situations when a naive application of the first variant of irr fails . however , the price to pay for this advantage is the necessity of explicit calculation of diagrams with counter - term insertions .
this is due to the fact that one needs to introduce counter - terms for divergences contributing to the auxiliary masses for vector and scalar bosons . for further details
see
@xcite .
it is worth mentioning that we can still exploit the symmetries of the unbroken sm .
for example , all the components of the higgs boson doublet should have the same auxiliary mass counter - term .
the same is true for the su(2 ) gauge bosons .
moreover , as it is stated in ref .
@xcite , the auxiliary mass appearing in the numerator in rhs of eq . can be safely neglected since it can only contribute to the unphysical mass counter - terms .
in addition , one can also skip feynman diagrams with vacuum subdiagrams . in spite of the fact that these subdiagrams are non - zero when the auxiliary mass is introduced , they still can be neglected due to the same reasons . as it was noticed above
there are a lot of diagrams which should be generated and evaluated in order to find three - loop contributions to the considered quantities . in our calculation
we made use of two popular codes , ` feynarts`@xcite and ` diana ` @xcite/`qgraf ` @xcite , which generate necessary diagrams and produce corresponding analytic expressions .
both packages require a model file prepared in a special format to do their job . since we were wanting to simplify the calculation as much as possible we prepared a model file for the unbroken sm in the background field gauge ( bfg ) @xcite .
this kind of gauge allows one to find gauge coupling beta - functions solely from uv - divergences of corresponding gauge field propagators .
we used a very fast ` lanhep ` code @xcite by a. semenov to derive all the sm vertices from the considered sm lagrangian ( see .
@xcite ) in ` feynarts ` notation .
it is worth mentioning that the karlsruhe group made use of alternative package ` feynrules ` @xcite to solve similar problem .
latter on a simple script was written to convert the ` feynarts ` model file to that of ` diana ` .
a typical problem which arises in this kind of calculation is internal momenta identification . in order to evaluate a feynman diagram one needs to use the momenta notation of the chosen code ( ` mincer`/`matad`/`bamba ` ) . in the case of gauge and
yukawa couplings the problem was solved with the help of routine , ` mapmincer ` , which associate with every ` feynarts ` topology the corresponding ` mincer ` topology and distribute ` mincer ` momenta accordingly .
it is worth mentioning that ` mapmincer ` can deal with three - point vertices with one external leg carrying zero momentum ( i.e. , when internal lines have dots ) .
the corresponding routine for ` diana`/`qgraf ` , ` mapdiana ` , performs similar task , but maps every generated topology to fully massive vacuum integrals , which appear after the mentioned``exact '' decomposition of internal propagators . given a model file for ` feynarts`/`diana ` together with the correct mapping of internal momenta to the notation of the utilized three - loop codes
it is tenuous but straightforward to generate and calculate one- , two- , and three - loop contributions to the 1pi green - functions presented in fig .
[ fig:1 ] .
for the su(3 ) color algebra the ` color ` @xcite package was used .
the green functions , considered in the beta - function calculations , together with the corresponding renormalization constants .
left- and right - handed fermions , denoted by @xmath8 , renormalise differently in the sm .
the same is true for background @xmath9 and quantum @xmath10 gauge fields in bfg employed .
the su(2 ) invariance implies the presented equalities , which serve as an additional cross - check . ] from two - point green functions we extract the corresponding wave function renormalization constants in the @xmath1-scheme .
it is worth pointing that the self - energies of both background and quantum gauge fields are considered .
the renormalization constants of the former , @xmath11 , are directly connected to that of gauge couplings and the renormalization of the latter corresponds to the z - factors of three gauge - fixing parameters , @xmath12 , @xmath13 , and @xmath14 , which we keep non - zero during the whole stage of calculation .
the independence of the final results on these parameters serves as an important cross - check of the obtained expressions .
the three - point 1pi functions for the yukawa vertices with neutral higgs bosons @xmath15 and @xmath16 are also presented in fig .
[ fig:1 ] .
it is interesting to note that our calculation explicitly demonstrated that the semi - naive treatment of @xmath0 discussed below is only applicable if one takes into account the gauge anomaly cancellation condition @xmath17 . here
@xmath18 denotes the number of colors .
the beta - functions of the sm parameters are extracted from the corresponding renormalization constants . for the gauge and yukawa couplings we used the following relations @xmath19 where @xmath20 , @xmath21 are su(2 ) and u(1 ) gauge couplings correspondingly
, @xmath22 denotes the strong coupling , and @xmath23 is the yukawa coupling associated with the ( right - handed ) fermion @xmath24 .
both neutral components , @xmath25 , gave the same result , and , thus , provided us with a confirmation of the validity of the obtained expressions .
for the higgs self - coupling it is impossible to use ` mincer ` naively , so feynman diagrams for the four - point functions ( see fig .
[ fig:1 ] ) converted to the fully massive vacuum integrals were calculated with the help of private code ` bamba ` ( by velizhanin , who considered the @xmath26 vertex ) and public package ` matad ` ( by bednyakov and pikelner who considered the fully symmetric @xmath27 vertex ) .
these two independent evaluations lead to the same final expression for the vertex renormalization constants , i.e. , confirming the su(2 ) relation @xmath28 .
a comment is in order on the higgs mass parameter @xmath29 .
it is possible to obtain the corresponding anomalous dimension by considering the renormalization of the @xmath30 composite operator within the unbroken(=massless ) sm ( see , e.g. , @xcite ) .
this kind of result can be found at almost no cost from the calculation of @xmath26 vertex .
it is sufficient to select the diagrams , which have @xmath31 and @xmath32 external fields connected to the same four - point vertex ( see fig .
[ fig:2 ] ) , and weight different contributions with a correct combinatorial factor .
this restricted set of diagrams give rise to the @xmath33}$ ] renormalization constant .
the restricted set of feynman diagrams which is used to obtain the renormalization constant @xmath33}$ ] .
the diagrams of the first type have to be multiplied by @xmath34 . ] at the end of the day , the renormalization constants for @xmath35 and @xmath29 are obtained with the help of the following relations @xmath36}}{z_{h } } \label{eq : lambda_mass_rc}\ ] ] from renormalization constants @xmath37 for the dimensionless sm parameters , @xmath38 it is straightforward to obtain the corresponding beta - functions @xmath39 here @xmath40 is the @xmath1 renormalization scale , @xmath41 is the parameter of dimensional regularization , and @xmath42 denotes the coefficient for the single pole in @xmath43 in the expression for @xmath37 , which enters the relation between the bare parameters @xmath44 and the renormalized ones .
for the anomalous dimension of the higgs mass parameter @xmath29 one can use similar formulae @xmath45 the full analytical results for the considered quantities can be found in ancillary files of the arxiv versions of our papers .
the intermediate expressions , e.g. , all the renormalization constants , can also be obtained , if needed , from the authors .
it is worth mentioning that the beta - functions of all the fundamental sm parameters are free from gauge parameter dependence , which is a crucial test for our calculation .
in addition , the anomalous dimension of the higgs doublet can be of some interest , so we also include the corresponding expression in the ancillary files of ref . @xcite . before going to the results we would like to stop on the problem related to the definition of the @xmath0 matrix withing the dimensional regularization .
it is known from the literature ( see , e.g. , ref .
@xcite and recent explicit calculation @xcite ) that the traces with an odd number of @xmath0 appearing for the first time in the three - loop diagrams require special treatment .
we closely follow the semi - naive approach presented in refs .
@xcite .
first of all , we anticommute @xmath0 to the rightmost position in a fermion chain and use @xmath46 . in the `` even '' traces all @xmath0 are contracted with each other , so the corresponding expressions can be treated naively in dimensional regularization . in `` odd '' traces
we are left with only one @xmath0 .
these traces are evaluated as in four dimensions and produce totally antisymmetric tensors via the relation @xmath47 with @xmath48 . due to the fact that we use both the @xmath0 anticommutativity and the four - dimensional relation , the cyclicity of the trace
should be relinquished @xcite .
one has to choose a certain `` reading prescription '' for an `` odd '' dirac trace , i.e. , start reading a closed fermion chain from a proper place , in order to achieve the correct final result .
however , in our calculations the problem of @xmath0 positioning within the `` odd '' traces is solved implicitly since the diagram generation routines split the traces for us at certain points .
it should be stressed that a non - trivial contribution to the considered quantities can only appear when there are two `` odd '' traces in a diagram , since two @xmath43-tensors should be `` contracted '' with each outher to produce a non - zero effect .
we can distinguish two situations .
two `` odd '' traces in our three - loop calculations can appear either as two internal fermion loops in a diagram with external bosons or , if one considers green functions with two external fermions , one dirac trace from internal fermion loop can be combined with the trace appearing after contraction with an appropriate projector .
it is easy to convince oneself that in the case of two internal traces only triangle subloops with three external vector particles can potentially produce `` eps''-tensors .
however , it is known that in the sm these kind of traces cancel upon summation over all the fermion species due to the absence of gauge anomalies @xcite .
the same is true if one considers fermion self - energies up to three - loops .
it turns our that the non - trivial contribution due to the contraction of two @xmath49-tensor appears from the yukawa vertex ( see fig .
[ fig:3 ] ) .
this kind of diagrams was also considered in @xcite . the diagrams for the fermion - fermion - higgs vertex ( @xmath50 ) , which produce a non - zero contribution due to contraction of two @xmath49-tensors appearing from two dirac traces with @xmath0 .
the ambiguity in positioning of @xmath0 within the traces does not affect the uv - divergent part . ] a final remark about @xmath0 is again related to the fact that the ambiguity in the choice of `` reading '' point in the `` odd '' traces can only spoil our result for the three - loop uv - divergence in the case of the mentioned triangle subgraphs , for which one can set all the `` odd '' traces to zero from the very beginning . for the only non - trivial case all the `` reading '' points are equivalent since the difference gives the contribution @xmath51 which we neglect here . the scale dependence of the relative contributions of different terms in the three - loop corrections to the beta - functions of the ( squared ) gauge and yukawa couplings
. only the most sizable corrections are shown .
the boundary conditions at scale @xmath52 gev are given in . ]
it is obvious , that the resulting expression for the three - loop contributions to the considered renormalization group quantities are too lengthy to be presented here . for the demonstration purposes only
, we would like to show how different relative contributions to the three - loop corrections to the sm coupling beta - functions evolve with renormalization scale @xmath40 ( see fig .
[ fig:4 ] ) . for a reasonable choice of the sm initial running parameters at the scale @xmath52 gev @xmath53 we solve the corresponding rge numerically up @xmath54 gev . with the help of these solutions one can evaluate the three - loop beta - functions at any scale and
find how different terms contribute to to the total value of @xmath55 . from fig .
[ fig:4 ] one can see that the dominant contributions is due to the strong and top yukawa couplings .
however , with the increase of the renormalization scale the su(2 ) coupling can also play a role .
it is fair to say that the most interesting sm beta - function is that of the higgs self - coupling , since from the evolution of the latter one can deduce the so - called `` vacuum stability bound '' ( see recent papers @xcite and references therein ) . in fig .
[ fig:5 ] one can find the same evolution of the relative contributions to @xmath56 .
in addition , the slice @xmath57 of the phase space @xmath58 is presented together with trajectories , obtained with the help of one- , two- , and three - loop evolution from 100 gev to @xmath54 gev .
one can see that for the given set of initial conditions the running @xmath59 is driven to zero faster when one - loop rges are employed instead of two- or three - loop ones .
the difference between two- and three - loop evolution is not sizable , but the fact that with three - loop corrections the scale at which @xmath35 hits zero slightly higher , favours the latter . [
cols="^,^ " , ] to conclude , we obtained the three - loop beta - functions for the sm parameters .
the results for gauge and higgs - potential couplings coincide with that obtained by two karlsruhe groups .
the beta - functions for yukawa couplings were obtained for the first time .
moreover , we established a framework that allow us to carry out a similar calculation within an `` arbitrary '' qft model .
however , it should be stressed that in a self - consistent rge analysis of the chosen model the obtained rges should be accompanied by the so - called threshold ( matching ) corrections ( see , e.g.,refs .
@xcite for the recent sm results ) .
avb is grateful to the organizers and conveniers of the acat2013 workshop for the invitation and for arranging such a nice event .
in addition , we would like to thank m. kalmykov for drawing our attention to the problem and stimulating discussions . the work is supported in part by rfbr grants 11 - 02 - 01177-a , 12 - 02 - 00412-a , and by jinr grant no .
13 - 302 - 03 .
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we argue that the results for the vacuum forces on a slab and on an atom embedded in a magnetodielectric medium near a mirror , obtained using a recently suggested lorentz - force approach to the casimir effect , are equivalent to the corresponding results obtained in a traditional way .
we also derive a general expression for the atom - atom force in a medium and extend a few classical results concerning this force in vacuum and dielectrics to magnetodielectric systems .
this , for example , reveals that the ( repulsive ) interaction between atoms of different polarizability type is at small distances unaffected by a ( weakly polarizable ) medium .
although modifications of the casimir and van der waals forces due to the presence of a medium between the interacting objects were the subject of interest for a long time @xcite , this issue is still of great importance owing to the dominant role of these forces at small distances and to the rapid progress in micro and nanotechnologies .
a common way of extending the lifshitz theory @xcite of the casimir effect @xcite to material cavities is to ( eventually ) employ the minkovski stress tensor @xmath0 when calculating the force @xcite .
recently , however , a lorentz - force approach to the casimir effect was suggested @xcite ( see also @xcite ) in which the relevant stress tensor is of the form ( brackets denote the average with respect to fluctuations and we use the standard notation for the macroscopic field operators ) @xmath1 in this work , we reinterpret the results for the vacuum forces on a slab and on an atom embedded in a semi - infinite magnetodielectric cavity ( see figure 1 ) as recently obtained using @xmath2 @xcite and argue their equivalence to the corresponding results obtained using @xmath3 . also , we derive a general expression for the atom - atom force in a medium and extend a few well - known results for this force to magnetodielectric systems . when calculating @xmath2 for planar geometry @xcite , the zero - temperature force on the slab per unit area in the configuration of figure [ sys ] can be written as @xcite @xmath4 @xmath5(i\xi , k),\ ] ] @xmath6(i\xi , k)\nonumber\\ \fl & + \frac{\hbar}{8\pi^2c^2}\int_0^\infty d\xi\xi^2\mu(n^2 - 1)\int^\infty_0\frac{dkk}{\kappa } \sum_{q = p , s}\delta_q \left[\frac{(1+r^q)^2-{t^q}^2}{(r^q\rme^{-2\kappa d})^{-1 } -r^q}\right](i\xi , k ) , \end{aligned}\ ] ] corresponding to the decomposition of the stress tensor in ( [ t ] ) . here
@xmath7 is the perpendicular wave vector in the cavity at the imaginary frequency , @xmath8 , @xmath9 and @xmath10 are fresnel coefficients for the ( symmetrically bounded ) slab and @xmath11 are those for the mirror .
as seen , the first term in @xmath12 may be combined with the traditional ( minkowski ) force @xmath13 to form a medium - screened casimir force , with the contributions of tm and te polarized waves scaled , respectively , by @xmath14 and @xmath15 , whereas the remaining term in @xmath12 may be regarded as a medium - assisted force @xcite . however , owing to this ( additional ) screening of the force , it turns out that the medium - screened van der waals and casimir forces have a rather unusual dependence on the material parameters of the system @xcite and , therefore , this formal combination demands reconsideration . here ,
we interpret ( 2 ) by recalling that , according to the @xmath16 quantum - field - theoretical approach to the casimir force , the minkowski stress tensor @xmath3 corresponds to the effective stress tensor in a medium which is in mechanical equilibrium @xcite .
if so , the ( unbalanced ) second term in ( [ t ] ) , which leads to @xmath12 , gives a force on the medium .
therefore , the force on the slab is given solely by the traditional force @xmath13 in ( 2 ) , whereas @xmath12 describes a force on the medium .
indeed , note that @xmath12 vanishes when there is no medium , @xmath17 , and is nonzero when there is no slab @xmath18 [ @xmath19 and @xmath20 in ( [ tf ] ) ] , in this case clearly representing the force on a layer of the medium .
the force on an atom near a mirror can be obtained from ( 2 ) by assuming that the slab consists of a thin ( @xmath21 ) layer of the cavity medium with a small number of atoms @xmath22 embedded in it @xcite .
then , from the lorentz - lorenz ( clausius - mossotti ) relation @xcite , it follows that @xmath23,\ ] ] where @xmath24 is the number density and @xmath25 the electric vacuum polarizability of embedded atoms .
assuming a similar relation between @xmath26 , @xmath15 and the effective atomic magnetic polarizability @xmath27 , for the force on an _ embedded _ atom at distance @xmath28 from the mirror we find ( for details , see the derivation of equation ( 14 ) in @xcite and @xmath29 must be made in the last term of this equation . ] ) [ fal ] @xmath30 @xmath31r^p\right .
\nonumber\\ \fl & + \left.\left[\alpha^a_m \left(2\frac{\kappa^2c^2}{\mu\xi^2}-\varepsilon\right ) -\alpha^a_e\mu\right]r^s\right\}(i\xi , k),\end{aligned}\ ] ] @xmath32r^p\right.\\ \fl & \left.+(\mu-1)\left[\alpha_m \left(2\frac{\kappa^2c^2}{\mu\xi^2}-\varepsilon\right ) -\alpha_e\mu\right]r^s+\left(\mu-\frac{1}{\varepsilon}\right ) \left[\alpha_e\mu r^p-\alpha_m\varepsilon r^s\right]\right\}(i\xi , k).\nonumber\end{aligned}\ ] ] as before , this can be combined to give a medium - screened and a medium - assisted force on the atom @xcite .
however , according to the interpretation adopted here , the true force on the atom is @xmath33 , whereas @xmath34 may be regarded as an atom - induced force on the medium ( per atom ) .
we note that ( [ fa ] ) extends ( in different directions ) previous results for the atom - mirror force in various circumstances @xcite by fully accounting for the magnetic properties of the system .
enables one to calculate the force between two atoms in a magnetodielectric medium
. this force can be found by assuming a single - medium mirror consisting of the cavity medium with a small number of , say , type @xmath35 atoms embedded in it , so that @xmath36 and @xmath37 . in this case , the reflection coefficients of the mirror can be approximated by @xcite @xmath38\ ] ] and @xmath39 .
also , the potential energy of the atom @xmath40 is given by @xmath41 , where @xmath42 and @xmath43 is the interaction energy between the atoms @xmath22 and @xmath35 .
using the identity @xmath44 and combining these relations , from ( [ fa ] ) and ( [ rp ] ) we obtain the extension of the feinberg and sucher formula @xcite to material systems @xmath45 where @xmath46 and @xmath47 . at small distances between the atoms ,
@xmath48 ( @xmath49 is an effective upper limit in the integration over @xmath50 @xcite ) , we may let @xmath51 , @xmath52 and @xmath53 in ( [ uab ] ) .
this gives for the atom - atom force @xmath54 at small distances @xmath55 which generalizes the well - known results for the van der waals - london force @xcite .
we observe that , whereas the force between the atoms of the same polarizability type is strongly affected by the surrounding medium , the ( repulsive ) force between the atoms of different polarizability type ( in weakly - polarizable media ) remains the same as in vacuum . at large distances between the atoms ,
the main contribution to the integral in ( [ uab ] ) comes from the @xmath56 region .
approximating the @xmath50-dependent quantities by their static values ( denoted by the subscript @xmath57 ) and then performing the integration , we arrive at @xmath58_0 , \label{faal}\ ] ] which gives the medium corrections to the retarded atom - atom force @xcite and extends the previous considerations of these corrections @xcite to magnetodielectric systems .
as seen , the repulsive component of the force is in a ( weakly - polarizable ) medium simply scaled by @xmath59 , whereas the modification of its attractive part is more complex .
we end this short discussion by noting that the symmetry of ( [ fa ] ) under the transformation : @xmath60 , @xmath61 and @xmath62 [ and consequently that of ( [ uab])-([faal ] ) with respect to the replacements @xmath60 and @xmath61 ] is a consequence of the invariance of the minkowski stress tensor @xmath3 with respect to a duality transformation , e.g. @xmath64 , @xmath65 , @xmath66 and @xmath67 @xcite .
this symmetry is lost in the second term of @xmath2 , so that the force @xmath34 is not invariant under the replacement of the electric and magnetic quantities .
since the medium - screened atom - mirror and atom - atom forces are combinations of the corresponding @xmath13 s and @xmath12 s , this explains the unusual ( asymmetric ) medium effects on these forces found in @xcite . in conclusion ,
when properly interpreted , the result for the vacuum force on an object ( a slab or an atom ) embedded in a medium near a mirror obtained using the lorentz - force approach to the casimir effect agrees with that obtained in a traditional way .
extensions of the well - known results for the atom - atom force to magnetodielectric systems reveals that the ( repulsive ) interaction between atoms of different polarizability type remains ( in weakly - polarizable media ) the same as in vacuum at small and is scaled by @xmath59 at large distances .
medium corrections to the force between atoms of the same polarizability type are more complex and depend on the polarizability type of atoms .
+ the author thanks to c. raabe and anonymous referees for constructive criticism and suggestions .
this work was supported in part by the ministry of science and technology of the republic of croatia under contract no . 0098001 .
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upon a rotation of angle @xmath12 of the spin quantization axis in the @xmath66 plane , the rotated of up and down density operators take the following form in the basis of the z - axis eigenstates .
@xmath67 @xmath68 thus , these are the local mean field operators in a collineal calculation only allowing the development of magnetism along this particular quantization axis .
this choice of mean field operators restrics the avaible hilbert space to the collinear solutions , whereas the inclusion of the exchange term will lead to the full non - collinear solutions .
even though the anisosotrpy calculations were performed on a zigzag edge , localized edge states also appear on chiral nanoribbons @xcite .
thus , this anisotropy effect will appear as well in chiral ( n , m ) nanoribbons due to the local lattice imbalance . as a particular example
, we show the selfconsistent band structure for in - plane and off - plane edge ferromagnetic configurations for a ( 8,1 ) chiral ribbon , as reported in @xcite .
it is observed that , in the same fashion as in the zigzag ribbon , the off - plane magnetic solution host gapless edge states whereas the in - plane solution opens up a gap in the topological edge states , being this last one the lowest energy configuration of the system .
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the independent predictions of edge ferromagnetism and the quantum spin hall phase in graphene have inspired the quest of other two dimensional honeycomb systems , such as silicene , germanene , stanene , iridiates , and organometallic lattices , as well as artificial superlattices , all of them with electronic properties analogous to those of graphene , but much larger spin - orbit coupling .
here we study the interplay of ferromagnetic order and spin - orbit interactions at the zigzag edges of these graphene - like systems .
we find an in - plane magnetic anisotropy that opens a gap in the otherwise conducting edge channels , that should result in large changes of electronic properties upon rotation of the magnetization .
magnetic anisotropy , a technologically crucial property , is driven by spin - orbit interaction , which is normally the underdog in the competition with the other two terms that control ferromagnetism , namely , kinetic and coulomb energy@xcite . as a result ,
magnetic anisotropy energy in conventional ferromagnets is at least 2 orders of magnitude smaller than the curie temperature and the fermi energies ( or the band - gap , in the case of insulators ) . for the same reason ,
transport properties in ferromagnetic metals are only weakly dependent on the magnetic orientation , and typical values for the anisotropic magnetoresistance ( amr ) are below 3 percent@xcite . here
we study magnetic anisotropy in a class of systems for which the balance between these three energy scales is very different from the usual , which leads to two dramatic consequences , very different from conventional ferromagnetism .
first , the conducting properties change from metal to insulator , depending on the magnetization orientation , an effect that , to the best of our knowledge , has never been reported .
second , the magnetic moment magnitude depends strongly on the magnetic orientation , and it can change even vanish in some directions , a phenomenon dubbed colossal magnetic anisotropy@xcite .
the class of systems in question are the zigzag edges of two dimensional honeycomb crystals@xcite whose electronic properties can be described with a tight - binding model with a single orbital per site and kane - mele spin - orbit interactions@xcite .
this includes several materials , such as group iv two dimensional crystals ( graphene@xcite , as silicene@xcite , germanene@xcite , and stanene@xcite ) , the double layer perovskyte iridates @xcite , and metal organic frameworks ( mof ) @xcite .
in addition , given that the existence of non - dispersive edge states occurs at the zigzag edge of any system described with the dirac equation@xcite , the results discussed here should also be valid for the so - called designer dirac fermions formed in `` artificial graphene '' formed by decoration of two dimensional electron gases with honeycomb arrangements@xcite . ignoring spin - orbit and coulomb interactions altogether , these 2d crystals are zero band - gap semiconductors with dirac - like dispersion close to the fermi energy .
zigzag edges in these systems are known to host localized edge states that , when both coulomb and spin - orbit coupling are neglected , are non - dispersive , sub lattice polarized , and lie precisely at the fermi energy , at half - filling@xcite . the ensuing large density of states results in a stoner instability that leads to ferromagnetic order at the edge@xcite .
on the other hand , kane - mele spin - orbit interaction , a second - neighbor spin dependent hopping that conserves the spin component @xmath0 perpendicular to the two dimensional crystal@xcite , has dramatic consequences in these honeycomb crystals .
it opens a topologically non - trivial gap in bulk and the emergence of in - gap spin - filtered dispersive edge states : for a given spin projection @xmath0 , electrons propagate along one direction only , preventing back - scattering even in the presence of time - reversal symmetric disorder .
importantly , the slope of the edge bands is proportional to the kane - mele spin - orbit coupling , which controls thereby the density of states at the fermi energy .
the interplay of spin - orbit and coulomb repulsion on the otherwise non - dispersive edge states leads to the strong magnetic anisotropy effects anticipated above . and
@xmath1 , scaledwidth=50.0% ] to model this kind of systems , we use the so called kane - mele - hubbard hamiltonian@xcite , which provides a minimal model to study the effect of the coulomb interactions on the topologically protected edge states : @xmath2 where @xmath3 are the spin projections of the spin along the axis perpendicular to the to the two dimensional crystal , @xmath4 stands for first neighbor and @xmath5 for second , @xmath6 for clockwise or anti - clockwise second neighbor hopping @xcite . for simplicity
, we neglect the rashba coupling,@xcite . in the case of planar honeycomb systems , such as graphene ,
the rashba term is null .
for buckled group iv crystals , such as silicene , germanene and stanene , the magnitude of the rashba is one order of magnitude lower than the pure spin - orbit.@xcite the hubbard term reads : @xmath7 where @xmath8 denotes the occupation operator of site @xmath9 with spin @xmath10 along an arbitrary quantization axis .
we treat the hubbard interaction in the collinear mean field approximation , enforcing the magnetization to lie along the axis @xmath11 , that we take as the quantization axis ( see fig.[f2]a ) .
this approach permits to study solutions with different @xmath12 and compare their properties .
rotations in the @xmath13 plane leave the results invariant , due to the symmetry of the kane - mele spin - orbit coupling . in general , the coulomb interaction term evaluated in the mean field approximation leads to two self - consistent potentials terms , direct and exchange . in the case of the hubbard model in the collinear approximation ,
only the direct term survives : @xmath14\ ] ] where the notation explicitly shows the spin quantization axis is taken along @xmath15 and @xmath16 stand for the average of the occupation operator calculated within the ground state of the mean field hamiltonian : @xmath17 as usual , this defines a self - consistent problem that we solve by iteration . because of the spin - orbit kane - mele term in @xmath18 , mean field solutions with different @xmath12 are not equivalent .
notice as well that the @xmath19 term is non - diagonal when represented in the basis of eigenstates of @xmath20 and @xmath21@xcite .
we pay special attention to the atomic magnetization , along the @xmath15 in site @xmath9 : @xmath22}{2}\ ] ] and we take @xmath23 . in order to study the zigzag edges it is convenient to study ribbons , that define a one dimensional crystal ( see fig .
1 ) with two edges . a given unit cell of the one dimensional crystal is formed by @xmath24 units of 4 atoms . in the following
we characterize the width of the ribbons by @xmath24 . for finite @xmath25 , and as long as @xmath26 is not too large , we find solutions with ferromagnetic order at the edges .
the magnetic moment calculated self - consistently is non - negligible only at the edge atoms . attending to their mutual magnetization orientation ,
ribbons yield two types of solutions with ferromagnetic edges : parallel ( fm ) and antiparallel ( af ) . for sufficiently wide ribbons
the inter - edge coupling is negligible and both solutions have identical properties .
the first important result of the paper is shown in figure 1 .
whereas off - plane magnetization @xmath27 leads to a conducting solution , found in previous works@xcite , the in - plane magnetization opens a gap .
therefore , transport properties of zigzag edges will change dramatically upon rotation of the magnetization direction , in contrast with conventional metallic ferromagnets .
this metal - insulator transistion will is developed as well in chiral edge ribbons @xcite , which have been widely reported @xcite .
the second important result of the manuscript is shown in fig.[f2]b .
the ground state energy @xmath28 is minimal for @xmath29 , i.e. , for in - plane magnetization , which means that spontaneous magnetic order in this system leads to insulating behavior . .
( a ) scheme of the edge magnetization for two different angles @xmath30 and @xmath12 .
ground state total energy ( per unit cell , with 2 magnetic atoms per cell ) ( b ) , gap(c ) and magnetization ( d ) as a function of @xmath12 .
( e ) and ( f ) evolution of the band structure for different values of @xmath12 for the fm ( e ) and af(f ) configurations .
@xmath31 , u = t , @xmath1.,scaledwidth=50.0% ] the results of figure 1 can be understood as follows . in the absence of magnetic order ,
two spin - filtered in - gap edge states with opposite velocities exist at each edge@xcite , resulting in a two - fold degeneracy ( not shown ) .
ferromagnetic order with @xmath32 breaks time reversal symmetry but does not mix spins .
thus , magnetic order merely yields a spin - dependent shift that breaks the two - fold degeneracy , as seen in fig.[f1]a , for the fm case . for the af configuration
, there is an extra symmetry that restores the double degeneracy : the combined action of spatial inversion , that results in a exchange of the atoms of the two interpenetrating triangular sub lattices @xmath33 and @xmath34 that form the honeycomb , and time reversal , that exchange @xmath10 and @xmath35 , leave the system invariant .
thus the spin @xmath10 band localized at the at the @xmath33 type edge is degenerate with the spin @xmath35 band localized in the opposite edge .
the situation is radically different when the magnetization lies in plane .
representing the self - content potential in the basis of the @xmath36 spin filtered edge states , with spin quantized along the @xmath37 axis , the effect of the in - plane magnetization is to mix bands with opposite spins . as a result ,
a band gap opens at the @xmath38 point where the non - interacting edge bands cross .
the evolution of the bands as the magnetization is rotated from almost off - plane ( left ) to almost in - plane ( right ) is shown if fig .
[ f2])(e , f ) .
it is apparent that the band gap ( fig .
[ f2])(c ) ) is maximal for in - plane magnetization ( @xmath29 ) and null for off - plane @xmath39 .
the preference for in - plane magnetization can also be connected with the variation of the magnitude of the edge magnetic moment with @xmath12 ( fig.[f2]d ) .
these two results naturally explain the fact that the ground state energy is minimal for in - plane magnetization . at half - filling ,
all the valence bands are occupied and the conduction bands are empty .
therefore , increasing the band - gap decreases the total energy .
the gap opening as long as magnetization is not off - plane will certainly have dramatic consequences on the transport properties along the edges .
a result similar to this has been obtained recently@xcite , using a kane - mele model where magnetic order is externally driven , and modeled by a magnetic exchange potential that arises from proximity rather than spontaneously , as discussed here .
the results of figures 1 and 2 are for a specific choice of @xmath40 and @xmath41 , and for a ribbon with @xmath31 sites , wide enough to decouple the two edges .
we now discuss how the results depend , quantitatively , on the specific values of the spin - orbit coupling , ribbon width and @xmath25 .
the evolution of several energy differences between af / fm and in - plane off - plane configurations , as a function of the ribbon width @xmath24 is shown in fig.[f3]a . for large @xmath24
it is apparent that fm and af have the same ground state and anisotropy energy .
in addition , the edge gap ( [ f3]b ) also becomes independent on the magnetic configuration at large width .
.energy scales for different graphene - like honeycomb materials . [
cols="<,^,^,^",options="header " , ] [ table ] in figure ( fig.[f3](c)(e ) ) ) we plot the dependence of the magnetic anisotropy and magnetic moment ( both in and off - plane ) on the magnitude of the spin - orbit coupling @xmath42 , for two different values of @xmath25 .
attending to the difference between the magnitude of the magnetic moment in the off - plane and in - plane cases ( fig.[f3](e ) ) , three different regions are found .
for very small @xmath42 the magnetic moment is the same for out of plane and in - plane magnetization and the magnetic anisotropy energy depends quadratically on @xmath42 . from this standpoint ,
the behavior of the zigzag edge is similar to conventional magnets , although a small gap , open for in - plane magnetization . for wide enough ribbons , the value of this gap is given by @xmath43 where @xmath44 is the bulk gap opened by spin - orbit interaction and @xmath45 is the exchange splitting gap , which is a decreasing function of @xmath46 , giving rise to the curve seen in figure [ f3](d ) .
for intermediate values of @xmath42 it is apparent that the magnetic moment magnitude is different for in - plane and off - plane orientations , but in both cases finite . in this region
the anisotropy energy scales approximately linear with @xmath42 , and the band - gap of in - plane magnetization is still a linear function of @xmath42 . finally , above a
given critical value @xmath47 ( @xmath48 ) for @xmath49 ( @xmath50 ) , the system enters in the so - called colossal@xcite magnetic anisotropy regime , for which magnetic order is only possible in - plane , and the magnetic solutions off - plane do not exist .
increasing spin - orbit coupling beyond this point starts to reduce the band - gap and the magnetic order altogether , which leads to a reduction of the magnetic anisotropy energy ( fig.[f3](e ) ) we thereby expect that graphene , silicene and germanene are in the small @xmath42 region , mof is in the intermediate region and the stanene zigzag edge could show the colossal magnetic anisotropy effect .
notice that in the intermediate region the magnetic anisotropy energy per magnetic atom can be extremely large .
for instance , for stanene , taking @xmath50 , intermediate @xmath51mev and @xmath52 , we obtain of @xmath53mev , significantly larger than record materials such as @xmath54@xcite .
na@xmath55iro@xmath56 and related systems@xcite offer also a fascinating possibility of real tuning of the effective @xmath46 by strain,@xcite , which would make it possible to build devices with strain - tunable anisotropy .
given the spread of estimates of the actual values of @xmath25 for a given material , as well as the fact that it different substrates can result in different values of @xmath25 , we address the question of how the results above depend on the strength of the on - site coulomb repulsion @xmath25 . at finite @xmath46
there is a critical @xmath57 below which the edges are non - magnetic@xcite , and a second @xmath58 above which the entire honeycomb lattice becomes antiferromagnetic@xcite . for @xmath59
only the edge is magnetic , and its magnetic anisotropy energy is a non - monotonic function of @xmath25 .
it increases first , reflecting the increase of the magnetic moment , and then it decreases slightly , reflecting the reduction of the ratio @xmath26 . as @xmath25 approaches @xmath60 the magnetic anisotropy overshoots because the bulk becomes magnetic as well . .
( b ) gap , for in - plane magnetization solution , for fm and af solutions , as a function of @xmath24 .
( c , d , e ) evolution with the strength of the spin - orbit coupling @xmath46 : anisotropy energy ( c ) , gap ( d ) and edge magnetization ( e ) .
( f ) evolution of the anisotropy energy with the on - site hubbard interaction . in ( a - b ) @xmath50 and @xmath61 . in ( c - d ) @xmath31 .
, scaledwidth=50.0% ] we now discuss the physical effects not covered within our two main approximations , namely , treating the interactions at the mean field level and ignoring the rashba spin - orbit term . in one dimension ,
collective spin fluctuations are expected to destroy the infinitely long - range order described by mean field theory .
still , for ribbons shorter than the spin correlation length @xmath62 the mean field theory provides a fair description , very much like density functional theory describes properly the magnetization of clusters and nanomagnets .
the spin correlation length @xmath62 in graphene edges , calculated within the spin wave approximation and ignoring spin - orbit coupling@xcite , is @xmath63 for @xmath64 .
the magnetic anisotropy barrier to rotate the spins out of plane for the approximately 15 tin atoms of a zigzag stanene edge that long would be @xmath6560 mev .
inclusion of the rashba coupling would have two consequences .
first , lack of inversion symmetry would split the bands in the case of af configurations .
second , it would break the in - plane @xmath13 magnetic symmetry at the edges . in conclusion
, we have studied the magnetic anisotropy of the ferromagnetic phase of the zigzag edges of graphene and graphene - like systems , that can be described with a single orbital hubbard model model on a honeycomb lattice with spin - orbit coupling described with the kane - mele hamiltonian .
this includes a large class of two dimensional crystals , such as silicene@xcite , germanene@xcite , stanene@xcite , iridates@xcite and metal organic frameworks@xcite .
since the electronic dispersion of the non - interacting edge states is fully determined by the spin - orbit coupling , the resulting magnetic anisotropy effects , computed within a mean field approximation , turn out to be very strong : the system undergoes a metal to insulator transition when the magnetization is rotated out of the normal and for large values of @xmath46 the magnetic solutions are only stable for in - plane magnetization . for all values of the spin - orbit interaction
we find that the ground state energy occurs for in - plane magnetization and the edge states are gapped .
jfr acknowledges financial supported by mec - spain ( fis2010 - 21883-c02 - 01 ) and generalitat valenciana ( acomp/2010/070 ) , prometeo .
this work has been financially supported in part by feder funds .
we acknowledge financial support by marie - curie - itn 607904-spinograph .
30 natexlab#1#1bibnamefont # 1#1bibfnamefont # 1#1citenamefont # 1#1url # 1`#1`urlprefix[2]#2 [ 2][]#2 , _ _ ( , , ) .
, , , , , * * , ( ) . , * * , ( ) . , , , , * * , ( ) . , * * , ( ) . , , , , , , , , , * * , ( ) . , , , , , , * * , ( ) . , , , , , , , , * * , ( ) . , , , , * * , ( ) . , , , , , , , , * * , ( ) . , , , , , ,
* * , ( ) . , , , * * , ( ) .
, , , * * , ( ) . , * * , ( ) . , , , , , * * , ( ) . , , , , , * * , ( ) , issn , . , , ,
, * * , ( ) . , , , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) .
see supplemental material at [ url ] which includes ref.@xcite for details of the collineal mean field antsaz and the anisotropy effect in chiral edges .
d. gonsalbez - martinez , d. soriano , j. j. palacios , j. fernandez - rossier , solid state .
communications 151 , 10751083 , ( 2011 ) , , , , , , , , , , , * * , ( ) .
, , , , , , , , , , , * * , ( ) . , * * , ( ) . , , , , ( ) , . , , , * * , ( ) , . , , ,
* * , ( ) . ,
( ) , . , , , , , , * * , ( ) . , , , , , * * , ( ) . , , , * * , ( ) . , , , * * , ( ) . , * * , ( ) . * supplemental material * in this supplemental material we present the mean field operators needed to perform an angle dependent collineal mean field hubbard calculation , as well as the angle dependent ground states of chiral hubbard kane - mele honeycomb ribbons
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we thank d.m . brink for useful discussions . this research was supported by the monbusho scholarship and the international scientific research program : joint research : contract number 09044051 from the japanese ministry of education , science and culture , and by the u.s .
department of energy under grant no .
de - fg03 - 00-er41132 .
w. reisdorf , f.p .
hessberger , k.d .
hildenbrand , s. hofmann , g. mnzenberg , k.h .
schmidt , j.h.r .
schneider , w.f.w .
schneider , k. smmerer , g. wirth , j.v .
kratz and k. schlitt , nucl .
a * 438 * , 212 ( 1985 ) .
fig . 2 + comparison of the experimental data and the results of the orientation average ( the dashed line ) and the exact coupled channels ( the solid line ) calculations for the @xmath32o+@xmath0sm fusion reaction .
the top and bottom panels are the fusion excitation function and the fusion barrier distribution , respectively .
the data are taken from ref .
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in heavy - ion fusion reactions involving a well deformed nucleus , one often assumes that the orientation of the target nucleus does not change during the reaction .
we discuss the accuracy of this procedure by analyzing the excitation function of the fusion cross section and the fusion barrier distribution in the reactions of @xmath0sm target with various projectiles ranging from @xmath1c to @xmath2ar .
it is shown that the approximation gradually looses its accuracy with increasing charge product of the projectile and target nuclei because of the effects of finite excitation energy of the target nucleus .
the relevance of such inaccuracy in analyzing the experimental data is also discussed .
it is now well established that nuclear intrinsic degrees of freedom strongly influence the fusion cross section in heavy - ion reactions at energies near and below the coulomb barrier @xcite .
typical examples include the rotational excitation of a deformed target nucleus which leads to a large enhancement of the fusion cross section at low energies .
a characteristic in this case is that the excitation energy of the rotational motion is often much smaller than the curvature of the fusion barrier , which determines the time scale of the fusion process .
this is the case when , e.g. , one of the deformed rare earth nuclei or actinides is the target nucleus .
one then often calculates the fusion probability for each partial wave @xmath3 following @xcite @xmath4 where @xmath5 is the orientation of the deformed target , which will be specified later more precisely .
we call this the orientation average formula . the upper index @xmath6 stands for the sudden tunneling approximation .
@xmath7 is the fusion probability for a given orientation .
it is determined by solving a one - dimensional schrdinger equation for the relative distance between the projectile and target @xmath8 , where the potential consists of the nuclear and coulomb components given by @xmath9 } \label{nucl } \\ v_c(r,\theta)&=&\frac{z_pz_te^2}{r}+\sum_{\lambda } \left[\beta_{\lambda}+ \frac{2}{7}\sqrt\frac{5}{\pi } \beta_{\lambda}^2\,\delta_{\lambda,2}\right ] \frac{3z_pz_te^2}{2\lambda+1}\frac{r_t^\lambda}{r^{\lambda+1 } } y_{\lambda 0}(\theta ) \label{coul}\end{aligned}\ ] ] with the angle dependent radius of the target nucleus @xmath10 . \label{radius}\end{aligned}\ ] ] in these equations @xmath11 is the depth parameter of the nuclear potential between the projectile and target , @xmath12 ( @xmath13 ) and @xmath14 ( @xmath15 ) are the radius and the atomic number of the projectile ( target ) , @xmath16 is the deformation parameter of the target nucleus of multipolarity @xmath17 , and @xmath18 the surface diffuseness parameter .
the second order term is retained only for the quadrupole coupling in the coulomb interaction . as in almost all analyses of heavy - ion fusion reactions at sub - barrier energies ,
we adopt the no - coriolis approximation and work in the rotating coordinate frame @xcite , where the z - axis is taken to be parallel to the radial vector of the relative motion between the projectile and the target nuclei .
also , we assume an axial symmetry for the target nucleus .
the parameter @xmath5 is then the angle between the symmetry axis and the z - axis in the rotating frame .
eq .
( [ orient ] ) is quite appealing and provides a simple understanding of the large enhancement of the fusion cross section at sub - barrier energies .
it states that the deformation and the associated rotational excitation of the target nucleus result in a distributed fusion barriers , where some of them will be lower than that of the bare potential in the absence of these degrees of freedom .
the same idea of the orientation average is still correct if one replaces the gauss integral in eq .
( [ orient ] ) by an appropriate summation even when the rotational band is truncated at a certain angular momentum @xmath19 as is always the case in nuclear physics .
if only @xmath20 is present , the corresponding sum is given by the @xmath21 points gauss quadrature @xcite .
the orientation average procedure is exact only in the degenerate spectrum limit , i.e. , when the excitation energy of the rotational motion is zero , which is not the case in actual nuclei .
it is , therefore , important to examine the applicability of this formula before one makes a quantitative analysis of the experimental data .
a step towards this direction has been undertaken in ref .
@xcite , where an analytic formula has been derived to modify the fusion probability in the sudden tunneling approximation by taking the finite excitation energy of the rotational motion into account as @xmath22p^s_j(e ) \label{analy}\end{aligned}\ ] ] where @xmath23 is the excitation energy of the first excited 2@xmath24 state of the rotational band .
@xmath25 , @xmath26 being the curvature of the fusion barrier , is the tunneling time and @xmath27 is a measure of the strength of the channel coupling which causes the rotational excitation of the target nucleus during the collision . in ref .
@xcite , it was identified with the coupling strength at the fusion barrier @xmath28 .
eq .
( [ analy ] ) is useful to qualitatively discuss the conditions to apply the orientation average formula .
it clearly shows that the excitation energy of the rotational motion should be much smaller than the curvature of the fusion barrier .
it also shows that the coupling strength also governs the validity of the formula . despite these advantages ,
as we see later , eq .
( [ analy ] ) is not tolerable for quantitative discussions .
this is partly because eq .
( [ analy ] ) has been derived in a perturbation theory , and also by ignoring the radial dependence of the coupling form factor . in heavy - ion fusion reactions ,
the latter is a very crude approximation , because the nuclear and coulomb couplings interfere leading to a strong radial dependence of the coupling form factor@xcite . nevertheless , for future reference
, we show in table i the exponential factor in eq .
( [ analy ] ) , which we call the dissipation factor , for several systems to be examined later .
table ii lists the bare nuclear potential parameters we used and the coulomb barrier properties , where the radius parameter @xmath29 has been introduced as @xmath30 . in this paper
we compare the results of the orientation average formula with the numerical solutions of the corresponding coupled channels equations , which are obtained using the computer code ccfull@xcite by keeping the finite excitation energy , i.e. 0.082 mev , of the target nucleus @xmath0sm .
we call the latter as the exact coupled channels calculations .
we keep only the quadrupole deformation in the target , i.e. , @xmath20 , for simplicity .
all the projectiles are treated as inert .
we truncate the rotational band of @xmath0sm at @xmath19=20@xmath31 member for all reactions . according to the table of isotopes @xcite ,
the highest member which has been so far experimentally observed is 16@xmath24 .
we have confirmed that the comparison between the exact coupled - channels and the orientation average calculations does not almost change beyond @xmath19=12@xmath24 .
we also remark that the coupled - channels calculations almost converge at @xmath19=12@xmath31 member@xcite for the @xmath32o and @xmath0sm reactions , but higher levels introduce non - negligible effects for heavy projectiles . both the orientation average and the exact coupled channels calculations give the fusion cross section which is a monotonically increasing function of energy . in order to facilitate to see the accuracy of the orientation average formula , we plot in fig .
1 the ratio of the fusion cross section calculated by solving the exact coupled channels equations to that obtained by the orientation average formula .
the figure clearly shows that the deviation of the results of the orientation average formula from those of the exact calculations gradually increases with the charge product of the projectile and target nuclei .
this behaviour is consistent with table i , which has been obtained based on eq .
( [ analy ] ) , though the actual deviation is much larger than what eq .
( [ analy ] ) predicts .
we also note that the deviation is significant even for light projectiles at energies below the coulomb barrier .
an important question is whether this significantly affects the understanding of the mechanism of heavy - ion fusion reactions and the structural informations such as the deformation parameters extracted from the data analyses . in this connection , we compare in fig .
2 the experimental data of the excitation function of the fusion cross section and the fusion barrier distribution @xcite with the theoretical results calculated by the orientation average formula ( the dashed line ) and by the exact treatment of the corresponding coupled - channels equations ( the solid line ) for @xmath32o+@xmath0sm reaction .
the second derivative of the cross section times the bombarding energy has been calculated with the point difference method with the interval @xmath33 mev .
similar comparison is done in fig .
3 for @xmath2ar+@xmath0sm reaction .
the bare coulomb barrier is at @xmath34 mev and 127.57 mev for @xmath32o+@xmath0sm and @xmath2ar+@xmath0sm reactions , respectively . in these calculations
the deformation parameter @xmath35 of @xmath0sm was determined to be 0.32 by fitting the data of @xmath32o+@xmath0sm fusion reactions .
the same deformation parameter has been used for @xmath2ar+@xmath0sm reactions as well , though the effective optimum values can differ in two reactions .
fig .
2 shows that the difference between two theoretical lines is much smaller than their deviation from the experimental data .
this indicates that one can safely use the orientation average formula to data analyses of @xmath32o+@xmath0sm fusion reactions , even though its deviation from the exact calculation can be noticeable at low energies as shown in fig .
1 . the top panel of fig .
3 shows that the deviation between the exact and the orientation average calculations is not so drastic even for @xmath2ar+@xmath0sm reaction in this semilogarithmic plot . on the other hand ,
the situation is different for the fusion barrier distribution shown in the lower panel .
the difference between the two calculations can be more easily recognizable than the case for @xmath32o+@xmath0sm fusion reactions .
the present theoretical calculations still underestimate the fusion cross sections compared with the experimental data for the @xmath2ar+@xmath0sm reaction at low energies
. this will be partly because we ignored the projectile excitations , whose excitation energy is as small as 1.46 mev . in summary
, we have studied the accuracy of the orientation average formula for fusion cross sections between a spherical projectile and a well deformed target by comparing its results with those of the exact coupled channels calculations .
we found that the results of the orientation average formula significantly differ from those of the exact numerical solution of the corresponding coupled channels equations for systems with a large charge product of the projectile and target nuclei .
this suggests the necessity of the proper coupled - channels calculations beyond the orientation average formula for these systems in order to properly identify the role of various channel coupling effects and to extract reliable informations on nuclear structure , especially through the analysis of the fusion barrier distribution of high precision data .
unexpectedly , we observed a significant deviation of the orientation average formula from the exact calculations even for light projectiles in the energy region well below the coulomb barrier .
this deviation is , however , much smaller than the deviation of these two calculations from the experimental data . in this sense ,
the orientation average formula is safely applicable to light systems .
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in the * dense @xmath0 loop model * , a square lattice is tiled with the following two kinds of square tiles known as * plaquettes * , denoted symbolically by @xmath1 and @xmath2 :
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the dense @xmath0 loop model is a statistical physics model with connections to the quantum xxz spin chain , alternating sign matrices , the six - vertex model and critical bond percolation on the square lattice .
when cylindrical boundary conditions are imposed , the model possesses a commuting family of transfer matrices .
the original proof of the commutation property is algebraic and is based on the yang - baxter equation . in this paper
we give a new proof of this fact using a direct combinatorial bijection .
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we would like to thank w. vogelsang for valuable discussions .
the work of j.k . was supported by the grant - in - aid for scientific research no .
the work of k.t . was supported by the grant - in - aid for scientific research no .
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we calculate qcd corrections to transversely polarized drell - yan process at a measured @xmath0 of the produced lepton pair in the dimensional regularization scheme . the @xmath0 distribution is discussed resumming soft gluon effects relevant for small @xmath0 .
hard processes with polarized nucleon beams enable us to study spin - dependent dynamics of qcd and the spin structure of nucleon .
the helicity distribution @xmath1 of quarks within nucleon has been measured in polarized dis experiments , and @xmath2 of gluons has also been estimated from the scaling violations of them .
on the other hand , the transversity distribution @xmath3 , i.e. the distribution of transversely polarized quarks inside transversely polarized nucleon , can not be measured in inclusive dis due to its chiral - odd nature,@xcite and remains as the last unknown distribution at the leading twist .
transversely polarized drell - yan ( tdy ) process is one of the processes where the transversity distribution can be measured , and has been undertaken at rhic - spin experiment .
we compute the 1-loop qcd corrections to tdy at a measured @xmath0 and azimuthal angle @xmath4 of the produced lepton in the dimensional regularization scheme . for this purpose
, the phase space integration in @xmath5-dimension , separating out the relevant transverse degrees of freedom , is required to extract the @xmath6 part of the cross section characteristic of the spin asymmetry of tdy.@xcite the calculation is rather cumbersome compared with the corresponding calculation in unpolarized and longitudinally polarized cases , and has not been performed so far .
we obtain the nlo @xmath7 corrections to the tdy cross section in the @xmath8 scheme .
we also include soft gluon effects by all - order resummation of logarithmically enhanced contributions at small @xmath0 ( `` edge regions of the phase space '' ) up to next - to - leading logarithmic ( nll ) accuracy , and obtain the first complete result of the @xmath0 distribution for all regions of @xmath0 at nll level .
we first consider the nlo @xmath7 corrections to tdy : @xmath9 , where @xmath10 denote nucleons with momentum @xmath11 and transverse spin @xmath12 , and @xmath13 is the 4-momentum of dy pair . the spin dependent cross section @xmath14 is given as a convolution @xmath15 where @xmath16 is the factorization scale , and @xmath17\ ] ] is the product of transversity distributions of the two nucleons , and @xmath18 is the corresponding partonic cross section .
note that , at the leading twist level , the gluon does not contribute to the transversely polarized process due to its chiral odd nature .
we compute the one - loop corrections to @xmath18 , which involve the virtual gluon corrections and the real gluon emission contributions , e.g. , @xmath19 , with @xmath20 .
we regularize the infrared divergence in @xmath21 dimension , and employ naive anticommuting @xmath22 which is a usual prescription in the transverse spin channel.@xcite in the @xmath8 scheme , we eventually get,@xcite to nlo accuracy , @xmath23 , \label{cross section}\end{aligned}\ ] ] where @xmath24 with @xmath25 , @xmath26 is the rapidity of virtual photon , and @xmath4 is the azimuthal angle of one of the leptons with respect to the initial spin axis . for later convenience
, we have decomposed the cross section into the two parts : the function @xmath27 contains all terms that are singular as @xmath28 , while @xmath29 is of @xmath7 and finite at @xmath30 .
writing @xmath31 as the sum of the lo and nlo contributions , we have@xcite @xmath32 , and @xmath33 \nonumber\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+ & & \!\!\!\!\!\!\!\!\left ( \frac{1}{(q_t^2)_+ } + \delta ( q_t^2 ) \ln \frac{q^2}{\mu_f^2 } \right)\!\!\ ! \left [ \int^1_{x_1 ^ 0 } \frac{d z}{z } \delta p_{qq}^{(0 ) } ( z)\ \delta h \left ( \frac{x_1 ^ 0}{z } , x_2 ^ 0 ; \
\mu_f \right ) + ( x_1 ^ 0 \leftrightarrow x_2 ^ 0 ) \right ] \biggr\ } , \label{eq : x}\end{aligned}\ ] ] where @xmath34 are the relevant scaling variables with @xmath35 , and @xmath36 is the lo transverse splitting function.@xcite in ( [ eq : x ] ) , the terms involving @xmath37 come from the virtual gluon corrections , while the other terms represent the recoil effects due to the real gluon emissions . for the analytic expression of @xmath29 , see ref.@xcite .
eq .
( [ cross section ] ) gives the first nlo result in the @xmath8 scheme .
we note that there has been a similar nlo calculation of tdy cross section in massive gluon scheme.@xcite we also note that , integrating ( [ cross section ] ) over @xmath0 , our result coincides with the corresponding @xmath0-integrated cross sections obtained in previous works employing massive gluon scheme@xcite and dimensional reduction scheme,@xcite via the scheme transformation relation.@xcite the cross section ( [ cross section ] ) becomes very large when @xmath38 , due to the terms behaving @xmath39 and @xmath40 in the singular part @xmath27 .
it is well - known that , in unpolarized and longitudinally polarized dy , large `` recoil logs '' of similar nature appear in each order of perturbation theory as @xmath41 , @xmath42 , and so on , corresponding to ll , nll , and higher contributions , respectively , and that the resummation of those `` double logarithms '' to all orders is necessary to obtain a well - defined , finite prediction of the cross section.@xcite because the ll and nll contributions are universal,@xcite we can work out the all - order resummation of the corresponding logarithmically enhanced contributions in ( [ cross section ] ) up to the nll accuracy , based on the general formulation@xcite of the @xmath0 resummation .
this can be conveniently carried out in the impact parameter @xmath43 space , conjugate to the @xmath0 space . as a result
, the singular part @xmath27 of ( [ cross section ] ) is modified into the corresponding resummed part , which is expressed as the fourier transform , @xmath44 here @xmath45 , and the large logarithmic corrections are resummed into the sudakov factor @xmath46 with @xmath47 .
the functions @xmath48 , @xmath49 as well as the coefficient functions @xmath50 are calculable in perturbation theory , and at the present accuracy of nll , we get:@xcite @xmath51 , @xmath52 , @xmath53 . we have utilized a relation@xcite between @xmath48 and the dglap kernels in order to obtain the two - loop term of @xmath48 .
the other contributions have been determined so that the expansion of the above formula ( [ resum ] ) in powers of @xmath54 reproduces @xmath27 of ( [ cross section ] ) , ( [ eq : x ] ) to the nlo accuracy .
eq .
( [ cross section ] ) with ( [ resum ] ) presents the first result of the nll @xmath0 resummation formula for tdy .
the nlo parton distributions in the @xmath8 scheme have to be used .
one more step is necessary to make the qcd prediction of tdy . similarly to other all - order resuumation formula
, our result ( [ resum ] ) is suffered from the ir renormalons due to the landau pole at @xmath55 in the sudakov factor , and it is necessary to specify a prescription to avoid this singularity . here
we deform the integration contour in ( [ resum ] ) in the complex @xmath43 space , following the method introduced in the joint resummation.@xcite obviously prescription to define the @xmath43 integration is not unique reflecting ir renormalon ambiguity , e.g. , `` @xmath56 prescription '' to `` freeze '' effectively the @xmath43 integration along the real axis is frequently used.@xcite the renormalon ambiguity should be eventually compensated in the physical quantity by the power corrections @xmath57 ( @xmath58 ) due to non - perturbative effects .
correspondingly , we make the replacement @xmath59 in ( [ resum ] ) with the `` minimal '' ansatz for non - perturbative effects , @xcite @xmath60 with a non - perturbative parameter @xmath61 .
fig.1 shows the @xmath0 distribution of tdy at @xmath62 gev , @xmath63 gev , @xmath64 , and with a model for the transversity @xmath3 that saturates the soffer s inequality at a low scale.@xcite solid line shows the nlo result using ( [ cross section ] ) , and the dashed and dot - dashed lines show the nll result using ( [ cross section ] ) , ( [ resum ] ) , @xmath60 , with @xmath65 gev@xmath66 and @xmath67 , respectively .
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this work was supported by the jsps core - to - core program
non - equilibrium dynamics of soft matter and information " and jsps kakenhi grant number 24340100 .
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during polymer translocation driven by e.g. voltage drop across a nanopore , the segments in the cis - side is incessantly pulled into the pore , which are then pushed out of it into the trans - side .
this pulling and pushing polymer segments are described in the continuum level by nonlinear transport processes known , respectively , as fast and slow diffusions . by matching solutions of both sides through the mass conservation across the pore
, we provide a physical basis for the cis and trans dynamical asymmetry , a feature repeatedly reported in recent numerical simulations .
we then predict how the total driving force is dynamically allocated between cis ( pulling ) and trans ( pushing ) sides , demonstrating that the trans - side event adds a finite - chain length effect to the dynamical scaling , which may become substantial for weak force and/or high pore friction cases .
# 1 the dynamics of translocation , i.e. , the polymer passage through a narrow pore , has been actively studied more than a decade @xcite .
in addition to its relevance to cellular biological processes , i.e. , biopolymer transports in cells , the phenomenon has found promising applications in genome sequencing and related technology as a nanopore sensor @xcite . being a unique mode of the molecular transport inherent in long flexible polymers ,
there have been numerous attempts to characterize the process and to uncover the underlying physics behind it @xcite .
such efforts have led to the consensus that the _ tension _ is a key physical quantity , which is created in the polymer by the spontaneous segment motions and/or the action of the driving force @xcite .
now we have a fairly good description of the translocation process , which may be categorized according to the magnitude @xmath0 of the bias as follows .
( i ) unbiased regime : the tension imbalance across the pore arising from the segment exchange between cis and trans sides creates a long term power - law decaying memory @xcite .
this retards the process , leading to the anomalous ( sub- ) diffusion of the so - called translocation coordinate @xcite .
( ii ) weakly driven regime : with weak enough force , the polymer conformation is essentially in equilibrium , and the same physics as that in the unbiased regime applies .
the linear response theory then leads to the anomalous drift of the translocation coordinate @xcite .
( iii ) driven regime : when the rate of the driving operation is faster in comparison with the terminal time of the polymer , the genuine nonequilibrium dynamics shows up @xcite .
the action of the driving force is transmitted along the chain in the cis - side , which induces the sequential conformational deformation . here to make the connection to the regime ( ii ) clearer , it is convenient to introduce the length scale @xmath1 and the corresponding time scale @xmath2 @xcite with @xmath3 , @xmath4 being the segment size , thermal energy , respectively , and @xmath5 is a dynamical exponent .
the shortest ( segment scale ) time scale is denoted as @xmath6 , where @xmath7 is the solution viscosity .
initially , the process takes place locally in the close proximity to the pore , and the equilibrium treatment is valid up to @xmath8 , during which the tension is transmitted up to @xmath9-th monomer according to the mechanism in regime ( ii ) .
the subsequent larger scale process is governed by the driving force , thus , described by the nonequilibrium regime ( iii ) , which eventually dominates the scaling limit .
this consideration provides the border between the regimes ( ii ) and ( iii ) as @xmath10 , the threshold force being rather weak for long chains , where @xmath11 is the polymerization index of the chain . for strong enough force ( @xmath12 )
, one only has the regime ( iii ) in the entire process .
we remark here an important distinction between regimes ( ii ) and ( iii ) .
while in equilibrium regime ( ii ) , one can treat the cis and trans sides on even ground , a qualitative difference should unclose as a characteristic feature in the nonequilibrium driven regime ( iii ) . here ,
compared to the sequential stretching of the cis - side chain owing to the propagating tension , the chain portion , which is pushed out of the pore faster than it relaxes in the trans - side , is compressed and resides dynamically in the crowded state . while the proper description of the cis - side dynamics constitutes a basis for our current understanding of the dynamical scaling in the driven translocation @xcite , the trans - side event has been mostly overlooked so far .
although the occurrence of the dynamical crowding has repeatedly been addressed in recent literature e.g. through the observation of simulation snapshots @xcite or the numerical results @xcite , it is poorly understood , and one is left with the question as to how the process in the trans - side can be described and when it becomes important .
below , we attempt to provide the answer to it .
the driving force @xmath0 sucks the cis - side polymer segments into the pore , then pushes them into the trans - side .
@xmath13 the polymer is stretched in the cis - side by the force @xmath14 , while it is dynamically compressed in the trans - side by @xmath15 , and the pore plays a role of the sink and the source in the cis- and trans - sides , respectively .
there is also a frictional force @xmath16 associated with the pore . to handle such a dynamically asymmetric situation on an equal footing ,
we introduce a length scale @xmath17 ; in the cis - side , this length is a tensile blob inversely proportional to the local chain tension , while it corresponds to the concentration blob in the trans - side , from which we can identify the thermodynamic force and the transport coefficient in the respective domains . below the length scale @xmath18 ,
the chain conformation is essentially in equilibrium , i.e. , letting @xmath19 be the segment number constituting the blob , @xmath20 @xcite . in larger scale , the asymmetry shows up in such a way that the problem is essentially one - dimensional in the cis - side , while it is three - dimensional in the trans - side .
keeping such a difference in the effective dimensionality @xmath21 in mind , let us introduce the dimensionless segment density @xmath22 , where @xmath23 and @xmath24 for the cis- and trans - sides , respectively , and @xmath25 .
given the free energy density @xmath26 and the transporting coefficient @xmath27 evaluated as a stokes friction for each blob ( non - draining case ) @xcite , the constitutive relation for the segment flux @xmath28 and the thermodynamic force can be standardized as @xmath29 where @xmath30 is a @xmath21-dimensional nabla operator , and @xmath31 is the segmental diffusion coefficient , and we introduce an exponent @xmath32 , which enables us to treat the effect of hydrodynamic interactions collectively , i.e. , setting @xmath33 reduces to the above non - draining case , while @xmath34 represents the free - draining ( rouse ) dynamics @xcite . with the continuity equation of segments
, this leads to the nonlinear diffusion equation of the type of porous media equation @xcite .
note that the exponent @xmath25 has different signs in the cis- and trans - sides @xcite . from eq .
( [ flux ] ) , the case with plus ( minus ) sign indicates the enhanced @xcite ( suppressed @xcite ) flux in less dense region .
such processes are called fast ( slow ) diffusions , which turns to be crucial in the following .
it is also useful to introduce the velocity potential @xmath35 the gradient of which leads to the segment velocity @xmath36 , hence its naming .
take @xmath37 coordinate perpendicular to a thin wall with a small hole , which is located at the origin ( fig . 1 ) .
we set the cis - side in @xmath38 region , where the polymer is initially placed taking an equilibrium conformation . at @xmath39 ,
the translocation process begins , when one of the chain ends finds the pore .
the segments are sequentially labeled from the first arriving end . at time @xmath40 , @xmath41-th segment is located at the pore , which is traditionally called a translocation coordinate in literature @xcite , while @xmath42-th segment is at the rear end of the moving domain , which represents the dynamics of the tension propagation in the cis - side ( see below and fig . 1 ) . _ cis - side dynamics ( @xmath23 ) : fast diffusion_ our dynamical equation in @xmath23 dimension describes the stretching process @xcite .
the essential nonequilibrium feature here can be captured by what we call the two - phase picture , in which the entire cis - side polymer is divided into the moving domain with a characteristic velocity @xmath43 and the yet quiescent domain ( fig . 1 ) .
integration of the velocity potential in the moving domain range , we have @xmath44 , \nonumber\\\end{aligned}\ ] ] where @xmath45 is the location of the tension propagation front ( see fig . 1 ) , and the positive numerical constant @xmath46 of order unity is absorbed in @xmath47 .
given the nonuniform stretched conformation @xmath48 , one can obtain the force - velocity relation for the cis - side polymer @xmath49 where the boundary conditions @xmath50 ( see ref . @xcite and @xcite arxiv ) and @xmath51 are used ( see fig . 2 ) .
in addition , from the definition of the tension front , we have the relation @xmath52 which traces back to the initial equilibrium conformation at @xmath39 . _ trans - side dynamics ( @xmath24 ) : slow diffusion_ the trans - side decompression process can be described by our dynamical equation in @xmath24 dimension @xcite . taking the spherical symmetry into account
, the integration of the velocity potential leads to @xmath53,\end{aligned}\ ] ] where @xmath54 is the radial position of the decompressed front of the trans - side chain ( see fig . 1 ) , and
the positive numerical constant @xmath55 of order unity is absorbed in @xmath47 .
to proceed , we assume that the velocity field rapidly decreases away from the pore so that the above integral can be dominated by the pore vicinity .
we then find @xmath56 where the boundary conditions @xmath57 and @xmath58 are used .
equation ( [ v_t ] ) states that the force - velocity relation in the trans - side is regularized locally as the condition for the blob closest to the pore ( see fig . 2 )
this may be justified by the physical observation that the injection rate is controlled by the local condition in the pore vicinity , but not aware of the position of the decompression front @xmath59 .
the contrast to eq .
( [ v_c ] ) , which includes the global information @xmath60 , reflects a qualitative difference between the pulling and pushing operations .
indeed , this first blob is pushed by @xmath61 , which is balanced by the drag force against it , i.e. , @xmath62 .
putting @xmath63 into it leads to eq .
( [ v_t ] ) . in eq .
( [ decoupled_a ] ) .
the flory and dynamical exponents are adopted as follows ; ideal chain ( i d ) @xmath64 , excluded volume chain ( ev ) @xmath65 , free - draining ( fd ) @xmath66 and non - draining ( nd ) @xmath33 . ]
_ mass conservation through pore_ we now patch the above described cis- and trans - sides dynamics by requiring the conservation of mass across the pore . the rate of segments sucked from the cis - side into pore is @xmath67 v_1(0 ^ -)$ ] , which is equal to the rate of segments pushed into the trans - side @xmath68 v_3(0^+ ) \xi(0^+)^2 $ ] . using the boundary conditions at the pore
, it can be rewritten as @xmath69 eliminating @xmath43 and @xmath70 using eqs .
( [ v_c ] ) and ( [ v_t ] ) , we obtain the expression of the tension front @xmath71 , hence @xmath42 via eq .
( [ init ] ) , in terms of the force allocation .
@xmath72 where , as already defined , @xmath73 is the number of segments in the blob immediate vicinity of the pore in the cis - side . to get a clear - cut time dependence of the fraction of the cis - side force @xmath74 , where @xmath75 , one can rewrite eq .
( [ fc_ft ] ) as @xmath76 with @xmath77 and the function @xmath78 which is monotone increasing given the domain @xmath79 ( in practice @xmath80 , however .
see below . ) in fig . 3 , we plot @xmath81 as a function of @xmath42 . at @xmath82 , which serves as the initial condition of the nonequilibrium driven regime ( iii ) , the chain in the cis - side forms an initial tensed blob .
this is connected to our remark in the introduction that , until this moment , cis- and trans - sides can be treated on even ground , thus , @xmath83 .
along with translocation process advanced , the tension propagates in cis - side , and @xmath14 monotonically increases and eventually dominates in magnitude over @xmath15 .
since the normalization factor of @xmath42 is @xmath84 , the growth of @xmath81 is unconcerned with the total chain length @xmath11 , thereby , the trans - side effect is irrelevant in the dynamical scaling scenario with the identification @xmath85 in the long chain limit @xcite .
it also predicts the smaller the driving force and/or the larger the pore friction , the larger the finite - chain length effect , since both of these make @xmath86 larger .
the translocation time in the asymptotic limit can be derived as follows . in this limit
, the tension propagation dynamics is described from eqs .
( [ v_c ] ) , ( [ init ] ) and the first equality in eq .
( [ flux_pore ] ) @xmath87 this equation can be closed by an additional relation @xmath88 valid in the driven regime ( iii ) ( see ref .
@xcite and @xcite arxiv ) .
then , the tension propagation time @xmath89 is identified as @xmath90 ; @xmath91 which coincides in leading order to the translocation time . as is evident from fig .
3 , the approach to the asymptote @xmath92 is rather slow , which may cause the subtlety and difficulty to measure the force exponent in eq .
( [ tau ] ) .
finally , note @xmath93 in the limit of the weak force @xmath94 , thus , we have a crossover to the weakly driven regime ( ii ) at this force , where the cis- and trans - sides can be treated on even ground during the whole translocation process . in summary
, we have provided a lucid description on the cis and trans dynamical asymmetry in the driven translocation .
the imbalance is further promoted under higher driving force ( @xmath95 ) , which leads to the almost full stretching in the cis - side . in such situations ,
we only need to modify exponents in the transport equation in cis - side as @xmath96 and @xmath97 @xcite . for our purpose
, this requires replacing the counterparts in eq .
( [ v_c ] ) and the first equation of eqs .
( [ flux_pore ] ) .
we can then show a similar result with @xmath98 and @xmath99 .
note that @xmath86 no longer has the meaning of the initial tensed blob in this strong force regime ; @xmath100 at @xmath101 and getting smaller with the force , which indicates the faster approach to the asymptote @xmath102 . a basic equation ( eq .
( [ flux ] ) ) can also be applied to the polymer transport in different space dimensions with appropriate exponents @xmath103 and @xmath5 .
therefore , the same asymmetric features would be verified in the translocation process , for instance , conducted in the slit geometry .
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we conduct a series of experiments with a special interest on a penetration process and instabilities arisen on a liquid jet impinged to a liquid of the same kind flowing in a channel .
the impinged jet penetrates into the flowing bath accompanying with entrainment of the ambient immiscible gas , which results in the impinged jet wrapped by the entrained gas as a sheath. this sheath formation enables the impinged jet to survive in the fluid in the channel without coalescing until the entrained - air sheath breaks down .
occasionally a cap of the entrained air is formed at the tip of the penetrated jet , and the jet elongates like a long balloon .
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dynamic behaviors of the penetrated jet and the departure of the bubble of wrapping gas at the tip of the collapsing jet observed by use of a high - speed camera are included in this fluid dynamics video .
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