document: lanthanide based heavy fermion ( hf ) metals constitute a major , long studied class of correlated electron materials  @xcite . 
   their behaviour is quite distinct from conventional clean metals , the basic physics being driven by strong spin - flip scattering from essentially localised @xmath3-levels , generating the large effective mass and attendant low - energy scale indicative of strong interactions . 
   the low - temperature ( @xmath4 ) state is a lattice - coherent fermi liquid with well defined quasiparticles and coherently screened @xmath3-spins , crossing over with increasing @xmath4 to essentially incoherent screening via independent kondo scattering , before attaining characteristic clean metallic behaviour . 
   physical properties of hf are in consequence typically ` anomalous ' : e.g.  the resistivity @xmath5 shows a strong non - monotonic @xmath4-dependence , while optics often exhibit rich structure from the microwave to the near infrared , and pronounced evolution on low temperature scales  @xcite . 
   theoretical treatments of hf centre on the periodic anderson model ( pam ) , in which a non - interacting conduction band hybridizes locally with a correlated @xmath3-level in each unit cell of the lattice ; or on its strong coupling limit , the kondo lattice model . 
   the absence of exact results ( save for some in one dimension , see e.g.  @xcite ) has long spurred the search for suitable approximation schemes . 
   one such framework , which has had a major impact in recent years , is provided by dynamical mean field theory ( dmft , for reviews see  @xcite ) . formally exact in the large - dimensional limit 
   , the self - energy within dmft becomes momentum independent and hence spatially local , but still retains full temporal dynamics ; such that all lattice models map onto an effective single - impurity model with a self - consistently determined host  @xcite . 
   that raises an immediate question , easier asked than answered : to what extent are the properties of real hf materials captured within a dmft approach to the pam ? to answer this clearly requires direct quantitative comparsion of theory to experiment . and 
   a prerequisite to that in turn is a method to solve the pam  which dmft does not _ 
   per se _ provide . 
   the latter has of course been studied extensively using a wide variety of techniques . 
   full scale numerical methods include the numerical renormalization group ( nrg )  @xcite , quantum monte carlo  @xcite and exact diagonalization  @xcite , while theoretical approaches encompass finite - order perturbation theory in the interaction @xmath6  @xcite , iterated perturbation theory  @xcite , the lattice non - crossing approximation  @xcite and the average @xmath7-matrix approximation  @xcite , large-@xmath8 mean - field theory / slave bosons  @xcite , the gutzwiller variational approach  @xcite and the recently developed local moment approach  @xcite . 
   all of these methods naturally have their own virtues . 
   but most possess significant , well known limitations  @xcite , be it the general inability of perturbative approaches ( and in practice quantum monto carlo ) to handle strong interactions ; failure to recover fermi liquid behaviour at low - energies as arises in nca - based approaches , restriction to the lowest - energy fermi liquid behaviour as in large-@xmath8/slave boson mean - field theories , finite - size effects limiting exact diagonalization , and so on . 
   to enable viable comparison to experiment requires an approach that can adequately handle all experimentally relevant energy and/or temperature scales in the strongly correlated hf regime of primary interest ; and indeed ideally also across the full spectrum of interaction strengths , such that intermediate valence and related behaviour can likewise be treated . 
   one such is employed here , the local moment approach ( lma )  @xcite . via study of the generic asymmetric pam , 
   our essential aims are ( i ) to provide a many - body description of dynamical and transport properties of paramagnetic hf , notably single - particle dynamics , d.c . 
   transport and optical conductivities ; as considered here . 
   ( ii ) to make direct quantitative comparison with experiment . that is taken up in the following paper where comparison to transport / optical properties of @xmath9 , @xmath10 , @xmath11 and @xmath12 is made . 
   some remarks on the lma are apposite at this point since the paper will focus mainly on results obtained using the approach , with minimal technical details . 
   intrinsically non - perturbative and as such capable of handling strong interactions , the lma  @xcite introduces the physically intuitive notion of local moments  @xcite from the outset . 
   this leads directly to a ` two - self - energy ' description in which , post mean - field level , the key correlated spin - flip dynamics is readily captured ; corresponding in physical terms to dynamical tunneling between initially degenerate local moment configurations , which lifts the erstwhile spin degeneracy and restores the local singlet symmetry characteristic of a fermi liquid state . as with all techniques for lattice models within dmft , the lma originated in study of the single - impurity anderson model ( aim )  @xcite , where results for dynamics are known to give good agreement with nrg calculations  @xcite , and for static magnetic properties with known exact results  @xcite 
   the approach has recently been developed to encompass the anderson lattice ( pam ) ; initially for the particle - hole symmetric limit  @xcite appropriate to the kondo insulating sub - class of heavy electron materials , where for all interaction strengths the system is an ` insulating fermi liquid ' that evolves continuously from its simple non - interacting limit of a hybridization - gap insulator  @xcite . from this 
   a rich description of transport and optical properties of kondo insulators arises  @xcite , particularly in strong coupling where the system is characterized by an exponentially small indirect gap scale @xmath13 , such that dynamics / transport exhibit scaling as functions of @xmath14 . exploiting that scaling enables direct comparison to experiment with minimal use of ` bare ' material / model parameters ; and in particular for three classic kondo insulators @xmath15 , @xmath16 and @xmath17 , leads to what we regard as excellent agreement between theory and experiment on essentially all relevant energy and temperature scales  @xcite . 
   the particle - hole symmetric pam is of course special , confined as it is to the case of kondo insulators . 
   most recently the lma has been non - trivially extended to handle the generic asymmetric pam  @xcite and hence hf metals ( with the insulating symmetric limit recovered simply as a particular case ) . 
   single - particle dynamics at @xmath18 were considered in  @xcite , with a natural emphasis on the strongly correlated kondo lattice regime of localised @xmath3-electrons but general conduction ( ` @xmath19 ' ) band filling @xmath20 . 
   the problem was found to be characterized by a _ 
   single _ low - energy coherence scale @xmath1  the precise counterpart of the insulating indirect gap scale @xmath21 , and likewise exponentially small in strong coupling  in terms of which dynamics exhibit one - parameter universal scaling as a function of @xmath22 , independently of either the interaction strength or local @xmath23 hybridization . with increasing @xmath24 dynamics cross over from the low - energy quasiparticle behaviour required by and symptomatic of the coherent fermi liquid state , to essentially incoherent single - impurity kondo scaling physics at high-@xmath24 
   but still in the @xmath24-scaling regime and as such incompatible  @xcite with a two - scale ` exhaustion ' scenario  @xcite .    in this paper 
   we extend the work of  @xcite to finite temperature , thereby enabling access to d.c . 
   transport and optics . 
   our primary focus is again the strongly correlated hf regime and attendant issues of scaling / universality ( that play a key role in comparing to experiment ) , the paper being organised as follows . the model and a bare bones description of background theory is introduced in section 2 , together with preliminary consideration of transport / optics . 
   results for the thermal evolution of single - particle dynamics and scattering rates , and the connection between the two , are given in section 3 . 
   the d.c . 
   resistivity is considered in section 4 , with particular emphasis in this context on the crossover from the low-@xmath25 coherent fermi liquid to the high-@xmath26 incoherent regime , and explicit connection to single - impurity scaling behaviour . 
   optical conductivities on all relevant @xmath27- and @xmath4-scales are investigated in section 5 ; and the paper concludes with a brief summary . 
   the hamiltonian for the pam is given by @xmath28 : = _ c_i , c^_i c^_i & -&t_(i , j),c^_i c^_i + _ 
   i,(_f+ f^_i - f^_i- ) f^_if^_i + & + & v_i , ( f^_i c^_i + ) [ 2.1 ] the first two terms represent the uncorrelated conduction ( @xmath19 ) band , + @xmath29 ( @xmath30 ) ; with @xmath19-orbital site energies @xmath31 and nearest - neighbour hopping matrix element @xmath32 , rescaled as @xmath33 in the large dimensional limit where the coordination number @xmath34  @xcite . 
   the third term describes the correlated @xmath3-levels , @xmath35 , with site energies @xmath36 and on  site coulomb repulsion @xmath6 ; while the final term @xmath37 hybridizes the @xmath19- and @xmath3-levels locally via the matrix element @xmath38 , rendering the otherwise localised @xmath3-electrons itinerant . 
   the model is thus characterized by four independent dimensionless parameters , @xmath39 and @xmath40 ( @xmath41 sets the scale for the width of the free conduction band and is taken as the basic unit of energy , @xmath42 ) . an equivalent and somewhat more convenient set of ` bare'/material parameters ( with @xmath43 ) is @xmath44 and @xmath45 , where @xmath46 . 
   this parameter space is large , and as such encompasses a wide range of physical behaviour for the paramagnetic phases we consider . 
   the system is of course generically metallic , with non - integral @xmath3-level and @xmath19-band occupancies ( @xmath47 and @xmath48 respectively ) . 
   that in turn extends from the trivial case of weakly correlated , perturbative behaviour , through intermediate valence to the strongly correlated heavy fermion ( hf ) regime . 
   it is naturally the latter , characterized by a low - energy coherence scale @xmath49 , that is of primary interest . 
   the hf ( or kondo lattice ) regime corresponds to essentially localised @xmath3-electrons , @xmath50 , but with arbitrary conduction band filling @xmath51 , the latter being controlled by @xmath52 ( which determines the centre of gravity of the free ( @xmath53 ) conduction band relative to the fermi level ) . 
   it arises when @xmath54 , for @xmath55 and @xmath56 ( whence @xmath57 ) ; where @xmath58 , with @xmath59 the free conduction electron density of states as specified below and @xmath60 the fermi level . 
   the heavy fermion regime forms our main focus here ; intermediate valence behaviour will be discussed in an experimental context in the following paper . 
   the exception to the above behaviour arises when @xmath61 . here 
   the system is generically a kondo insulator ( see _ eg  @xcite ) , with an indirect gap in both its @xmath18 single - particle spectrum _ and _ optical conductivity  @xcite ; the canonical example being the particle - hole symmetric pam with @xmath62 and @xmath63 , where @xmath64 for all @xmath6 . just like its metallic counterpart arising for @xmath65 , 
   the kondo insulator is however a fermi liquid , evolving continuously with increasing interaction strength from its non - interacting limit ( in this case a ` hybridization gap insulator '  @xcite ) . as such , the kondo insulating state 
   is obtained simply as a particular limit of the underlying theory . _ 
   a knowledge of local single - particle dynamics and their thermal evolution is well known to be sufficient within dmft  @xcite to determine transport properties ( see section 2.2 below ) . 
   our initial focus is thus on the local retarded green functions @xmath66 and likewise @xmath67 for the @xmath19-levels , with corresponding spectra @xmath68 ( and @xmath69 or @xmath3 ) . 
   some brief comments on the free conduction band are first required ( @xmath53 in equation ( [ 2.1 ] ) , where the @xmath19- and @xmath3-subsystems decouple ) ; specified by the local propagator @xmath70 with corresponding density of states ( dos ) @xmath59 . 
   this is given by g_0^c ( ) & = & ( ^+-_c ) [ 2.2a ] + & = & [ 2.2b ] with @xmath71 , where for arbitrary complex @xmath72 h(z ) = ^+_- d [ 2.3 ] denotes the hilbert transform with respect to @xmath73 ; such that from equation ( [ 2.2a ] ) , @xmath74 corresponds simply to a rigid shift of @xmath75 by @xmath52 . 
   equation ( [ 2.2b ] ) defines the feenberg self - energy @xmath76  @xcite as used below , with @xmath77 $ ] alone ( since @xmath78 from equations ( 2.2 ) ) . the free conduction band is thus determined by the non - interacting dos @xmath79 which , modulo the rigid @xmath52-shift , reflects the underlying host bandstructure , @xmath80 . while the formalism below holds for an arbitrary @xmath73 , explicit results will later be given for the hypercubic lattice ( hcl ) , for which within dmft  @xcite @xmath81 is an unbounded gaussian ; and the bethe lattice ( bl ) , with compact spectrum @xmath82 for @xmath83  @xcite . 
   the hcl will in fact be the primary case , because the bloch states characteristic of it ultimately underlie the lattice coherence inherent to low - temperature metallic hf behaviour . 
   the major simplifying feature of dmft is that the self - energy becomes momentum - independent and hence site - diagonal  @xcite ; and since we are interested in the homogeneous paramagnetic phase , the local green functions @xmath84 ( @xmath85 ) are also site - independent . 
   straightforward application of feenberg renormalized perturbation theory  @xcite , then gives the @xmath86 as g^c()&=&[2.4a ] + g^f()&= & [ 2.4b ] + & = & [ 2.4c ] where @xmath87 is the retarded @xmath3-electron self - energy ( @xmath88 such that @xmath89 ) . in equations ( 2.4 ) 
   , @xmath90 is the feenberg self - energy for the fully interacting case , with @xmath91 $ ] the _ same _ functional of @xmath92 as it is of @xmath93 in the @xmath53 limit . in consequence 
   , @xmath92 is given using equations ( [ 2.4a ] ) , ( 2.2 ) , ( [ 2.3 ] ) as g^c()=h ( ) [ 2.5a ] where @xmath94 ( @xmath95 ) is given by ( ; t)=^+-_c- . 
   [ 2.5b ]    let us first point up the physical interpretation of equations ( 2.4 ) , ( 2.5 ) . 
   @xmath92 is a _ local _ propagator , and as such familiarly expressed as @xmath96 ; with the @xmath97-resolved conduction electron propagator @xmath98^{-1}$ ] and the usual conduction electron self - energy @xmath99 thus defined . 
   since @xmath80 , it follows directly that g^c ( ) = ^+_-d  _ 0()g^c ( ; )   g^c ( ; ) _ [ 2.6 ] with ( @xmath100 ) @xmath101^{-1}$ ] . but equation ( [ 2.6 ] ) is precisely the form equation ( [ 2.5a ] ) ( with equation ( [ 2.3 ] ) for @xmath102 ) , showing that g^c ( ; ) = [ ( ; t)-]^-1 [ 2.7 ] with @xmath94 related to the conduction electron self - energy by ( ; t)=^+-_c - _ c(;t ) ; [ 2.8 ] and hence ( via equation ( [ 2.5b ] ) ) that @xmath103^{-1}$ ] in terms of the @xmath3-electron self - energy alone ( because the @xmath3-levels alone are correlated ) .    for an arbitrary conduction band ( specified by @xmath104 ) equations ( 2.4 ) , ( 2.5 ) 
   are central ; for given the self - energy @xmath87 , and hence @xmath94 from equation ( [ 2.5b ] ) , @xmath92 follows directly from the hilbert transform equation ( [ 2.5a ] ) , and @xmath105 in turn from equation ( [ 2.4c ] ) . 
   that statement hides however the truly difficult part of the problem : obtaining the self - energy @xmath87 . 
   this is not merely a calculational issue , e.g.  the need to solve the problem iteratively and self - consistently ( any credible approximation to @xmath87 will in general be a functional of self - consistent propagators ) . 
   it reflects by contrast the longstanding problem of obtaining an approximate @xmath87 that , ideally : ( i ) handles non - perturbatively the full range of interaction strengths , from weak coupling ( itself accessible by perturbation theory or simple variants thereof  @xcite ) all the way to the strongly correlated kondo lattice regime that is dominated by spin - fluctuation physics and typified by an exponentially small coherence scale @xmath1 . 
   ( ii ) respects the asymptotic dictates of fermi liquid behaviour on the lowest energy ( @xmath27 ) and/or @xmath4 scales  on the order of @xmath106 itself  yet can also handle the _ full _ 
   @xmath27 and/or @xmath4 range ; including the non - trivial dynamics that arise on energy scales up to many multiples of @xmath1 yet which remain universal ( and the existence of which we find to dominate transport and optics ) , as well as the non - universal energy scales prescribed by the bare material parameters of the problem . 
   the success of any theory naturally hinges on the inherent approximation to @xmath87 . in this paper 
   we employ the local moment approach ( lma )  @xcite , for it is known to satisfy the above desiderata and to our knowledge is currently the only theory that does . 
   it is based on an underlying two - self - energy description  a natural consequence of the mean - field approach from which it starts , and from which the conventional single self - energy @xmath107 follows together with the concept of _ symmetry restoration _ that is central to the lma generally  @xcite . 
   full details of the lma for the pam , including discussion of its physical basis and content , are given in  @xcite . 
   in particular the generic asymmetric pam ( as considered here ) is detailed in  @xcite for @xmath18 ; and extension of it to finite-@xmath4 , required to consider transport and optics , follows the approach of  @xcite where the particle - hole symmetric pam appropriate to the case of kondo insulators was considered . 
   for that reason further discussion of the approach is omitted here . 
   the reader is instead directed to  @xcite on the pam , from which appropriate results will be used when required ; and to  @xcite for anderson impurity models _ per se _ where details of the lma , including its stengths and limitations in relation to other approaches , are fully discussed .      as mentioned above 
   , a knowledge of single - particle dynamics is sufficient within dmft to determine @xmath108 transport properties  @xcite . 
   this arises because the strict absence of vertex corrections in the skeleton expansion for the current - current correlation function means only the lowest - order conductivity bubble survives  @xcite , and a formal result for it is thus readily obtained . 
   denoting the trace of the conductivity tensor by @xmath109 ( @xmath110 of which , denoted by @xmath111 , provides an approximation to the isotropic conductivity of a 3-dimensional system ) , this may be cast in the form ( ; t)=_0 f(;t ) [ 2.9 ] with @xmath112 merely an overall scale factor ( @xmath113 is the lattice constant and @xmath114 is typically of order @xmath115 ) . 
   the dimensionless dynamical conductivity @xmath116 naturally depends on the lattice type , and for a bloch decomposable lattice such as the hcl is given ( with @xmath43 ) by  @xcite f_hcl(;t)=^_- d_1d^c(;_1 ) d^c(;_1 + ) _ [ 2.10 ] where @xmath117^{-1}$ ] is the fermi function . here 
   ( as in equation ( [ 2.6 ] ) ) , the notation @xmath118 denotes an average with respect to the non - interacting conduction band dos @xmath73 ; and the spectral density @xmath119 with @xmath120^{-1}$ ] from equation ( [ 2.7 ] ) . 
   physically , @xmath121 ( @xmath122 ) represents the @xmath27-dependent conduction electron scattering rate ( inverse scattering time ) arising from electron interactions , @xmath123 ( @xmath124 from equation ( [ 2.8 ] ) ) . 
   it is given using equation ( [ 2.5b ] ) by = _ 
   i(;t ) = [ 2.11 ] in terms of the @xmath3-electron self - energy ; a knowledge of which thus determines the scattering rates ( considered explicitly in section 3.1 ) , and in consequence the dynamical conductivity equation ( [ 2.10 ] ) ( noting that @xmath125 ^ 2 + [ \gamma_i(\om;t)]^2)$ ] ) 
   .    results for @xmath126 obtained using the lma will be considered in sections 4,5 . 
   here we simply point out an exact result , not apparently well known , for the weight of the drude peak in the @xmath18 conductivity . at @xmath18 , scattering at the fermi level is absent since the system is a fermi liquid , i.e.  @xmath127 and hence @xmath128 . 
   the leading low - frequency behaviour of @xmath129 is given by ^r_f(;0 ) ~^r_f(0;0 ) - ( -1)[2.12 ] where @xmath130^{-1}$ ] is the usual quasiparticle weight / inverse mass renormalization ; hence ( from equation ( [ 2.5b ] ) ) @xmath131 , where ^*_f = _ f + ^r_f(0;0 ) [ 2.13 ] is the renormalized @xmath3-level energy . 
   a straightforward evaluation of equation ( [ 2.10 ] ) for @xmath18 and @xmath132 then shows that @xmath133 contains a @xmath134 drude ` peak ' ( as it must , reflecting the total absence of fermi level scattering and a vanishing @xmath18 resistivity ) . 
   denoted by @xmath135 , it is given explicitly by f_drude(;0 ) = ( )   _ 0(-_c+ ) [ 2.14a ] or equivalently f_drude(;0 ) = ( )   _ 0(-_c+ ) [ 2.14b ] where @xmath136 and _ l = zv^2 . [ 2.15 ]    equations ( 2.14 ) are exact , and bear comment . 
   in the trivial limit @xmath53 where ( equation(2.1 ) ) the @xmath3-levels decouple from the conduction band , the total drude weight is naturally @xmath137 , the free conduction band dos at the fermi level ( recall @xmath138 ) . for any @xmath139 , 
   the luttinger integral theorem requires ( n_c+n_f ) = ^-_c+1/_f^*_- _ 0 ( )  d   +   ( -_f^ * ) [ 2.16 ] ( with @xmath140 merely the unit step function ) . 
   this again is an exact result , proven in  @xcite . 
   it holds for _ any _ interaction @xmath6 , reflecting the adiabatic continuity to the non - interacting limit that is intrinsic to a fermi liquid ; and shows in general that ( any ) fixed total filling @xmath141 determines @xmath142 entering equations ( 2.14 ) . 
   of particular interest is of course the strongly correlated hf regime , where @xmath143 . here 
   @xmath1 in equation ( [ 2.15 ] ) ( @xmath144 with @xmath145 ) is the coherence scale : exponentially small in strong coupling ( because @xmath146 is ) , it is the _ single _ low - energy scale in terms of which all properties of the system exhibit universal scaling ( as shown in  @xcite and pursued below ) . in the hf regime , @xmath20 itself is moreover given ( see  @xcite ) by n_c = ^-_c_-_0 ( )  d[2.17a ] showing that @xmath147 and @xmath20 are in essence synonymous , @xmath148 being determined by @xmath147 alone . 
   conjoining this with equation ( [ 2.16 ] ) gives n_f = ^-_c+1/_f^*_-_c _ 0 ( )  d   +   ( -_f^ * ) [ 2.17b ] so as @xmath143 ( the hf regime ) , @xmath149 is _ also _ determined by @xmath147 alone , and is typically of order unity . 
   it is this that determines @xmath150 entering equation ( [ 2.14b ] ) for @xmath135 ; showing in turn that the net drude weight is itself @xmath151 , and hence exponentially diminished compared to the free conduction band limit . 
   we add that kondo insulators , arising generically for @xmath152 as mentioned earlier , are also encompassed by the above . using @xmath153 , the luttinger theorem equation ( [ 2.16 ] ) 
   shows that @xmath152 arises either for @xmath154 ( for an unbounded @xmath104 ) or for @xmath142 outside the band edges of a compact @xmath104 ; such that in either case the drude weight in equations ( 2.14 ) vanishes , symptomatic of the vanishing @xmath18 d.c .  conductivity characteristic of the kondo insulating state . 
   our focus above has naturally been on the canonical case of a bloch decomposable lattice . for a bethe lattice by contrast , @xmath116 is given  @xcite by ( _ cf _ equation ( [ 2.10 ] ) ) f_bl(;t)=^_- d_1d^c(_1 ) d^c(_1 + ) [ 2.18 ] where @xmath155 ( @xmath156 ) is the local conduction band spectrum . 
   in particular the d.c . 
   conductivity at @xmath18 follows as @xmath157 ^ 2 $ ] ; which , using @xmath158 together with equations ( [ 2.2a ] ) , ( 2.5 ) , ( [ 2.12 ] ) , is given by f_bl(0;0 ) = [ _ 0(-_c+)]^2 . [ 2.19 ] in contrast to equations ( 2.14 ) there is thus no drude @xmath134-peak and the @xmath18 d.c . 
   resistivity is in general finite , reflecting of course that the underlying one - particle states of the bl are not coherent bloch states . 
   hence , aside from the case of kondo insulators where the bl ( like the hcl ) does capture the vanishing @xmath18 d.c . 
   conductivity and indirect - gapped optics characteristic of the insulator  @xcite , the ` joint density of states ' type formula equation ( [ 2.19 ] ) should not be taken seriously when considering transport / optics of real materials on sufficiently low @xmath4 and/or @xmath27 scales ( as discussed further in section 4 ) . 
   we turn now to lma results for single - particle dynamics at finite-@xmath4 . our natural focus will be the strong coupling kondo lattice regime ( where @xmath143 ) , characterized by the low - energy lattice scale @xmath159 . 
   this scale is of course a complicated function of the bare / material parameters , @xmath160 ( detailed lma results for it are given in  @xcite , and nrg results in  @xcite ) . 
   that dependence is however a subsidiary issue in comparison to the fact that , because @xmath1 becomes exponentially small in strong coupling , physical properties exhibit scaling in terms of it ; i.e.  depend universally on @xmath161 , independently of the interaction strength . 
   universality in strong coupling single - particle dynamics at @xmath18 has been considered in  @xcite for the generic pam ; the essential findings of which are first reprised for use below . 
   ( i ) both the @xmath19-electron spectrum @xmath155 ( @xmath162 with @xmath43 ) and the @xmath3-electron spectrum @xmath163 ( with @xmath164 introduced in section 2 ) , exhibit universal scaling as a function of @xmath22 in a manner that is _ independent _ of both the interaction strength @xmath6 _ and _ hybridization matrix element @xmath38 . 
   ( ii ) that scaling depends in general only on @xmath147 ( or equivalently the conduction band filling @xmath20 , see equation ( [ 2.17a ] ) ) which embodies the conduction band asymmetry ; and on @xmath165 reflecting the @xmath3-level asymmetry . more specifically , 
   ( iii ) in the coherent fermi liquid regime arising for @xmath166 , the @xmath3- scaling spectra depend only on @xmath147 and are in fact independent of @xmath45 as well as @xmath6 and @xmath38 . in this low-@xmath24 regime 
   the scaling spectra amount in essence to the quasiparticle behaviour ( equations ( 3.11 ) of  @xcite ) required by the asymptotic dictates of low - energy fermi liquid theory . 
   ( iv ) for @xmath167 by contrast the @xmath3- scaling spectra depend on the @xmath3-level asymmetry @xmath45 ( albeit rather weakly ) , but are now independent of @xmath147 and indeed also of the lattice type ; and the spectrum contains a long , logarithmically slowly decaying spectral tail . 
   ( v ) the latter behaviour , which sets in progressively for @xmath168 , reflects in turn the crossover to incoherent effective single - impurity physics that one expects to arise for sufficiently high @xmath27 ( and/or @xmath4 ) : for @xmath167 the _ scaling form _ of the @xmath3-spectrum is found to be precisely that of an anderson impurity model ( aim ) . with increasing @xmath24 , 
   dynamics thus cross over from the low - energy quasiparticle behaviour symptomatic of the lattice coherent fermi liquid state to single - impurity kondo scaling physics at high @xmath24 ( and that this crossover occurs in a single @xmath161 scaling regime is thus incompatible with the occurrence of ` two - scale exhaustion '  @xcite as explained in  @xcite ) . 
   figure [ fig1 ] summarises representative results for @xmath18 scaling dynamics ( irrelevant non - universal energy scales such as @xmath6 , @xmath41 ( @xmath169 ) or @xmath170 are of course projected out in scaling spectra  @xcite ) . 
   the main figure shows @xmath23 scaling spectra for the hcl as functions of the scaled frequency @xmath171 , for @xmath172 with @xmath173 ( dashed , and @xmath174 ) and @xmath175 ( solid , with @xmath176 ) . 
   the @xmath177 example corresponds to the particle - hole ( p - h ) symmetric kondo insulator , whose spectra are thus gapped at the fermi level @xmath178 ( with @xmath159 here corresponding to the insulating gap scale  @xcite ) . for the asymmetric conduction band @xmath179 by contrast , 
   the gap ( which is well developed in strong coupling  @xcite ) moves above the fermi level ; and a sharp lattice - kondo resonance symptomatic of the hf metal , straddling the fermi level and of width @xmath180 , takes its place in the @xmath3-spectra . 
   the inset shows the @xmath3-spectra on a much larger @xmath24 scale ; displaying the @xmath31-independence of the slow logarithmic tails  @xcite and reflecting the crossover to effective single - impurity behaviour ( which we emphasise arises whether the system is a hf metal or a kondo insulator ) .    at finite temperatures , what one expects for the strong coupling scaling spectra 
   is clear : they should now depend universally on @xmath181 _ and _ @xmath182 . 
   that this arises correctly within the present lma is shown in figure [ fig2 ] . for a fixed @xmath183 , the @xmath3- and @xmath19-spectra are shown for progressively increasing interaction strengths @xmath184 and @xmath185 with @xmath186 ; for @xmath187 and @xmath172 ( corresponding results for the p - h symmetric limit have been obtained in  @xcite ) . 
   the inset shows the @xmath3-spectra on an absolute scale ( _ vs _ @xmath188 ) , where the exponential reduction of the @xmath1-scale with increasing @xmath6 is clearly seen from the change in the width of the resonance . 
   the main figures by contrast show the spectra as functions of @xmath24 , from which the @xmath6-independent scaling collapse is evident ; repeating the calculations with different @xmath189 likewise shows the scaling to be independent of @xmath38 . 
   this behaviour is not of course confined to the chosen @xmath26 , and figure [ fig3 ] shows the resultant lma scaling spectra for a range of @xmath26 ( again for the representative @xmath190 ) . 
   figures [ fig3 ] and [ fig2 ] show clearly the thermal broadening and ultimate collapse of the @xmath3-resonance with increasing @xmath26 ; which is naturally accompanied by a redistribution of spectral weight leading to infilling of the ( @xmath191 ) spectral gap seen in figure [ fig1 ] for @xmath18 . in fact by @xmath192 this gap is already obliterated , and the lattice kondo resonance also significantly eroded . 
   this behaviour is typical of the metallic hf state . 
   by contrast , corresponding results for the p - h symmetric kondo insulator ( @xmath193 ) are shown in figure 4 of  @xcite . in that case 
   the insulating gap at the fermi level fills up with increasing temperature , and the fermi level @xmath194 in particular increases monotonically with increasing temperature ; in contrast to the the asymmetric hf spectra shown above where @xmath195 diminishes with @xmath26 . 
   two further points regarding figure [ fig3 ] should be noted . 
   first , the thermal evolution of the @xmath3- and @xmath19-spectra differ somewhat in terms of the persistence of a pseudogap  the @xmath3-spectrum shows no sign of the gap by @xmath196 , while a weak pseudogap structure persists in the @xmath19-spectrum up to @xmath197 ; this reflects the rapid spread of spectral weight caused by the meltdown of the sharp resonance in the @xmath3-spectra , of which there is no counterpart in the @xmath19-spectra . 
   second , the inset to figure [ fig3 ] shows the @xmath3-spectra on an enlarged frequency scale out to @xmath198 , from which it is seen that the high frequency behaviour of the finite-@xmath26 scaling spectra coincide with that for @xmath18 . 
   this is physically natural , since one expects the dominant influence of temperature to be confined to frequencies @xmath199 . 
   the corollary of course is that non - universal frequencies are affected only on non - universal , and thus in general physically irrelevant , temperature scales ( as shown in figure 5 of  @xcite for the p - h symmetric case ) . 
   we consider now the scattering rates @xmath200 that underlie the evolution of the conductivity , and are given explicitly in terms of the @xmath3-electron self - energy by equation ( [ 2.11 ] ) . since the system is a fermi liquid with @xmath201 , at @xmath18 there is of course no scattering at the fermi level , @xmath202 . 
   the low frequency behaviour of the @xmath18 scattering rate can be understood qualitatively by using the low-@xmath27 expansion of @xmath203 ( equation ( [ 2.12 ] ) ) and simply neglecting the imaginary part @xmath204 ; leading to ^-1(;t=0 ) ( - _ f^ * ) [ 3.1 ] with @xmath205 the renormalized level . 
   restoring the small but strictly non - vanishing @xmath206 naturally implies a narrow resonance centred on @xmath207 instead of a pure @xmath208-function . at finite temperature 
   , we likewise expect the scattering rate to increase from zero in the neighbourhood of the fermi level , reflecting the finite-@xmath4 contribution to @xmath209 ; and that this will simultaneously lead to further , thermal broadening of the resonance at @xmath150 . 
   the above picture is corroborated by lma results as shown in figure [ fig4 ] , displaying the @xmath22 dependence of @xmath200 ( in units of @xmath42 ) arising in strong coupling for @xmath187 and @xmath172 , for a range of temperatures @xmath210 between @xmath211 and @xmath212 .    in this case 
   the renormalized level is found to be @xmath213 , precisely where @xmath214 has a narrow resonance . with increasing temperature the resonance 
   is indeed seen to broaden and decrease in intensity ; and we reiterate that this occurs for temperatures @xmath4 set by the scale @xmath1  the sole low - energy scale characteristic of the problem in strong coupling . excepting the lowest @xmath26 we also note that scattering rates in the vicinity of the fermi level are on the order of @xmath215 of the bandwidth @xmath41 , values some two or so orders of magnitude higher than for conventional clean metals ( and indicative of the higher d.c . 
   resistivities that are typical of heavy fermion materials  @xcite ) . 
   neither is this behaviour confined to a narrow @xmath26 regime since even for @xmath216 the scattering rates decay very slowly with @xmath26 ; the fermi level scattering rate for example is readily shown to decay as @xmath217 . 
   the scattering rates are also related to the @xmath3-electron scaling spectra considered above . for the kondo insulating p - h symmetric pam , 
   it was shown in  @xcite that the dimensionless scattering rate defined as = _ 
   i(;t ) [ 3.2 ] coincides asymptotically with the @xmath3-spectral function , specifically ~_0 d^f ( ) [ 3.3 ] in the regime @xmath218 for any @xmath26 ( the spectral ` tails ' ) , and for all @xmath219 for sufficiently large @xmath216 . equation ( [ 3.3 ] ) is in fact readily shown to be quite general , and not dependent on p - h symmetry . 
   that it holds for hf metals embodied in the asymmetric pam is illustrated in figure [ fig5 ] , where for @xmath187 and @xmath172 the strong coupling @xmath220 and @xmath221 _ vs _ @xmath24 are compared , for @xmath222 in the left panel and @xmath183 and @xmath223 in the right panel . 
   the high - frequency behaviour of the scaling spectrum @xmath221 is itself known , being given ( here for @xmath224 explicitly ) by  @xcite _ 0 d^f ( ) ( + ) [ 3.4 ] with @xmath113 a pure constant @xmath225 . 
   these slowly decaying logarithmic tails are evident in figure [ fig5 ] , and as mentioned in section 3 embody the connection to effective incoherent single - impurity physics on high energy scales . 
   they are independent of the interaction @xmath6 , local hybridization @xmath38 , underlying conduction band asymmetry @xmath31 , and even of the lattice type ; depending , albeit weakly , only on the @xmath3-level asymmetry  @xcite . 
   the above discussion of scattering rates leads naturally to consideration of transport ; beginning with the d.c . 
   limit where ( section 2.2 ) the static conductivity @xmath226 , with @xmath116 given for the hypercubic lattice by equation ( [ 2.10 ] ) . in the strong coupling regime we expect static transport to exhibit universal scaling in terms of @xmath25 , and our aim here is to understand its thermal evolution across the full @xmath26 range . 
   transport on non - universal temperature scales @xmath227 ( @xmath228 ) or @xmath229 , will be discussed briefly at the end of the section .    for the p - h symmetric kondo insulator , 
   lma results for the @xmath26-dependence of the scaling resistivity have been considered in  @xcite ( in this case @xmath230 is equivalently the insulating gap scale ` @xmath13 ' ) . 
   the @xmath18 resistivity is naturally infinite reflecting the gapped ground state , the scaling resistivity @xmath231 has an activated form @xmath232 for @xmath233 ( with @xmath234 a pure constant @xmath235 and hence a ` transport gap ' of @xmath236 ) ; and @xmath5 decreases monotonically with increasing @xmath26 , tending asymptotically to incoherent single - impurity scaling behaviour (  @xcite and figures [ fig7],[fig8 ] below ) .    for the general case of heavy fermion metals 
   the situation is of course quite different , and what one expects in qualitative terms well known  @xcite . 
   the @xmath18 resistivity vanishes , reflecting the absence of fermi level scattering and the underlying coherence generic to any bloch decomposable lattice . 
   with increasing temperature @xmath5 increases ( initially as @xmath237 for @xmath233  @xcite ) , passes through a maximum at @xmath238  a classic signature of hf compounds  @xcite  and decreases thereafter in the strong coupling , kondo lattice regime of interest . 
   figure [ fig6 ] shows lma results for @xmath5 _ vs _ 
   @xmath26 for fixed @xmath239 , and with increasing interaction @xmath240 and @xmath185 for @xmath186 . 
   the scaling collapse is clearly evident : while the low - energy scale @xmath241 itself diminishes exponentially on increasing @xmath6 , universal scaling of @xmath5 as a function of @xmath25 indeed arises in strong coupling , independent of interaction strength ( and likewise readily shown to be @xmath38-independent on repeating the calculations varying @xmath189 ) . 
   this leads us first to comment briefly on the issue of ` the coherence scale ' , characterising the crossover from low - temperature lattice coherent behaviour to high - temperature effective single - impurity behaviour . 
   experimentally , many such identifications of the low - energy scale are commonly employed . 
   some groups use @xmath238 at which @xmath5 peaks , others identify the scale via the inflection points ( @xmath242 , marked by arrows in figure [ fig6 ] ) , via the leading @xmath237 behaviour of @xmath5 at low-@xmath4 , or via the onset of the ` log - linear ' regime  @xcite ( shown in inset ( a ) to figure [ fig6 ] and seen in many experimental systems  @xcite ) ; the inverse of the @xmath243 paramagnetic susceptibility , or the width of the lattice kondo resonance , are other possibilities . 
   this leads to what at first sight might seem a plethora of low - energy scales . 
   the key point however is that , because physical properties in strong coupling scale universally in terms of _ one _ low - energy scale , all the above definitions of ` the coherence scale ' are fundamentally equivalent : all are proportional to @xmath1 , and hence to each other  in figure [ fig6 ] for example , the inflection points in @xmath5 lie at @xmath244 and @xmath245 , and the peak maximum at @xmath246 .    as for single - particle dynamics and scattering rates considered in section 3 , 
   the @xmath26-dependent scaling resistivity is independent of @xmath6 or @xmath38 ( as above ) but depends in general on @xmath147 ( reflecting the conduction band asymmetry and determining @xmath20 via equation ( [ 2.17a ] ) ) and @xmath45 ( reflecting the @xmath3-level asymmetry ) . to consider this figure [ fig7 ] shows the resultant scaling resistivities @xmath5 _ vs _ 
   @xmath26 for @xmath172 , and a range of different @xmath147 @xmath247 and @xmath248 , corresponding respectively to conduction band fillings @xmath249 and @xmath250 . 
   the @xmath177 example is the kondo insulator  @xcite , with its characteristic diverging @xmath5 as @xmath251 . 
   the others are all hf metals , and exhibit the same qualitative behaviour for all @xmath147  a positive slope for @xmath252 , going through the maximum and then decreasing monotonically for @xmath253 ; the coherence peak itself increasing monotonically with @xmath147 , albeit slowly such that @xmath254 for the @xmath147-range shown . qualitatively similar behaviour is found on varying the @xmath3-level asymmetry @xmath45 for fixed conduction band asymmetry embodied in @xmath147 , although quantitatively this effect is appreciably less . 
   the significant @xmath147-dependence of @xmath255 seen in figure [ fig7 ] for @xmath256 is intuitively natural : the strong coupling kondo lattice regime corresponds to @xmath257 , but with variable conduction band filling ( @xmath20 ) controlled by @xmath147 ( equation ( [ 2.17a ] ) ) ; and on decreasing @xmath20 ( increasing @xmath147 ) one expects the static conductivity to diminish and hence an increased @xmath5 , as found . to understand the @xmath147-dependence , and in turn to enable connection to incoherent effective single - impurity behaviour at high-@xmath26 
   , we first consider an approximate evaluation of @xmath258 ( equation ( [ 2.10 ] ) ) ; in which the energy dependence of the free conduction band dos @xmath259 is neglected , @xmath260 being replaced by its fermi level value . employing this 
   ` flat band ' approximation in equation ( [ 2.10 ] ) ( where it enters via the @xmath261 average ) leads to f_hcl(0;t ) [ _ 0(-_c)]^2 ^+_-d   ( ; t )    [ _ 0(-_c)]^2 [ 4.1 ] expressed as a physically intuitive thermal average of the dimensionless scattering time @xmath262 ( equation ( [ 3.2 ] ) ) . for kondo insulators 
   this approximation is qualitatively inadequate at low-@xmath26  @xcite , but as illustrated in figure [ fig7 ] ( inset ) it is entirely respectable for the hf metals and in particular recovers precisely the high-@xmath26 asymptotics of @xmath5 . 
   as shown in section 3.1 , the large @xmath24 and/or @xmath26 dependence of the reduced scattering rate @xmath220 coincides with the @xmath3-spectral function @xmath221 ( equation ( [ 3.3 ] ) ) ; and in section 3 ( see also  @xcite ) the latter were shown to have common spectral tails , independently of @xmath147 . 
   this suggests that the primary effect of @xmath147 seen in figure [ fig7 ] for @xmath255 is contained in the @xmath263 ^ 2 $ ] of equation ( [ 4.1 ] ) . 
   that this is so is seen in figure [ fig8 ] where the results of figure [ fig7 ] are now shown as @xmath264 _ vs _ @xmath26 , where ^(t)=. [ 4.2 ] for @xmath265 or so in practice , @xmath264 is seen in particular to be _ independent _ of the conduction band filling embodied in @xmath147 ; including we note the kondo insulator , whose ` high ' temperature resistivity is thus seen to be that of a regular heavy fermion metal . indeed as readily demonstrated , and evident in part from the above discussion , the behaviour seen in figure [ fig8 ] is barely dependent on the details ( @xmath266-dependence ) of the host bandstructure embodied in @xmath104 . 
   [ h ]    the obvious final question here concerns the high-@xmath26 form of @xmath264 for the pam . to that end 
   we consider the anderson single - impurity model ( aim ) , with @xmath267 denoting as usual the change of resistivity due to addition of the impurity to the non - interacting host , and @xmath268 . 
   this is given by  @xcite ( _ _ cf__equations ( [ 4.1]),([4.2 ] ) ) = ^+_-d   _ imp(;t ) [ 4.3 ] with the impurity scattering rate @xmath269 ; where @xmath270 is the impurity spectral function such that @xmath271 follows from the friedel sum rule  @xcite in the singly occupied , strong coupling kondo regime of the aim . the lma scaling resistivity @xmath272 _ vs _ @xmath26 is also shown in figure [ fig8 ] , where @xmath273 and @xmath274 is the aim kondo scale ( with @xmath275 the impurity quasiparticle weight ) . from this 
   it is seen that the high-@xmath26 scaling behaviour of @xmath264 for the pam is precisely that of the aim ; in particular the leading @xmath216 behaviour of the lma @xmath264 is readily shown analytically to be given by @xmath276 , which is exact in the kondo limit of the impurity model  @xcite . 
   this reflects again the crossover in the strong coupling pam from low - temperature lattice coherent behaviour to incoherent effective single - impurity scaling physics , here in the context of d.c .  transport . as for its counterpart in the case of single - particle dynamics  @xcite 
   , we point out ( a ) that since this connection is established from scaling considerations it is entirely independent of how the scales @xmath1 and @xmath277 for the two distinct models ( pam and aim ) depend on the underlying bare / material parameters of the respective problems ; and ( b ) the fact that it arises in the @xmath278 scaling regime precludes a two - scale description of the crossover from lattice - coherent to incoherent effective single - impurity physics . 
   our focus above has naturally been on the strong coupling , kondo lattice regime . 
   we now look briefly at d.c . 
   transport on non - universal scales . 
   what one expects here is that when the temperature is a not insignificant fraction of the hybridization @xmath170 or bandwidth scale @xmath279 , kondo screening will be washed out , and hence @xmath5 should cross over from the logarithmically decreasing single - impurity form at @xmath280 ( figures [ fig7 ] and [ fig8 ] ) to conventional metallic behaviour @xmath281 at non - universal temperatures ; and thus as such must go through a minimum . 
   that this indeed happens can be seen in the inset to figure [ fig8 ] where we show @xmath282 _ vs _ 
   @xmath25 for @xmath187 , @xmath283 and @xmath284 ( solid line ) and @xmath285 ( point - dash ) . 
   for the lower @xmath6 example , a minimum is seen at @xmath286 , which corresponds in ` absolute ' units ( @xmath41 ) to a temperature @xmath287  an appreciable fraction of the hybridization @xmath288 . 
   the corresponding minimum does of course exist for the higher @xmath6 , but is pushed beyond @xmath289 ( and to concomitantly lower vales of @xmath5 ) ; and @xmath5 in this case lies on the universal scaling curve throughout the @xmath26-range shown in figure [ fig8 ] . 
   a final point is worth noting here . for @xmath290 , the @xmath5 _ vs _ 
   @xmath25 for the two @xmath6 s shown in figure [ fig8 ] ( inset ) are in essence coincident ; each lies on the universal scaling curve . 
   what distinguishes different interaction strengths is of course the location of the minimum , occurring as it does on non - universal temperature scales . 
   no real hf material is however in the universal scaling regime ` for ever '  with increasing @xmath4 the scaling regime will be exited sooner or later . 
   and the temperature for which the experimental @xmath5 is a minimum ( once phonon contributions have been subtracted out ) can provide valuable information on the interaction strength , as we shall see in action in the following paper . 
   we have considered almost exclusively the hypercubic lattice , for the obvious reason that its one - particle bloch states ultimately underlie the low - temperature lattice coherence of the interacting problem . for the bethe lattice , the strong coupling scaling resistivity @xmath291 ( with @xmath292 from equation ( [ 2.18 ] ) ) is shown _ vs _ @xmath26 in inset ( b ) to figure [ fig6 ] , for @xmath293 . 
   in contrast to its counterpart for the hcl shown in the main figure , the @xmath222 resistivity is non - vanishing ( given by equation ( [ 2.19 ] ) ) , reflecting the absence of coherent bloch states for the bl . 
   further , the _ high_-@xmath26 asymptote of the bl @xmath5 in the scaling regime is likewise non - zero ; being given by the @xmath18 value of the free ( @xmath53 ) conduction band resistivity , namely @xmath294 ^ 2 = \pi^2/[4(1-\epsilon_c^2)]$ ] as marked by a cross in figure [ fig6 ] inset ( and arising for the same physical reasons discussed for kondo insulators in  @xcite ) . 
   the qualitative contrast between @xmath5 for the canonically bloch decomposable hcl , and that for the bl , illustrates why the latter  more specifically the associated ` joint density of states ' type formula equation ( [ 2.18 ] ) for @xmath295 that is not uncommonly employed in the literature  gives a poor caricature of d.c . 
   transport for hf metals in which the lattice coherence is of central importance . 
   we turn now to the optical conductivity @xmath296 ( with @xmath126 given by equation ( [ 2.10 ] ) ) . in the strong coupling kondo lattice regime 
   @xmath126 is of course independent of @xmath6 and @xmath189 , and a universal function of @xmath22 and @xmath25 for fixed @xmath147 and @xmath45 . 
   lma results for @xmath126 are shown in figure [ fig9 ] , for @xmath179 and @xmath224 . 
   the right panel shows the thermal evolution of the optical conductivity ( on a linear @xmath24-scale ) for temperatures @xmath297 and @xmath223 ; while the left panel ( on a log - log scale ) shows the behaviour for a lower range of temperatures up to @xmath298 . 
   the latter in particular illustrates the thermal evolution of the optical drude peak , which at @xmath299 consists of an @xmath300 @xmath208-function given by equations ( 2.14 ) ( with net weight @xmath180 in strong coupling ) . on increasing @xmath26 from @xmath211 
   the drude peak naturally broadens , and is well fit by a lorentzian up to its half - width or so , after which it decays more slowly in @xmath24 . at 
   the lowest @xmath26 shown the drude peak is well separated from the ` optical edge ' in @xmath126 seen at @xmath301 ( although we add that @xmath126 is strictly non - zero for all @xmath24 ) , and with increasing @xmath26 is seen to persist as an essentially separate entity up to @xmath302 or so ; after which it is progressively destroyed as expected , merging into an optical pseudogap in the neighbourhood of @xmath303 , which is reasonably well filled up by @xmath304 and all but gone by @xmath305 ( see figure [ fig9 ] , right panel ) . 
   similar behaviour is naturally found on varying @xmath147 and/or @xmath45 . 
   figure [ fig10 ] shows in particular the influence of @xmath147 ( varying conduction band filling ) on the optical pseudogap for a fixed temperature @xmath306 , from which it is seen that the pseudogap becomes shallower with increasing @xmath147 . 
   the above behaviour should be compared to the p - h symmetric kondo insulator ( ki ) @xmath307 considered in  @xcite . in that case 
   the @xmath18 optical conductivity is characterized by an indirect gap @xmath308 , and there is of course no @xmath18 drude peak . 
   instead a drude - like peak in the optical conductivity actually builds up on initially increasing @xmath26 from zero ( see figure 15 of  @xcite ) , before being thermally broadened and subsumed into the optical pseudogap . for @xmath309 or so 
   the low - frequency optics of the ki are thus very different from those of the hf metal , as expected . 
   but for @xmath310 the optical behaviour of the two is qualitatively similar as shown by comparison of figure [ fig9 ] ( right panel ) and its counterpart for the ki , figure 14 of  @xcite . 
   this too is physically natural , since the infilling of the indirect optical gap on temperature scales @xmath192 means that the ki behaves to all intents and purposes as a hf metal ; as seen also in figures [ fig7 ] or [ fig8 ] for the static transport . 
   a second point should be emphasised here , obvious though it is from the preceding discussion : whether for hf metals or kondo insulators , it is the low - energy scale @xmath159 that sets the intrinsic scale for both the @xmath27-dependence of the low - energy optical conductivity and its thermal evolution . and in strong coupling that scale is wholly distinct from the optical _ direct _ gap , @xmath311 . 
   the latter arises at its simplest in the commonly employed renormalized band picture ( see e.g.  @xcite ) , as the minimum direct gap for which optical transitions are allowed . in this effective single - particle description the imaginary part of the @xmath3-electron self - energy  and hence all scattering 
   is neglected entirely , and the corresponding real part @xmath312 is replaced by its leading low-@xmath27 behaviour equation ( [ 2.12 ] ) ( as also inherent to a slave boson mean - field approximation  @xcite ) . 
   the two branches of the renormalized bandstructure , denoted by @xmath313 with @xmath314 , then follow from the zeros of @xmath315^{-1 } = [ \gamma(\om)-\epsilon_{\mathbf k}]$ ] ( see equation ( [ 2.7 ] ) ) with the approximate @xmath316 @xmath317^{-1}$ ] from equation ( [ 2.5b ] ) ; and the resultant @xmath266-dependent direct gap @xmath318 $ ] is given by @xmath319^{1/2}$ ] with @xmath320 the usual renormalized level . 
   the minimum direct gap , @xmath311 , occurs for @xmath321 ( @xmath322 in strong coupling ) and is thus _ 
   [ 5.1 ] the corresponding result for the optical conductivity @xmath126 is readily determined from equation ( [ 2.10 ] ) . denoted by @xmath323 
   it is given for @xmath18 ( and all @xmath324 ) by f_o(;0 ) =    [ 5.2 ] with @xmath325 the free conduction band dos and @xmath326 the unit step function ; and is thus non - zero only for frequencies @xmath327 _ above _ the direct gap ( which result is also readily shown to hold for _ all _ temperatures ) . 
   two points should be noted here . 
   first that the low - energy scale @xmath159 intrinsic to hfs or kis is qualitatively distinct from the direct gap @xmath311 . 
   in fact since @xmath328 it follows that in strong coupling where the quasiparticle weight @xmath146 and hence @xmath1 becomes exponentially small , optics on the direct gap scale do not even lie in the @xmath329 scaling regime ; although neither do they occur on truly non - universal scales ( because @xmath330 ) and in that sense belong to the ` low - frequency ' optical spectrum . 
   second , we emphasise the inherent naivet of interpreting optics in terms of renormalized single - particle interband transitions : it is scattering due to electron interactions that generates _ all _ the optical density below the direct gap scale . failure to include such , as in a renormalized band picture  and regardless of how sophisticated the underlying band structure employed in practice  inevitably leads to a qualitatively inadequate description of optics ( as illustrated explicitly in figure [ fig11 ] below ) . 
   neither is this situation ameliorated in materials application by the introduction of _ ad hoc _ @xmath27-dependent broadening factors , for that simply avoids the basic underlying physics . 
   lma results for optics on all frequency scales are given in figure [ fig11 ] , for @xmath190 ; where @xmath126 is shown _ vs _ 
   @xmath22 on a log scale spanning five orders of magnitude , for the same range of temperatures @xmath25 employed in figure [ fig9 ] ( right panel ) . to encompass all @xmath27 including non - universal energies , the bare parameters @xmath6 and @xmath189 must of course be specified , @xmath331 and @xmath332 here being chosen for illustration ; 
   although note that the optical conductivity as a function of @xmath22 remains ` universal ' ( independent of @xmath6 or @xmath189 ) up to large but finite values of @xmath24 determined by the particular @xmath6 and @xmath189 chosen , in this example @xmath333 ( a directly analogous situation for the resistivity @xmath5 is shown in the inset to figure [ fig8 ] ) . 
   the inset to figure [ fig11 ] shows the renormalized bandstructure @xmath334 _ vs _ the free conduction band energy @xmath266 ; determined as above from solution of @xmath335 ( with @xmath336 the full @xmath337 ) . 
   this enables the notional direct gap to be determined , @xmath338 here  well separated from the low - energy coherence scale @xmath1 in strong coupling  and indeed seen to occur for @xmath339 . 
   the essential points from figure [ fig11 ] are clear . 
   as expected and well known ( see e.g.  @xcite ) , significant optical absorption occurs in the vicinity of the direct gap ; strongly broadened to low - energies due to electron interactions as above , and all but ` dead ' on non - universal energy scales ( e.g.  the hybridization @xmath340 for the chosen bare parameters ) . 
   regarding the thermal evolution of the optical conductivity note also that temperatures on the order of a few multiples of the coherence scale @xmath1  which control the thermal evolution of the low - energy optics  have essentially no effect on frequencies of the order of the direct gap , reflecting the clean separation between @xmath1 and @xmath311 characteristic of strong coupling . as a corollary 
   the direct gap should be thermally eroded only for @xmath341 ; as indeed seen in figure [ fig12 ] where ( for the same parameters as figure [ fig11 ] ) the thermal evolution of @xmath126 is shown for temperatures up to @xmath342 . 
   significant thermal erosion sets in by about @xmath343 or so , and is well developed by the highest temperature shown . 
   the clear scale separation between @xmath1 and @xmath311 will not however be captured properly if one is restricted to relatively low interactions and high temperatures as e.g.  in quantum monte carlo  @xcite , or from theories in which the quasiparticle weight @xmath146 is algebraically rather than exponentially small in the interaction strength , such as iterated perturbation theory  @xcite . 
   we have considered here the periodic anderson lattice , the canonical model for understanding heavy fermion metals , kondo insulators , intermediate valence and related materials . 
   optical conductivities , d.c . 
   transport and single - particle dynamics of the paramagnetic phase have been investigated , using the local moment approach within a dmft framework . for obvious physical reasons 
   our main focus has been the strongly correlated kondo lattice regime , where we find the problem to be characterised by a single , exponentially small coherence scale @xmath1 ; in terms of which the frequency and temperature dependence of physical properties scale  being universally dependent on @xmath344 and/or @xmath25 regardless of the interaction or hybridization strengths . 
   all relevant energy / temperature scales are handled by the theory , from the low - energy coherent fermi liquid domain out to large ( and in the strict scaling limit arbitrarily large ) multiples of @xmath1 where incoherent many - body scattering dominates the physics ; followed by the crossover out of the scaling regime to non - universal , high energy / temperature scales dictated by ` bare ' model / material parameters . and 
   while our emphasis has been on strong correlations we add that all interaction strengths from weak to strong coupling are encompassed by the lma  @xcite , such that intermediate valence behaviour in particular can also be addressed . 
   the first question posed in the introduction nonetheless remains : to what extent does the model , and our theory for it , capture experiment ? 
   we turn to that in the following paper where direct comparison of theory and experiment is made for three heavy fermion materials and a classic intermediate valence compound . 
   we are grateful to the epsrc for supporting this research .
summary: the paramagnetic phase of heavy fermion systems is investigated , using a non - perturbative local moment approach to the asymmetric periodic anderson model within the framework of dynamical mean field theory . 
   the natural focus is on the strong coupling kondo - lattice regime wherein single - particle spectra , scattering rates , d.c . 
   transport and optics are found to exhibit @xmath0 scaling in terms of a single underlying low - energy coherence scale @xmath1 . 
   dynamics / transport on all relevant ( @xmath2)-scales are encompassed , from the low - energy behaviour characteristic of the lattice coherent fermi liquid , through incoherent effective single - impurity physics likewise found to arise in the universal scaling regime , to non - universal high - energy scales ; and which description in turn enables viable quantitative comparison to experiment .