document: transverse spin fluctuations are gapless , low - energy excitations in the broken - symmetry state of magnetic systems possessing continuous spin - rotational symmetry . therefore at low temperatures they play an important role in diverse macroscopic properties such as existence of long - range order , magnitude and temperature - dependence of the order parameter , nel temperature , spin correlations etc . specifically in the antiferromagnetic ( af ) ground state of the half - filled hubbard model 
   transverse spin fluctuations are important both in two and three dimensions , where antiferromagnetic long - range order ( aflro ) exists at @xmath4 . in the strong - coupling limit @xmath5 , where spin fluctuations are strongest 
   , they significantly reduce the zero - temperature af order parameter in two dimensions to nearly @xmath6 of the classical ( hf ) value.@xcite similarly the nel temperature in three dimensions is reduced to nearly @xmath7 of the mean - field result @xmath8 , for the equivalent @xmath9 quantum heisenberg antiferromagnet ( qhaf).@xcite    recently there has also been interest in spin fluctuations due to defects , disorder and vacancies in the quantum antiferromagnet . in the random-@xmath3 model , where @xmath3 is set to zero on a fraction @xmath10 of sites , the lattice - averaged af order parameter appears to be enhanced for small @xmath10 , as seen in qmc calculations,@xcite presumably due to an overall suppression of quantum spin fluctuations . on the other hand spin fluctuations 
   are enhanced by strong disorder in the hubbard model with random on - site energies . in the strong disorder regime , 
   overlap of the two hubbard bands leads to formation of essentially empty and doubly - occupied sites , which act like spin vacancies.@xcite the problem of spin vacancies in the quantum antiferromagnet is also relevant to the electron - doped materials like @xmath11 , where spin - dilution behavior is observed.@xcite while the problem of magnon energy renormalization due to spin vacancies has been addressed recently,@xcite these methods are limited to the low - concentration limit , and the vacancy - induced enhancement in transverse spin fluctuations has not been studied in the whole range of vacancy concentration .    in this paper 
   we describe a new method for evaluating transverse spin correlations and quantum spin - fluctuation corrections about the hf - level broken - symmetry state , in terms of magnon mode energies and spectral functions obtained in the random phase approximation ( rpa ) . 
   the method is applicable in the whole @xmath0 range of interaction strength , and is illustrated with three applications involving the af ground state of the half - filled hubbard model  ( i ) the pure model in @xmath12 , ( ii ) spin vacancies in the strong coupling limit in @xmath13 , and ( iii ) low-@xmath3 impurities in @xmath13 . 
   this method for obtaining quantum correction to sublattice magnetization solely in terms of transverse spin correlations is parallel to the spin - wave - theory ( swt ) approach,@xcite and differs from the method involving self - energy corrections.@xcite    the rpa approach has been demonstrated earlier to properly interpolate between the weak and strong coupling limits for the spin - wave velocity.@xcite by going beyond the rpa level within a systematic inverse - degeneracy expansion scheme , which preserves the spin - rotational symmetry and hence the goldstone mode order by order , it was also shown that in the strong coupling limit identical results are obtained for all quantum corrections , order by order , as from the swt approach for the qhaf.@xcite a renormalized rpa approach has also been devised recently to obtain the magnetic phase diagram for the three dimensional hubbard model in the whole @xmath0 range , and the @xmath14 vs. @xmath3 behaviour was shown to properly interpolate between the weak and strong coupling limits.@xcite 
   the method is based on a convenient way to perform the frequency integral in order to obtain spin correlations from spin propagators , and we illustrate it here for transverse spin correlations . 
   we write the time - ordered , transverse spin propagator for sites @xmath15 and @xmath16 , @xmath17|\psi_{\rm g}\rangle$ ] at the rpa level in frequency space as , @xmath18=\frac{[\chi^0(\omega)]}{1-u[\chi^0(\omega ) ] } = \sum_n \frac{\lambda_n(\omega)}{1-u\lambda_n(\omega ) }     @xmath19 and @xmath20 are the eigenvectors and eigenvalues of the @xmath21 $ ] matrix . here 
   @xmath21_{ij}=i\int ( d\omega'/2\pi ) g_{ij}^{\uparrow}(\omega')g_{ji}^{\downarrow}(\omega'-\omega)$ ] is the zeroth - order , antiparallel - spin particle - hole propagator , evaluated in the self - consistent , broken - symmetry state from the hf green s functions @xmath22 . 
   spin correlations are then obtained from , @xmath23_{ij}\ ; e^{-i\omega ( t - t ' ) } \nonumber \\ & = & \pm \sum_n \frac{\phi_n ^i ( \omega_n)\phi_n ^j ( \omega_n ) } { u^2 ( d\lambda_n / d\omega)_{\omega_n } }   e^{-i\omega_n ( t - t ' ) } \ ; , \end{aligned}\ ] ] where the collective mode energies @xmath24 are obtained from @xmath25 , and @xmath20 has been taylor - expanded as @xmath26 near the mode energies to obtain the residues . for convergence , 
   the retarded ( advanced ) part of the time - ordered propagator @xmath27 , having pole below ( above ) the real-@xmath28 axis , is to be taken for @xmath29 ( @xmath30 ) . 
   the frequency integral is conveniently replaced by an appropriate contour integral in the lower or upper half - plane in the complex-@xmath28 space for these two cases , respectively , which results in eq . 
   we first illustrate this method for the half - filled hubbard model in two and three dimensions on square and simple - cubic lattices , respectively . in this case 
   it is convenient to use the two - sublattice representation due to translational symmetry , and we work with the @xmath32 matrix @xmath33 $ ] in momentum space , which is given in terms of eigensolutions of the hf hamiltonian matrix.@xcite the @xmath34-summation is performed numerically using a momentum grid with @xmath35 and 0.05 , in three and two dimensions , respectively . 
   equal - time , same - site transverse spin correlations are then obtained from eq . 
   ( 2 ) by summing over the different @xmath36 modes , using a momentum grid with @xmath37 and @xmath38 in three and two dimensions , respectively . 
   we consider @xmath39 , so that the retarded part is used , with positive mode energies . from spin - sublattice symmetry , 
   correlations on a and b sublattice sites are related via @xmath40 . 
   thus the transverse spin correlations are obtained from magnon amplitudes on a and b sublattices , and from eq . 
   ( 2 ) we have @xmath41    from the commutation relation @xmath42=2s^z$ ] , the difference @xmath43 , of transverse spin correlations evaluated at the rpa level , should be identically equal to @xmath44 . 
   this is becasue both the rpa and hf approximations are o(1 ) within the inverse - degeneracy expansion scheme@xcite in powers of @xmath45 ( @xmath46 is the number of orbitals per site ) , and therefore become exact in the limit @xmath47 , when all corrections of order @xmath45 or higher vanish . 
   this is indeed confirmed as shown in figs . 1 and 2 . 
   the deviation at small @xmath3 is because of the neglect in eq . 
   ( 2 ) of the contribution from the single particle excitations across the charge gap , arising from the imaginary part of @xmath48 in eq . 
   the sum @xmath49 yields a measure of transverse spin fluctuations about the hf state , and in the strong coupling for spin @xmath50 , one obtains @xmath51.@xcite using the identity , @xmath52 in this limit , the sublattice magnetization @xmath53 is then obtained from , @xmath54^{1/2 } .\ ] ] to order @xmath55 , this yields the correction to the sublattice magnetization of @xmath56=0.156 $ ] and 0.393 for @xmath9 in three and two dimensions , respectively . 
   this same result at one loop level was also obtained from a different approach in terms of the electronic spectral weight transfer,@xcite and is in exact agreement with the swt result.@xcite    as @xmath57 is the maximum ( classical ) spin polarization in the z - direction , and therefore also the maximum eigenvalue of the local @xmath58 operator , therefore for arbitrary @xmath3 , the hf magnitude @xmath57 plays the role of the effective spin quantum number , @xmath50 . 
   the sublattice magnetization @xmath59 is therefore obtained from @xmath60 , where the first - order , quantum spin - fluctuation correction @xmath61 is obtained from eq . ( 4 ) with @xmath62 , @xmath63 in the strong coupling limit @xmath2 , @xmath64 for @xmath9 , so that the spin - fluctuation correction simplifies to , @xmath65 . 
   for a site on the b sublattice , where @xmath66 , we have @xmath67 .    the @xmath3-dependence of sublattice magnetization @xmath59 is shown in figs . 1 , and 2 for @xmath13 and @xmath68 , respectively . 
   in both cases it interpolates properly between the weak and strong coupling limits , approaching the swt results 0.607 and 0.844 , respectively , as @xmath2 . 
   a comparison of the @xmath59 vs. @xmath3 behaviour with earlier results is presented in fig . 
   3 for the well studied @xmath13 case . 
   earlier studies have employed a variety of methods including the variational monte carlo ( vmc),@xcite self - energy corrections ( se),@xcite quantum monte carlo ( qmc),@xcite functional - integral schemes,@xcite the generalized linear spin - wave approximation ( glswa),@xcite and mapping of low - energy excitations to those of a qhaf with @xmath3-dependent , extended - range spin couplings.@xcite    in addition to the two - sublattice basis , we have also used the full site representation in the strong coupling limit , in order to illustrate the scaling of the quantum correction with system size . here the @xmath69 matrix is evaluated and diagonalized in the site basis for finite lattices . 
   results for lattice sizes with @xmath70 are shown in fig . 
   4 . a quadratic least - square fit is used to extrapolate to infinite system size , which yields @xmath71 , in agreement with the result from eq . 
   the site representation has also been used to obtain transverse spin fluctuations for the problem of spin vacancies in the af in the limit @xmath2 . 
   as mentioned already this method is applicable in the whole range of vacancy concentration , and allows determination of the critical vacancy concentration at which the af order parameter vanishes . for the vacancy problem 
   we consider the following hamiltonian on a square lattice with nearest - neighbor ( nn ) hopping , @xmath72 where the hopping terms @xmath73 if sites @xmath15 or @xmath16 are vacancy sites , and @xmath74 otherwise . 
   thus , for a vacancy on site @xmath15 , all hopping terms @xmath75 connecting @xmath15 to its nn sites @xmath16 are set to zero . 
   the vacancy site is thus completely decoupled from the system . 
   half - filling is retained by having one fermion per remaining site . 
   we consider the @xmath2 limit , where the local moments are fully saturated , and the vacancy problem becomes identical to the    spin - vacancy problem in the qhaf . 
   this is also equivalent to the problem of non - magnetic impurities in the af in the limit of the impurity potential @xmath76.@xcite    the structure of the @xmath48 matrix in the host af , and the modification introduced by spin vacancies has been considered earlier in the context of static impurities.@xcite as the goldstone mode is preserved , it is convenient to work with the matrix @xmath77 $ ] , inverse of which yields the spin propagator in eq . 
   ( 1 ) near the mode energies . when expressed in units of @xmath78 , where @xmath79 is the af gap parameter , and @xmath80 , it has the following simple structure for the host af : @xmath81    vacancies introduce a perturbation in @xmath82 due to absence of hopping between the vacancy and nn sites , and we take @xmath83 to refer to this vacancy - induced perturbation , so that @xmath84 . if site @xmath15 is a vacancy site and @xmath16 the nn sites , then the matrix elements of @xmath85 are , @xmath86 and the magnitude of @xmath87 is irrelevant since the vacancy site @xmath15 is decoupled . 
   thus @xmath88 , reflecting the decoupling of the vacancy , and the static part of the diagonal matrix elements @xmath89 on nn sites are reduced by 1/4 . for @xmath90 vacancies on nn sites , 
   the static part is @xmath91 . 
   this ensures that the goldstone mode is preserved . 
   thus if a spin on site @xmath16 were surrounded by a maximum of four vacancies on nn sites , then the static part vanishes , and @xmath92 , representing an isolated spin , which yields a @xmath93 pole in the transverse spin propagator . 
   this is an isolated single - spin cluster , and with increasing vacancy concentration , larger isolated spin clusters are formed .    as these are decoupled from the remaining system , their spin - fluctuation contributions 
   are not included , as the op vanishes for finite spin clusters . 
   when @xmath94 exceeds the percolation threshold @xmath95 , the fraction of macroscopically large spin clusters in the system vanishes , and therefore no aflro is possible for @xmath96 .    for a given vacancy concentration and system size 
   , the appropriate number of vacancies are placed randomly across the lattice , and the matrix @xmath82 constructed accordingly . 
   exact diagonalization of @xmath82 is carried out , and the eigensolutions are used to compute the transverse spin correlations from eq . ( 2 ) . 
   the quantum , spin - fluctuation correction is then obtained from the strong - coupling limit of eq . 
   the transverse spin correlation @xmath97 is averaged over all spins within the a sublattice . as mentioned already , only the spins in the macroscopic cluster spanning the whole lattice are considered , and contributions from spins in isolated spin clusters are excluded . 
   configuration averaging over several realizations of the vacancy distribution is also carried out . 
   the quantum spin - fluctuation correction vs. vacancy concentration is shown in fig . 5 for three lattice sizes , @xmath98 . 
   best fits are obtained with an expression including a cubic term , @xmath99 . 
   for the three lattice sizes @xmath1000.236 , 0.260 and 0.278 , respectively , and as shown in fig . 
   3 , it extrapolates to 0.39 as @xmath101 . 
   the coefficient of the linear term is found to be nearly independent of system size , @xmath102 . 
   and the cubic term @xmath103 takes values approximately 3.6 , 4.0 , and 4.4 for the three lattice sizes , and extrapolates to about 6.5 as @xmath101 . with these coefficients , 
   the spin - fluctuation correction @xmath61 is nearly 1 for @xmath104 . beyond @xmath104 , the percolation limit 
   , there is no single , macroscopically large spin cluster left in the system . 
   therefore the point where the af order parameter vanishes and aflro is destroyed due to spin fluctuations nearly coincides with the percolation threshold . 
   this is in agreement with results from series expansion@xcite and quantum monte carlo simulations@xcite of the qhaf with spin vacancies . 
   we now consider a quenched impurity model with a random distribution of impurity sites characterized by a local coulomb interaction @xmath105 for the host sites . with h and 
   i referring to the sets of host and impurity sites respectively , we consider the following hubbard model in the particle - hole symmetric form at half - filling and on a square lattice , @xmath106    the motivations for studying this impurity model are threefold . in view of the observed _ enhancement _ of magnetic order at low concentration of impurities,@xcite we shall analyze the suppression of quantum spin fluctuations to examine whether this is due to a local suppression at the low-@xmath3 sites . 
   the rpa evaluation of transverse spin correlations is also extended to the case of site - dependent interactions . 
   furthermore , at half - filling this model also provides a simplistic representation for magnetic impurity doping in an af . 
   this may appear contradictory in view of the apparently nonmagnetic ( @xmath107 ) nature of the impurity sites . 
   however , this feature is expressed only away from half - filling.@xcite    the atomic limit @xmath108 provides a convenient starting point for further discussions . in the particle - hole 
   symmetric form of eq . 
   ( 9 ) , since not only local interaction terms , but the on - site energy terms are also modified ( from @xmath109 to @xmath110 ) at the impurity sites , therefore the energy levels for added hole and particle are @xmath111 and @xmath112 for host and impurity sites respectively . to order @xmath113 , this impurity model therefore canonically maps to the following @xmath9 heisenberg model @xmath114 where in the first term @xmath80 is the conventional exchange coupling between neighboring host spins , and 
   @xmath115 is the exchange coupling between impurity spins and neighboring host spins . in writing eq . 
   ( 10 ) we have assumed the dilute impurity limit , and discounted the possibility of two impurity spins occupying nn positions , in which case the impurity - impurity exchange coupling will be @xmath116 .    therefore , in the strong correlation limit , this random-@xmath3 model also describes magnetic impurities in the af within an impurity - spin model . 
   the magnetic - impurity doping is characterized by identical impurity and host spins @xmath117 , but with different impurity - host exchange coupling @xmath118 . 
   both cases @xmath119 or @xmath120 are possible , and in this paper we have considered the two cases : ( i ) @xmath121 so that @xmath122 , and @xmath123 so that @xmath124 . while this model is easily generalized to other magnetic - impurity models represented by locally modified impurity - host hopping terms @xmath125 , and/or different impurity energy levels , in fact , the essential features are already contained here as the impurity exchange coupling @xmath126 is the relevant quantity in determining the spin fluctuation behavior . 
   we recast the rpa expression for the transverse spin propagator in a form suitable for site - dependent interactions . in terms of a diagonal interaction matrix 
   @xmath127 $ ] , with elements @xmath127_{ii}=u_i $ ] , the local coulomb interaction at site @xmath15 , the time - ordered transverse spin propagator at the rpa level can be rewritten , after simple matrix manipulations , as @xmath18= \frac{[\chi^{0}(\omega ) ] } { 1 - [ u ] [ \chi^{0}(\omega ) ] } = \frac{1}{[a(\omega ) ] } - \frac{1}{[u]}\ ] ] where @xmath128=[u ] - [ u][\chi^0 ( \omega)][u]$ ] is a symmetric matrix . 
   as @xmath127 $ ] is non - singular , the singularities in @xmath129 $ ] , which yield the magnon modes , are then given completely by the vanishing of the eigenvalues of the matrix @xmath130 $ ] . in terms of @xmath131 and @xmath132 , the eigenvalues and eigenvectors of the matrix @xmath130 $ ] , we have @xmath128^{-1 } = \sum_n \lambda_n(\omega)^{-1 }    magnon - mode energies @xmath24 are then given by @xmath133 , and in analogy with eq . 
   ( 2 ) the transverse spin correlations are obtained from , @xmath134    as we are interested in the dilute behaviour , we examine the correction to sublattice magnetization due to two impurities , one on each sublattice for symmetry . 
   since @xmath135 , corrections to both @xmath136 and @xmath61 are expressed in the dilute limit ( impurity concentration @xmath94 ) as @xmath137 , and @xmath138 . 
   the overall @xmath139 behavior therefore depends on the relative magnitude of the coefficients @xmath140 and @xmath141 . for @xmath142 and @xmath143 
   we find , at the hf level , that @xmath144 and @xmath145 , both quite independent of system size .    the ( site - averaged ) spin fluctuation correction @xmath61 is obtained from eq . 
   ( 5 ) with and without impurities , and the impurity contribution extracted . for @xmath143 
   we find a net _ reduction _ in @xmath61 . 
   divided by @xmath146 , the impurity concentration , this yields the coefficient @xmath141 defined above , and also the per - impurity contribution to the total spin fluctuation correction over the whole lattice . 
   the spin - fluctuation correction @xmath61 for the pure case , and the per - impurity contribution @xmath147 are shown in fig . 6 for different lattice sizes , along with least - square fits . 
   it is seen that in the infinite size limit , the per - impurity reduction is nearly 0.2 , which is more than half of the correction per site in the pure case ( 0.35 ) . 
   thus , there is a substantial reduction in the averaged spin fluctuation correction due to the low@xmath1 impurities .    as the two coefficients @xmath140 and @xmath141 are very nearly the same , the sublattice magnetization @xmath148 shows negligible concentration dependence . 
   thus the ( relatively small ) reduction in the hf value due to the low-@xmath3 impurities is almost fully compensated by the ( relatively substantial ) reduction in the spin fluctuation correction . 
   to first order in @xmath94 , we thus find that there is nearly no loss of af order due to the low-@xmath3 impurities . 
   as mentioned earlier , even a slight enhancement in the af order was recently seen for the case @xmath149.@xcite    we next examine the site - dependence of the local spin fluctuation corrections @xmath150 near the impurities . table i shows that spin fluctuation is actually _ enhanced _ on the low-@xmath3 impurity sites . 
   the suppression of @xmath150 in the vicinity more than compensates for this local enhancement , resulting in an overall reduction on the average . on the other hand , for @xmath151 , we find that the correction is suppressed on the high-@xmath3 impurity site , while it is enhanced on the average . 
   thus , to summarize , when the impurity spin is coupled more strongly ( weakly ) , the spin - fluctuation correction is enhanced ( suppressed ) locally at the impurity site , but the average correction to sublattice magnetization is suppressed ( enhanced ) . 
   this local enhancement can be understood in terms of the correlations @xmath152 as follows . since the impurity spin is more strongly coupled to the neighboring spins , the nn matrix elements @xmath153 are enhanced . 
   this puts more magnon amplitude @xmath154 on the impurity site , so that from eq . 
   ( 2 ) the transverse spin correlations @xmath152 , and therefore the spin - fluctuation correction , are enhanced for low-@xmath3 impurities . the overall decrease in the averaged fluctuation correction , however , is due to the stiffening of the magnon spectrum in the important low - energy sector , following from the increased average spin coupling . 
   .the local spin - fluctuation corrections @xmath155 for a @xmath156 system ( @xmath142 ) , with two impurities at ( 11,4 ) and ( 4,14 ) . 
   for the two cases @xmath143 ( @xmath157 ) and @xmath151 ( @xmath158 ) , quantum corrections are enhanced / suppressed locally at the impurity sites ( indicated in boldfaces ) , but are suppressed / enhanced on the average . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ]     2    while these results also follow from the impurity - spin picture , the small charge gap at the impurity site does have an impact on the magnon spectrum . within the localized spin picture 
   , the highest - energy magnon mode corresponding to a local spin deviation at the impurity site would cost energy @xmath159 . 
   however , the highest energy in the magnon spectrum is actually seen to be @xmath160 , which is substantially smaller than @xmath161 . 
   this shows the compression effect of the low charge gap ( @xmath162 ) on the magnon spectrum.@xcite    in conclusion , using a convenient numerical method for evaluating transverse spin correlations at the rpa level , quantum spin - fluctuation corrections to sublattice magnetization are obtained for the half - filled hubbard model in the whole @xmath0 range . 
   results in two and three dimensions are shown to interpolate properly between both the weak and strong correlation limits , and approach the swt results as @xmath2 . 
   the method is readily extended to other situations of interest involving defects in the af , such as vacancies / impurities / disorder . 
   numerical diagonalization for finite lattices , along with finite - size scaling tested with the pure hubbard model , allows for exact treatment of defects at the rpa level . 
   this is illustrated with a study of the defect - induced enhancement / suppression in transverse spin fluctuations for spin vacancies and low-@xmath3 impurities in two dimensions . 
   while the quantum spin fluctuation correction to sublattice magnetization is sharply enhanced by spin vacancies , it is strongly suppressed by the low-@xmath3 impurities , although the fluctuation correction is enhanced at the low@xmath1 sites . 
   helpful conversations with d. vollhardt and m. ulmke , and support from the alexander von humboldt foundation through a research fellowship at the universitt augsburg are gratefully acknowledged .
summary: a numerical method is described for evaluating transverse spin correlations in the random phase approximation . 
   quantum spin - fluctuation corrections to sublattice magnetization are evaluated for the antiferromagnetic ground state of the half - filled hubbard model in two and three dimensions in the whole @xmath0 range . 
   extension to the case of defects in the af is also discussed for spin vacancies and low@xmath1 impurities . in the @xmath2 limit 
   , the vacancy - induced enhancement in the spin fluctuation correction is obtained for the spin - vacancy problem in two dimensions , for vacancy concentration up to the percolation threshold . for low-@xmath3 impurities , 
   the overall spin fluctuation correction is found to be strongly suppressed , although surprisingly spin fluctuations are locally enhanced at the low@xmath1 sites .    2