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the quantum heisenberg antiferromagnet ( qhafm ) is a prototype model to describe the magnetic properties in the parent compounds of high temperature superconducting cuprates.@xcite non - linear sigma model is shown that its ground state has long - range order and its low energy excitations are in a renormalized classical state.chakravartyprl,chakravartyprb,chubukov these results are confirmed by the quantum monte carlo methods@xcite and are consistent with experimental data.@xcite moreover , the schwinger boson mean field theory@xcite and the variational spin - wave theorytakahashi have also been developed for the two - dimensional qhafm , leading to similar conclusions . previous theoretical studies have concentrated in the low temperature region . however , the low - energy interesting physics of the two - dimensional qhafm can persist up to the temperature region @xmath0 . moreover , the anomalous normal state in the charged doped qhafm in the cuprates@xcite relies on a theory for the magnetic fluctuations in the intermediate temperature range . therefore , a theory is needed to be well defined in the full temperature range . although the schwinger boson mean field theory@xcite and takahashi s variational spin - wave theory@xcite for a qhafm in square lattices can be extended to the finite temperature range , the mean field ansatz results in a finite temperature phase transition at @xmath1 , which explicitly violates the mermin - wagner theorem@xcite . to overcome this artificial mean filed divergence of the magnetic susceptibility , we will propose a perturbation spin - wave theory .    in the takahashi s variational spin - wave theory , the dyson - maleev spin - boson representation is used for the qhafm on a square lattice with a nel ordered ground state : @xmath2for the spins in the spin - up sublattice , and @xmath3for the spin - down sublattice . the dyson - maleev or the holstein - primakoff spin - wave theory works well for the ground state . the corresponding results , such as the spin - wave velocity , the spin stiffness constant , the sublattice magnetization , and the perpendicular susceptibility , igarashi , canali , zheng agree well to the quantum monte carlo simulations.@xcite    to develop a finite temperature spin - wave theory , takahashi introduced a constraint for the vanishing local magnetization , i.e. , @xmath4at the @xmath5th lattice site , which automatically fulfills the mermin - wagner theorem . takahashi also introduced a mean field order parameter @xmath6 to account for the spin - wave interaction in mean field level . this mean field ansatz can describe reliably the low temperature physics , but fails at high temperatures , where it produces an artificial phase transition .    in this paper , we present a perturbation spin - wave theory with the constraint eq . ( [ eqn1.3 ] ) . the spin - wave interactions are considered within a many - body perturbation method . the mean field divergence in the previous variational spin - wave theory is removed . to verify our perturbation spin - wave theory , we calculate the uniform static susceptibility . its temperature dependence in the zeroth order can catch most features given by the quantum monte carlo simulations and high temperature series expansion . it shows a linear temperature dependence at low temperatures , a smooth crossover in the intermediate range and the curie law at high temperatures . by including the first order corrections from the spin wave interactions , our perturbation spin - wave theory agrees quantitatively well with the numerical simulations . this paper is organized as follows . in sec . [ sec2 ] , we present the linearized spin wave results up to the zeroth order . in sec . [ sec3 ] , a perturbation theory with the first order corrections from the spin - wave interactions are developed . we also give the comparison of the static magnetic susceptibility with the results of quantum monte carlo simulations and high - temperature series expansions . in sec . [ sec4 ] , we give the discussion and summary . in this paper we mainly discuss a qhafm on a quare lattice with hamiltonian defined by @xmath7where @xmath8 and @xmath9 denotes nearest - neighbor sites . since the ground state of this qhafm hamiltonian is an nel ordered state , we separate the square lattice with @xmath10 sites into sublattice @xmath11 for the up spins and sublattice @xmath12 for the down spins .    the dyson - maleev spin representation given by eq.([eqn1.2-a ] ) and ( eqn1.2-b ) are defined upon the nel ordered state . in terms of boson operators , the model can be divided into @xmath13where @xmath14 contains the classical energy and the quadratic terms , @xmath15and @xmath16 contains the quartic terms , describing the spin - wave interactions , @xmath17    in this section , we neglect the spin - wave interactions and focus on the hamiltonian @xmath14 . we refer this approximate to a linearized spin - wave theory ( lsw ) . the spin - wave interaction in the hamiltonian @xmath16 will be studied in sec . .    the constraint eq . ( [ eqn1.3 ] ) , from the dyson - maleev representation , can be described by introducing a lagrangian hamiltonian as@xcite @xmath18where the lagrangian multipliers @xmath19 and @xmath20 are assumed to be site independent , i.e. @xmath21 . physically the lagrangian multiplier @xmath22 plays a role of an effective chemical potential for the spin waves and leads to a finite gap , thus an effective mass , for these bosonic excitations at finite temperatures . for convenience in our following discussion , we define @xmath23 with the coordinate number @xmath24 .    by using the fourier transformations and performing the bogoliubov transformations , the hamiltonian @xmath25 can be expressed as @xmath26where @xmath27 and @xmath28 . here @xmath29 and @xmath30 are operators describing two branches of bogoliubov excitations . the free energy per site is given by @xmath31minimizing the free energy with respect to @xmath32 leads to a self - consistent equation @xmath33as a consequence of the constraint eq . ( [ eqn1.3 ] ) . the uniform static susceptibility per site can be calculated from the static spin - spin correlation function , @xmath34here the thermal average is defined in the zeroth order linearized spin - wave approximation as @xmath35 . it can be easily shown that the transverse spin - spin correlation function @xmath36 . therefore , only the longitudinal spin - spin correlation function @xmath37 has a finite value . the vanishing transverse spin - spin correlation function can also be shown in the takahashi s variational spin - wave theory.@xcite physically it stems from the local constraint given in eq . ( [ eqn1.3 ] ) .    the longitudinal spin - spin correlation function can be calculated by introducing a weak external magnetic field @xmath38 along @xmath39-axis which introduces an additional term @xmath40 to the model hamiltonian . the longitudinal static susceptibility thus reads @xmath41 , where the free energy @xmath42 includes the contribution from the external field @xmath43 . then the uniform static susceptibility within the linearized spin - wave theory has the form : @xmath44a similar expression has also been obtained by takahashi in his variational spin - wave theory.@xcite      the ground state of the qhafm on a square lattice has a long - range ordertakahashi , hirsch due to the presence of bose - einstein condensation ( bec ) at the momentum @xmath45 . since @xmath46 at zero temperature , the spin - wave spectrum is given by @xmath47comparing to the takahashi s spin wave spectrum@xcite @xmath48 with @xmath49 , we find that it can be obtained by rescaling our spin - wave result as @xmath50the additional factor @xmath51 is the main difference at low temperatures between our linearized spin - wave theory and takahashi s variational theory . the spin - spin correlation function in the long distant limit can be derived as @xmath52where @xmath53 if @xmath54 are in the same sublattice , and @xmath55 if @xmath54 belong to different sublattices . @xmath56 is the magnetization at zero temperature , which is the same as that in the schwinger boson mean field theory@xcite . both the spin - spin correlation function and the local spontaneous magnetization @xmath57 in our linearized spin - wave theory agree with those of the variational spin - wave theory.@xcite these results imply that , to the order of @xmath58 , the spin - wave interactions have no contributions to the spontaneous magnetization and static spin - spin correlation at @xmath59k , consistent to the previous results.@xcite       in the linearized spin - wave theory for the qhafm on a square lattice with different spin magnitude @xmath60 and @xmath61 . ]     for @xmath60 and @xmath61 in our linearized spin wave theory of the qhafm on a square lattice . ]    at finite temperatures , according to the mermin - wagner theorem , the long - range order is destroyed in the qhafm model by quantum fluctuations.@xcite fig . [ fig2.1 ] shows the temperature dependence of the lagrangian multiplier @xmath62 with different quantum spins @xmath63 . at low temperatures @xmath64 , we follow the takahashi s method and show analytically that @xmath32 approaches exponentially to its bec value @xmath65 as @xmath66the uniform static spin susceptibility in this low temperature region can also be derived analytically , leading to a linear - temperature dependence , @xmath67the first term gives a finite spin susceptibility at zero temperature . the linear term results from the linear density of states of the spin - wave excitations in the low energy limit . the low - energy spin - wave excitations at low temperatures have an approximate dispersion @xmath68where a dimensionless bosonic gap is defined by @xmath69 . at low temperatures , @xmath32 decays exponentially to @xmath65 as eq.([eqn2b.1 ] ) , therefore the energy spectrum eq.([eqn2b.3 ] ) is approximately as @xmath70 . the density of states @xmath71 of the spin - wave excitations is @xmath72 . it is this linear density of states that leads to the linear - temperature dependence of the static magnetic susceptibility at low temperatures . note that by scaling the linear temperature susceptibility @xmath73 by the factor @xmath51 , we can obtain the low - temperature susceptibility given by takahashi,@xcite @xmath74 .    at low temperatures , the spin - spin correlation function follows @xmath75where @xmath76 is defined in eq.([eqn2a.3 ] ) and @xmath77 is the correlation length defined by @xmath78 . from eq.([eqn2b.1 ] ) , it can be easily shown that @xmath79takahashi s result for the spin - spin correlation length @xmath80 can be obtained by rescaling with the factor @xmath51 defined in eq.([eqn2a.2 ] ) , @xmath81 . this exponential temperature dependence of the spin correlation length agrees with that obtained from one - loop renormalization group approach of the non - linear sigma model.kopietz,manousakis compared to the experimental data and the numerical simulations , the two - loop approximation is necessary to convert the @xmath82 prefactor of the exponential into a constant.chakravartyprl,chakravartyprb    when increasing the temperature , the lagrangian multiplier @xmath62 in fig . [ fig2.1 ] evolves from the low - temperature exponential behavior to the linear temperature dependence at high temperatures . at the same time , the uniform susceptibility as shown in fig . [ fig2.2 ] increases firstly to a broad peak at the intermediate temperature @xmath83 , and then decreases at high temperatures .    at high temperatures @xmath84 , the dispersion of the energy spectrum becomes relatively weak and @xmath85 . the self - consistent equation eq.([eqn2.7 ] ) gives rise to @xmath86 in the high - temperature region , the uniform static magnetic susceptibility obeys the curie law , @xmath87such a high - temperature curie - law behavior in the linearized spin - wave theory is exactly the same as that in the takahashi s theory.@xcite why our linearized spin - wave theory can give rise to a curie - law behavior , in contrast to the general spin - wave theory is a very interesting question . the basic reason is that there is an effective chemical potential introduced for our spin waves by the local constraint eq.([eqn1.3 ] ) . this gives rise to a temperature dependent finite gap for the spin waves which behave as bosons with effective mass at finite temperatures . it should be noted that in our numerical solution to the self - consistent equation eq.([eqn2.7 ] ) and in calculating the uniform magnetic susceptibility eq.([eqn2.9 ] ) , there is a trick to include the contribution of the low - energy spin - wave excitations at low temperatures . although no exact bec occurs at finite temperatures , in the actual numerical calculation eq.([eqn2b.1 ] ) shows us that the exponential decay of the bosonic gap @xmath88 at @xmath45 will lead to a difficulty in numerically calculating the momentum integral from the low - energy region @xmath89 . we call this difficulty as a bec - like pseudo - singularity . a trick to deal with the bec - like pseudo - singularity is to separate the momentum space into two regions : region ( i ) @xmath90 and region ( ii ) is the rest part in the first brillouin zone . here the momentum @xmath91 satisfies @xmath92 . @xmath93 is an irrelevant parameter and we choose @xmath94 in our calculations . the momentum integral in region ( i ) can be firstly calculated with an approximate spin - wave energy dispersion eq.([eqn2b.3 ] ) .     linearized spin - wave theory ( lsw ) and the takahashi s variational theory@xcite for the qhafm on a square lattice . ]    for comparison , the uniform static magnetic susceptibility from our linearized spin - wave theory and the takahashi s variational results are shown in fig . [ fig2.3 ] . the sharp peak value at @xmath95 in the takahashi s theory comes from an artificial phase transition due to vanishing of the mean - field ansatz . at low temperatures , the static magnetic susceptibility from our linearized spin - wave theory is slightly larger than the takahashi s result . at high temperatures , the curie law occurs exactly in both theories . clearly , the artificial phase transition has been removed in the linearized spin - wave theory . one special feature in the uniform magnetic susceptibility shown in fig . fig2.2 is a kink structure around the temperature @xmath96 , where the slop of the susceptibility changes from a smaller value into a larger one . this special feature is also observed in the quantum monte carlo data . @xcite it was argued to originate from a crossover from a renormalized classical regime into a quantum critical regime . however , the monte carlo simulation shows that a similar abrupt change also appears in a classical heisenberg model.@xcite therefore , whether there is a crossover is still controversial . however , this kink structure implies obviously different scenarios occurring in these two temperature regions . in this section , we will go beyond the linearized approximation by including the first order corrections from the spin - wave interactions of the qhafm model . the method is based on the many - body perturbation theory . after the fourier and bogoliubov transformations , the total hamiltonian @xmath97 can be approximated by@xcite @xmath98 ,   \notag \\ e_{c } & = & \sum_{\mathbf{k}}\varepsilon _ { \mathbf{k}}+njzs\left [ s-\lambda \left ( 1 + 2s\right ) \right ] -\frac{njz}{4}c_{1}^{2 } ,   \label{eqn4.2}\end{aligned}\]]where @xmath99 , @xmath100 , @xmath101 , @xmath102 , and @xmath103 represents the spin - wave energy spectrum with corrections from the normal ordering of @xmath16 . at zero temperature , @xmath104 and the spin - wave energy dispersion is given by @xmath105 with @xmath106 for @xmath107 . the parameters @xmath108 and @xmath109 are defined by @xmath110    in the above approximation , we have ignored the off - diagonal parts in the fock space and picked up all the terms of the two spin - wave interactions . this is enough when we only consider the first order corrections from the spin - wave interactions for the free energy , since the linked cluster theorem tells that @xmath111 , where the thermal average is defined by @xmath112 and @xmath113 is defined for the free bosons with energy spectrum @xmath114 . the free energy per lattice site reads @xmath115 -\frac{jz}{2}c_{2}^{2}+\frac{1}{2n}e_{c } ,   \label{eqn3.2}\]]where @xmath116 and @xmath117 . minimizing this free energy , we can obtain the following self - consistent equation for the lagrangian multiplier @xmath32 , @xmath118where @xmath119 and @xmath120it should be noted that the self - consistent equation eq.([eqn3.2 ] ) is physically equivalent to the first order perturbation expansion for the constraint eq.([eqn1.3 ] ) , @xmath121    the uniform static susceptibility can also be calculated from the spin - spin correlation function given in eq.([eqn2.8 ] ) . when the first order corrections from the spin - wave interactions are included , the transverse static susceptibility has a small but finite contribution , @xmath122 in the derivation for @xmath123 , we have used the perturbation constraint equation eq.([eqn3.3 ] ) and ignored the higher order terms of the spin - wave interactions . the longitudinal static susceptibility can also be calculated from the free energy by introducing a weak external field hamiltonian @xmath43 , yielding @xmath124 , \label{eqn3.5}\]]where the second term comes from the two spin - wave interaction compared to the linearized spin - wave expression eq.([eqn2.9 ] ) , and @xmath125the uniform static susceptibility is thus given by @xmath126 . \label{eqn3.6}\ ] ]      the self - consistent equation eq.([eqn3.2 ] ) and the static magnetic susceptibility eq.([eqn3.6 ] ) have been numerically calculated . when the first order corrections of the spin - wave interactions are included , the temperature dependence of both the lagrangian multiplier @xmath32 and the uniform static magnetic susceptibility have similar behaviors to the linearized spin - wave theory . the lagrangian multiplier @xmath32 is shown in fig . [ fig3.1 ] for different spin @xmath60 and @xmath61 . at low temperatures , it decays exponentially to its zero - temperature value @xmath65 , indicating the gapless spin wave excitations . at high temperatures , @xmath32 has a linear - temperature variation @xmath127 , where @xmath128 and @xmath129 have weak temperature dependence . for @xmath107 , @xmath130 and @xmath131 in the temperature region @xmath132 . the static magnetic susceptibility is displayed in fig . [ fig3.2 ] . it has three main features : ( i ) the linear - temperature behavior at low temperatures , ( ii ) a broad peak at an intermediate characteristic temperature @xmath83 , ( iii ) the curie - weiss law at high temperatures . for @xmath107 , a low - temperature linear behavior can be fitted by @xmath133 , with @xmath134 and @xmath135 . compared to the takahashi s theory where a similar linear behavior with @xmath136 and @xmath137,@xcite our zero temperature susceptibility @xmath138 is slightly smaller than that of takahashi s . also the slope @xmath139 is also slightly larger than that of the takahashi s .     and @xmath61 in our perturbation spin - wave theory with the first order corrections from the spin - wave interactions . ]    at high temperatures , the spin - wave interactions modify the exact curie law of the static magnetic susceptibility eq.([eqn2b.7 ] ) into a curie - weiss behavior . as shown in fig . [ fig3.3 ] , it evolves into the behavior obtained from the numerical high - temperature series expansions . physical origin of the high - temperature curie - weiss law also comes from the linear - temperature dependence of lagrangian multiplier @xmath32 in the energy spectrum of the spin waves . it should be noted that the kink structure in the static magnetic susceptibility in the linearized spin - wave theory shown in fig . [ fig2.2 ] also occurs in the perturbation theory . this implies that the kink feature is a characteristic property in the qhafm model .     comes from ref . and qmc@xmath140 comes from ref . . htse comes from eq.(@xmath141 ) of ref . . for comparison , qmc@xmath140 have been multiplied by a factor @xmath142 . ]    in fig . [ fig3.3 ] , we compare the static magnetic susceptibility of our spin - wave theory to the quantum monte carlo data@xcite and the high - temperature series expansions . it shows that our linearized spin - wave theory can reproduce all the main features in the numerical simulations : the low temperature linear behavior , the smooth crossover in the intermediate temperature region and the curie - weiss law behavior at high temperature . this good agreement encourages us to study the frustrated @xmath143 model proposed to describe the magnetic fluctuation in the newly discovered iron - pnictide superconductors.@xcite the first order correction from the spin - wave interactions makes the uniform static susceptibility both at low and high temperatures agrees quantitatively with the numerical quantum monte carlo and high temperature series expansions . this good fitting in fig . [ fig3.3 ] indicates that the physics of qhafm model , at least to the static magnetic response , has been captured by our perturbation spin - wave theory in the full temperature region . the good agreement of our perturbation spin - wave theory with the numerical simulations shows the validity of our theory at finite temperatures . it seems surprising because the spin - wave theory developed in the pioneering papers@xcite is generally thought to work well only at low temperatures in the ordered phase . the main difference of our finite temperature spin - wave theory is the requirement of the vanishing local magnetization . it is this local constraint that restricts the number of the spin wave excitations which thus behaves as bosons with an effective mass at finite temperatures . the extension of the spin - wave theory to the low - dimensional spin systems have been shown to work well . in one - dimensional quantum magnets , no magnetic order can survive at zero temperature due to the strong quantum fluctuations . although there is no long range order in the ground state , the free energy and the static magnetic susceptibility of the one - dimensional ferromagnet from a modified spin - wave theory show same behaviors to bethe - ansatz solutions.@xcite the haldane gap for the one - dimensional quantum antiferromagnet with @xmath144 from a spin - wave theory is @xmath145 ( ref . ) which well agrees with the numerical density - matrix renormalization group ( @xmath146 from ref . ) . the spin - wave theory for the one - dimensional quantum antiferromagnet with integral spin can reproduce the static magnetic susceptibility which agrees well with the quantum monte carlo simulations.@xcite these previous results , together with our study , indicate that the formulation of the spin - wave theory is a valid formalism to apply to the low - dimensional quantum magnets .    in summary , we have studied the qhafm on a square lattice by using a perturbation spin - wave theory . the lagrangian constraint for the absence of the local magnetization plays an important role and leads to an effective mass for the spin wave excitations at finite temperatures . in linearized spin - wave theory , the calculation of the uniform static magnetic susceptibility can reproduce almost all features obtained from the quantum monte carlo and the high - temperature series expansions . it has a linear - temperature dependence at low temperatures and a broad peak feature at an intermediate characteristic temperature @xmath83 . at high temperatures , the uniform magnetic susceptibility follows a curie - weiss law . the static magnetic susceptibility from our perturbation theory with the first corrections from the spin - wave interactions agrees well with the numerical simulations .
a perturbation spin - wave theory for the quantum heisenberg antiferromagnets on a square lattice is proposed to calculate the uniform static magnetic susceptibility at finite temperatures , where a divergence in the previous theories due to an artificial phase transition has been removed . to the zeroth order , 
 the main features of the uniform static susceptibility are produced : a linear temperature dependence at low temperatures and a smooth crossover in the intermediate range and the curie law at high temperatures . 
 when the leading corrections from the spin - wave interactions are included , the resulting spin susceptibility in the full temperature range is in agreement with the numerical quantum monte carlo simulations and high - temperature series expansions .