The Spin-Half XXZ Antiferromagnet on the Square Lattice Revisited: A High-Order Coupled Cluster Treatment

# The Spin-Half Xxz Antiferromagnet on the Square Lattice Revisited: A High-Order Coupled Cluster Treatment

R.F. Bishop P.H.Y. Li R. Zinke R. Darradi J. Richter D.J.J. Farnell J. Schulenburg School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, UK Institute for Theoretical Physics, Otto-von-Guericke University Magdeburg, P.O.B. 4120, 39016 Magdeburg, Germany School of Dentistry, Cardiff University, Cardiff CF14 4XY, Wales, UK Computing Center, Otto-von-Guericke University Magdeburg, P.O.B. 4120, 39016 Magdeburg, Germany
###### Abstract

We use the coupled cluster method (CCM) to study the ground-state properties and lowest-lying triplet excited state of the spin-half XXZ antiferromagnet on the square lattice. The CCM is applied to it to high orders of approximation by using an efficient computer code that has been written by us and which has been implemented to run on massively parallelized computer platforms. We are able therefore to present precise data for the basic quantities of this model over a wide range of values for the anisotropy parameter in the range of interest, including both the easy-plane and easy-axis regimes, where represents the Ising limit. We present results for the ground-state energy, the sublattice magnetization, the zero-field transverse magnetic susceptibility, the spin stiffness, and the triplet spin gap. Our results provide a useful yardstick against which other approximate methods and/or experimental studies of relevant antiferromagnetic square-lattice compounds may now compare their own results. We also focus particular attention on the behaviour of these parameters for the easy-axis system in the vicinity of the isotropic Heisenberg point (), where the model undergoes a phase transition from a gapped state (for ) to a gapless state (for ), and compare our results there with those from spin-wave theory (SWT). Interestingly, the nature of the criticality at for the present model with spins of spin quantum number that is revealed by our CCM results seems to differ qualitatively from that predicted by SWT, which becomes exact only for its near-classical large- counterpart.

###### keywords:
antiferromagnet, Square lattice, Easy-plane and easy-axis, Low-energy parameters, Spin gap, Coupled cluster method
journal: Journal of magnetism and magnetic materials\biboptions

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## 1 Introduction

The antiferromagnetic XXZ model on the square lattice is an important model that is used to describe antiferromagnetic insulators. The Hamiltonian for this system is given by

 H = ∑⟨i,j⟩[sxisxj+syisyj+Δsziszj]  , (1)

where the index runs over all sites on the infinite () square lattice and the sum on runs over all nearest-neighbour bonds on this lattice (counting each bond once only). Each site of the lattice carries a quantum spin , with , and where the spin components obey the usual SU(2) commutation relations. We shall be interested here specifically in the case only. For the classical version () of the model, it is trivial to see that for the energy is minimized (in this easy-axis case) when the spins align in the spin-space direction, to give a ferromagnetic ground state for and an antiferromagnetic Néel ground state for . Conversely, for values in the easy-plane regime, the classical ground state is again a Néel state, but now with the spins aligned parallel or antiparallel to some arbitrary direction in the spin plane. The classical ground-state energy per spin, , is thus

 ecl0={−2s2  ;|Δ|<1−2s2|Δ|  ;|Δ|>1  , (2)

for classical spins of length . Whereas the ferromagnetic state is also an eigenstate of the quantum Hamiltonian for any value of the spin quantum number , this is not the case for either of the Néel states, and the role of quantum fluctuations now becomes important for finite value of .

Increasing experimental effort has been expended to investigate layered quantum magnets, and precise theoretical results for the fundamental quantities, such as the ground-state energy, the sublattice magnetization, the spin stiffness, and the uniform transverse magnetic susceptibility, are therefore desirable for the antiferromagnetic XXZ model on the square lattice. In particular, the spin-half XXZ antiferromagnet on the square lattice has attracted much attention in relation to the magnetic properties of the parent compounds of high-temperature cuprate superconductors manousakis91 (); 002 ().

The properties of two-dimensional (2D) bipartite (i.e., geometrically unfrustrated) lattice quantum spin systems may be investigated by using a variety of approximate techniques (see, e.g., Refs. manousakis91 (); lnp04 ()). Foremost among these for 2D unfrustrated quantum spin systems are various quantum Monte Carlo (QMC) simulation methods (see, e.g., Refs. cavo (); qmc1 (); qmc2 (); qmc3 (); Beard:1996_SqLatt (); qmc4 (); Kim:1998_SqLatt_QMC (); qmc5 ()). Other approximate techniques that may be applied in order to simulate the properties of 2D quantum magnets include spin-wave theory (SWT) swt1 (); swt2 (); Igarashi:1992_SqLatt (); swt3 (); thirdorderswt (); series (); Hamer:1994_SqLatt (); Stinchcombe:1971_SqLatt (), exact diagonalizations (ED) qmc5 (); schulz (); ed (); ED40 (); richter2010 (); lauchli2011 () on small finite-sized lattices, and series expansion (SE) methods series (); Hamer:1994_SqLatt (); Singh:1989 (). Another versatile method of ab initio quantum many-body theory that has been shown over the last two decades to give consistently reliable and accurate results for 2D quantum magnetic systems at zero temperature is provided by the coupled cluster method (CCM) ccm1 (); ccm2 (); ccm5 (); ccm9 (); ccm10 (); ccm12 (); ccm13a (); ccm15 (); ccm17 (); bishop04 (); ccm24 (); ccm26 (); ccm27 (); ccm29 (); ccm32 (); ccm34 (); ccm35 (); ccm42 (); richter2010 (); honey2011 (); xian2011 (); LPSUB2011 (); goetze2011 (); bishop2012 (); wir_XXZ_2015 (); jiang2015 (); gap2015 (); bishop2015 (); trian2016 (); Bishop:2016_honey_grtSpins (). In particular, the use of computer-algebraic implementations ccm12 (); ccm15 (); ccm26 () of the CCM for spin-lattice problems has increased the accuracy of the method greatly. It has been demonstrated conclusively in a series of recent studies (see, e.g., ccm24 (); ccm32 (); ccm34 (); ccm35 (); ccm42 (); richter2010 (); honey2011 (); goetze2011 (); bishop2012 (); wir_XXZ_2015 ()) that the CCM gives reliable results even in the vicinity of quantum phase transition points for a host of quantum magnetic systems. Hence, the CCM applied to high orders of approximation is a good choice in order to provide accurate results for 2D quantum magnetic systems. In this paper we present CCM results for the ground-state energy, the sublattice magnetization, the zero-field, uniform transverse magnetic susceptibility, the spin stiffness, and the spin gap over a wide range of values of the anisotropy parameter for the Hamiltonian given in Eq. (1).

We start with a brief description of the CCM formalism in Sec. 2, and then we go on to describe the application of the method to the spin-half XXZ model on the square lattice in Sec. 3. We present our results in Sec. 4, where we also provide a discussion of their implications. All results are presented in graphical and tabular formats in order to provide a straightforward quantitative “reference” data set, against which results from other approximate methods or from experiment for relevant magnetic materials may be compared. We conclude with a summary and discussion in Sec. 5.

## 2 Method

The details of both the fundamental and practical aspects involved in applying the high-order CCM formalism to lattice quantum spin systems are given, e.g., in Refs. ccm2 (); ccm12 (); ccm13a (); ccm15 (); bishop04 (); ccm26 (); ccm27 (); ccm42 (). For the sake of brevity, we outline here only some important features of the CCM. First we mention that the CCM provides results in the infinite-lattice limit from the outset, since it obeys the important Goldstone linked-cluster theorem at any level of approximate implementation. The ket and bra ground-state eigenvectors, and , are parametrized within the single-reference CCM as follows

 |Ψ⟩=eS|Φ⟩ ; S=∑I≠0SIC+I, ⟨~Ψ|=⟨Φ|~Se−S ; ~S=1+∑I≠0~SIC−I, (3)

where is a suitably chosen single normalized model or reference state. The ground-state ket- and bra-state Schrödinger equations for a general Hamiltonian are given by and . State normalizations are chosen so that . The reference state is required to have the property of being a cyclic vector with respect to two well-defined Abelian subalgebras of multi-configurational creation operators and their Hermitian-adjoint destruction counterparts , such that . These conditions ensure the automatic fulfillment of the above normalization conditions. The set-index denotes here a set of single-spin configurations, and the states span the Hilbert space. By definition, , the identity operator. The correlation coefficients are calculated by minimizing the ground-state energy expectation value functional with respect to , thus leading to a coupled set of ket-state equations given by , . The correlation coefficients are similarly found by minimizing with respect to , thus leading to , . An equivalent form of this latter equation is given by .

An excited state is parametrized within the CCM by applying an excitation operator linearly to the ground state , such that

 |Ψe⟩=XeeS|Φ⟩  ;  Xe=∑I≠0XeIC+I  . (4)

From the Schrödinger equation, , it follows that

 e−S[H,Xe]eS|Φ⟩=εXe|Φ⟩  , (5)

where is the excitation energy. We now project Eq. (5) on the left with the state , and use that the states labeled by the indices are, as usual, orthormalized, , to yield the generalized set of eigenvalue equations

 ⟨Φ|C−Ie−S[H,Xe]eS|Φ⟩=εXeI  , (6)

which we solve in order to obtain . In the present case we will be interested specifically in the case when is the lowest-lying triplet excited state, above the spin-singlet ground state , and is hence the (triplet) spin gap.

The CCM formalism is exact in the limit of inclusion of all possible multi-spin clusters within the ground- and excited-state operators [i.e., by inclusion of all multi-spin configurations in the sums in Eqs. (2) and (4)], although this is usually impossible to achieve practically. The so-called LSUB approximation scheme is used here for both the ground and excited states. This approximation scheme uses all multi-spin correlations over all distinct cluster locales on the lattice defined by or fewer contiguous sites. Such locales (or lattice animals) of size are said to be contiguous if every site in the cluster is nearest-neighbour to at least one other. We select equivalent levels of LSUB approximation for both the ground and excited states. However, we remark that for our calculation of the (triplet) spin gap the choice of clusters for the lowest-lying (triplet) excited state is different from those for the ground state because we know that the ground state lies in the subspace, whereas the lowest-lying triplet excited state in terms of energy must have . Hence, we only use configurations in the excited-state operator that change the total spin by one. We find that the number of configurations for the excited state is larger than for the ground state at all levels of LSUB approximation. The number of terms in the corresponding equation systems is correspondingly larger, and so the calculation of the excited state is more difficult computationally than that of the ground state.

The LSUB approximation scheme allows the systematic analysis of CCM data as a function of the level of approximation , without any further approximations being made. We extrapolate the individual LSUB data to the limit in order to form accurate estimates of all expectation values. The general form for extrapolating LSUB results in the limit is given by , where the (fixed) leading exponents and () may be different for the different quantities to be extrapolated (and see Sec. 3 for details). Finally, we note that at any LSUB level of approximation the CCM exactly fulfills both the Goldstone linked-cluster theorem and the very important Hellmann-Feynman theorem.

## 3 The CCM applied to the Xxz Model

We recall that the spin-half XXZ antiferromagnetic model on the square lattice with nearest-neighbour interactions is given by Eq. (1). Here we use the quasiclassical -aligned Néel state as the model state for values of the anisotropy parameter in the range , whereas we use a Néel state aligned in the plane for . Both reference states give identical results for the rotationally invariant model at . It is convenient to carry out a transformation of the local spin axes on each site such that all spins in each reference state align along the negative axis. A complete set of multi-spin creation operators may then be formed with respect to every model state, and we note that this set of multi-configurational creation operators with respect to the rotated coordinate frame is defined by , where . As we are henceforth interested only in the case , we note that no site index contained in any retained cluster index may appear more than once. In the LSUB approximation for the present case therefore, we retain in the sums over multi-spin configurations in Eqs. (3) and (4) only those terms involving the set-indices where , and where each site index is nearest-neighbour to at least one other site index .

For the -aligned Néel model state we perform a rotation of all “up-pointing” spins (say, on the sublattice) by 180 about the -axis. The transformation of the local axes of the -sublattice spins is given by

 sxj → −sxj , syj → syj , szj → −szj  . (7)

The local spin axes of the “down-pointing” spins (say, on the sublattice) do not need to be rotated. The Hamiltonian of Eq. (1) within the rotated coordinate frame is given by

 H=−12∑⟨i,j⟩[s+is+j+s−is−j+2Δsziszj]  , (8)

for the Néel model state with spins aligned in the direction and with respect to the rotated spin axes.

We use the Néel state with spins aligned along the axis as the model state in the regime given by . We rotate the axes of the left-pointing spins (i.e., those pointing along the negative -direction on, say, sublattice ) by 90 about the axis, whereas we rotate the axes of the right-pointing (i.e., those pointing along the positive -direction on, say, sublattice ) spins by 270 about the axis. The corresponding transformation of the local spin axes on sublattice is given by

 sxi → −szi , syi → syi , szi → sxi  ; (9)

and the corresponding transformation of the local spin axes on sublattice is given by

 sxj → szj , syj → syj , szj → −sxj  . (10)

The Hamiltonian of Eq. (1) is then given by

 H = −14∑⟨i,j⟩[(Δ+1)(s+is+j+s−is−j)+(Δ−1)(s+is−j+s−is+j)+4sziszj]  , (11)

for the Néel model state with spins aligned in the plane and with respect to the rotated spin axes.

We are able to evaluate ground-state expectation values of arbitrary operators once the values for the bra- and ket-state correlation coefficients, and respectively, have been determined (at a given level of approximation), as described in Sec. 2. The ground-state energy per spin is given, uniquely, in terms of the coefficients alone, by

 e0≡E0N=1N⟨Φ|e−SHeS|Φ⟩  . (12)

The sublattice magnetization is given in terms of the rotated spin coordinates for both model states by

 M = −1N⟨~Ψ|N∑i=1szi|Ψ⟩ = −1N⟨Φ|~Se−S(N∑i=1szi)eS|Φ⟩  . (13)

The classical () version of the model has a sublattice magnetization for each of the ground-state phases. For the quantum version, when takes a finite value, remains equal to its classical value only in the ferromagnetic phase. For each of the two Néel phases one expects that quantum fluctuations will reduce the value of below its classical counterpart.

The transverse uniform magnetic susceptibility may be calculated within the CCM by using the method outlined in Refs. ccm42 (); trian2016 () for the square- and triangular-lattice Heisenberg antiferromagnet. However, it is useful to note here briefly that we add an appropriate transverse magnetic field term to the Hamiltonian of Eq. (1), namely: for the -aligned Néel reference state (); or, for the -aligned Néel reference state (), both in units where the gyromagnetic ratio . Spins are now allowed to cant at an angle, and this angle tends to zero in the limit . The precise nature of the canted model states and the solution of the associated CCM problem is described in detail in Refs. ccm42 (); trian2016 (). The uniform transverse magnetic susceptibility is then defined as usual by the relation

 χ(λ)=−1Nd2E0dλ2  , (14)

where we now calculate the ground-state energy, , in the presence of the applied magnetic field. The zero-field susceptibility, , may be calculated from the small- expansion,

 E0(λ)N=E0(λ=0)N−12χλ2+O(λ4)  . (15)

For the classical version of the model it is easy to show that takes the same value,

 χcl=14(1+Δ)  ;  −1<Δ<∞  , (16)

in both ground-state Néel phases, independent of the length of the classical spins.

The calculation of the spin stiffness using the CCM is described in Refs. ccm27 (); ccm29 (); ccm35 (); bishop2015 (); trian2016 (). The spin stiffness measures the increase in the amount of energy for a magnetically long-range ordered system when a helical “twist” of magnitude per unit length is imposed on the spins, in a given direction. In this case the ground-state energy per spin is given by

 E0(θ)N=E0(θ=0)N+12ρsθ2+O(θ4)  , (17)

where is the ground-state energy as a function of the imposed twist, (see, e.g., Refs. schulz1995 (); lhuillier1995 (); trumper1998 () for details). Again, we use a rotation of the local spin at site by an appropriate angle such that the local spin axes for the now helical reference state appear mathematically to align along the (negative) axis (for details see Refs. ccm27 (); ccm29 ()). The helical state lies in the plane for , and is thus well-defined to give a unique determination of . For the classical version of the model it is simple to show that takes the classical value,

 ρcls=s2  ;  −1<Δ<1  , (18)

for classical spins of length , in units where the nearest-neighbour spacing on the square lattice has been set to unity. By contrast, the spin stiffness is ill-defined for because the helical state lies in the plane. The easy-axis anisotropy therefore adds an energy contribution proportional to , and so the energy depends on the individual angles relative to the easy axis.

As already outlined briefly in Sec. 2, as a final step we need to extrapolate our LSUB estimates for all physical quantities to the limit where the method becomes exact. Although exactly provable rules are not known for these extrapolations, robust empirical rules do exist, and these rules have successfully been tested for a wide range of quantum magnetic systems ccm15 (); bishop04 (); ccm27 (); ccm29 (); ccm42 (); gap2015 (); trian2016 (). We use the “standard” rules in order to extrapolate all expectation values, namely: the ground-state energy per spin using ; the sublattice magnetization using ; the zero-field, uniform transverse magnetic susceptibility using ; the spin stiffness using ; and the spin gap using .

The numbers, , of distinct (fundamental) configurations that are retained in the summations for both the ground state in Eq. (3) and the excited state in Eq. (4) at a given LSUB level of approximation are reduced by utilizing the space- and point group symmetries of the Hamiltonian and the model state, together with any conservation laws that pertain to both the Hamiltonian and the specific model state being used (viz., specifically here for ). We are able to compute data up to the order LSUB12 for the ground-state energy , the sublattice magnetization , and the spin gap using the high-order CCM code thecode (). The maximum number of fundamental ground-state configurations used in our calculations is , and this calculation was carried out for the planar Néel model ground state at the LSUB12 level of approximation. The solution of the LSUB equations is more challenging for the susceptibility and the spin stiffness because less symmetries can be used in these cases. As a result we can calculate the magnetic susceptibility and the spin stiffness only up to the LSUB10 level of approximation. Finally, we extrapolate our LSUB results for the ground-state energy , the sublattice magnetization , and the spin gap by using data for and then separately also by using data for . In this manner, we provide two sets of extrapolated values for , , and . By comparing these two sets of estimates, we obtain an estimate of the precision of these extrapolated quantities. We refer to extrapolated results using LSUB results for and as LSUB and LSUB, respectively.

We remark that the results presented in this article are carried out to much higher levels of LSUB approximation than those presented in previous CCM investigations of the XXZ model ccm5 (); ccm9 (); ccm10 (); xian2011 (), where the highest order of approximation was the LSUB8 approximation. The consequent accuracy of our results is thus significantly higher than those presented in Refs. ccm5 (); ccm9 (); ccm10 (); xian2011 (). Moreover, a systematic study of the magnetic susceptibility and the spin stiffness of the XXZ model was not presented in these earlier studies.

## 4 Results

We first show in Figs. 1 and 2 our CCM results for the ground-state energy per site, , and the ground-state sublattice magnetization pertaining to the spin- Hamiltonian of Eq. (1) on the square lattice. In both figures we show results obtained in LSUB approximations with , using as CCM model states an -aligned Néel state in the range and a -aligned Néel state in the range of the anisotropy parameter.

We note that these model states provide exact ground states of the Hamiltonian of Eq. (1) in the respective limits and (the Ising limit). Thus, exact results for all ground-state quantities are achieved for these two limiting cases at all LSUB levels of approximation (viz., and at , and and at ). In each of Figs. 1 and 2 we also show two sets of extrapolated (LSUB) results, based on the respective schemes described in Sec. 3, and using the two appropriate LSUB input data sets with and .

Figure 1 shows that our CCM results for the ground-state energy converge very rapidly as the order of the LSUB approximation is increased towards the exact () limit. Indeed, both the raw LSUB results and the two LSUB extrapolations, based on the two different input LSUB data sets as described above, are difficult to resolve by eye in the main panel of Fig. 1, which shows the high accuracy achieved within the CCM LSUB framework for the energy. The first-order transition at between the two Néel forms of long-range order (viz., that aligned in the plane for and that aligned along the axis for ) is clearly visible in the curves shown in Fig. 1. The inset to Fig. 1 presents the results near the critical point at in more detail.

An estimate of the accuracy of our extrapolated results can be obtained by a comparison of the two different extrapolation schemes, LSUB and LSUB. For example, our LSUB results at the isotropic Heisenberg () point () are using the LSUB data set and using the LSUB data set . Corresponding results at the isotropic () point () are using the LSUB data set and using the LSUB data set . It is clear that the results for the ground-state energy are very insensitive to the extrapolation procedure. We estimate that over the whole range of values of , our accuracy is better than 1 part in .

Our corresponding results for the sublattice magnetization are shown in Fig. 2. As is fully to be expected the results for the order parameter are both more strongly dependent on the order of the LSUB approximation and converge more slowly as . Just as for the ground-state energy the two LSUB extrapolations, based on LSUB results with and respectively, are almost indiscernible in Fig. 2. The maximum difference in the two extrapolations is at the isotropic Heisenberg point, , where from Fig. 2 we see that the effect of quantum fluctuations is largest at reducing the order parameter from its classical value . Thus, our LSUB results at the isotropic Heisenberg () point () are using the LSUB data set and using LSUB data set . The relative error between the two results is thus of the order of 3 parts in . By comparison, the corresponding LSUB results at the isotropic point () are using the LSUB data set with and using the LSUB data set with . The relative error between the two extrapolations is now only of the order of 3 parts in .

Our CCM results shown in Fig. 2 imply that the classical Ising limit, , is approached rather rapidly as the anisotropy parameter is increased. For example, even at a value , the order parameter already attains a value of about of the classical value, and for all values the value of is greater than of the classical limit.

It is interesting to compare our results for the spin- model in the vicinity of the isotropic Heisenberg point, , with those of SWT, which are applicable in the high-spin () classical limit. Thus, SWT predicts series (); swt2 (); Stinchcombe:1971_SqLatt () that in the vicinity of the isotropic point all of the physical ground-state parameters are analytic functions of the quantity for . Hence SWT predicts that any physical parameter of the model that pertains to the scaled Hamiltonian of Eq. (1) would have an expansion in the region . In particular, the ground-state energy and order parameter are predicted (and see, e.g., Ref. series ()) to behave as

 ESWT0NΔ=ϵ0+ϵ2(1−Δ−2)+ϵ3(1−Δ−2)32+⋯  , (19)
 MSWT=μ0+μ1(1−Δ−2)1/2+μ2(1−Δ−2)+⋯  . (20)

Naively, one might expect that the phenomenology of SWT, which is strictly valid only in the limit, including these functional forms, could remain correct for finite values of , at least so long as long-range antiferromagnetic Néel order persists (i.e., ) at in the quantum model. That is certainly the case here, since we find at . Thus, it is tempting to hypothesize that since the SWT singularities in the physical parameters near [i.e., the odd powers in in the expansions] are caused by the Goldstone modes and not by critical fluctuations, the associated leading critical exponents for finite values of the spin quantum number should therefore be the same as predicted by SWT, even for the case considered here.

In order to test this hypothesis we have carefully examined our CCM results for the magnetic order parameter in the narrow range . We show in Fig. 3 our LSUB extrapolations based on the LSUB data set in this range, plotted as a function of the parameter .

In order to find the leading (critical) exponent we have fitted the data to the totally unbiased form , where each of the parameters , and is fitted. The best fit to the data points shown in Fig. 3 is obtained with , and . Since the leading exponent takes the value , we thus attempt a fit of the form

 M=m0+m1(Δ−1)+m2(Δ−1)2  , (21)

with fixed at the value appropriate to the LSUB value for , obtained as described above using the LSUB data set with . The best fit, shown as the solid line in Fig. 3, is obtained with and . Thus, perhaps surprisingly, the SWT hypothesis is not confirmed by our results. The square-root cusp in that is predicted by SWT appears to be entirely absent. Of course it is possible that for this model the parameter in Eq. (20) vanishes (or takes a very small value) accidentally. More likely, however, is the scenario that the series for for the spin- model is actually analytic in , possibly multiplied by some additional slowly varying non-algebraic (e.g., logarithmic) term, near the isotropic Heisenberg point, rather than in the parameter predicted by SWT, as is appropriate in the classical () limit.

We turn next to our results for the zero-field, uniform transverse magnetic susceptibility of the model. Thus, we show in Fig. 4 the CCM LSUB results with and the corresponding LSUB extrapolation based on this set, for the same range of values for the anisotropy parameter, , as shown in Figs. 1 and 2 above for the ground-state energy and sublattice magnetization respectively.

Once again we remark that the results become exact in both limits and (the Ising limit). It is clear that the LSUB sequence of results for converges extremely rapidly, with the curves difficult to resolve by eye over most of the range shown, except for a small region around , where quantum fluctuations are again greatest. The inset to Fig. 4 again presents the results near the critical point at in more detail. SWT again predicts (and see, e.g., Ref. series ()) a square-root cusp for near the Heisenberg point for values ,

 ΔχSWT=ζ0+ζ1(1−Δ−2)1/2+ζ2(1−Δ−2)+⋯  , (22)

which appears also not to be borne out by our results in Fig. 4 for the spin- model.

Hence, once again we show in Fig. 5

our extrapolated LSUB results for the zero-field, uniform transverse magnetic susceptibility in the narrow range , based on our LSUB results with . The leading (critical) exponent is again obtained by fitting the LSUB data to the unbiased form , where each of the parameters , and is fitted. The best fit to the data points shown in Fig. 5 is obtained with , and . Just as for the previous fit for the staggered magnetization , the leading exponent again takes a value very close to unity. We thus attempt now a fit of the form

 χ=x0+x1(Δ−1)+x2(Δ−1)2  , (23)

with fixed at the value appropriate to the LSUB value for , obtained as described previously using the LSUB data set with . The best fit, shown in Fig. 5 by the solid line, is obtained with the values and .

Our CCM results for the spin stiffness coefficient are shown in Fig. 6 in LSUB approximation levels , together with the corresponding LSUB extrapolation based on this data set, over the range of values of the anisotropy parameter.

Again, as expected, the results are exact in the limit. Figure 6 shows the extremely rapid convergence of the LSUB sequence of values for in the range , followed by a slower convergence in the range . The effect of quantum fluctuations is again greatest in the vicinity of the isotropic Heisenberg point (), where the difference from the classical result is largest.

Finally, in Fig. 7 we show our CCM results for the spin gap for a range of values , where the system is expected to be gapped.

Theoretically, we expect that as the isotropic Heisenberg limit is approached and the excitations become gapless Goldstone modes. These modes then persist for all values of the anisotropy parameter in the range , in which remains zero. From Fig. 7 we see that both LSUB extrapolations, based on the two LSUB data sets and , give values of at which are zero within small numerical errors associated solely with the extrapolations. The actual LSUB extrapolated values at are using the LSUB data set and using the LSUB data set . One also observes from Fig. 7 that the LSUB sequence of values for converges appreciably more rapidly as for larger values of , and hence one expects that the associated extrapolated values will be even more accurate than those obtained at the limit. Figure 7 shows that in the Ising limit, , becomes proportional to , exactly as expected classically.

Once again, SWT predicts (and see, e.g., Ref. series ()) however that vanishes near as

 εSWTΔ=η1(1−Δ−2)1/2+η2(1−Δ−2)+η3(1−Δ−2)3/2+⋯  . (24)

This behaviour, just as before for the ground-state parameters, appears not to be borne out by our results shown in Fig. 7 for the spin- model. To investigate further we show in Fig. 8 our extrapolated LSUB results for the spin gap

in the narrow range based on our LSUB results with . The leading (critical) exponent is again obtained by fitting the extrapolated LSUB data points to the unbiased form , where each of the parameters , and is fitted. The best fit to the data points shown in Fig. 8 is obtained with , and . Once again, just as for the previous fits for the staggered magnetization and the zero-field, uniform transverse magnetic susceptibility , the leading exponent takes a fitted value very close to unity. Hence, we now attempt a fit of the form,

 ε=γ0+γ1(Δ−1)+γ2(Δ−1)2  , (25)

with fixed at the value appropriate to the LSUB value for , obtained as described above using the LSUB date set with . The best fit, shown in Fig. 8 by the solid line, is obtained with the values and .

In Table 1 we present our best CCM extrapolated (LSUB) results for each of the ground-state parameters , , and , together with the spin gap , for various values of the anisotropy parameter , in both the easy-axis and easy-plane regimes, as well as at the isotropic Heisenberg point .

This tabulation should hence allow a direct comparison of our results both to those obtained in appropriate experiments on systems to which the model is applicable and in other theoretical approaches or simulations using alternative techniques.

Before proceeding it is useful to compare our results to those obtained by other approximate techniques for the two special cases and of the anisotropy parameters. Several different techniques have been applied to study the spin- model on the square lattice for (see, e.g., Refs. qmc5 (); series (); thirdorderswt (); Hamer:1994_SqLatt (); weihong1995 (); witte1997 (); Takahasi1997 ()). Both ED and QMC methods have also been applied to it in the range (see, e.g., Ref. qmc5 ()). Furthermore, other techniques have also been applied for the specific case () of the isotropic Heisenberg model. Our result for the ground-state energy at is . This may be compared firstly, for example, with corresponding results from three different QMC simulations. Thus, a zero-temperature () Green’s function Monte Carlo (GFMC) calculation qmc3 () directly for the ground state gave , while another finite-temperature () calculation using the stochastic series expansion QMC (SSE-QMC) method qmc4 () gave . Both of these calculations were performed on square lattices with , and the results extrapolated to the thermodynamic limit (). Two other QMC simulations of the model, based on a continuous Euclidean time version of a loop cluster algorithm for evaluating path integrals (PIMC) Beard:1996_SqLatt (); Kim:1998_SqLatt_QMC (), extracted the low-energy parameters by fitting the data to finite-temperature scaling forms derived from chiral perturbation theory Hasenfratz:1993_magnon-chiral-PT (). Using very large-scale simulations on lattices with , for example, Kim and Troyer Kim:1998_SqLatt_QMC () found . The spin-