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Progress of Theoretical Physics Supplement

Critical Phenomena of theS=1/2 Heisenberg-Ising Model Studied by CVM and CAM

Giichi Tanaka, Minoru Kimura

Published Jan 1, 1994• 2 citations

Abstract

The critical temperatures Tc and the mean-field critical coefficients of the susceptibility χ are calculated for the S=1/2 Heisenberg-Ising model on the simple cubic lattice by the Kikuchi approximation and its simplified versions. χ̄ shows a coherent anomaly. It means that the coherent-anomaly method (CAM) is applicable to evaluate the true critical temperature T*c and the non-classical critical exponent γ for the susceptibility. As a result, the dependence of T*c on an anisotropy parameter η is given. It is shown that the behavior of γ versus η approaches a fiat line step by step as the corrections up to the higher order approximation are taken into account for 0≦η<1. It implies that the anisotropy parameter η of the Heisenberg-Ising model is irrelevant for critical exponents, if η is less than 1, as expected by the renormalization-group argument.

DOI: 10.1143/ptps.115.207

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The critical temperatures Tc and the mean-field critical coefficients of the susceptibility χ are calculated for the S=1/2 Heisenberg-Ising model on the simple cubic lattice by the Kikuchi approximation and its simplified versions. χ̄ shows a coherent anomaly. It means that the coherent-anomaly method (CAM) is applicable to evaluate the true critical temperature Tc and the non-classical critical exponent γ for the susceptibility. As a result, the dependence of Tc on an anisotropy parameter η is given. It is shown that the behavior of γ versus η approaches a fiat line step by step as the corrections up to the higher order approximation are taken into account for 0≦η<1. It implies that the anisotropy parameter η of the Heisenberg-Ising model is irrelevant for critical exponents, if η is less than 1, as expected by the renormalization-group argument.

The coherent-anomaly scaling is investigated for the approximations derived on the basis of the cluster-variation method (CVM). All of the mean-field critical coefficients are calculated in the Weiss, the Bethe and the Kikuchi approximations for the three-dimensional Ising model on the simple cubic lattice. The dynamical coefficients of the relaxation time are also obtained in the corresponding approximations for the kinetic Ising model. The non-classical values of critical exponents as well as the true value of critical temperature are estimated by extrapolating these approximants by the coherent-anomaly method. The results suggest that the approximations by the CVM possess the property of coherent-anomaly scaling.

A systematic procedure of constructing a series of cluster-mean-field approximations is established to investigate the coherent anomalies of the mean-field critical coefficients \barχ( L , T c ( L )) in the two- and three-dimensional Ising models. The values of critical exponents as well as the critical temperatures are estimated by means of the new method (Coherent Anomaly Method) as T c * =2.230(0.038) J / k B , γ=1.785(0.116) for the two-dimensional square lattice and T c * =4.529(0.058) J / k B , γ=1.258(0.068) for the three-dimensional simple cubic lattice. The results suggest that the present approach to study critical phenomena is very promising.

Critical temperatures T c and mean-field critical coefficients [Formula: see text] for susceptibilities are calculated by a cluster-variation method (CVM) for a three-dimensional Heisenberg model on a simple cubic lattice. Both the full CVM based on a 4-spin cluster and the SCVM are applied and the same values for T c and [Formula: see text] are obtained by three different approximation in the SCVM. The results are compared with those obtained by the same approach for the three-dimensional Ising model on the simple cubic lattice.

The coherent-anomaly method (CAM) is applied to the Heisenberg model to test the applicability of the CAM for quantum spin systems. The Weiss, Bethe and constant coupling approximations are tried for the Heisenberg model on the simple cubic lattice and estimate the critical exponents of the susceptibility and spontaneous magnetization using the CAM. The results show that the CAM is also powerful for quantum spin systems. The detailed results of the Bethe approximation of the spin-1/2 isotropic Heisenberg model are presented.

The critical exponent ν of correlation length is calculated on the basis of the coherent anomaly method (CAM) by using the pair, the cactus-square and the square approximations of the cluster variation method (CVM) in the D (≥2)-dimensional cubic lattice of the Ising model. These simple analytical approximations give good values of critical exponent ν and critical temperature in the two- and three-dimensional lattices. However, further consideration is needed to determine a critical dimension in the D -dimensional Ising model by the CAM.