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Journal of the Physical Society of Japan

Study of Coherent Anomalies and Critical Exponents Based on the Cluster-Variation Method

Makoto Katori, Masuo Suzuki

Published Nov 15, 1988• 26 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.57.3753

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Progress of Theoretical Physics Supplement•Jan 1, 1994•2 citations
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.
On the basis of the coherent-anomaly method (CAM) proposed by Suzuki, the static and dynamic critical exponents of the Ising model in the triangle and the square lattice are calculated using the cluster variation method (CVM) and the path probability method (PPM) devised by Kikuchi within analytical tractable approximations. The values of critical temperatures and critical exponents estimated from three approximations with different levels are generally reasonable. The two-dimensional results are also compared with those of the simple cubic lattice.
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.
The coherent-anomaly scaling property of the susceptibility for the Ising model is investigated using the Tanoji (a Chinese character which means a rice field) and cube cluster-approximations in the cluster-variation method on the square and simple cubic lattices, respectively. Estimates of the critical temperature and critical exponent are improved considerably by the inclusion of these high-level cluster-variation approximations in the coherent-anomaly extrapolation. It is also found that the dimensionality of the basic cluster is sensitive to the evaluation of critical exponents in higher dimensions.
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.
The critical exponents of the bond percolation model are calculated in the D (=2, 3,···)-dimensional simple cubic lattice on the basis of Suzuki's coherent anomaly method (CAM) by making use of a series of the pair, the square-cactus and the square approximations of the cluster variation method (CVM) in the s -state Potts model. These simple approximations give reasonable values of critical exponents α, β, γ and ν in comparison with ones estimated by other methods. It is also shown that the results of the pair and the square-cactus approximations can be derived as exact results of the bond percolation model on the Bethe and the square-cactus lattice, respectively, in the presence of ghost field without recourse to the s →1 limit of the s -state Potts model.
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.