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Journal of the Physical Society of Japan

The CAM Calculation of Critical Exponent ν by the Cluster Variation Method

Koh Wada, Naotoshi Watanabe

Published Aug 15, 1990• 9 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.59.2610

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Journal of the Physical Society of Japan•Aug 15, 1991•15 citations
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.
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 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 correlation length of the Ising model on the simple cubic lattice is calculated directly from the magnetization profiles which are determined in the pair, the square, and the cube approximations in the simplified cluster variation method. The critical exponent ν is obtained by applying the coherent anomaly method to the correlation length. The wave-number dependent magnetic susceptibility or the spin-pair correlation function is not required in obtaining the expression for the correlation length.
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.
The cluster-variation method (CVM) proposed by Kikuchi is a general theory to give approximations, which are useful to discuss phase transitions qualitatively in many systems. Recently one of the present authors (M. S.) proposed the coherent-anomaly method (CAM). If we have a well-behaved series of approximations, which is called a canonical series, we can estimate critical exponents following the CAM. In this paper it is demonstrated by using the ferromagnetic Ising models that the CVM will provide a canonical series which shows coherent anomaly. Our result implies that combining the CVM with the CAM will give a powerful method to study phase transitions and critical phenomena.