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Progress of Theoretical Physics Supplement

Structure of the Density Matrices in the Cluster Variation Method

Tomoyasu Tanaka, Kouzi Hirose, Kenpou Kurati

Published Jan 1, 1994• 8 citations
Publisher

Abstract

Two of the formulations of the cluster variation method are compared. The Heisenberg model of ferromagnetism in four-spin approximation is used as a working example. The numerical results obtained are used in the CAM analysis and an improved critical temperature is found. Formulations of the cluster variation method in the momentum space are also reviewed briefly.

DOI
10.1143/ptps.115.41

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Two of the formulations of the cluster variation method are compared. The Heisenberg model of ferromagnetism in four-spin approximation is used as a working example. The numerical results obtained are used in the CAM analysis and an improved critical temperature is found. Formulations of the cluster variation method in the momentum space are also reviewed briefly.
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 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.
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.