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Journal of the Physical Society of Japan

Application of the Cluster Variation Method to the Ferromagnetic Six-State Clock Model on the Triangular Lattice and Its CAM Analysis

Sumiyoshi Fujiki

Published Feb 15, 1993• 7 citations
Publisher

Abstract

The cluster variation method is applied to the ferromagnetic six-state clock model on the triangular lattice up to the hexagonal cluster approximation. The Kosterlitz-Thouless type phase transition of the model is studied by using the coherent anomaly method (CAM).
DOI
10.1143/jpsj.62.556

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A Monte Carlo study is carried out on the six-state clock model on the triangular lattice. Physical quantities which characterize the Kosterlitz-Thouless (KT) phase transition are calculated. The temperature dependence of the exponent of the spin-pair correlation function, η( T ), in the KT phase is obtained precisely. The exponential divergence of the correlation length, ξ∼ exp ( a t - b ), with the theoretical value, b =0.5, gives a good fit to the data by the present MC simulations by choosing a =1.6∼1.8, η=0.23∼0.25 at the upper critical temperature T U =1.46∼1.49. However, the choice of critical parameters, a ∼3.1, b ∼0.3 and η∼0.28 at T U ∼1.53 gives the most consistent result by a finite-size scaling analysis on several physical quantities. The lower critical temperature, T L , is estimated to be 1.1.
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 phase transitions and thermodynamic properties of the clock model with q = 5 spin states on a triangular lattice have been investigated using the Wang–Landau Monte Carlo algorithm. The phase transitions have been analyzed with the histogram method and the fourth-order Binder cumulant method. Two Berezinskii–Kosterlitz–Thouless phase transitions are shown to be observed in the ferromagnetic clock model, while a second-order phase transition has been detected in the antiferromagnetic clock model.
Abstract The phase transitions and thermodynamic properties of the clock model with q = 5 spin states on a triangular lattice have been investigated using the Wang–Landau Monte Carlo algorithm. The phase transitions have been analyzed with the histogram method and the fourth-order Binder cumulant method. Two Berezinskii–Kosterlitz–Thouless phase transitions are shown to be observed in the ferromagnetic clock model, while a second-order phase transition has been detected in the antiferromagnetic clock model.
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.