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Journal of the Physical Society of Japan

Study of Critical Phenomena by Spin-Cluster Approximation Series: Ising Spin System

Hiroshi Mano, Katsumi Nakao

Published Feb 15, 1991• 6 citations
Publisher

Abstract

A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.

DOI
10.1143/jpsj.60.548

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A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.
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.
A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.
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.
Two kinds of systematic mean-field transfer-matrix methods are formulated in the 2-dimensional Ising spin system, by introducing Weiss-like and Bethe-like approximations. All the critical exponents as well as the true critical point can be estimated in these methods following the CAM procedure. The numerical results of the above system are T c * ≃2.271 ( J / k B ), γ=γ'≃1.749, β≃0.131 and δ≃15.1. The specific heat is confirmd to be continuous and to have a logarithmic divergence at the true critical point, i.e., α=α'=0. Thus, the finite-degree-of-approximation scaling ansatz is shown to be correct and very powerful in practical estimations of the critical exponents as well as the true critical point.
The coherent-anomaly method (CAM) is applied to the kinetic Ising model. Dynamical cluster-mean-field approximations are formulated to obtain a series of the dynamical mean-field critical coefficients. The coherent-anomaly scaling relations are derived on the basis of the scaling form of the generating function in nonequilibrium systems to estimate the exponent Δ of the critical slowing down. The dynamical critical exponent is estimated as Δ⋍2.15(±0.02){\varDelta}{\backsimeq}2.15({\pm}0.02)Δ⋍2.15(±0.02) for the kinetic Ising model on the two-dimensional triangular lattice.
A cluster-effective-field theory, called the double-cluster approximation, is formulated for quantum spin systems at T=0. Combining this approximation with the coherent-anomaly method (CAM), critical phenomena in such systems can be analysed. To test their theory, the authors have estimated the critical points and critical exponents of the one-dimensional S=1/2 transverse Ising model and the one-dimensional S=1/2 XZ model. The estimates are in good agreement with the exact solutions. These results show that the CAM approach is useful in treating zero-temperature phase transitions in quantum spin systems.