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Progress of Theoretical Physics

Cluster Variation Method for Non-Uniform Ising and Heisenberg Models and Spin-Pair Correlation Function

Tohru Morita

Published Feb 1, 1991• 27 citations
Publisher

Abstract

A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.

DOI
10.1143/ptp.85.243

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Progress of Theoretical Physics Supplement•Jan 1, 1994•2 citations
A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.
The cluster variation method for the cooperative phenomena proposed by Kikuchi and reformulated and generalized by Morita, is applied to the Heisenberg model with arbitrary spin and range of exchange. A general expression for the two-body reduced density matrix is obtained in the approximation in which the clusters of pairs of lattice sites are retained correctly. The constant-coupling approximation for the Heisenberg model of $S\ensuremath{\geqq}1$ is shown to be derived by satisfying the reducibility conditions ${{\mathrm{tr}}_{k\ensuremath{\rho}}}^{(2)}(j, k)={\ensuremath{\rho}}^{(1)}(j)$ only partly, requiring the consistency for the zeroth and first moments of Sjz{S}_{\mathrm{jz}}Sjz​ and ignoring the consistency for the second to 2Sth2S\mathrm{th}2Sth moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.
The method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin Sz=±1S^z=\pm1Sz=±1) the expressions for pair and ternary correlation functions within two-particle approximation in { q }\set{q}{q}-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility χ({ q },E)χ(\set{q},E)χ({q},E) is found. In the 1D case and in the absence of external field the expression for χ({ q },E)χ(\set{q},E)χ({q},E) is exact. For the Emery-Blume-Griffiths model (Sz=−2,0,2S^z=-2,0,2Sz=−2,0,2) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility χ({ q })χ(\set{q})χ({q}) of KD2PO4KD_2PO_4KD2​PO4​ ferroelectrics.
I show that the cluster variation method, long used as a powerful hierarchy of approximations for discrete (Ising-like) two-dimensional lattice models, yields exact results on the disorder varieties which appear when competitive interactions are put into these models. I consider, as an example, the plaquette approximation of the cluster variation method for the square lattice Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, and, after rederiving known results, report simple closed-form expressions for the pair and plaquette correlation functions.
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 spin-pair correlation functions in the Ising model with nearest-neighbor interaction J>0 and next-nearest-neighbor interaction J′ on the square lattice are obtained in three approximations of the simplified cluster variation method. We estimate the critical indices ην, γ and ν, by applying the coherent-anomaly method to the obtained spin-pair correlation functions. We conclude that the system belongs to the Onsager universality class when J′/J > −0.5 and does not when J′/J < −0.5.