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Journal of Physics C Solid State Physics

Correlation reduction theory for Ising model with spin 1/2

H I Zhang

Published Jan 10, 1981• 11 citations
Publisher

Abstract

A new method of correlation reduction near the critical temperature is introduced for a classical spin system. This method is based on an assumption much less restrictive than the usual mean-field approximation. It permits an important parameter to be determined realistically. The transition temperature of the Ising ferromagnet is expressed very simply in a closed form. The values of the transition temperature evaluated with various approximations in this method are presented for three-dimensional Ising lattices.

DOI
10.1088/0022-3719/14/1/008

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arXiv:0812.4139•Physical Review B•Apr 24, 2009•69 citations
A new method of correlation reduction near the critical temperature is introduced for a classical spin system. This method is based on an assumption much less restrictive than the usual mean-field approximation. It permits an important parameter to be determined realistically. The transition temperature of the Ising ferromagnet is expressed very simply in a closed form. The values of the transition temperature evaluated with various approximations in this method are presented for three-dimensional Ising lattices.
An analytic method for finding the highly accurate values of the critical point for cubic Ising models with arbitrary spin is presented. The thermodynamic quantities of interest are expressed through some higher-order correlations with a reference spin excluded. A reduction scheme for these higher-order correlations is constructed. It leads to two types of resulting equations for correlation functions. The coincidence of two different representations of critical correlations is ensured by the introduced parameter of the reduction scheme. The estimation of critical points for all cubic lattices is compatible with series works to within 1.69%, 0.70% for the limiting values of spin, ½ and ∞, respectively.
For pt.II see ibid., vol.14, p.1789 (1981). The phase transition of the anisotropic Ising lattices has been studied by the recently developed correlation reduction theory. The transition temperature and some other quantities at transition temperature for the anisotropic Ising lattices with spin 1 / 2 , 1, 3 / 2 are evaluated as functions of anisotropy parameter. The effect of anisotropy on these values is similar for the systems with three different spin values. Comparison with other calculations is made, and some physical significances are discussed.
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.
For pt.I see ibid., vol.14, p.57 (1981). The recently developed theory of correlation reduction method near transition temperature is generalised to the Ising model with arbitrary spin. The generalisation from the case of spin 1 / 2 is natural and straightforward, although some non-trivial technicalities are involved for the detailed applications. The transition temperatures for the spin 1 and spin 3 / 2 Ising ferromagnets are obtained explicitly, and are shown to be consistent with the general trend exhibited by other calculations.
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.
A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of ``correlations'' between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.