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Journal of Physics C Solid State Physics

Correlation reduction theory for Ising model. II. Generalisation to arbitrary spin

Hwe Ik Zhang, Byung Il Min

Published Apr 30, 1981• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/14/12/010

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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.
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 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.
A generalised non-uniform magnetic field method is used to derive, from the linearisation assumption of a new theory for the Ising ferromagnet, a set of critical correlation-function approximations (CCFA) which express the higher-order correlations in terms of the second- and fourth-order correlations. This set is found to coincide at least for the first five relations, with an earlier CCFA set which was suggested by a term-by-term comparison of the second- and fourth-order equations of the theory, and which was justified a posteriori by the accurate results for the transition temperature to which it leads.
A stringent test is applied to a set of approximations used, in an earlier work (CCFA) see Frank et al. (ibid., vol.11, p.208 (1978) to calculate T c for the spin- 1 / 2 Ising ferromagnet. In CCFA, all cumulant averages M 2n (2n>or=6) were approximated, for all the cubic lattices, and justified a posteriori. Here, a spin operator reduction relation allows one, for the SC lattice, to approximate only M 6 , and moreover, to understand the approximation used. The extreme closeness at every stage (within 0.01%) of the present T c values to those of CCFA gives one confidence method of approximating the M 2n for the BCC and FCC lattices as well. Here also, as in CCFA, the M 60 value of T c is within 0.5% of the series approximation value of Sykes et al. (1972).