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Journal of Physics C Solid State Physics

Correlation reduction theory for Ising model. III. Application to anisotropic lattices

Byung Il Min, Hwe Ik Zhang

Published Apr 30, 1981• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/14/12/011

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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.
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.
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 system with Hamiltonian H0 is assumed to undergo a phase transition at transition temperature T0. When a small perturbation term λH′ is added to H0, the transition temperature will depend on the coupling parameter λ and the critical indices will in general be changed. The change induced by the application of λH′ is classified into three types, and these types discussed for the cases of anisotropic Ising ferromagnets by means of thermodynamic perturbation theory and scaling law equation for spin correlation functions.
The phase transition occurring in a square two-dimensional spin lattice governed by an anisotropic Heisenberg Hamiltonian has been studied according to two recently proposed methods. The first one, the Dressed Cluster Method, provides excellent evaluations of the cohesive energy. The discontinuity of its derivative around the critical (isotropic) value of the anisotropy parameter confirms the first-order character of the phase transition. Nevertheless, the method introduces two distinct reference functions (either N\'eel or XYXYXY) which may, in principle, force the discontinuity. The Real-Space Renormalization Group with Effective Interactions does not reach the same numerical accuracy, but it does not introduce a reference function and the phase transition appears qualitatively, due to the existence of two domains with specific fixed points. The method confirms the dependence of the spin gap on the anisotropy parameter occurring in the Heisenberg-Ising domain.
The spin 3/2 transverse Ising model with anisotropic crystal field on honeycomb lattice (z=3) has been studied by using the effective field theory with correlations. The general expressions for evaluating the second order phase transition temperatures are obtained. The results show that both the transverse field and crystal field have important effects on the phase diagram of the Ising system. The transition temperatures do not always fall to zero when Ω/|J|0 457. They approach a constant value 0 525 for large negative D(D→-∞) without transverse field (Ω=0). On the other hand, it is shown that all phase transformations are second order.