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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlated molecular-field theory for Ising models

Gary Mathew Wysin, Jerome I. Kaplan

Published Jun 1, 2000• 30 citations
Publisher

Abstract

The critical temperature T(c) of Ising models is obtained quite accurately by simple improvements over the standard molecular-field theory. The important physical effect we include is that the effective field of neighboring spins is influenced by the spin state of the central spin. When used in combination with a self-consistency condition, this correlated molecular-field theory leads to estimates of T(c) more accurate than those obtained from the Bethe-Peierls-Weiss approximation.

DOI
10.1103/physreve.61.6399

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The critical temperature T(c) of Ising models is obtained quite accurately by simple improvements over the standard molecular-field theory. The important physical effect we include is that the effective field of neighboring spins is influenced by the spin state of the central spin. When used in combination with a self-consistency condition, this correlated molecular-field theory leads to estimates of T(c) more accurate than those obtained from the Bethe-Peierls-Weiss approximation.
We use an effective field model which explicitly incorporates correlations to determine the critical temperatures for Ising ferromagnets on the common lattices. We specifically retain only the first nontrivial odd correlation function, i.e., the triplet correlation function, while neglecting higher order correlations. Within this approximation, assuming the equivalence of triplet correlations involving both central and noncentral sites in the nearest neighbor cluster yields values for the critical temperature below the Bethe approximations for z⩾4. Likewise, assuming the inequivalence of triplet correlations involving the central and noncentral sites yields values of the critical temperature only slightly higher than the Bethe approximation for z⩾4. For lattices with four- and six-fold coordination this equivalence and inequivalence of triplet correlation functions is sufficient to cause the critical temperatures to approach their two- and three-dimensional values. For instance, for quadratic lattices we obtain values for the critical temperature of 2.5284 and 2.9796 as compared to the exact and series results of 2.2692 and 2.7044 for the simple quadratic and diamond lattices, respectively, and the Bethe approximation of 2.8854.
Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with zzz nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.
We propose a method to incorporate pair correlations of an Ising lattice system in molecular field theory to higher orders. The theory is applied to ferromagnetism, where a system of equations is obtained for the response function (here the susceptibility). We obtain important corrections on the Weiss model and the spherical model. This is illustrated by an explicit calculation of the critical temperature and the spurious phase, appearing for T>TcT>{T}_{c}T>Tc​, near the critical point in the case of a simple cubic lattice.
The self-consistency conditions inherent in the mean-field approximation are extended to take into account some pair correlations.Ii is shown that this extension is systematically improvable and applicable to the spin systems with short-range interactions in the entire temperature region.With two correlations included and the renormalization-group ideas implemented, the critical properties of the Ising model are satisfactorily revealed.Moreover, the improved estimates of the thermodynamical quantities for the quantum linear XY model from the finite-size calculations are found down to very low temperatures.
A new method incorporating the effects of many-body static spin correlations into an effective-field theory is discussed. The method is based on the introduction of a differential operator and the concept of the correlated effective field into two exact Callen identities of the Ising model. The resulting statistical theory is shown to have an accuracy equivalent to that of the Bethe-Peierls method. It is shown that the correlated-effective-field parameter at the transition temperature has a universal function of $\frac{1}{(z\ensuremath{-}1)}$, where zzz is the number of nearest neighbors.
We study a modified mean-field approximation for the Ising Model in arbitrary dimension. Instead of taking a "central" spin, or a small "drop" of fluctuating spins coupled to the effective field of their nearest neighbors as in the Mean-Field or the Bethe-Peierls-Weiss methods, we take an infinite chain of fluctuating spins coupled to the mean field of the rest of the lattice. This results in a significative improvement of the Mean-Field approximation with a small extra effort.