A New Approximation Method for Order-Disorder Problems

Abstract

This paper presents an approximation method for the treatment of Ising lattices that is essentially an extension of Bethe's method, designed to take more direct account of the short-range order in the lattice and to give a better description of this order near the Curie point. Attention is fixed on spin configurations on a small group of sites. Full account is taken of the coupling between these spins, and the effect of coupling to the rest of the lattice is described by means of one or more long-range order parameters, and a correlation parameter characteristic of the method. These parameters are determined as functions of temperature by means of a corresponding number of consistency relations. The method is among the most accurate of the approximation methods in this field, giving about the same accuracy as Kikuchi's method, for a comparable amount of labor; its main advantage over Kikuchi's method lies in the treatment of orientational correlations between relatively remote neighbors in the lattice. The method is here applied to the plane triangular, square, and hexagonal Ising lattices, and to the simple cubic lattice. Curie points are determined, and plots of long- and short-range order and the specific heat are given for all but the hexagonal lattice. Like all other approximation methods, this one indicates only a finite discontinuity in the specific heat of the plane lattices at the Curie point, instead of the logarithmic singularity given by exact solutions. Correlation functions for the orientations of up to fifth neighbors are also given for the simple cubic lattice.