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Progress of Theoretical Physics

Pair Correlation Function in the Ising Square Lattice: --Generalized Wronskian Form--

Keiji Yamada

Published Sep 1, 1986• 17 citations
Publisher

Abstract

The pair correlation function X(m, n) between lattice sites (0,0) and (m, n) (for any m'i';,n without loss of generality) in the Ising square lattice without magnetic field is expressed in a simple determinantal form with dimension m, which is called a "generalized Wronskian" form, as an extension of the Toeplitz determinantal form with dimension m in the cases of X(m, 0), X(m, 1) and X(m, m). § 1. Introduction

DOI
10.1143/ptp.76.602

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The pair correlation function X(m, n) between lattice sites (0,0) and (m, n) (for any m'i';,n without loss of generality) in the Ising square lattice without magnetic field is expressed in a simple determinantal form with dimension m, which is called a "generalized Wronskian" form, as an extension of the Toeplitz determinantal form with dimension m in the cases of X(m, 0), X(m, 1) and X(m, m). § 1. Introduction
The zero·field susceptibility x ( T) of the antiferromagnet in the Ising square lattice with the symmetric interaction was evaluated by carrying out the lattice sum of the short · range correlation function X ( m, n) (O~m, n~4), which is exactly and explicitly calculated on the basis of a "generalized Wronskian " form with dimension m. It is naturally derived from the singularity of X(m, n) that the highest singularity of x ( T) (per site) near the critical temperature Tc is logarithmic as kB Tz ( T) =A + B(l- T/Tc)lnI1- T/Tcl, with A =0.1596264029 and B=0.2009359779 in the best approximation of the present calculation. Both A and B are completely equal below and above the critical temperature.
Expressions for many-site correlation functions of a square-lattice Ising model have been derived for the case in which all sites lie in the same row. The results involve expressions for minors of Toeplitz matrices which are obtained by the Wiener-Hopf technique. We derive an expression for the magnetization at any site in a row if a finite number of defect bonds are distributed at given positions in that row; this expression can be used to study the effect of a dislocation. Some remarks about the decoupling of correlation functions are made.
A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.