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Physical Review

Row Correlation Functions of the Two-Dimensional Ising Model

David A. Pink

Published Dec 10, 1969• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.188.1032

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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.
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In this paper we rederive, by simpler methods, the Onsager-Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two-dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager-Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.