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Progress of Theoretical Physics

Pair Correlation Function in the Ising Square Lattice: Determinantal Form

K. Yamada

Published Nov 1, 1984• 13 citations
Publisher

Abstract

An integral equation, which is related to the pair correlation function X(m,n) in the Ising square lattice, is simplified by making use of an elliptic transformation. On the bases of that equation, X(m,n) is obtained in the form of a determinant with dimension m (≥n).
DOI
10.1143/ptp.72.922

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Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors — matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.