Periodic Ising Correlations

TL;DR

This paper revisits Kaufman's 1949 analysis of the periodic Ising model, employing representation theory to simplify the derivation of the transfer matrix spectrum and to formulate spin correlation functions.

Contribution

It introduces a representation-theoretic approach that simplifies the derivation of the transfer matrix spectrum and provides new formulas for spin correlations in the periodic Ising model.

Findings

Simplified derivation of the transfer matrix spectrum.

New formulas for spin correlation functions.

Identification of holomorphic factors in correlation kernels.

Abstract

In this paper, we first rework B. Kaufman's 1949 paper, "Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis", by using representation theory. Our approach leads to a simpler and more direct way of deriving the spectrum of the transfer matrix for the finite periodic Ising model. We then determine formulas for the spin correlation functions that depend on the matrix elements of the induced rotation associated with the spin operator in a basis of eigenvectors for the transfer matrix. The representation of the spin matrix elements is obtained by considering the spin operator as an intertwining map. We exhibit the "new" elements V+ and V- in the Bugrij-Lisovyy formula as part of a holomorphic factorization of the periodic and anti-periodic summability kernels on the spectral curve associated with the induced rotation for the transfer matrix.