Gap Labels for Zeros of the Partition Function of the 1D Ising Model via the Schwartzman Homomorphism

TL;DR

This paper explains the distribution of Lee--Yang zeros in the 1D Ising model using the gap labelling theorem, connecting transfer matrices, orthogonal polynomials, and CMV matrices.

Contribution

It introduces a novel link between the zeros of the partition function and the gap labelling theorem through transfer matrix and orthogonal polynomial formalisms.

Findings

Distribution of Lee--Yang zeros follows the gap labelling pattern.

Connection established between Ising model zeros and CMV matrices.

Provides a theoretical framework explaining empirical observations.

Abstract

Inspired by the 1995 paper of Baake--Grimm--Pisani, we aim to explain the empirical observation that the distribution of Lee--Yang zeros corresponding to a one-dimensional Ising model appears to follow the gap labelling theorem. This follows by combining two main ingredients: first, the relation between the transfer matrix formalism for 1D Ising model and an ostensibly unrelated matrix formalism generating the Szeg\H{o} recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices.