The critical Ising model via Kac-Ward matrices

TL;DR

This paper explores the properties of Kac-Ward determinants for isoradially embedded graphs with critical weights, revealing duality relations, connections to Laplacians, and similarities to Riemann surface torsions.

Contribution

It demonstrates remarkable properties of Kac-Ward determinants in the critical Ising model on isoradial graphs, including duality relations and links to Laplacian determinants.

Findings

Determinants satisfy a generalized Kramers-Wannier duality.

They are proportional to critical Laplacian determinants for genus 0 or 1.

They share properties with Ray-Singer ar ext{d}-torsions.

Abstract

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer \bar\partial-torsions of the Riemann surface in which G embeds.