The critical Z-invariant Ising model via dimers: the periodic case

TL;DR

This paper explores the critical Z-invariant Ising models on periodic graphs, establishing a detailed correspondence with dimer models, and deriving explicit formulas for free energy and Gibbs measures.

Contribution

It provides a complete description of the dimer model associated with critical Z-invariant Ising models, linking the characteristic polynomials and deriving explicit thermodynamic quantities.

Findings

Dimer characteristic polynomial equals the critical Laplacian polynomial (up to a constant).

The characteristic polynomial defines a genus 0 Harnack curve.

Explicit formulas for free energy and Gibbs measure are obtained.

Abstract

We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical square, triangular and honeycomb lattice at the critical temperature. Fisher introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model. We prove that the dimer characteristic polynomial is equal (up to a constant) to the critical Laplacian characteristic polynomial, and defines a Harnack curve of genus 0. We prove an explicit expression for the free energy, and for the Gibbs measure obtained as weak limit of Boltzmann measures.