Phase transition free regions in the Ising model via the Kac-Ward operator

TL;DR

This paper analyzes the spectral properties of the Kac-Ward operator for the Ising model on planar graphs to identify regions where the free energy is analytic, revealing criticality conditions.

Contribution

It provides new bounds on the spectral radius of the Kac-Ward operator, establishing phase transition free regions and criticality conditions for isoradial graphs.

Findings

Identifies regions in the complex plane with analytic free energy density

Provides optimal bounds for isoradial graphs

Establishes criticality of self-dual Z-invariant couplings

Abstract

We investigate the spectral radius and operator norm of the Kac-Ward transition matrix for the Ising model on a general planar graph. We then use the obtained results to identify regions in the complex plane where the free energy density limits are analytic functions of the inverse temperature. The bound turns out to be optimal in the case of isoradial graphs, i.e. it yields criticality of the self-dual Z-invariant coupling constants.