The critical temperature for the Ising model on planar doubly periodic graphs

TL;DR

This paper characterizes the critical temperature of the Ising model on any planar doubly periodic graph using a linear equation involving hyperbolic tangents of the couplings, based on high-temperature expansion and Kac-Ward matrices.

Contribution

It provides a simple, explicit linear equation to determine the critical temperature for the Ising model on planar doubly periodic graphs, extending previous understanding.

Findings

Critical inverse temperature is the unique solution of a linear equation.

The characterization applies to arbitrary planar doubly periodic graphs.

Uses Kac-Ward matrices to analyze high-temperature expansion.

Abstract

We provide a simple characterization of the critical temperature for the Ising model on an arbitrary planar doubly periodic weighted graph. More precisely, the critical inverse temperature \beta for a graph G with coupling constants (J_e)_{e\in E(G)} is obtained as the unique solution of a linear equation in the variables (\tanh(\beta J_e))_{e\in E(G)}. This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices.