The signed loop approach to the Ising model: foundations and critical point

TL;DR

This paper rigorously develops the signed loop method for the 2D Ising model, deriving explicit expressions for key quantities and establishing the self-dual point as critical for phase transition behavior.

Contribution

It provides a detailed foundation for the signed loop approach and applies it to derive explicit formulas up to the critical point, clarifying its role in phase transition analysis.

Findings

Explicit formulas for free energy and two-point functions derived

Self-dual point identified as critical for phase transition

Method clarifies the role of loops in Ising model analysis

Abstract

The signed loop method is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that have so far been neglected or overlooked in the literature. We demonstrate how the method can be applied to the Ising model on the square lattice to derive explicit formal expressions for the free energy density and two-point functions in terms of sums over loops, valid all the way up to the self-dual point. As a corollary, it follows that the self-dual point is critical both for the behaviour of the free energy density, and for the decay of the two-point functions.