Thermodynamic limit of the first Lee-Yang zero

TL;DR

This paper verifies that the thermodynamic singularities of the Ising model correspond to the limits of finite-volume zeros in the complex plane, establishing a precise connection between finite and infinite systems.

Contribution

It proves that the first zero modulus of the Ising partition function converges to a well-defined limit as the system size grows, confirming the Lee-Yang theory in the thermodynamic limit.

Findings

The first zero modulus decreases to a finite limit as volume increases.

The limit is positive if and only if the inverse temperature is below critical.

The free energy remains analytic within the disk of the limiting zero modulus.

Abstract

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in ${\mathbb R}$ of finite-volume singularities in ${\mathbb C}$. For the Ising model defined on a finite $\Lambda\subset\mathbb{Z}^d$ at inverse temperature $\beta\geq0$ and external field $h$, let $\alpha_1(\Lambda,\beta)$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable $h$). We prove that $\alpha_1(\Lambda,\beta)$ decreases to $\alpha_1(\mathbb{Z}^d,\beta)$ as $\Lambda$ increases to $\mathbb{Z}^d$ where $\alpha_1(\mathbb{Z}^d,\beta)\in[0,\infty)$ is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that $\alpha_1(\mathbb{Z}^d,\beta)$ is strictly positive if and only if $\beta$ is strictly less than the critical inverse temperature.