Thermodynamic limit of the first Lee-Yang zero

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in R${\mathbb {R}}$ of finite-volume singularities in C${\mathbb {C}}$. For the Ising model defined on a finite Lambda subset of Zd$\Lambda \subset \mathbb {Z}<^>d$ at inverse temperature beta >= 0$\beta \ge 0$ and external field h, let alpha 1(Lambda,beta)$\alpha _1(\Lambda ,\beta )$ 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)$\alpha _1(\Lambda ,\beta )$ decreases to alpha 1(Zd,beta)$\alpha _1(\mathbb {Z}<^>d,\beta )$ as Lambda increases to Zd$\mathbb {Z}<^>d$ where alpha 1(Zd,beta)is an element of[0,infinity)$\alpha _1(\mathbb {Z}<^>d,\beta )\in [0,\infty )$ 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(Zd,beta)$\alpha _1(\mathbb {Z}<^>d,\beta )$ is strictly positive if and only if beta is strictly less than the critical inverse temperature.

Automated summary

This research completes the verification of the relationship between thermodynamic singularities and finite-volume singularities, showing that the modulus of the singularities decreases as the volume increases and approaches the radius in the thermodynamic limit.