Density of Yang-Lee zeros in the thermodynamic limit from tensor network methods

TL;DR

This paper uses tensor network methods to analyze the distribution of Yang-Lee zeros in the Ising and Potts models, revealing their behavior in the thermodynamic limit and resolving debates about their location.

Contribution

It introduces a tensor network approach to directly compute Yang-Lee zeros in the thermodynamic limit for complex field planes in Ising and Potts models.

Findings

Yang-Lee zeros lie on the unit circle below and at critical temperature

Zeros do not lie on the unit circle for q>2 in Potts models except at zero temperature

The radius of zeros shrinks exponentially with inverse temperature in 2D Potts models

Abstract

The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle theorem. Distinct features are observed at different temperatures---below, above and at the critical temperature. Application of the tensor-network approach is also made to the $q$-state Potts models in both two and three dimensions and a previous debate on whether, in the thermodynamic limit, the Yang-Lee zeros lie on a unit circle for $q>2$ is resolved: they clearly do not lie on a unit circle except at the zero temperature. For the Potts models (q=3,4,5,6) investigated in two dimensions, as the temperature is lowered the radius of the zeros at a fixed angle from the real axis shrinks exponentially towards unity with the inverse temperature.