Lee-Yang zeros for DHL and 2D rational dynamics, I. Foliation of the physical cylinder

TL;DR

This paper rigorously analyzes the distribution of Lee-Yang zeros for the Ising model on a hierarchical lattice, revealing smooth and singular behaviors related to phase transitions through the dynamics of a rational function.

Contribution

It provides the first complete rigorous description of Lee-Yang zero distribution beyond one-dimensional models using dynamical systems techniques.

Findings

Zeros form a smooth density below critical temperature.

Density vanishes on a Cantor set above critical temperature.

Distribution is absolutely continuous and organized by foliation of a rational map.

Abstract

In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising models always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\RR$ in two variables (the renormalization transformation). We prove that $\RR$ is partially hyperbolic on an invariant cylinder $\CC$. The Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of $\RR|\, \CC$. Their distribution is absolutely continuous. Its density is $C^\infty$ (and non-vanishing) below the critical temperature. Above the critical temperature, it is $C^\infty$ on a open dense subset, but it vanishes on the complementary Cantor set of positive measure. This seems to be the first occasion of a complete rigorous description of the Lee-Yang distributions beyond 1D models.