Partition function zeros at first-order phase transitions: A general analysis

TL;DR

This paper develops a rigorous theory describing the distribution of partition function zeros in lattice spin models, revealing their concentration on phase boundaries and extending the Lee-Yang Circle Theorem to more general settings.

Contribution

It provides a general framework for locating partition function zeros in spin models, including a local Lee-Yang theorem extension for models with symmetry.

Findings

Zeros concentrate on phase boundary curves for large systems

Exponential accuracy in locating zeros with system size

Extension of Lee-Yang Circle Theorem to asymmetric cases

Abstract

We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [BBCKK2, math-ph/0304007]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Ising-like plus-minus symmetry, we also establish a local version of the Lee-Yang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves.