A Generalized Circle Theorem on Zeros of Partition Function at Asymmetric First Order Transitions

TL;DR

This paper generalizes the circle theorem for partition function zeros, extending Lee-Yang results to asymmetric first-order transitions and providing explicit calculations for the Blume-Capel model.

Contribution

It introduces a generalized circle theorem for zeros of the partition function that encompasses asymmetric transitions, with explicit calculations for specific models.

Findings

Zeros lie on a circle in symmetric cases

Zeros are tangent to the unit circle at the positive real axis in asymmetric cases

Finite-size zeros can lie off the unit circle when phases are combined with unequal weights

Abstract

We present a generalized circle theorem which includes the Lee-Yang theorem for symmetric transitions as a special case. It is found that zeros of the partition function can be written in terms of discontinuities in the derivatives of the free energy. For asymmetric transitions, the locus of the zeros is tangent to the unit circle at the positive real axis in the thermodynamic limit. For finite-size systems, they lie off the unit circle if the partition functions of the two phases are added up with unequal prefactors. This conclusion is substantiated by explicit calculation of zeros of the partition function for the Blume-Capel model near and at the triple line at low temperatures.