Partition function zeros of the one-dimensional Potts model: The recursive method

TL;DR

This paper investigates the distribution and properties of partition function zeros in the one-dimensional Potts model using a dynamical systems approach, revealing their relation to fixed points and deriving explicit expressions.

Contribution

It introduces an exact recurrence relation for the partition function and classifies the zeros, providing analytical expressions and insights into their density and singularities.

Findings

Zeros are associated with neutral fixed points of the recurrence relation.

Fisher zeros in a magnetic field lie on multiple lines in the complex temperature plane.

Densities of zeros are singular at edge points with critical exponent -1/2.

Abstract

The Yang-Lee, Fisher and Potts zeros of the one-dimensional Q-state Potts model are studied using the theory of dynamical systems. An exact recurrence relation for the partition function is derived. It is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation. Further, a general equation for zeros of the partition function is found and a classification of the Yang-Lee, Fisher and Potts zeros is given. It is shown that the Fisher zeros in a nonzero magnetic field are located on several lines in the complex temperature plane and that the number of these lines depends on the value of the magnetic field. Analytical expressions for the densities of the Yang-Lee, Fisher and Potts zeros are derived. It is shown that densities of all types of zeros of the partition function are singular at the edge singularity points with the same critical exponent \sigma=-1/2.