Partition Function Zeros of a Restricted Potts Model on Self-Dual Strips of the Square Lattice

TL;DR

This paper computes the exact partition function zeros of a restricted Potts model on self-dual square-lattice strips, analyzes their accumulation loci, and discusses properties for large widths, providing insights into phase transitions and model behavior.

Contribution

It provides exact calculations of partition function zeros for a restricted Potts model on self-dual strips, revealing properties of their accumulation loci and conjecturing behaviors for large widths.

Findings

Determined the locus of zeros in the $v$ and $Q$ planes for various strip widths.

Identified features of the accumulation locus ${\\cal B}$ and its properties.

Proposed a conjecture for the behavior of the locus as width increases.

Abstract

We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model exactly for self-dual cyclic square-lattice strips of various widths $L_y$ and arbitrarily great lengths $L_x$, with $Q$ and $v$ restricted to satisfy the relation $Q=v^2$. From these calculations, in the limit $L_x \to \infty$, we determine the continuous accumulation locus ${\cal B}$ of the partition function zeros in the $v$ and $Q$ planes. A number of interesting features of this locus are discussed and a conjecture is given for properties applicable for arbitrarily great width. Relations with the loci ${\cal B}$ for general $Q$ and $v$ are analyzed.