Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions

TL;DR

This paper investigates the zeros of the partition function for the antiferromagnetic Potts model on a square lattice with special boundary conditions, revealing dense zeros outside the limiting curve but maintaining analytic free energy.

Contribution

It introduces transfer-matrix analysis for a novel boundary condition setup and provides numerical evidence of zero density and analytic free energy in the complex plane.

Findings

Partition-function zeros become dense outside the limiting curve.

Infinite-volume free energy remains analytic despite dense zeros.

Special boundary conditions affect zero distribution without causing non-analyticity.

Abstract

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.