Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions

TL;DR

This paper analyzes the chromatic polynomial for square and triangular lattice strips with cyclic boundary conditions, revealing phase structures, zeros distribution, and critical properties of the antiferromagnetic Potts model.

Contribution

It introduces a transfer matrix construction in the Fortuin--Kasteleyn representation for these lattices and characterizes phase behavior via topological parameters.

Findings

Identified accumulation sets of chromatic zeros in the complex plane.

Characterized phases using a topological parameter.

Computed bulk and surface free energies and the central charge.

Abstract

We study the chromatic polynomial P_G(q) for m \times n square- and triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.