Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial

TL;DR

This paper analyzes the chromatic polynomial for triangular-lattice strips to understand the zero-temperature behavior of the Potts antiferromagnet, revealing new features and their physical implications.

Contribution

It computes transfer matrices for triangular-lattice strips and explores the accumulation sets of chromatic zeros, extending Baxter's thermodynamic limit results with new features.

Findings

Recomputed Baxter's limiting curve for infinite lattice size.

Identified new features in the accumulation sets of zeros.

Analyzed the relation between isolated limiting points and Beraha numbers.

Abstract

We study the chromatic polynomial P_G(q) for m \times n triangular-lattice strips of widths m <= 12_P, 9_F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin--Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m,n\to\infty and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.