Phase diagram of the chromatic polynomial on a torus

TL;DR

This paper analyzes the chromatic polynomial of the Potts antiferromagnet on a torus using transfer matrices, providing exact expressions for certain widths and revealing phase diagrams for infinite-length strips.

Contribution

It presents new exact formulas for the chromatic polynomial on square and triangular lattices with specific widths and derives phase diagrams for these lattice strips.

Findings

Exact chromatic polynomial expressions for widths L=5,6,7

Phase diagrams for infinite-length lattice strips

Insights into the infinite-volume phase structure

Abstract

We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the square and triangular lattices. On the physical side, we obtain the exact ``phase diagrams'' for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases.