Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips

TL;DR

This paper develops transfer matrices for the zero-temperature Potts antiferromagnet on various lattice strips, providing exact solutions and analyzing the zeros of the chromatic polynomial in the complex plane to understand phase transitions.

Contribution

It introduces new transfer matrix results for lattice strips of width up to 5, generalizes formulas for M"obius strips, and analyzes the zero distributions for complex $q$.

Findings

Exact transfer matrices for specific lattice strips.

General expression for M"obius strip chromatic polynomial.

Location of zeros and nonanalytic points in the complex $q$ plane.

Abstract

We present transfer matrices for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and M\"obius strips of the square, triangular, and honeycomb lattices of width $L_y$ and arbitrarily great length $L_x$. We relate these results to our earlier exact solutions for square-lattice strips with $L_y=3,4,5$, triangular-lattice strips with $L_y=2,3,4$, and honeycomb-lattice strips with $L_y=2,3$ and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a M\"obius strip of a lattice $\Lambda$ and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width $L_y$. New results are presented for the $L_y=5$ strip of the triangular lattice and the $L_y=4$ and $L_y=5$ strips of the honeycomb lattice. Using these results and taking the infinite-length limit $L_x \to \infty$, we determine the continuous accumulation locus of the zeros of the above partition function in the complex $q$ plane, including the maximal real point of nonanalyticity of the degeneracy per site, $W$ as a function of $q$.