Exact Partition Functions for Potts Antiferromagnets on Cyclic Lattice Strips

TL;DR

This paper provides exact calculations of the zero-temperature partition function for the Potts antiferromagnet on various lattice strips, revealing nonzero ground-state entropy and analyzing the properties of the associated chromatic polynomial zeros.

Contribution

It presents the first exact solutions for the partition function of Potts antiferromagnets on cyclic lattice strips with specific widths and boundary conditions, including new insights into ground-state entropy.

Findings

Exact partition functions for square, triangular, and kagome lattice strips.

Demonstration of nonzero ground-state entropy for large q.

Analysis of chromatic polynomial zeros and their relation to $W(q)$.

Abstract

We present exact calculations of the zero-temperature partition function of the $q$-state Potts antiferromagnet on arbitrarily long strips of the square, triangular, and kagom\'e lattices with width $L_y=2$ or 3 vertices and with periodic longitudinal boundary conditions. From these, in the limit of infinite length, we obtain the exact ground-state entropy $S_0=k_B \ln W$. These results are of interest since this model exhibits nonzero ground state entropy $S_0 > 0$ for sufficiently large $q$ and hence is an exception to the third law of thermodynamics. We also include results for homeomorphic expansions of the square lattice strip. The analytic properties of $W(q)$ are determined and related to zeros of the chromatic polynomial for long finite strips.