Ground State Entropy of Potts Antiferromagnets: Cases with Noncompact W Boundaries Having Multiple Points at 1/q = 0

TL;DR

This paper provides exact calculations of the ground-state degeneracy for Potts antiferromagnets on specific graph families, revealing conditions affecting large-q series expansions and the complex analytic structure of the partition function.

Contribution

It introduces new exact results for Potts antiferromagnets with noncompact boundary regions, highlighting the complex analytic structure and multiple points at 1/q=0.

Findings

Exact calculations of $W(G,q)$ for specific graph families.

Identification of noncompact boundary regions with multiple points at 1/q=0.

Insights into the validity of large-q series expansions.

Abstract

We present exact calculations of the zero-temperature partition function, $Z(G,q,T=0)$, and ground-state degeneracy (per site), $W({G},q)$, for the $q$-state Potts antiferromagnet on a number of families of graphs ${G}$ for which the boundary ${\cal B}$ of regions of analyticity of $W$ in the complex $q$ plane is noncompact and has the properties that (i) in the $z=1/q$ plane, the point $z=0$ is a multiple point on ${\cal B}$ and (ii) ${\cal B}$ includes support for $Re(q) < 0$. These families are generated by the method of homeomorphic expansion. Our results give further insight into the conditions for the validity of large--$q$ series expansions for the reduced function $W_{red.}=q^{-1}W$.