Ground State Entropy of Potts Antiferromagnets: Homeomorphic Classes with Noncompact W Boundaries

TL;DR

This paper calculates the zero-temperature partition function and ground-state degeneracy for Potts antiferromagnets on specific graph families, revealing nonanalytic behavior at certain complex points and exploring conditions for large-q expansions.

Contribution

It introduces methods to generate graph families with noncompact boundary regions in the complex q-plane, extending understanding of Potts model ground states.

Findings

Exact calculations of $W$ for various graphs.

Identification of nonanalytic points at $z=0$ in complex q-plane.

Insights into the limitations of large-q expansions.

Abstract

We present exact calculations of the zero-temperature partition function $Z(G,q,T=0)$ and ground-state degeneracy $W(\{G\},q)$ for the $q$-state Potts antiferromagnet on a number of families of graphs $G$ for which (generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$) the boundary ${\cal B}$ of regions of analyticity of $W$ in the complex $q$ plane is noncompact, passing through $z=1/q=0$. For these types of graphs, since the reduced function $W_{red.}=q^{-1}W$ is nonanalytic at $z=0$, there is no large--$q$ Taylor series expansion of $W_{red.}$. The study of these graphs thus gives insight into the conditions for the validity of the large--$q$ expansions. It is shown how such (families of) graphs can be generated from known families by homeomorphic expansion.