$q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs

TL;DR

This paper analyzes the zeros of the Potts model partition function in the complex q-plane on diamond hierarchical graphs, revealing detailed crossing points and accumulation loci, with implications for phase transitions.

Contribution

It provides exact results and complex dynamics analysis of the Potts partition function zeros on diamond hierarchical graphs, including the limit as graph size approaches infinity.

Findings

The locus ${\\mathcal B}_q(v_0)$ crosses the real-q axis at specific points including 0, 3, 32/27, and a cubic root.

For $v=-1$, the crossing points include a minimal, maximal, and an infinite sequence converging to 32/27.

For $v > 0$, the locus crosses the real-q axis only at two points.

Abstract

We report exact results concerning the zeros of the partition function of the Potts model in the complex $q$ plane, as a function of a temperature-like Boltzmann variable $v$, for the $m$'th iterate graphs $D_m$ of the Diamond Hierarchical Lattice (DHL), including the limit $m \to \infty$. In this limit we denote the continuous accumulation locus of zeros in the $q$ planes at fixed $v = v_0$ as ${\mathcal B}_q(v_0)$. We apply theorems from complex dynamics to establish properties of ${\mathcal B}_q(v_0)$. For $v=-1$ (the zero-temperature Potts antiferromagnet, or equivalently, chromatic polynomial), we prove that ${\mathcal B}_q(-1)$ crosses the real-$q$ axis at (i) a minimal point $q=0$, (ii) a maximal point $q=3$ (iii) $q=32/27$, (iv) a cubic root that we give, with the value $q = q_1 = 1.6388969..$, and (v) an infinite number of points smaller than $q_1$, converging to $32/27$ from above. Similar results hold for ${\mathcal B}_q(v_0)$ for any $-1 < v < 0$ (Potts antiferromagnet at nonzero temperature). The locus ${\mathcal B}_q(v_0)$ crosses the real-$q$ axis at only two points for any $v > 0$ (Potts ferromagnet). We also provide computer-generated plots of ${\mathcal B}_q(v_0)$ at various values of $v_0$ in both the antiferromagnetic and ferromagnetic regimes and compare them to numerically computed zeros of $Z(D_4,q,v_0)$.