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

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 $Z(D_4,q,v_0)$.