(q\)-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs

• Published in: arXiv.org 2019-11
• Main Authors: Chang, Shu-Chiuan, Roeder, Roland K W, Shrock, Robert
• Format: Article
• Language: English
• Subjects: Antiferromagnetism; Diamonds; Ferromagnetism; Graphs; Loci; Mathematical analysis; Partitions; Partitions (mathematics); Polynomials
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1911.04012

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 \(Z(D_4,q,v_0)\).