$q$-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 $q=q_1=\mathrm{1.6388969..}$, 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 $Z(D_4,q,v_0)$.

41 pages, 16 figures. Comments welcome!

References in corpus (26)

• Spanning trees on the Sierpinski gasket
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial
• Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets
• Exact Potts Model Partition Functions on Ladder Graphs
• Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
• A Little Statistical Mechanics for the Graph Theorist
• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Square Lattice Strips with (Twisted) Periodic Boundary Conditions
• Chromatic Polynomials for $J(\prod H)I$ Strip Graphs and their Asymptotic Limits
• Complex-Temperature Properties of the Ising Model on 2D Heteropolygonal Lattices
• The Julia sets and complex singularities in hierarchical Ising models
• T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
• Ground-State Degeneracy of Potts Antiferromagnets on Two-Dimensional Lattices: Approach Using Infinite Cyclic Strip Graphs
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models II. Extended Results for Square-Lattice Chromatic Polynomial
• Ground State Entropy of the Potts Antiferromagnet on Cyclic Strip Graphs
• Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Couplings on Strips of the Square Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
• Yang-Lee Zeros of the Q-state Potts Model on Recursive Lattices
• Zeros of the Potts Model Partition Function on Sierpinski Graphs
• On the dynamics of a family of renormalization transformations
• Distribution and density of the partition function zeros for the diamond-decorated Ising model
• Zeros of the Potts Model Partition Function in the Large-$q$ Limit

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