Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs

Abstract

We study properties of the Potts model partition function $Z(H_m,q,v)$ on $m$'th iterates of Hanoi graphs, $H_m$, and use the results to draw inferences about the $m\to \infty$ limit that yields a self-similar Hanoi fractal, $H_{\infty }$. We also calculate the chromatic polynomials $P(H_m,q)=Z(H_m,q,-1)$. From calculations of the configurational degeneracy, per vertex, of the zero-temperature Potts antiferromagnet on $H_m$, denoted $W(H_m,q)$, estimates of $W(H_{\infty },q)$, are given for $q=3$ and $q=4$ and compared with known values on other lattices. We compute the zeros of $Z(H_m,q,v)$ in the complex $q$ plane for various values of the temperature-dependent variable $v=y-1$ and in the complex $y$ plane for various values of $q$. These are consistent with accumulating to form loci denoted ${\mathcal{B}}_q(v)$ and ${\mathcal{B}}_v(q)$, or equivalently, ${\mathcal{B}}_y(q)$, in the $m\to \infty$ limit. Our results motivate the inference that the maximal point at which ${\mathcal{B}}_q(-1)$ crosses the real $q$ axis, denoted $q_c$, has the value $q_c=(1/2)(3+\sqrt{5})$ and correspondingly, if $q=q_c$, then ${\mathcal{B}}_y(q_c)$ crosses the real $y$ axis at $y=0$, i.e., the Potts antiferromagnet on $H_{\infty }$ with $q=(1/2)(3+\sqrt{5})$ has a $T=0$ critical point. Finally, we analyze the partition function zeros in the $y$ plane for $q\gg 1$ and show that these accumulate approximately along parts of the sides of an equilateral triangular with apex points that scale like $y\sim q^{2/3}$ and $y\sim q^{2/3}{e}^{\pm 2\pi i/3}$. Some comparisons are presented of these findings for Hanoi graphs with corresponding results on $m$'th iterates of Sierpinski gasket graphs and the $m\to \infty$ limit yielding the Sierpinski gasket fractal.

43 pages, 17 figures

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
• 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
• Exact Potts Model Partition Function for Strips of the Square Lattice
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial
• Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
• Dimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices
• Exact Potts Model Partition Functions for Strips of the Triangular Lattice
• Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q\to \infty$ Limit
• Dimer-monomer model on the Sierpinski gasket
• $q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs
• The number and degree distribution of spanning trees in the Tower of Hanoi graph
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
• Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips
• The Tutte polynomial of the Sierpinski and Hanoi graphs
• Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice
• Zeros of the Potts Model Partition Function on Sierpinski Graphs
• General Structural Results for Potts Model Partition Functions on Lattice Strips
• Dimer-monomer Model on the Towers of Hanoi Graphs
• Zeros of the Potts Model Partition Function in the Large-$q$ Limit
• Acyclic orientations on the Sierpinski gasket
• The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities

Cited by in corpus (1)

• Chromatic Zeros on the Limit $G_{\infty }^{(p,\ell )}$ of the Family $G_m^{(p,\ell )}$ of Hierarchical Graphs