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 \(m \rightarrow \infty \) 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 \(m \rightarrow \infty \) 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 \(m \rightarrow \infty \) limit yielding the Sierpinski gasket fractal.