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 ${\cal B}_q(v)$ and ${\cal B}_v(q)$, or equivalently, ${\cal B}_y(q)$, in the $m \to \infty$ limit. Our results motivate the inference that the maximal point at which ${\cal B}_q(-1)$ 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 ${\cal B}_y(q_c)$ 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.