Ground State Entropy of Potts Antiferromagnets on Homeomorphic Families of Strip Graphs

Abstract

We present exact calculations of the zero-temperature partition function, and ground-state degeneracy (per site), $W$, for the $q$-state Potts antiferromagnet on a variety of homeomorphic families of planar strip graphs $G=(Ch{)}_{k_1,k_2,\Sigma ,k,m}$, where $k_1$, $k_2$, $\Sigma$, and $k$ describe the homeomorphic structure, and $m$ denotes the length of the strip. Several different ways of taking the total number of vertices to infinity, by sending (i) $m\to \infty$ with $k_1$, $k_2$, and $k$ fixed; (ii) $k_1$ and/or $k_2\to \infty$ with $m$, and $k$ fixed; and (iii) $k\to \infty$ with $m$ and $p=k_1+k_2$ fixed are studied and the respective loci of points $\mathcal{B}$ where $W$ is nonanalytic in the complex $q$ plane are determined. The $\mathcal{B}$'s for limit (i) are comprised of arcs which do not enclose regions in the $q$ plane and, for many values of $p$ and $k$, include support for $Re(q)<0$. The $\mathcal{B}$ for limits (ii) and (iii) is the unit circle $|q-1|=1$.

Comments: 42 pages, Latex, 11 encapsulated postscript figures, Physica A, in press