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

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 ${\cal B}$ where $W$ is nonanalytic in the complex $q$ plane are determined. The ${\cal B}$'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 ${\cal B}$ for limits (ii) and (iii) is the unit circle $|q-1|=1$.