Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice

Abstract

We present exact solutions for the zero-temperature partition function (chromatic polynomial $P$) and the ground state degeneracy per site $W$ (= exponent of the ground-state entropy) for the $q$-state Potts antiferromagnet on strips of the square lattice of width $L_y$ vertices and arbitrarily great length $L_x$ vertices. The specific solutions are for (a) $L_y=4$, $(FB{C}_y,PB{C}_x)$ (cyclic); (b) $L_y=4$, $(FB{C}_y,TPB{C}_x)$ (Möbius); (c) $L_y=5,6$, $(PB{C}_y,FB{C}_x)$ (cylindrical); and (d) $L_y=5$, $(FB{C}_y,FB{C}_x)$ (open), where $FBC$, $PBC$, and $TPBC$ denote free, periodic, and twisted periodic boundary conditions, respectively. In the $L_x\to \infty$ limit of each strip we discuss the analytic structure of $W$ in the complex $q$ plane. The respective $W$ functions are evaluated numerically for various values of $q$. Several inferences are presented for the chromatic polynomials and analytic structure of $W$ for lattice strips with arbitrarily great $L_y$. The absence of a nonpathological $L_x\to \infty$ limit for real nonintegral $q$ in the interval $0<q<3$ ($0<q<4$) for strips of the square (triangular) lattice is discussed.

Comments: 37 pages, latex, 4 encapsulated postscript figures