T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions

Abstract

We present exact calculations of the zero-temperature partition function for the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) for families of arbitrarily long strip graphs of the square and triangular lattices with width $L_y=4$ and boundary conditions that are doubly periodic or doubly periodic with reversed orientation (i.e. of torus or Klein bottle type). These boundary conditions have the advantage of removing edge effects. In the limit of infinite length, we calculate the exponent of the entropy, $W(q)$ and determine the continuous locus ${\cal B}$ where it is singular. We also give results for toroidal strips involving ``crossing subgraphs''; these make possible a unified treatment of torus and Klein bottle boundary conditions and enable us to prove that for a given strip, the locus ${\cal B}$ is the same for these boundary conditions.