Transfer matrices for the zero-temperature Potts antiferromagnet on cyclic and Möbius lattice strips

Abstract

We present transfer matrices for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and Möbius strips of the square, triangular, and honeycomb lattices of width L y and arbitrarily great length L x . We relate these results to our earlier exact solutions for square-lattice strips with L y = 3 , 4 , 5 , triangular-lattice strips with L y = 2 , 3 , 4 , and honeycomb-lattice strips with L y = 2 , 3 and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a Möbius strip of a lattice Λ and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width L y . New results are presented for the L y = 5 strip of the triangular lattice and the L y = 4 and L y = 5 strips of the honeycomb lattice. Using these results and taking the infinite-length limit L x → ∞ , we determine the continuous accumulation locus of the zeros of the above partition function in the complex q plane, including the maximal real point of nonanalyticity of the degeneracy per site, W as a function of q.