Exact Potts Model Partition Functions for Strips of the Square Lattice

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths Lt and for arbitrarily great length Ll, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v) = ∑NZ,G,λ j=1 cZ,G,j(λZ,G,j) Ll . We give general formulas for NZ,G,j and its specialization to v = −1 for arbitrary Lt in the case of free boundary conditions, as well as other general structural results on Z. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus B, arising as the accumulation set of partition function zeros as Ll → ∞, in the q plane for fixed v and in the v plane for fixed q. email: [email protected] email: [email protected] email: [email protected]