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 L_t and for arbitrarily great length L_ℓ, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v) = ∑_{j = 1}^{NZ, G, λ} C_{Z, G, j}(λ_{Z, G, j})^Lℓ. We give general formulas for N_{Z, G, j} and its specialization to v = -1 for arbitrary L _t for both types of 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 L_ℓ → ∞, in the q plane for fixed v and in the v plane for fixed q.