Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice

Abstract

We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii) (FBC_y,TPBC_x)= M\"obius, (iii) (PBC_y,PBC_x)= toroidal, and (iv) (PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the L_y=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the {\mathbb C}^2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.