Exact Potts model partition functions on ladder graphs

We present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex ladder graphs, i.e., strips of the square lattice with width L_y = 2 and arbitrary length L_x, with free, cyclic, and Mobius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed. By comparison with strip graphs of other widths, we analyze how the singularities at the zero-temperature critical point of the ferromagnet on infinite-length, finite-width strips depend on the width. We point out and study the following noncommutativity at certain special values q_s: lim_n→∞ lim_q→q(s) Z^1/n ≠ lim_q→q(s) lim_n→∞ Z^1/n. It is shown that the Potts antiferromagnet on both the infinite-length line and ladder graphs with cyclic or Mobius boundary conditions exhibits a phase transition at finite temperature if 0 < q < 2, but with unphysical properties, including negative specific heat and non-existence, in the low-temperature phase, of an n → ∞ limit for thermodynamic functions that is independent of boundary conditions. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus B in the corresponding C² space, arising as the accumulation set of partition function zeros as n → ∞. In particular, we study the connection with the T = 0 limit of the Potts antiferromagnet where B reduces to the accumulation set of chromatic zeros. Certain properties of the complex-temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice, showing that exact solutions on infinite-length strips provide a way of gaining insight into these complex-temperature phase diagrams.