The Riemann surface of the chiral Potts model free energy function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice “inversion relation” method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site Tpq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of Tpq. In terms of the tp, tq variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N − 1)-dimensional lattice (for N > 2). The function Tpq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N +4)/4] parameters to be determined.