The Riemann surface of the chiral Potts model free energy function

Details The Riemann surface of the chiral Potts model free energy function The Riemann surface of the chiral Potts model free energy function

2002-12-03

arxiv.org/abs/cond-mat/0212076v1

J.Statist.Phys. 112 (2003) 1-26

Physics

Condensed Matter

Statistical Mechanics

16 pages, 1 figure

Scientific paper

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 $T_{pq}$ of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of $T_{pq}$. In terms of the $t_p, t_q$ 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 $T_{pq}$ 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.

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