Exact Potts Model Partition Functions for Strips of the Triangular Lattice

TL;DR

This paper provides exact calculations of the Potts model partition function for strip graphs of the triangular lattice, analyzing thermodynamics, boundary conditions, and the distribution of zeros in the complex plane.

Contribution

It introduces exact formulas for the partition function of the Potts model on triangular lattice strips with various boundary conditions and analyzes the singular locus of zeros in the complex plane.

Findings

Exact partition functions for various strip widths and boundary conditions.

Connection between cylindrical boundary energy and critical properties.

Determination of the singular locus of zeros in the complex q and v planes.

Abstract

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 strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and its specialization to v=-1 for arbitrary L. 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 triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus {\cal B}, arising as the accumulation set of partition function zeros as m\to\infty, in the q plane for fixed v and in the v plane for fixed q.