Character decomposition of Potts model partition functions. I. Cyclic geometry

TL;DR

This paper analyzes the decomposition of Potts model partition functions on finite 2D lattices with mixed boundary conditions, showing that certain character functions coincide and can be expressed as transfer matrix traces, with extensions to constrained cases.

Contribution

It demonstrates the equivalence of character functions in the Potts model and expresses them as transfer matrix traces, extending the decomposition to constrained and fixed boundary conditions.

Findings

Characters K_{1+2l} coincide and are trace of transfer matrices.

Decomposition extends to constrained partition functions with non-contractible clusters.

Provides a unified transfer matrix framework for various boundary conditions.

Abstract

We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L x N, with boundary conditions that are free in the L-direction and periodic in the N-direction. The decomposition of the partition function in terms of the characters K\_{1+2l} (with l=0,1,...,L) has previously been studied using various approaches (quantum groups, combinatorics, transfer matrices). We first show that the K\_{1+2l} thus defined actually coincide, and can be written as traces of suitable transfer matrices in the cluster picture. We then proceed to similarly decompose constrained partition functions in which exactly j clusters are non-contractible with respect to the periodic lattice direction, and a partition function with fixed transverse boundary conditions.