On the Potts model partition function in an external field

TL;DR

This paper explores the Potts model's partition function in an external field, providing new expansion formulas linking it to zero-field functions and spanning tree structures, extending known Tutte polynomial connections.

Contribution

It introduces a deletion-contraction formulation for the external field Potts model and expresses its partition function as sums over spanning trees and forests, extending Tutte polynomial relations.

Findings

Partition function expanded in terms of zero-field partition function

Representation as sum over spanning trees and forests

Extension of Tutte polynomial connections to external field case

Abstract

We study the partition function of Potts model in an external (magnetic) field, and its connections with the zero-field Potts model partition function. Using a deletion-contraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zero-field partition function. We also show that Z can be written as a sum over the spanning trees, and the spanning forests, of a graph G. Our results extend to Z the well-known spanning tree expansion for the zero-field partition function that arises though its connections with the Tutte polynomial.