A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model

TL;DR

This paper offers a combinatorial interpretation of the free fermion condition in the six-vertex model, extending the Gessel-Viennot theorem to intersecting walks and enabling explicit solutions of the model.

Contribution

It introduces a combinatorial explanation for the free fermion condition using a generalized Gessel-Viennot involution, expanding its application to intersecting walks.

Findings

Extended Gessel-Viennot theorem to intersecting walks

Expressed partition functions in terms of single walk functions

Provided combinatorial insight into the free fermion condition

Abstract

The free fermion condition of the six-vertex model provides a 5 parameter sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple, hence allowing explicit solutions. Such conditions arose originally in early field-theoretic S-matrix approaches. Here we provide a combinatorial explanation for the condition in terms of a generalised Gessel-Viennot involution. By doing so we extend the use of the Gessel-Viennot theorem, originally devised for non-intersecting walks only, to a special weighted type of \emph{intersecting} walk, and hence express the partition function of $N$ such walks starting and finishing at fixed endpoints in terms of the single walk partition functions.