The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance

Paul Melotti

doi:10.1007/s00220-020-03901-2

Abstract

We study the eight-vertex model at its free-fermion point. We express a new “switching” symmetry of the model in several forms: partition functions, order-disorder variables, couplings, Kasteleyn matrices. This symmetry can be used to relate free-fermion 8V-models to free-fermion 6V-models, or bipartite dimers. We also define new solution of the Yang–Baxter equations in a “checkerboard” setting, and a corresponding Z-invariant model. Using the bipartite dimers of Boutillier et al. (Probab Theory Relat Fields 174:235–305, 2019), we give exact local formulas for edge correlations in the Z-invariant free-fermion 8V-model on lozenge graphs, and we deduce the construction of an ergodic Gibbs measure.

Highlights

The eight-vertex model, or 8V-model for short, was introduced by Sutherland [63] andFan and Wu [30] as a generalization of the 6V-model, which itself finds its origins in the study of square ice [53,61]
In the most classical setting, the model is characterized by four local weights a, b, c, d such that opposite local configurations are given the same weight, see [6]
In what follows we provide a way to transform these non-bipartite dimers into bipartite ones

Summary

Introduction

The eight-vertex model, or 8V-model for short, was introduced by Sutherland [63] and. The following 8V-weights of lozenge graphs, expressed in terms of Jacobi’s elliptic functions (see [1,52]) at a face f with half-angle θ , satisfy the Yang–Baxter equations: A( f ) = sn (θk|k) + sn (θl |l) When k = l we get a Z -invariant 6V model whose corresponding dimer model can be found in [16] At this point it is unknown if this new parametrization of checkerboard Yang–Baxter equations could be extended outside of the free-fermion manifold. We prove that this choice of local weights provides a Gibbs measure with the locality property, confirming in that case the prediction of Baxter. Gibbs measure in the Z -invariant case in Sect. 5.4 and prove Theorem 6

Definitions

Couplings of 8V-Models

Kasteleyn Matrices

Toric case

Z-Invariant Regime

Proof of Lemma 35