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

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.

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“…where the sum ⟨ij⟩ is over all nearest-neighbours and the field H ex can be 0 or i(π/2)k B T . The approach we employ is the mapping into the free-fermion model [41][42][43][44][45], which was proposed in the vertex model problem. Vertex model is also an important and intriguing issue in statistical mechanics.…”