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The free-fermion eight-vertex model: couplings, bipartite dimers and Z-invariance

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, de Tilière and Raschel, 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.