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

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Communications in

Mathematical Physics

The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z -Invariance Paul Melotti Université de Fribourg, Fribourg, Switzerland. E-mail: [email protected] Received: 6 November 2018 / Accepted: 18 September 2020 © The Author(s) 2020

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 freefermion 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. Contents 1. 2.

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Introduction . . . . . . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . 2.1 Eight-vertex-model . . . . . . . . . 2.2 Ising model . . . . . . . . . . . . . 2.3 Dimer model . . . . . . . . . . . . 2.4 Order and disorder variables . . . . 2.4.1 Ising correlators . . . . . . . . . 2.4.2 8V correlators . . . . . . . . . . Couplings of 8V-Models . . . . . . . . . 3.1 Spin-vertex correspondence . . . . . 3.2 Modifications of weights . . . . . . 3.3 Free-fermion 8V correlators . . . . 3.4 Coupling of free-fermion 8V-models Kasteleyn Matrices . . . . . . . . . . . . 4.1 Free-fermion 8V to dimers . . . . . 4.2 Skew-symmetric real matrix . . . .

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P. Melotti

4.3 Skew-hermitian complex matrix . . . . . . . . 4.4 Eight-vertex partition function and correlations −1 4.5 Relations between matrices K α,β . . . . . . . . 4.5.1 Spherical and planar cases . . . . . . . . . 4.5.2 Toric case . . . . . . . . . . . . . . . . . . 5. Z -Invariant Regime . . . . . . . . . . . . . . . . . 5.1 Checkerboard Yang–Baxter equations . . . . . 5.2 Lozenge graphs . . . . . . . . . . . . . . . . . −1 5.3 Local expression for K k,l . . . . . . . . . . . . 5.3.1 Inverse of Kk . . . . . . . . . . . . . . . . 5.3.2 Inverse of K k,l . . . . . . . . . . . . . . . 5.3.3 Asymptotics of coefficients . . . . . . . . . 5.4 Free energy and Gibbs measure . . . . . . . . . A. 8V-Configurations as 1-Forms . . . . . . . . . . . . A.1 Setup . . . . . . . . .

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