Classification of (non)-frustrated 2D Ising models in genus 1 on isoradial graphs

We prove a complete classification of 2D Ising models defined on isoradial graphs, frustrated or not, whose underlying spectral curve has genus 1. As a specific case, we recover Baxter's Z-invariant Ising model, thus extending his class of models to real coupling constants. We identify two additional families of models, both having non-Harnack spectral curves. We show that in all cases the spectral curve is maximal. Moreover, each family undergoes an algebraic phase transition, in the sense that the genus changes from one to zero, explaining the different behaviors observed in the physics literature. In our proof, we use properties of the spectral curve and Fock's approach. This yields a natural framework for a further systematic study of the frustrated Ising model, in particular for proving local formulas. In the course of the proof, we also identify Fock's dimer models corresponding to real algebraic curves of genus 1, and to real dimer models. As an example of application of our main result, we prove a full classification of the frustrated Ising model on the triangular lattice.