Discretely holomorphic parafermions and integrable loop models

Abstract

We define parafermionic observables in various lattice loop models, including examples where no Kramers–Wannier duality holds. For a particular rhombic embedding of the lattice in the plane and a value of the parafermionic spin these variables are discretely holomorphic (they satisfy a lattice version of the Cauchy–Riemann equations) as long as the Boltzmann weights satisfy certain linear constraints. In the cases considered, the weights then also satisfy the critical Yang–Baxter equations, with the spectral parameter being related linearly to the angle of the elementary rhombus.