Discretely Holomorphic Parafermions and Integrable Loop Models

TL;DR

This paper introduces parafermionic observables in lattice loop models, demonstrating their discrete holomorphicity under specific conditions, and links these to integrable weights satisfying Yang-Baxter equations.

Contribution

It defines discretely holomorphic parafermions in various lattice models, including non-dual cases, and connects their properties to integrability via Yang-Baxter equations.

Findings

Parafermionic observables are discretely holomorphic under certain linear weight constraints.

The weights satisfying these constraints also fulfill critical Yang-Baxter equations.

Spectral parameter relates linearly to the lattice's rhombus angle.

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.