Discretely Holomorphic Parafermions in Lattice Z(N) Models

TL;DR

This paper constructs lattice parafermions in Z(N) models that are discretely holomorphic at critical points, extending to anisotropic and inhomogeneous models with specific coupling conditions.

Contribution

It introduces a method to construct discretely holomorphic parafermions in Z(N) lattice models at critical points, including anisotropic and inhomogeneous cases.

Findings

Discretely holomorphic parafermions exist at critical FZ points.

Such parafermions are found for anisotropic models with correct planar embedding.

Results extend to inhomogeneous models with rhombic lattice embedding.

Abstract

We construct lattice parafermions - local products of order and disorder operators - in nearest-neighbor Z(N) models on regular isotropic planar lattices, and show that they are discretely holomorphic, that is they satisfy discrete Cauchy-Riemann equations, precisely at the critical Fateev-Zamolodchikov (FZ) integrable points. We generalize our analysis to models with anisotropic interactions, showing that, as long as the lattice is correctly embedded in the plane, such discretely holomorphic parafermions exist for particular values of the couplings which we identify as the anisotropic FZ points. These results extend to more general inhomogeneous lattice models as long as the covering lattice admits a rhombic embedding in the plane.