Holomorphic Parafermions in the Potts model and SLE

TL;DR

This paper explores the properties of parafermionic operators in the Q-state Potts model, constructing lattice observables, comparing scaling dimensions with conformal field theory, and linking them to Schramm-Loewner Evolution (SLE) in the continuum limit.

Contribution

It provides explicit lattice constructions of holomorphic parafermionic observables and connects these to SLE, offering new insights into the model's continuum and lattice descriptions.

Findings

Lattice holomorphic observables constructed for the Potts model.

Comparison of parafermionic scaling dimensions with CFT and Coulomb gas results.

Parafermion expectation values relate to SLE local observables.

Abstract

We analyse parafermionic operators in the Q-state Potts model from three different perspectives. First, we explicitly construct lattice holomorphic observables in the Fortuin-Kasteleyn representation, and point out some special simplifying features of the particular case Q=2 (Ising model). In particular, away from criticality, we find a lattice generalisation of the massive Majorana fermion equation. We also compare the parafermionic scaling dimensions with known results from CFT and Coulomb gas methods in the continuum. Finally, we show that expectation values of these parafermions correspond to local observables of the SLE process which is conjectured to describe the scaling limit of the Q-state Potts model.