Parafermionic conformal field theory on the lattice

TL;DR

This paper establishes a near-complete correspondence between lattice operators and continuum parafermion conformal fields in the three-state Potts model, advancing understanding of strongly interacting critical points.

Contribution

It provides a systematic method to identify lattice analogues of continuum fields in non-free conformal field theories, demonstrated on the three-state Potts model.

Findings

Constructed lattice versions of nearly all relevant and marginal fields.

Clarified operator product expansion structures between order and disorder fields.

Confirmed theoretical predictions through extensive numerical simulations.

Abstract

Finding the precise correspondence between lattice operators and the continuum fields that describe their long-distance properties is a largely open problem for strongly interacting critical points. Here we solve this problem essentially completely in the case of the three-state Potts model, which exhibits a phase transition described by a strongly interacting 'parafermion' conformal field theory. Using symmetry arguments, insights from integrability, and extensive simulations, we construct lattice analogues of nearly all the relevant and marginal physical fields governing this transition. This construction includes chiral fields such as the parafermion. Along the way we also clarify the structure of operator product expansions between order and disorder fields, which we confirm numerically. Our results both suggest a systematic methodology for attacking non-free field theories on the lattice and find broader applications in the pursuit of exotic topologically ordered phases of matter.