Domain walls, fusion rules and conformal field theory in the quantum Hall regime

TL;DR

This paper introduces a straightforward method to derive fusion rules for quasi-holes in quantum Hall states using domain walls, aiding the identification of the associated conformal field theories, including well-known and non-unitary cases.

Contribution

It provides a novel, simplified approach to determine fusion rules and explicitly constructs conformal field theories for a broad family of quantum Hall wave functions.

Findings

Fusion rules derived for various quantum Hall states.

Explicit CFT descriptions provided for the (k,r) family.

Identification of non-unitary CFTs for certain wave functions.

Abstract

We provide a simple way to obtain the fusion rules associated with elementary quasi-holes over quantum Hall wave functions, in terms of domain walls. The knowledge of the fusion rules is helpful in the identification of the underlying conformal field theory describing the wave functions. We obtain the fusion rules, and explicitly give a conformal field theory description, for a two-parameter family (k,r) of wave functions. These include the Laughlin, Moore-Read and Read-Rezayi states when r=2. The `gaffnian' wave function is the prototypical example for r>2, in which case the conformal field theory is non-unitary.