Topological invariants for the fractional quantum Hall states

TL;DR

This paper introduces a topological invariant for fractional quantum Hall states that generalizes the Chern number, aiding in distinguishing different states, especially when traditional invariants are insufficient.

Contribution

It defines a new topological invariant for fractional quantum Hall states, applicable to both Abelian and non-Abelian cases, extending the concept of the Chern number.

Findings

Invariant equals the trace of the K-matrix for Abelian states

Invariant can be computed from conformal field theory for non-Abelian states

Invariant helps distinguish between different fractional quantum Hall states

Abstract

We calculate a topological invariant, whose value would coincide with the Chern number in case of integer quantum Hall effect, for fractional quantum Hall states. In case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.