Topological invariants for fractional quantum Hall states

V. Gurarie, A. M. Essin

Department of Physics, University of Colorado, CO 80309 Boulder, USA

**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 the case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the
trace of the K-matrix. In the 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.