Eur. Phys. J. B (2012) 85: 15
https://doi.org/10.1140/epjb/e2011-20857-6

Regular Article

From fractional Chern insulators to a fractional quantum spin hall effect

Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR 8502, 91405 Orsay, France

^a e-mail: goerbig@lps.u-psud.fr

Received: 13 July 2011
Received in final form: 20 October 2011
Published online: 11 January 2012

Abstract

We investigate the algebraic structure of flat energy bands a partial filling of which may give rise to a fractional quantum anomalous Hall effect (or a fractional Chern insulator) and a fractional quantum spin Hall effect. Both effects arise in the case of a sufficiently flat energy band as well as a roughly flat and homogeneous Berry curvature, such that the global Chern number, which is a topological invariant, may be associated with a local non-commutative geometry. This geometry is similar to the more familiar situation of the fractional quantum Hall effect in two-dimensional electron systems in a strong magnetic field.