From Fractional Chern Insulators to a Fractional Quantum Spin Hall Effect

TL;DR

This paper explores the algebraic structure of flat energy bands with fractional fillings, revealing conditions under which fractional quantum anomalous Hall and spin Hall effects emerge, linked to local non-commutative geometry.

Contribution

It introduces a framework connecting flat energy bands and topological effects to local non-commutative geometry, extending understanding of fractional quantum Hall phenomena.

Findings

Fractional quantum anomalous Hall effect can arise in flat bands with homogeneous Berry curvature.

Fractional quantum spin Hall effect is linked to similar algebraic structures in flat energy bands.

Global Chern number relates to local non-commutative geometry in topological insulators.

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.