Topological Triviality of Flat Hamiltonians

TL;DR

This paper proves that in two-dimensional systems, flat energy bands in certain Hamiltonians are necessarily topologically trivial, with projectors onto these bands being strictly local, regardless of translational symmetry.

Contribution

It establishes a general topological triviality result for flat bands in 2D Hamiltonians without assuming lattice translational invariance.

Findings

Flat bands in 2D Hamiltonians are topologically trivial.

Projectors onto flat bands are strictly local.

Results do not depend on translational symmetry.

Abstract

Landau levels play a key role in theoretical models of the quantum Hall effect. Each Landau level is degenerate, flat and topologically non-trivial. Motivated by Landau levels, we study tight-binding Hamiltonians whose energy levels are all flat. We demonstrate that in two dimensions, for such Hamiltonians, the flat bands must be topologically trivial. To that end, we show that the projector onto each flat band is necessarily strictly local. Our conclusions do not need the assumption of lattice translational invariance.