Delocalization of boundary states in disordered topological insulators

TL;DR

This paper proves that disordered topological insulators always have at least one delocalized boundary state at zero energy, with some having a whole band of such states, in any number of dimensions, without using sigma models.

Contribution

It provides a general proof that disordered topological insulators possess delocalized boundary states, extending previous conjectures and numerical results to arbitrary dimensions without sigma model reliance.

Findings

At least one delocalized boundary state at zero energy in disordered topological insulators.

Insulators without chiral symmetry have a whole band of delocalized boundary states.

The proof is general and applies in any number of dimensions.

Abstract

We use the method of bulk-boundary correspondence of topological invariants to show that disordered topological insulators have at least one delocalized state at their boundary at zero energy. Those insulators which do not have chiral (sublattice) symmetry have in addition the whole band of delocalized states at their boundary, with the zero energy state lying in the middle of the band. This result was previously conjectured based on the anticipated properties of the supersymmetric (or replicated) sigma models with WZW-type terms, as well as verified in some cases using numerical simulations and a variety of other arguments. Here we derive this result generally, in arbitrary number of dimensions, and without relying on the description in the language of sigma models.