Bulk-boundary correspondence of topological insulators from their Green's functions

TL;DR

This paper demonstrates that the bulk-boundary correspondence in topological insulators can be understood through Green's functions, establishing a direct link between bulk topological invariants and edge states across all classes and dimensions.

Contribution

It generalizes the bulk-boundary correspondence to all topological insulators using Green's functions, applicable to interacting systems with some modifications.

Findings

Edge states follow from Green's function topological invariants.

Edge states have their own topological invariants equal to the bulk.

Applicable to all ten classes of topological insulators.

Abstract

Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different ways, including in terms of Green's functions. Here we show that the existence of the edge states directly follows from the existence of the topological invariant written in terms of the Green's functions, for all ten classes of topological insulators in all spatial dimensions. We also show that the resulting edge states are characterized by their own topological invariant, whose value is equal to the topological invariant of the bulk insulator. This can be used to test whether a given model Hamiltonian can describe an edge of a topological insulator. Finally, we observe that the results discussed here apply equally well to interacting topological insulators, with certain modifications.