Spectral flows associated to flux tubes

TL;DR

This paper links spectral flow caused by flux tubes in 2D topological insulators to topological invariants, providing proofs and criteria for zero modes and edge states using Fredholm operators.

Contribution

It introduces a mathematical framework connecting spectral flow, topological invariants, and physical phenomena in 2D topological insulators, including proofs of bulk-edge correspondence.

Findings

Spectral flow equals the index of a Fredholm operator.

Spectral flow exhibits a $Z_2$ signature in time-reversal symmetric systems.

Criteria for zero-energy modes attached to half-flux tubes are established.

Abstract

When a flux quantum is pushed through a gapped two-dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin's Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic $Z_2$ signature, while for particle-hole symmetric systems it leads to a criterion for the existence of zero energy modes attached to half-flux tubes. Combined with other results, this allows to explain all strong invariants of two-dimensional topological insulators in terms of a single Fredholm operator.