Continuous Topological Insulators Classification and Bulk Edge Correspondence

TL;DR

This paper reviews the mathematical classification of topological insulators and their interfaces, focusing on robust asymmetric transport phenomena and the relationship between interface currents and bulk invariants.

Contribution

It introduces a general classification framework for elliptic interface pseudo-differential operators and establishes a bulk-edge correspondence using spectral flow and index theory.

Findings

Explicit computation of interface current observable via spectral flow.

Classification of bulk phases for Landau and Dirac operators.

Bulk-edge correspondence established for elliptic operators.

Abstract

This paper reviews recent results on the classification of partial differential operators modeling bulk and interface topological insulators in Euclidean spaces. Our main objective is the mathematical analysis of the unusual, robust-to-perturbations, asymmetric transport that necessarily appears at interfaces separating topological insulators in different phases. The central element of the analysis is an interface-current-observable describing this asymmetry. We show that this observable may be computed explicitly by spectral flow when the interface Hamiltonian is explicitly diagonalizable. We review the classification of bulk phases for Landau and Dirac operators and provide a general classification of elliptic interface pseudo-differential operators by means of domain walls and a corresponding bulk-difference invariant (BDI). The BDI is simple to compute by the Fedosov-H\"ormander formula implementing in a Euclidean setting an Atiyah-Singer index theory. A generalized bulk-edge correspondence then states that the interface current observable and the BDI agree on elliptic operators, whereas this is not necessarily the case for non-elliptic operators.