Continuous bulk and interface description of topological insulators

TL;DR

This paper develops a mathematical framework using Dirac equations and Fredholm operator indices to analyze the topological properties of insulators, accounting for spatial heterogeneities and extending to various dimensions.

Contribution

It introduces a spectral calculus-based method to describe bulk and interface topological invariants in topological insulators, including stability under heterogeneities and higher-dimensional generalizations.

Findings

Topological invariants are stable under spatial perturbations.

The models quantify the interplay between topology and spatial fluctuations.

Extension of the theory to arbitrary dimensions with chiral symmetry.

Abstract

We analyze continuous partial differential models of topological insulators in the form of systems of Dirac equations. We describe the bulk and interface topological properties of the materials by means of indices of Fredholm operators constructed from the Dirac operators by spectral calculus. We show the stability of these topological invariants with respect to perturbations by a large class of spatial heterogeneities. These models offer a quantitative tool to analyze the interplay between topology and spatial fluctuations in topological phases of matter. The theory is first presented for two-dimensional materials, which display asymmetric (chiral) transport along interfaces. It is then generalized to arbitrary dimensions with the additional assumption of chiral symmetry in odd spatial dimensions.