Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

Topological protection of perturbed edge states

Pages: 193 – 225



Guillaume Bal (Departments of Statistics and Mathematics and CCAM, University of Chicago, Illinois, U.S.A.)


This paper proposes a quantitative description of the low energy edge states at the interface between two-dimensional topological insulators. They are modeled by systems of massive Dirac equations, which are amenable to a large class of random perturbations. We consider general as well as fermionic time reversal symmetric models. In the former case, Hamiltonians are classified by means of the index of a Fredholm operator. In the latter case, the classification involves a $\mathbb{Z}_2$ index. These indices dictate the number of topologically protected edge states.

A remarkable feature of topological insulators is the asymmetry (chirality) of the edge states, with more modes propagating, say, up than down. In some cases, backscattering off imperfections is prevented when no mode can carry signals backwards. This is a desirable feature from an engineering perspective, which raises the question of how backscattering is protected topologically. A major motivation for the derivation of continuous models is to answer such a question.

We quantify how backscattering is affected but not suppressed by the non-trivial topology by introducing a scattering problem along the edge and describing the effects of topology and randomness on the scattering matrix. Explicit macroscopic models are then obtained within the diffusion approximation of field propagation to show the following: the combination of topology and randomness results in unhindered transport of randomness-dependent protected modes while all other modes (Anderson) localize.


topological insulators, edge states, Fredholm operators, index theory, Dirac equations, $\mathbb{Z}_2$ index, scattering theory, diffusion approximation, Anderson localization

2010 Mathematics Subject Classification

34L25, 35Q41, 47A53, 60J70

This work was partially supported by the National Science Foundation and the Office of Naval Research.

Received 3 April 2018

Accepted 3 November 2018

Published 30 May 2019