Topological protection of perturbed edge states

Abstract

This paper proposes a quantitative description of the low energy edge states at the interface between two-dimensional topological insulators. They are modeled by continuous Hamiltonians as systems of Dirac equations that 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 using 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 back-scattering 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 un-hindered transport of the protected modes while all other modes (Anderson) localize.