Impurity induced current in a Chern insulator

TL;DR

This paper demonstrates that impurities in Chern insulators induce bound states with circulating currents, revealing a topological response that persists even for infinitesimal impurities and differs from trivial insulators.

Contribution

It provides an analytic and numerical analysis of impurity-induced currents in Chern insulators, highlighting their topological nature and contrasting behavior with trivial insulators.

Findings

Impurities induce bound states with circulating currents in Chern insulators.

The impurity-induced current decays polynomially with distance in topological phases.

In trivial insulators, bound states occur without circulating currents.

Abstract

Chern insulators arguably provide the simplest examples of topological phases. They are characterized by a topological invariant and can be identified by the presence of protected edge states. In this article, we show that a local impurity in a Chern insulator induces a twofold response: bound states that carry a chiral current and a net current circulating around the impurity. This is a manifestation of broken time symmetry and persists even for an infinitesimal impurity potential. To illustrate this, we consider a Coulomb impurity in the Haldane model. Working in the low-energy long-wavelength limit, we show that an infinitesimal impurity strength suffices to create bound states. We find analytic wavefunctions for the bound states and show that they carry a circulating current. In contrast, in the case of a trivial analogue, graphene with a gap induced by a sublattice potential, bound states occur but carry no current. In the many body problem of the Haldane model at half-filling, we use a linear response approach to demonstrate a circulating current around the impurity. Impurity textures in insulators are generally expected to decay exponentially; in contrast, this current decays polynomially with distance from the impurity. Going beyond the Haldane model, we consider the case of coexisting trivial and non-trivial masses. We find that the impurity induces a local chiral current as long as time reversal symmetry is broken. However, the decay of this local current bears a signature of the overall topology - the current decays polynomially in a non-trivial system and exponentially in a trivial system. In all cases, our analytic results agree well with numerical tight-binding simulations.