Linear response of a Chern insulator to finite-frequency electric fields

TL;DR

This paper derives a microscopic, analytical expression for the charge and current response of a Chern insulator to finite-frequency electric fields, revealing a topological term linked to the quantum anomalous Hall effect.

Contribution

It introduces a general response formula for bulk Chern insulators under dynamic electric fields, extending previous static and finite-size theories with a microscopic approach.

Findings

Identifies a topological term involving the Chern number in the response

Reduces to known Kubo response in trivial insulators

Provides a framework for inhomogeneous and dynamic electromagnetic responses

Abstract

We derive the macroscopic charge and current densities of a Chern insulator initially occupying its electronic ground state as it responds to a finite-frequency electric field; we use a previously developed formalism based on microscopic polarization and magnetization fields in extended media. In a topologically trivial insulator, our result reduces to the familiar expression for the induced current density in linear response obtained from a Kubo analysis. But for a Chern insulator we find an extra "topological" term involving the (first) Chern number associated with the occupied bands, encoding the quantum anomalous Hall effect in the presence of a frequency-dependent electric field. While an analogous term has been introduced in the "modern theories of polarization and magnetization" for the linear response of finite-sized systems to static electric fields, our expression is valid for bulk Chern insulators in the presence of both static and finite-frequency electric fields, being derived analytically from a microscopic treatment of the electronic degrees of freedom, and can be generalized in a straightforward way to describe the response of a Chern insulator to electromagnetic fields that are not only frequency-dependent but also spatially inhomogeneous.