Chern insulators in two and three dimensions: A global perspective

TL;DR

This paper develops a comprehensive theoretical framework for Chern insulators in 2D and 3D, introducing new expressions for topological invariants and exploring their electromagnetic responses across frequencies.

Contribution

It presents a second-quantized field theory with a periodic vector potential, deriving globally defined topological invariants and analyzing optical responses of Chern insulators.

Findings

Derived novel expressions for Chern invariants valid across the Brillouin zone

Analyzed the electromagnetic response of Chern insulators at finite frequencies

Extended the quantum anomalous Hall effect to the optical regime

Abstract

We introduce a second-quantized field theory for Chern insulators in which the Hamiltonian features a static vector potential that has the periodicity of the crystal's lattice and spontaneously breaks time-reversal symmetry in the system's ground state. Such a vector potential generates a magnetic field at the microscopic level that may be thought of as arising from local moments associated with one or more magnetic ions in each unit cell. Considering spinor electrons, we study the Chern invariants characterizing the topology of the occupied valence bands of Chern insulators in both two and three dimensions - the Chern number and the Chern vector, respectively - and we derive novel expressions for these topological invariants that are globally defined across the Brillouin zone and involve the full band structure of the system. We also study the long-wavelength response of a Chern insulator to electromagnetic fields at finite frequency, generalizing the quantum anomalous Hall effect in the static limit to the optical regime.