Optical response of a Chern insulator

A defining feature of Chern insulators is that one cannot construct exponentially-localized Wannier functions from the occupied bands alone; however, by including sufficiently many unoccupied bands, such a construction is possible. Recently we have implemented a formalism based on microscopic polarization and magnetization fields, defined using these localized Wannier functions, to study the zero-temperature ground state of a Chern insulator. The moments of these microscopic fields are identified with the multipole moments of the charge-current distribution, and their spatial averages correspond to the macroscopic polarization and magnetization featured in the ``modern theories of polarization and magnetization." Here we implement this formalism to study optical response of a Chern insulator to general electromagnetic fields in the linear regime. In particular, we derive the response of the macroscopic electric dipole and quadrupole moments and the magnetic dipole moment to applied electric and magnetic fields. Unlike in the ``modern theories," which are typically focused on DC response to uniform electric fields, our approach is valid for spatially-varying and frequency-dependent fields, and can be extended to nonlinear response of arbitrary order.