Spatially Inhomogeneous Electromagnetic Response in a Moiré-Chern Insulator*

Nonlinear electromagnetic response functions have emerged as a significant tool to study quantum materials due to their connections with optical response, quantum geometry, and band topology. Until recently, most attention has focused on responses to spatially uniform electric fields, which is a good approximation for optical response in solids since effects from spatial inhomogeneities are small compared to typical lattice spacings. However, in the emerging fields of moiré materials and superlattice heterostructures, where characteristic lattice scales are very long, the potential impact of spatial variation in optical electric fields on experiments becomes evident. To address these considerations, we develop a gauge-invariant formalism for computing responses to spatially inhomogeneous electromagnetic fields. Our formalism is universally applicable, irrespective of the microscopic Hamiltonian's specific form. We apply our formalism to compute wavevector-dependent conductivities, leveraging Ward identities to determine diamagnetic current operators in the process. To demonstrate the practical significance of spatially inhomogeneous fields, we utilize our formalism to calculate the Kerr effect's magnitude under oblique incidence for a moiré-Chern insulator model. In the realm of nonlinear response, our formalism facilitates the computation of second-order transverse responses to spatially varying transverse electric fields in a moiré-Chern insulator model, setting the stage for the next generation of experiments in these materials.