Regularized lattice theory for spatially dispersive nonlinear optical conductivities

Nonlinear optical responses are becoming increasingly relevant for characterizing the symmetries and quantum geometry of electronic phases in materials. Here, we develop an expanded diagrammatic scheme for calculating spatially dispersive corrections to nonlinear optical conductivities, which we expect to enhance or even dominate even-order responses in materials of recent interest. Building upon previous work that enforces gauge invariance of spatially uniform nonlinear optical responses, we review the cancellation of diagrams required to ensure the equivalence between velocity gauge and length gauge formulations, and provide a simple vertex rule for extending optical responses to first order in the light wave vector q. We then demonstrate the method with calculations on a prototypical centrosymmetric model where spatial dispersion admits anomalous secondharmonic generation, a response that is symmetry-forbidden under the dipole approximation.