Basic formulation and first-principles implementation of nonlinear magneto-optical effects

First-principles calculation of nonlinear magneto-optical effects has become an indispensable tool to reveal the geometric and topological nature of electronic states and to understand light-matter interactions. While intriguingly rich physics could emerge in magnetic materials, further methodological developments are required to deal with time-reversal symmetry breaking, due to the degeneracy and gauge problems caused by symmetry and the low-frequency divergence problem in the existing calculation formalism. Here we present a gauge-covariant and low-frequency convergent formalism for the first-principles computation. Remarkably, this formalism generally works for both nonmagnetic and magnetic materials with or without band degeneracy. Reliability and capability of our method are demonstrated by studying example materials (i.e., bilayers of ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ and ${\mathrm{CrI}}_{3}$) and comparing with published results. Moreover, an importance correction term that ensures gauge covariance of degenerate states is derived, whose influence on physical responses is systematically checked. Our method enables computation of nonlinear magneto-optical effects in magnetic materials and paves the way for exploring rich physics created by the interplay of light and magnetism.