Nonlinear response in a noncentrosymmetric topological insulator

Nonlinear phenomena are inherent in most systems in nature. Second- or higher-order harmonic generations, three-wave mixing, and four-wave mixing are typical phenomena in nonlinear optics. To obtain a nonzero signal for second-harmonic generation in the long-wavelength limit ($q\ensuremath{\rightarrow}0$), the breaking of inversion symmetry is required. In topological materials, a hexagonal warping term that breaks the rotation symmetry of the Fermi surface is observed by angular-resolved photoemission spectroscopy. If a gap opens (e.g., by doping with magnetic impurities), the inversion symmetry will be broken. Here we use a nonlinear response theory based on a generalized Kubo formula to explain the frequency up-conversion in topological materials.