  Nonlinear phenomena are inherent in most systems in nature. Second or
higher-order harmonic generations, three-wave and four-wave mixing are typical
phenomena in nonlinear optics. To obtain a nonzero signal for second-harmonic
generation in the long-wavelength limit ($q\rightarrow0$), the breaking of
inversion symmetry is required. In topological materials, a hexagonal warping
term which breaks the rotation symmetry of the Fermi surface is observed by
angular-resolved photo-emission spectroscopy (ARPES). 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.
