Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals

We study nonlinear magneto-optical responses of metals by a semiclassical\nBoltzmann equation approach. We derive general formulas for linear and second\norder nonlinear optical effects in the presence of magnetic fields that include\nboth Berry curvature and orbital magnetic moment. Applied to Weyl fermions, the\nsemiclassical approach (i) captures the directional anisotropy of linear\nconductivity under magnetic field as a consequence of an anisotropic $B^2$\ncontribution, which may explain the low-field regime of recent experiments;\n(ii) predicts strong second harmonic generation proportional to $B$ that is\nenhanced as the Fermi energy approaches the Weyl point, leading to large\nnonlinear Kerr rotation. Moreover, we show that the semiclassical formula for\nthe circular photogalvanic effect arising from the Berry curvature dipole is\nreproduced by a full quantum calculation using a Floquet approach.\n