On the equivalence of the semiclassical theory and the response theory

It is long believed that the response theory can give quantum correction (or interband coherent effects) to the semiclassical theory, while both formulations essentially are perturbatively solving the time-dependent Sch\"odinger equation in a periodic potential probed by the electric field within the independent-particle approximation. In this work, by extending the semiclassical theory under a uniform electric field to the nonlinear AC regime, we show explicitly that up to the second order of the electric field, the AC semiclassical theory gives exactly the same results as the response theory in the absence of relaxation. Remarkably, this equivalence can be expected even in DC limit when the zero-frequency divergence is regulated on the same footing. Particularly, at the linear order of the electric field, we show that both formulations deliver the same Drude current, intrinsic anomalous Hall current, and the displacement current, where the last one arising from the time variation of the first-order positional shift can only appear in AC transport and is usually overlooked. Furthermore, at the second order of the electric field, we show that all the well-known transport effects, including the nonlinear Drude and Hall effects (that can be driven by the Berry curvature dipole or/and quantum metric dipole) can be derived by both formulations. Additionally, we show that the nonlinear displacement current due to the time variation of the second-order positional shift and the nonlinear intrinsic Ohmic current due to the second-order energy correction (which is determined by the first-order positional shift) in AC semiclassical theory can also be obtained by response theory.