Equivalence of semiclassical and response theories for second-order nonlinear ac Hall effects

It has been known that the semiclassical theory and the response theory can equivalently give the Drude and the intrinsic anomalous Hall conductivities in the linear order of electric field. However, recent theoretical advances implied that the second-order nonlinear conductivities calculated with both approaches are no longer equivalent, which leads to various experimental explanations even in a similar experimental setup conducted in \href{https://www.science.org/doi/10.1126/science.adf1506}{[\textit{Science \textbf{381}, 181 (2023)}]} and \href{https://www.nature.com/articles/s41586-023-06363-3}{[\textit{Nature \textbf{621}, 487 (2023)}]}, respectively. Herein, by extending the AC semiclassical theory up to the second order of electric field, we show that the semiclassical theory is still equivalent to the response theory in the second order of electric field when the relaxation is taken into account on the same footing. In particular, we show that the familiar second-order nonlinear current responses, including the nonlinear Drude current and the Berry curvature (quantum metric) dipole driven extrinsic (intrinsic) nonlinear Hall current, can be derived by both approaches. Further, we show that the quantum-corrected intrinsic nonlinear longitudinal current, as recently proposed by the response theory or in a similar manner, can also be reproduced by the semiclassical theory. Beyond those known second-order current responses, with both approaches, we uncover two previously overlooked nonlinear displacement currents unique to the AC electric field. As a consequence of this equivalence,...