Nonlinear Hall effect driven by spin-charge-coupled motive force

Parity-time-reversal symmetric ($\mathcal{PT}$-symmetric) magnets have garnered much attention due to their spin-charge coupled dynamics enriched by the parity-symmetry breaking. By real-time simulations, we study how localized spin dynamics can affect the nonlinear Hall effect in $\mathcal{PT}$-symmetric magnets. To identify the leading-order term, we derive analytical expressions for the second-order optical response and classify the contributions by considering their transformation properties under $\mathcal{PT}$ symmetry. Notably, our results reveal that the sizable contribution is attributed to the mixed dipole effect, which is analogous to the Berry curvature dipole term.