Dissipationless Photovoltaic Spin Hall Effect from Spin-current Vorticity

Spin-current vorticity (SCV) can generate the linear magnetic spin Hall effect [\href{https://doi.org/10.1038/s41586-018-0853-0}{Nature \textbf{565}, 627 (2019)}], yet its role in nonlinear spin Hall transport has been much less explored. Here, we show that, under a dc electric field, SCV can deflect optically excited electrons to drive a dissipationless photovoltaic spin Hall effect (PSHE), in which optical excitation by circularly and linearly polarized light is governed by the Berry curvature and quantum metric, respectively. Because the Berry curvature is $\mathcal{T}$-odd whereas the quantum metric is $\mathcal{T}$-even, their respective combinations with the $\mathcal{T}$-odd SCV give rise to $\mathcal{T}$-even and $\mathcal{T}$-odd PSHEs, where $\mathcal{T}$ denotes time-reversal symmetry. Remarkably, we find that the spin current of the $\mathcal{T}$-even PSHE can be reversed by switching the light helicity, as illustrated in monolayer WTe$_2$. By contrast, the $\mathcal{T}$-odd PSHE in altermagnets changes sign upon N\'eel-vector reversal, as demonstrated in a $d$-wave altermagnetic model. Beyond the PSHE, we show that the SCV dipole governs both the Drude and intrinsic nonlinear spin Hall effects proposed recently. Our results reveal two switchable spin Hall mechanisms and establish SCV as a unifying concept for understanding dissipationless nonlinear spin Hall transport.