Intrinsic spin Hall effect from spin quantum metric (2406.02257v2)

Abstract: The intrinsic spin Hall effect (ISHE), as proposed in [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.126603} {Phys. Rev. Lett. \textbf{92}, 126603 (2004)}], stems from the spin Berry curvature. Herein, we propose the concept of \textit{spin quantum metric}, which is established as the quantum geometric counterpart of the spin Berry curvature within the \textit{spin quantum geometric tensor}, defined in a manner analogous to the conventional quantum geometric tensor. In contrast to the well-known $\mathcal{T}$-even ($\mathcal{T}$, time reversal) spin Berry curvature, the \textit{spin quantum metric} is a $\mathcal{T}$-odd tensor. Remarkably, by symmetry analysis we show that the $\mathcal{T}$-odd \textit{spin quantum metric} can also drive an ISHE particularly under a high-frequency electric field. We investigate this $\mathcal{T}$-odd ISHE in the magnetically tilted surface Dirac cone and ferromagnetic monolayer MnBi$_2$Te$_4$. We find that this $\mathcal{T}$-odd ISHE dominates when the Fermi level is close to the band crossing or anticrossing point and can be as large as the $\mathcal{T}$-even ISHE when a THz or an infrared driving field is applied. Our work not only reveals an indispensable member in emergent quantum geometry physics but also offers a novel response function for ultrafast spintronics.