Intrinsic spin Hall effect from spin quantum metric

The intrinsic spin Hall effect (ISHE) proposed in [{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.126603} {Sinova et al. Phys. Rev. Lett. 92, 126603 (2004)}] is driven by the spin Berry curvature. Herein, we establish the concept of spin quantum metric, which is the counterpart of the spin Berry curvature in the spin quantum geometric tensor defined in a similar way to the quantum geometric tensor. Dual to the T-even (T, time reversal) spin Berry curvature, the spin quantum metric features a T-odd characteristic. Notably, we show that the T-odd spin quantum metric can also drive an ISHE (T-odd ISHE) under a high-frequency electric field. Guided by symmetry, we evaluate this T-odd ISHE in the magnetically tilted surface Dirac cone and in ferromagnetic monolayer MnBi₂Te₄. We find that this T-odd ISHE dominates when the Fermi level is close to the band (anti)crossing point, where its magnitude can be as large as the T-even ISHE when an infrared driving field is applied. Our work not only uncovers an indispensable ingredient in the emergent community of quantum geometric physics but also offers a novel mechanism for ultrafast spintronics.