  The intrinsic spin Hall effect (ISHE) proposed in
[\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.126603}
{Sinova \textit{et al.} Phys. Rev. Lett. \textbf{92}, 126603 (2004)}] is driven
by the spin Berry curvature. Herein, we establish the concept of \textit{spin
quantum metric}, which is the counterpart of the spin Berry curvature in the
\textit{spin quantum geometric tensor} defined in a similar way to the quantum
geometric tensor. Dual to the $\mathcal{T}$-even ($\mathcal{T}$, time reversal)
spin Berry curvature, the spin quantum metric features a $\mathcal{T}$-odd
characteristic. Notably, we show that the $\mathcal{T}$-odd spin quantum metric
can also drive an ISHE ($\mathcal{T}$-odd ISHE) under a high-frequency electric
field. Guided by symmetry, we evaluate this $\mathcal{T}$-odd ISHE in the
magnetically tilted surface Dirac cone and in ferromagnetic monolayer
MnBi$_2$Te$_4$. We find that this $\mathcal{T}$-odd ISHE dominates when the
Fermi level is close to the band (anti)crossing point, where its magnitude can
be as large as the $\mathcal{T}$-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.
