We theoretically study the intrinsic spin Hall effect in parity-time ($\mathcal{PT}$)-symmetric spin-orbit-coupled quantum gases confined in an optical lattice. The interplay of the $\mathcal{PT}$ symmetry and the spin-orbit coupling leads to a doubly degenerate noninteracting band structure in which the spin polarization and the Berry curvature of any Bloch state are opposite to those of its degenerate partner. Using experimentally available systems as examples, we show that such a system with a two-component Fermi gas exhibits an intrinsic spin Hall effect akin to that found in the context of electronic materials. For a two-component Bose gas, however, an unconventional spin Hall effect emerges in which the spin polarization and the currents are coplanar and the spin Hall conductivity displays a characteristic anisotropy. We propose to detect such an unconventional spin Hall effect in harmonically trapped systems using dipole oscillations and perform extensive numerical simulations to validate the proposal. Our work paves the way for quantum simulation of the solid-state intrinsic spin Hall effect and experimental explorations of unconventional spin Hall effects in quantum gases.

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• Quantum simulation
• Spin Hall effect
• Spin-orbit coupling