Topological quantum spin-tensor Hall insulators protected by pseudo-time-reversal symmetry

The celebrated family of the Hall effect plays a fundamental role in modern physics. Starting from the anomalous Hall effect (AHE) and the quantum AHE (QAHE) with broken time-reversal symmetry (TRS) and proceeding to their spinful generalizations, including the spin Hall effect (SHE) and the quantum SHE (QSHE) protected by TRS, they reveal rich transport and topological phenomena. However, in larger-spin $S$ ($S>1/2$) systems, besides charge current and spin current, there arise higher-rank spin-tensor currents. Recent work has uncovered an interesting spin-tensor Hall effect with spin-tensor currents in these larger-spin systems. Taking a step further, this work describes a class of topological states of matter dubbed quantum spin-tensor Hall (QSTH) insulators with broken TRS, and their nontrivial topology is protected by a unique pseudo-TRS. Most strikingly, QSTH insulators exhibit a quantized rank-2 spin-tensor Hall conductivity, whereas both charge (rank-0) and spin (rank-1) conductivities vanish. We also fully characterize their topological properties and highlight the physical interpretations via the underlying connections to QSHE. Our work enriches the family of the famous Hall effects and sheds light on the intriguing topological state of matter in larger-spin systems. It further offers avenues toward spin-tensor-tronics and low-power atomtronics.