Universal intrinsic high-rank spin Hall effects

Spin Hall effect (SHE) is one of the key concepts in modern condensed-matter physics since its first discovery more than two decades ago. In this work, we introduce the concept dubbed high-rank spin Hall effect, in which the usual (charge) Hall effects and SHEs are incorporated as rank-0 and rank-1 SHEs. The first non-trivial example is then rank-2 SHE and we showcase a minimal intrinsic model in a spin-1 Fermionic system (a three-component Fermion or triply-degenerate point), which exhibits an universal rank-2 spin Hall conductivity e/4π. As a generalization to larger spin, we further provide another model in a spin-3/2 system. A simple experimental setup is proposed based on recently experimentally realized 2-dimensional spin-orbit coupling in cold atoms and the effects of Zeeman fields are investigated. Our work reveals interesting spin transport phenomena in large-spin systems and may lead to novel applications in spintronic and quantum-mechanical devices.