Vortex spin liquid with fractional quantum spin Hall effect in moir\'e Chern bands

Abstract

Integer and fractional quantum anomalous Hall (QAH) effects have been widely seen in moir\'e systems. Recently there is even observation of a time reversal invariant fractional quantum spin hall (FQSH) state at filling $n=3$ in twisted MoTe$_2$ bilayer. We consider a pair of half-filled $C=\pm 1$ Chern band in the two valleys, similar to the well-studied quantum Hall bilayer, but now with opposite chiralities. Due to the strong inter-valley repulsion, we expect a charge gap opening with low energy physics dominated by the neutral inter-valley excitons. However, the presence of an effective `flux' frustrates exciton condensation by proliferating vortices. Here we construct a vortex liquid of excitons dubbed as vortex spin liquid (VSL), from exciton pairing of the composite fermions in the decoupled composite Fermi liquids (CFL) phase. This insulator is a quantum spin liquid with gapless spin excitations carried by the flux of an emergent U(1) gauge field. Additionally, there exist neutral and spinless Fermi surfaces formed by fermionic vortices of a nearby inter-valley-coherent (IVC) order. Unlike a conventional Mott insulator, the VSL phase also exhibits FQSH effect with gapless helical charge modes along the edge. Our work suggests a new platform to search for quantum spin liquid enriched by fractional quantum spin Hall effect. We also point out the possibility of quantum oscillations and thermal Hall effect under Zeeman field in this exotic insulator.