Vortex Spin Liquid with Fractional Quantum Spin Hall Effect in Moiré Chern Bands

Recently there is a report of the experimental signatures of a fractional quantum spin hall (FQSH) state at hole filling n=3 in a twisted MoTe_{2} bilayer. Previous theories of FQSH phases simply considered a decoupled pair of a fractional quantum Hall phase and its time reversal partner. Here, we show the first construction of an FQSH phase beyond the decoupling picture. We consider a pair of half-filled C=±1 Chern bands in the two valleys, similar to the well-studied quantum Hall bilayer, but now with opposite chiralities. Because of the strong intervalley repulsion, we expect a charge gap to open with low-energy physics dominated by the neutral intervalley excitons. However, the presence of an effective "flux" frustrates exciton condensation by proliferating vortices. Here, we construct a vortex liquid of excitons dubbed a vortex spin liquid, formed from exciton pairing of the composite fermions in the decoupled composite Fermi liquid 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 intervalley-coherent order. Unlike a conventional Mott insulator, the vortex spin liquid phase also exhibits a quantized FQSH effect with gapless helical charge modes along the edge. Our work demonstrates the possibility of nontrivial FQSH phases and provides predictions to detect them in future experiments.