Figure 1

Phase diagram in the space of $t^{\prime }/t$ and $U/t$ in a $40\times 40$ lattice with $U_{-}=0$. The weakly interacting phase (small $U$) has $Z\ne 0$ and the first layer forms the conventional QSH phase while the second layer is in the semimetal (SM) phase with gapless Dirac nodes. The intermediate region has the fractionalized quantum spin Hall (FQSH) phase with $Z=0$ where chargeless spinons form the QSH phase in the first layer and the gapless spin liquid (SL) phase in the second layer. In both QSH/SM and FQSH/SL phases, the NN and NNN hopping order parameters are nonzero and site independent. The large $U$ region is a dimerized phase where $Z=0$ and the hopping order parameters along the bold lines have the maximum amplitude and all other bonds have zero amplitude. The solid line represents the second-order transition and the dotted line the first-order transition.Reuse & Permissions

Figure 2

(Color online) (a) Transverse spin response to an applied external magnetic field in the conventional quantum spin Hall phase. Upon threading a magnetic-flux quantum, a spin up propagates from one edge 1, say, to edge 2 and a spin down propagates from edge 2 to edge 1. (b) The response in the fractionalized quantum spin Hall phase. The external flux does not generate any transverse spin transport because the edge modes are charge neutral spinons.Reuse & Permissions