Multiple quantum spin Hall states and topological current divider in Twisted Bilayer WSe$_2$

It has been demonstrated that topological quantum spin Hall (QSH) state exist in twisted bilayers of transition metal dichalcogenides. However, a comprehensive theoretical characterization of the topological edge states remains a topic of interest and an unresolved issue. Here, the topological transport properties of the twisted WSe$_2$ bilayers are investigated. Beyond the conventional single QSH, we identify emergent double and quartuple quantum spin Hall states, hosting two and four pairs of counter-propagating helical edge channels respectively. Furthermore, the charge carriers in these edge states are not localized at edge but rather the high potential point of the moire superlattice boundary, undergoing interlayer transitions and propagating forward continuously. We term these edge states as moire edge states. These edge states can survive in non-magnetic disorder, with the robustness of double QSH states surpassing that of single QSH states. At a twisting angle of 2.45$^\circ$, the transition between the single and double QSH states can be achieved by adjusting the gate on the surface. Based on this, we propose a five-terminal device to as a topological current devider. Our findings provide support for the development of dissipationless spintronics.