Approximate SU(4) spin models on triangular and honeycomb lattices in twisted AB-Stacked WSe$_2$ homo-bilayer

In this paper, we derive lattice models for the narrow moir\'e bands of the AB-stacked twisted WSe$_2$ homobilayer through continuum model and Wannier orbital construction. Previous work has shown that an approximate SU(4) Hubbard model may be realized by combining spin and layer because inter-layer tunneling is suppressed due to spin $S_z$ conservation. However, Rashba spin-orbit coupling (SOC) was ignored in the previous analysis. Here, we show that a Rashba SOC of reasonable magnitude can induce a finite but very small inter-layer hopping in the final lattice Hubbard model. At total filling $n=1$, we derive a spin-layer model on a triangular lattice in the large-U limit where the inter-layer tunneling contributes as a sublattice-dependent transverse Ising field for the layer pseudospin. We then show that the $n=2$ Mott insulator is also captured by an approximate SU(4) spin model, but now on honeycomb lattice. We comment on the possibility of a Dirac spin liquid (DSL) and competing phases due to SU(4) anisotropy terms.