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In this paper, we derive lattice models for the narrow moir\'e bands of theAB-stacked twisted WSe$_2$ homobilayer through continuum model and Wannierorbital construction. Previous work has shown that an approximate SU(4) Hubbardmodel may be realized by combining spin and layer because inter-layer tunnelingis suppressed due to spin $S_z$ conservation. However, Rashba spin-orbitcoupling (SOC) was ignored in the previous analysis. Here, we show that aRashba SOC of reasonable magnitude can induce a finite but very smallinter-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 wherethe inter-layer tunneling contributes as a sublattice-dependent transverseIsing field for the layer pseudospin. We then show that the $n=2$ Mottinsulator is also captured by an approximate SU(4) spin model, but now onhoneycomb lattice. We comment on the possibility of a Dirac spin liquid (DSL)and competing phases due to SU(4) anisotropy terms.

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