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Recent experimental progress has established the twisted bilayer transitionmetal dichalcogenide (TMD) as a highly tunable platform for studying many-bodyphysics. Particularly, the homobilayer TMDs under displacement field arebelieved to be described by a generalized triangular-lattice Hubbard model witha spin-dependent hopping phase $\theta$. To explore the effects of $\theta$ onthe system, we perform density matrix renormalization group calculations forthe relevant triangular lattice t-J model. By changing $\theta$ at small holedoping, we obtain a region of quasi-long-range superconducting order coexistingwith charge and spin density wave within $0<\theta<\pi/3$. Thesuperconductivity is composed of a dominant spin singlet $d$-wave and asubdominant triplet $p$-wave pairing. Intriguingly, the $S_z=\pm 1$ tripletpairing components feature pair density waves. In addition, we find a region oftriplet superconductivity coexisting with charge density wave andferromagnetism within $\pi/3<\theta<2\pi/3$, which is related to the formerphase at smaller $\theta$ by a combined operation of spin-flip and gaugetransformation. Our findings provide insights and directions for experimentalsearch for exotic superconductivity in twisted TMD systems.

Singlet, Triplet and Pair Density Wave Superconductivity in the Doped Triangular-Lattice Moir\'e System