Recent experimental progress has established the twisted bilayer transition metal dichalcogenide (TMD) as a highly tunable platform for studying many-body physics. Particularly, the homobilayer TMDs under displacement field are believed to be described by a generalized triangular-lattice Hubbard model with a spin-dependent hopping phase $\theta$. To explore the effects of $\theta$ on the system, we perform density matrix renormalization group calculations for the relevant triangular lattice t-J model. By changing $\theta$ at small hole doping, we obtain a region of quasi-long-range superconducting order coexisting with charge and spin density wave within $0<\theta <\pi /3$. The superconductivity is composed of a dominant spin singlet $d$-wave and a subdominant triplet $p$-wave pairing. Intriguingly, the $S_z=\pm 1$ triplet pairing components feature pair-density waves. In addition, we find a region of triplet superconductivity coexisting with charge-density wave and ferromagnetism within $\pi /3<\theta <2\pi /3$, which is related to the former phase at smaller $\theta$ by a combined operation of spin-flip and gauge transformation. Our findings provide insights and directions for experimental search for exotic superconductivity in twisted TMD systems.

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• Superconducting order parameter
• Transition metal dichalcogenides
• Twisted heterostructures
• Density matrix renormalization group