  Moir\'e systems provide a rich platform for studies of strong correlation
physics. Recent experiments on hetero-bilayer transition metal dichalcogenide
(TMD) Moir\'e systems are exciting in that they manifest a relatively simple
model system of an extended Hubbard model on a triangular lattice. Inspired by
the prospect of the hetero-TMD Moir\'e system's potential as a
solid-state-based quantum simulator, we explore the extended Hubbard model on
the triangular lattice using the density matrix renormalization group (DMRG).
Specifically, we explore the two-dimensional phase space of the kinetic energy
relative to the interaction strength $t/U$ and the further-range interaction
strength $V_1/U$. We find competition between Fermi fluid, chiral spin liquid,
spin density wave, and charge density wave. In particular, our finding of the
optimal further-range interaction for the chiral correlation presents a
tantalizing possibility.
