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Quantum particles on a lattice with competing long-range interactions areubiquitous in physics. Transition metal oxides, layered molecular crystals andtrapped ion arrays are a few examples out of many. In the strongly interactingregime, these systems often exhibit a rich variety of quantum many-body groundstates that challenge theory. The emergence of transition metal dichalcogenidemoir\'e heterostructures provides a highly controllable platform to studylong-range electronic correlations. Here we report an observation of nearlytwo-dozen correlated insulating states at fractional fillings of aWSe$_{2}$/WS$_{2}$ moir\'e heterostructure. The discovery is enabled by a newoptical sensing technique that is built on the sensitivity to dielectricenvironment of the exciton excited states in single-layer semiconductorWSe$_{2}$. The cascade of insulating states exhibits an energy ordering whichis nearly symmetric about filling factor of half electron (or hole) persuperlattice site. We propose a series of charge-ordered states at commensuratefilling fractions that range from generalized Wigner crystals to charge densitywaves. Our study lays the groundwork for utilizing moir\'e superlattices tosimulate a wealth of quantum many-body problems that are described by thetwo-dimensional t-V model or spin models with long-range charge-charge andexchange interactions.

Abundance of correlated insulating states at fractional fillings of WSe$_{2}$/WS$_{2}$ moir\'e superlattices