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

Quantum particles on a lattice with competing long-range interactions are ubiquitous in physics. Transition metal oxides, layered molecular crystals and trapped ion arrays are a few examples out of many. In the strongly interacting regime, these systems often exhibit a rich variety of quantum many-body ground states that challenge theory. The emergence of transition metal dichalcogenide moire heterostructures provides a highly controllable platform to study long-range electronic correlations. Here we report an observation of nearly two-dozen correlated insulating states at fractional fillings of a WSe$_{2}$/WS$_{2}$ moire heterostructure. The discovery is enabled by a new optical sensing technique that is built on the sensitivity to dielectric environment of the exciton excited states in single-layer semiconductor WSe$_{2}$. The cascade of insulating states exhibits an energy ordering which is nearly symmetric about filling factor of half electron (or hole) per superlattice site. We propose a series of charge-ordered states at commensurate filling fractions that range from generalized Wigner crystals to charge density waves. Our study lays the groundwork for utilizing moire superlattices to simulate a wealth of quantum many-body problems that are described by the two-dimensional t-V model or spin models with long-range charge-charge and exchange interactions.