Origin and stability of generalized Wigner crystallinity in triangular moir\'e systems

Generalized Wigner crystals (GWC) on triangular moir\'e superlattices, formed from stacking two layers of transition metal chalcogenides, have been observed at multiple fractional fillings [Nature 587, 214 - 218 (2020), Nat. Phys. 17, 715 - 719 (2021), Nature 597, 650 - 654 (2021)]. Motivated by these experiments, tied with the need for accurate microscopic descriptions of these materials, we explore the origins of GWC at $n=1/3$ and $2/3$ filling. We demonstrate the general limitations of theoretical descriptions relying on finite-range, versus long-range interactions, however, we clarify why some properties are captured by an effective nearest-neighbor model. We study both classical and quantum effects at zero and finite temperatures, discussing the role of charge frustration, identifying a ``pinball" phase, a partially quantum melted GWC, with no classical analog. Our work addresses several experimental observations and makes predictions for how many of the theoretical findings can be potentially realized in future experiments.