Moiré systems offer an exciting playground to study many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings, exhibiting phenomena such as itinerant ferromagnetism and kinetic frustration [1]. Recently, generalized Wigner crystals (GWC) on triangular moiré superlattices, formed from stacking two layers of transition metal chalcogenides, have been observed at multiple fractional fillings of the moiré unit cell [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 an accurate microscopic description of these materials, we explore the theoretical origins of GWC at n=1/3 and 2/3 filling. We demonstrate the limitations of theoretical descriptions of these moiré GWCs relying on truncated (finite-range) electron-electron interactions instead of a long-range Coulomb interaction. We validate our findings by studying both classical and quantum mechanical effects at zero and finite temperatures. More generally, we discuss the role of charge frustration in the theoretical extended Hubbard-model phase diagram, identifying a "pinball" phase, a partially quantum melted generalized Wigner crystal with coexisting solid and liquid-like features, with no classical analog.  Our work addresses several experimental observations and makes concrete predictions for how many of the theoretical findings can be potentially realized in future experiments [2].

[1] K. Lee, P. Sharma, O. Vafek, H. J. Changlani, Phys. Rev. B 107, 235105 (2023)

[2] A. Kumar, C. Lewandowski, H. J. Changlani, arXiv: 2409.13814 (2024), under review