Generalized Wigner crystallization in moiré materials

Recent experiments on the twisted transition metal dichalcogenide (TMD) material WSe2/WS2 have observed insulating states at fractional occupancy of the moiré bands. Such states were conceived as generalized Wigner crystals (GWCs). In this paper, we investigate the problem of Wigner crystallization in the presence of an underlying (moiré) lattice. Based on the best estimates of the system parameters, we find a variety of homobilayer and heterobilayer TMDs to be excellent candidates for realizing GWCs. In particular, our analysis based on rs indicates that MoSe2 (among the homobilayers) and MoSe2/WSe2 or MoS2/WS2 (among the heterobilayers) are the best candidates for realizing GWCs. We also establish that due to larger effective mass of the valence bands, in general, hole crystals are easier to realize that electron crystals as seen experimentally. For completeness, we show that satisfying the Mott criterion nMott1/2a∗=1 requires densities nearly three orders of magnitude larger than the maximal density for GWC formation. This indicates that for the typical density of operation, bilayer moiré systems are far from the Mott insulating regime. These crystals realized on a moiré lattice, unlike the conventional Wigner crystals, are incompressible due to the gap arising from pinning with the lattice. Finally, we capture this many-body gap by variationally renormalizing the dispersion of the vibration modes. We show these low-energy modes, arising from the coupling of the WC with the moiré lattice, can be effectively modeled as a Sine-Gordon theory of fluctuations.