Moiré Superlattices at Fractional Band Fillings: Emergent Fermi Liquids, Charge Density Waves and Fractional Chern Insulators*

We consider the core problem of Coulomb interactions within fractionally filled Moiré flat bands and demonstrate that the dual description in terms of holes, which acquire a non-trivial hole-dispersion, provides key physical intuition and enables the use of standard perturbative techniques for this strongly correlated problem. We find that the single-hole dispersion has a profound impact on the phase diagram: in experimentally relevant examples such as ABC stacked trilayer and twisted bilayer graphene aligned with boron nitride, it leads to emergent Fermi liquid states at band filling fractions down to 1/3 and 2/3 respectively. Moreover, we investigate the possible instabilites to the emergent Fermi surfaces and find competing charge density wave states at different commensurate fillings in the Moiré system ABC stacked trilayer graphene aligned with boron nitride. In addition, we predict both twisted bilayer graphene aligned with boron nitride and gate-tunable twisted double bilayer graphene to be versatile platforms for the realization of fractional Chern insulator states like spin-singlet Halperin states and spin polarized states in bands with Chern number C =1 and C = 2 at zero external magnetic fields.