Moiré flatbands, occurring e.g. in twisted bilayer graphene at magic angles, have attracted ample interest due to their high degree of experimental tunability and the intriguing possibility of generating novel strongly interacting phases. Here we consider the core problem of Coulomb interactions within fractionally filled spin and valley polarized Moiré flatbands 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. In experimentally relevant examples such as ABC stacked trilayer and twisted bilayer graphene aligned with boron nitride, it leads to emergent interaction-driven Fermi liquid states at electronic filling fractions down to around 1/3 and 2/3 respectively. At even lower filling fractions, the electron density still faithfully tracks the single-hole dispersion while exhibiting distinct non-Fermi liquid behavior. Most saliently, we provide microscopic evidence that high temperature fractional Chern insulators can form in twisted bilayer graphene aligned with hexagonal boron nitride.Setup. Motived by the recent discoveries [11][12][13]15] along with theoretical predictions [16,21] of spin and valley polarized insulators at integer fillings in various Moiré bands, we consider electrons with polarized spin arXiv:1912.04907v1 [cond-mat.mes-hall]

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Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of fractional Chern insulator states. In fact, we show that an ``ideal flatband'' cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.