Quantum Metric Induced Phases in Moir\'e Materials

We show that, quite generally, quantum geometry plays a major role in determining the low-energy physics in strongly correlated lattice models at fractional band fillings. We identify limits in which the Fubini Study metric dictates the ground states and show that this is highly relevant for Moir\'e materials leading to symmetry breaking and interaction driven Fermi liquids. This phenomenology stems from a novel interplay between the quantum geometry and interactions which is absent in continuum Landau levels but generically present in lattice models where these terms tend to destabilize e.g. fractional Chern insulators. We explain this as a consequence of the fundamental asymmetry between electrons and holes for band projected normal ordered interactions, as well as from the perspective of a self-consistent Hartree-Fock calculation. These basic insights about the role of the quantum metric turn, when dominant, an extremely strongly coupled problem into an effectively weakly coupled one, and may also serve as a guiding principle for designing material setups.