Quantum geometry, particle-hole asymmetry and their applications in moiré materials with flat dispersion

Topological bands with a flat dispersion host strongly correlated states with or without intrinsic topological orders. The kinetic part of the system is trivial and the system is dominated by the interaction. At a first glance, electrons do not have any preference to occupy in the Brillouin zone. Despite the featureless kinetic dispersion, topological bands are usually equipped with nontrivial band geometry. We show that the nonuniform band geometry gives rise to emergent Fermi surfaces and it leads to a general particle-hole asymmetry. The electrons tend to fill regions in the Brillouin zone where their quantum distance is shorter. The emergent Fermi surface transforms the strongly interacting problem to a weakly interacting one. This dictates the low-energy physics and serves as a guiding principle for potential symmetry-breaking states. We show that in moiré materials, the quantum distance can be well approximated by a local quantity called the quantum metric. From this simple quantity, we can deduce what phases are favoured in different moiré systems at fractional fillings.