Quantum Geometry in 2D Materials: Flat Bands and Superlattices

Motivated by our numerical finding of the fractional quantum anomalous Hall (FQAH) effect in a singular flat band (SFB) with band touching [1], here we further explore strong correlation effects in gapless flat-band systems. By breaking the spatial symmetry and/or bipartite (or dark-state) symmetry, we generalize the previously studied fluxed dice lattice model [1] to realize various SFBs with band touching. We construct the correlated phase diagrams in the parameter space at fractional fillings of the flat band via the density matrix renormalization group and exact diagonalization calculations. Competition between FQAH, charge density waves and metallic phases is explored, and we characterize the specific parameter regions in which the FQAH states exist. To shed light on the various strongly correlated behaviors, Hartree-Fock mean-field calculations are performed to examine the quantum geometric properties of the quasi-particle bands at integer fillings. The quantum geometry homogeneity of the mean-field background is found to play a critical role in promoting the formation of FQAH states at fractional fillings. Our results demonstrate that SFB systems are promising platforms for exploring the FQAH effect beyond the Landau level and Chern insulator paradigms.