Superconductivity and competing orders in flat bands: The roles of geometry and topology

Electronic systems with nearly flat bands provide a natural platform for strongly correlated quantum phases, as the suppression of kinetic energy enhances the role of interactions. In this talk, I discuss how the geometry and topology of flat bands shape superconductivity and competing orders in interacting electron systems. Using numerically exact quantum Monte Carlo simulations of sign-problem-free lattice models, I first show that isolated flat bands with attractive interactions can support superconductivity with a transition temperature that scales linearly with the interaction strength, accompanied by a broad pseudogap regime. I then demonstrate that the quantum geometry of the band, encoded in the spatial structure of the Wannier functions, controls the competition between superconductivity and charge density wave order, giving rise to intertwined phases including a supersolid state. Finally, I turn to topological flat bands with valley-contrasting Chern numbers, where doping an insulating magnetic state leads to superconductivity carried by composite quasiparticles formed from electrons dressed by spin excitations. These results illustrate how band geometry and topology provide powerful guiding principles for understanding correlated phases in flat-band systems, with potential relevance to moiré materials.