Quantum Geometric Superfluidity in Ultracold Fermi Gases on 3D Extended Lieb Lattices

We investigate the superfluidity of ultracold Fermi gases with a short-range attractive interaction in a three-dimensional extended Lieb (perovskite) lattice, within the context of the BCS-BEC crossover. Using BCS mean-field theory for the ground state and pairing fluctuation theory for finite temperatures, we reveal exotic quantum phenomena arising from flat bands and the van Hove singularity. Quantum geometric effects dramatically enhance the effective pair hopping integral and superfluid density, thereby improving superfluidity. As the Fermi level enters the flat band, the pairing gap, superfluid transition temperature, and superfluid density exhibit unconventional power-law dependence on interaction strength, along with sharply increased compressibility near the BCS limit. At low temperatures in this region, the superfluid density further exhibits power-law temperature scaling. Crossing the van Hove singularity in the lower band induces a transition from particle-like to hole-like pairing, yielding nonmonotonic chemical potential variation with interaction strength. Our phase diagram uncovers a pair density wave state at high densities and strong interactions. These predictions can be tested in future experiments.