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The ground-state superfluid behavior of ultracold atomic Fermi gases with anon-site attractive interaction in a quasi-two-dimensional Lieb lattice isstudied using BCS mean-field theory, within the context of BCS-BEC crossover.We find that the flat band leads to nontrivial exotic effects. As the Fermilevel enters the flat band, both the pairing gap and the in-plane superfluiddensity exhibit an unusual power law as a function of interaction, withstrongly enhanced quantum geometric effects, in addition to a dramatic increaseof compressibility as the interaction approaches the BCS limit. As the Fermilevel crosses the van Hove singularities, the character of pairing changes fromparticle-like to hole-like or vice versa. We present the computed phasediagram, in which a pair density wave state emerges at high densities withrelatively strong interaction strength.

Flat band effects on the ground-state BCS-BEC crossover in atomic Fermi gases in a quasi-two-dimensional Lieb lattice