  The ground-state superfluid behavior of ultracold atomic Fermi gases with a
short-range attractive interaction in a quasi-two-dimensional Lieb lattice is
studied 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 Fermi
level enters the flat band, both the pairing gap and the in-plane superfluid
density exhibit an unusual power law as a function of interaction, with
strongly enhanced quantum geometric effects, in addition to a dramatic increase
of compressibility as the interaction approaches the BCS limit. As the Fermi
level crosses the van Hove singularities, the character of pairing changes from
particle-like to hole-like or vice versa. We present the computed phase
diagram, in which a pair density wave state emerges at high densities with
relatively strong interaction strength.
