  Flat bands may offer a route to high critical temperatures of
superconductivity. It has been predicted that the quantum geometry of the band
as well as the ratio of the number of flat bands to the number of orbitals
determine flat band superconductivity. However, such results have assumed at
least one of the following: an isolated flat band, zero temperature, mean-field
theory, and/or uniform pairing. Here, we explore flat band superconductivity
when these assumptions are relaxed. We consider an attractive Hubbard model for
different extensions of the Lieb lattice. The superconducting order parameter,
critical temperature, superfluid weight, and Berezinskii-Kosterlitz-Thouless
temperature are calculated within cluster dynamical mean-field theory. We find
that while the flat-band ratio and quantum geometry are good indicators of
superconductivity near zero temperature, at finite temperatures the behavior is
more complicated. Our results suggest that the properties of the other bands
near the flat band(s) are crucial.
