  To explore whether a flat-band system can accommodate superconductivity, we
consider repulsively interacting fermions on the diamond chain, a simplest
quasi-one-dimensional system that contains a flat band. Exact diagonalization
and the density-matrix renormalization group (DMRG) are used to show that we
have a significant binding energy of a Cooper pair with a long-tailed pair-pair
correlation in real space when the total band filling is slightly below $1/3$,
where the dispersive band interacts with the flat band that is empty but close
to $E_F$. Pairs selectively formed across the outer sites of the diamond chain
are responsible for the pairing correlation. At exactly $1/3$-filling an
insulating phase emerges, where the entanglement spectrum indicates the
particles on the outer sites are highly entangled and topological. These come
from a peculiarity of the flat band in which "Wannier orbits" are not
orthogonalizable.
