  Bi-layer Kagome compounds provide an exciting playground where the interplay
of topology and strong correlations can give rise to exotic phases of matter.
Motivated by recent first principles calculation on such systems (Phys. Rev.
Lett 125, 026401), reporting stabilization of a Chern metal with topological
nearly-flat band close to Fermi level, we build minimal models to study the
effect of strong electron-electron interactions on such a Chern metal. Using
approriate numerical and analytical techniques, we show that the topologically
non-trivial bands present in this system at the Fermi energy can realize
fractional Chern insulator states. We further show that if the time-reversal
symmetry is restored due to destruction of magnetism by low dimensionality and
fluctuation, the system can realize a superconducting phase in the presence of
strong local repulsive interactions. Furthermore, we identify an interesting
phase transition from the superconducting phase to a correlated metal by tuning
nearest-neighbor repulsion. Our study uncovers a rich set of non-trivial phases
realizable in this system, and contextualizes the physically meaningful regimes
where such phases can be further explored.
