  The Lieb lattice possesses three bands and with intrinsic spin orbit coupling
$\lambda$, supports topologically non-trivial band insulating phases. At half
filling the lower band is fully filled, while the upper band is empty. The
chemical potential lies in the flat band (FB) located at the middle of the
spectrum, thereby stabilizing a flat band insulator. At this filling, we
introduce on-site Hubbard interaction $U$ on all sites. Within a slave rotor
mean field theory we show that, in spite of the singular effect of interaction
on the FB, the three bands remain stable up to a fairly large critical
correlation strength ($U_{crit}$), creating a correlated flat band insulator.
Beyond $U_{crit}$, there is a sudden transition to a Mott insulating state,
where the FB is destroyed due to complete transfer of spectral weight from the
FB to the upper and lower bands. We show that all the correlation driven
insulating phases host edge modes with linearly dispersing bands along with a
FB passing through the Dirac point, exhibiting that the topological nature of
the bulk band structure remains intact in presence of strong correlation.
Furthermore, in the limiting case of $U$ introduced only on one sublattice
where $\lambda=0$, we show that the Lieb lattice can support mixed edge modes
containing contributions from both spinons and electrons, in contrast to purely
spinon edge modes arising in the topological Mott insulator.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics