  We obtain a phase diagram of the spin imbalanced Hubbard model on the Lieb
lattice, which is known to feature a flat band in its single-particle spectrum.
Using the BCS mean-field theory for multiband systems, we find a variety of
superfluid phases with imbalance. In particular, we find four different types
FFLO phases, i.e. superfluid phases with periodic spatial modulation. They
differ by the magnitude and direction of the centre-of-mass momentum of Cooper
pairs. We also see a large region of stable Sarma phase, where the density
imbalance is associated with zero Cooper pair momentum. In the mechanism
responsible for the formation of those phases, the crucial role is played by
the flat band, wherein particles can readjust their density at zero energy
cost. The multiorbital structure of the unit cell is found to stabilize the
Sarma phase by allowing for a modulation of the order parameter within a unit
cell. We also study the effect of finite temperature and a lattice with
staggered hopping parameters on the behaviour of these phases.
