  We report the appearance of multiple numbers of completely flat band states
in an extended Lieb lattice model in two dimensions with five atomic sites per
unit cell. We also show that this edge-centered square lattice can host
intriguing topologically nontrivial phases when intrinsic spin-orbit (ISO)
coupling is introduced in the microscopic description of the corresponding
tight-binding Hamiltonian of the system. This ISO coupling strength acts like a
complex next-nearest-neighbor hopping term for this model and can be, in
principle, tuned in a real-life experimental setup. In the presence of this ISO
coupling, the band spectrum of the system gets gapped out, leading to nonzero
integer values of the spin Chern number for different bands, indicating the
nontrivial topological properties of the system. Furthermore, we show that for
certain values of the ISO coupling, nearly flat bands with nonzero Chern
numbers emerge in this lattice model. This opens up the possibility of
realizing interesting fractional quantum spin Hall physics in this model when
interaction is taken into account. This study might be very useful in an
analogous optical lattice experimental setup. A possible application of our
results can also be anticipated in the field of photonics using single-mode
photonic waveguide networks.
