Flat bands and nontrivial topological properties in an extended Lieb lattice

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.