  The Lieb Lattice exhibits intriguing properties that are of general interest
in both the fundamental physics and practical applications. Here, we
investigate the topological Landau-Zener Bloch oscillation in a photonic
Floquet Lieb lattice, where the dimerized helical waveguides is constructed to
realize the synthetic spin-orbital interaction through the Floquet mechanism,
rendering us to study the impacts of topological transition from trivial gaps
to non-trivial ones. The compact localized states of flat bands supported by
the local symmetry of Lieb lattice will be associated with other bands by
topological invariants, Chern number, and involved into Landau-Zener transition
during Bloch oscillation. Importantly, the non-trivial geometrical phases after
topological transitions will be taken into account for constructive and
destructive interferences of wave functions. The numerical calculations of
continuum photonic medium demonstrate reasonable agreements with theoretical
tight-binding model. Our results provide an ongoing effort to realize designed
quantum materials with tailored properties.
