  Flat bands in lattice models have provided useful platforms for studying
strong correlation and topological physics. Recently, honeycomb superlattices
have been shown to host flat bands that persist in the presence of local
perturbations respecting lattice symmetries. We analytically derive the flat
band energies in the presence of longer range hopping and find that the
energies of flat bands are tunable by these perturbations. In real space, the
wave function is constructed from standing waves on each honeycomb edge,
allowing the construction of plaquette and loop eigenstates due to destructive
interference in real space that give rise to the flat bands robust against long
range hoppings.
