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Topological flat bands in optical checkerboard-like lattices

We present comparatively simple two-dimensional and three-dimensional checkerboard-like optical lattices possessing nontrivial topological properties. By simple tuning of the parameters these lattices can have a topological insulating phase, a topological semi-metallic phase, or a trivial insulating phase. This allows study of different topological phase transitions within a single cold atom system. In the topologically nontrivial phases flat bands appear at the surfaces of the system. These surface states possess short localization lengths such that they are observable even in systems with small lattice dimensions. Our proposed lattices neither need spin-orbit coupling nor non-Abelian gauge fields to reach topologically nontrivial states.