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Localization-delocalization transition in self-dual quasi-periodic lattices

Within the framework of the Aubry-André model, one kind of self-dual quasi-periodic lattice, it is known that a sharp transition occurs from all eigenstates being extended to all being localized. The common perception for this type of quasi-periodic lattice is that the self-duality excludes the appearance of a finite critical energy separating localized from extended states. In this work, we propose a multi-chromatic quasi-periodic lattice model retaining the self-duality identical to the Aubry-André model. In this model we find numerically a well-defined localization-delocalization transition at the mobility edges in contrast with the Aubry-André model. As a result, the diffusion of wave packet exhibits a transition from ballistic to diffusive motion, and back to ballistic motion. We point out that experimental realizations of the predicted transition can be accessed with light waves in photonic lattices and matter waves in optical lattices.