The fundamental localization phases in quasiperiodic systems: A unified framework and exact results

The disordered quantum systems host three types of quantum states, the extended, localized, and critical, which bring up various distinct fundamental phases, including the pure phases and coexisting ones with mobility edges. The quantum phases involving critical states are of particular importance, but are less understood compared with the other ones, and the different phases have been separately studied in different quasiperiodic models. Here we propose a unified framework based on a spinful quasiperiodic system which unifies the realizations of all the fundamental Anderson phases, with the exact and universal results being obtained for these distinct phases. Through the duality transformation and renormalization group method, we show that the pure phases are obtained when the chiral symmetry preserves in the proposed spin-1/2 quasiperiodic model, which provides a criterion for the emergence of the pure phases or the coexisting ones with mobility edges. Further, we uncover a new universal mechanism for the critical states that the emergence of such states is protected by the generalized incommensurate matrix element zeros in the spinful quasiperiodic model, as a nontrivial generalization of the mechanism in the spinless systems. We also show with the Avila's global theory the criteria of exact solvability for the present unified quasiperiodic system, with which we identify several new quasiperiodic models hosting exactly solvable Anderson phases. We particularly reach two novel models, with one hosting all basic types of mobility edges and the other hosting all the seven fundamental phases of Anderson localization. Finally, an experimental scheme is proposed to realize these models using quasiperiodic optical Raman lattices.