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

The disordered quantum systems host three classes of quantum states, the extended, localized, and critical, which bring up seven distinct fundamental phases in nature: three pure phases and four coexisting ones with mobility edges, yet a unified theory built on universal mechanism and full realization of all these phases has not been developed. Here we propose a unified framework based on a spinful quasiperiodic (QP) system which realizes all the fundamental localization phases, with the exact and universal results being obtained for their characterization. First, we show that the pure phases are obtained when the chiral(-like) symmetry preserves in the proposed spinful QP model, giving a criterion for emergence of the pure phases and otherwise the coexisting ones. Further, we uncover a novel mechanism for the critical states that their emergence is protected by the generalized incommensurate matrix element zeros in the spinful QP model, which considerably broadens rigorous realizations of the exotic critical states. We then show criteria of exact solvability for the present spinful QP system, with which we construct various exactly solvable models for all distinct localization phases. In particular, we propose two novel models, dubbed spin-selective QP lattice model and QP optical Raman lattice model, to achieve all basic types of mobility edges and all the seven fundamental phases of Anderson localization physics, respectively. The experimental scheme is proposed and studied in detail to realize these models with high feasibility. This study establishes a complete and profound theoretical framework which enables an in-depth exploration of the broad classes of all fundamental localization phenomena in QP systems, and offers key insights for constructing their exactly solvable models with experimental feasibility.