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

TL;DR

This work develops a unified spin-$1/2$ quasiperiodic framework that encompasses all seven fundamental localization phases in Anderson localization, using a synthesis of duality, renormalization group, and Avila's global theory to obtain exact results. It identifies a symmetry-based criterion for pure phases, proves a universal mechanism for the emergence of critical states via generalized incommensurate zeros in matrix elements, and provides an exact solvability condition that reduces the problem to effective spinless dressed-particle models. The authors construct new exactly solvable mosaic models, including a spin-selective QP lattice and a QP optical Raman lattice that realize all mobility edges and all seven phases, with experimentally feasible realization schemes. Together, these results unify existing spinful/spinless QP models, guide the design of solvable models, and offer analytic insight into localization phenomena and potential many-body extensions.

Abstract

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.

Figures (12)

• Figure 1: Illustration of generic framework. (a) The different processes of the spinful quasiperiodic systems. The hopping coupling matrix $\Pi_{j}^{s,s'}$ denote the momentum transfer coupling with the diagonal (off-diagonal) terms for the same (different) internal degrees of freedom. The on-site matrix $M_{j}^{s,s'}$ represents the on-site spin flipped and on-site modulation of the system. (b) The conditions under which Avila's global theory yields analytical characterization of the (reduced) 1D quasiperiodic chain. The nonzero part of derivative of the complexified Lyapunov exponent $\gamma_{\epsilon}$ is quantized to the unique integer: $d\gamma_{\epsilon}/d\epsilon=2\pi\mathbb{Z}$. Then the original Lyapunov exponent can be obtained by the expression obtained from the $\epsilon\rightarrow\infty$.
• Figure 2: Three universal results for the spinful quasiperiodic chains. (a) Criteria for the system to exhibit pure phases without mobility edges. Here, $\lambda$ is the tuning parameter and $E$ is the energy. States A and B refer to extended, localized, or critical states. The energy-dependent transition points $\lambda_c(E)$ become energy-independent, denoted as $\lambda_c$, indicates transitions from a regime with mobility edges (MEs) to a pure phase without MEs. (b) Mechanism for the emergence of critical states in the 1D spinful quasiperiodic (QP) system, generated by dual-invariant generalized incommensurate zeros (GIZs) in matrix elements, marked as the blue and red circles in real and dual spaces. (c) Exact solvability of QP systems from local constraint. The local constraint that reduces the generic 1D spinful QP chain into a 1D spinless QP chain of dressed particles (represented by orange spheres), with energy-dependent or energy-independent nearest-neighbor hopping coefficients $t_{j}^{\mathrm{eff}}(E)$ and on-site potentials $V_{j}^{\mathrm{eff}}(E)$, rendering the system exactly solvable through Avila's global theory.
• Figure 3: New exactly solvable models constructed using universal results and dual transformations. The fractal dimension (FD) versus energy $E$ and hopping strengths $\lambda/t$. (a) The quasiperiodic spin-flipped (QPSF) model, obtained by removing the $\sigma_0$ component of the type-II QP mosaic model, exhibits pure localized and critical phases. (b) The dual counterpart of the type-II QP mosaic model, featuring analytic mobility edges (MEs) that separate extended and critical states. The MEs occur at $E_c = \pm \lambda$ and are marked by solid lines. (c) The dual QPSF model, obtained by removing the spin-independent component in the dual model from (b). It exhibits pure extended and critical phases, with the phase transition points $\lambda = t$ marked by dashed lines. The system size is $L = 2584$.
• Figure 4: Generalized incommensurate zeros generated critical states. (a) The quasiperiodic (QP) spin-conserved hopping in this model gives rise to generalized incommensurate zeros (GIZs), which introduce critical orbitals. (b) The spin-selective QP (SSQP) lattice at the exactly solvable regime. Left panel: The model exhibits analytic mobility edges (MEs) at $E_c = \pm \lambda^2 / \mu$, marked by the solid lines, which separate the extended states ($E < |\lambda^2 / \mu|$) with fractal dimension $\mathrm{FD}\rightarrow 1$, and the rigorous critical states ($E > |\lambda^2 / \mu|$) with $\mathrm{FD}$ approaching a value between 0 and 1. Right panel: The corresponding $\mathrm{FD}$ for the dual model. The MEs $E_c = \pm \lambda^2 / \mu$ (solid lines) now separate the localized states ($E < |\lambda^2 / \mu|$) and the critical states ($E > |\lambda^2 / \mu|$). (c) The SSQP model at another exactly solvable regime that realizes all basic types of MEs. Upper panel: The energy-dependent effective potential resulted from the incommensurately distributed zeros in the shared component, which gives incommensurately distributed divergent potential. Lower panel: The $\mathrm{FD}$ of the eigenstates shown versus $V_B / t$ and $E/t$, with $\lambda_0 / t = 1$ and $V_0 / t = 1.5$. The MEs are indicated by the solid lines. All systems have a size of $L = 2584$.
• Figure 5: Seven fundamental phases realized in 1D quasiperiodic (QP) optical Raman lattice. (a) Model illustration: The hopping coupling matrix includes both spin-conserved and spin-flipped hopping terms, while the on-site matrix incorporates both spin-dependent ($\sigma_z$) and spin-independent ($\sigma_0$) QP potential. (b) Phase diagram: The phase diagram, shown as a function of the QP Zeeman potential $M_z$ and the chiral parameter $\eta$, reveals seven distinct phases: three pure phases--extended (E), localized (L), and critical (C)--and four coexistence regions. The latter include three two-coexistence regimes, (L+E), (L+C), and (C+E), where two types of eigenstates coexist at different energies, as well as a three-phase coexistence region (L+E+C). The phase diagram is obtained with the parameter set $t_{\mathrm{so}} = 0.8 t_0$. (c) Exact Solvable Points: (c1) When $\eta = 0$, the entire spectrum is localized. (c2) For $\eta = 0.5$, the spectrum splits into localized and critical states, with the mobility edges marked by solid lines at $E_c = \pm 2t_0$. States with $|E| < 2t_0$ are critical, while those with $|E| > 2t_0$ are localized. (c3) At $\eta = 1$, the system exhibits pure localized and critical phases, with the transition point marked by the dashed line at $M_z = 4t_0$. The system is in the localized (critical) phase when $M_z > 4t_0$ ($M_z < 4t_0$). The system size used in these calculations is $L = 2584$.