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Observation of a tripartite quantum phase for coexisting extended, localized, and critical states

Xiong-Jun Liu, Z W Hu, Yajing Guo, Yu-Dong Wei, Bing-Chen YAO, Zhentian Qian, Xin-Chi Zhou, Bo Wang, Jie Yang, X Y Chen, Shengjie Jin

Published Jul 13, 2026• 0 citations
Publisher

Abstract

Abstract The disordered quantum world hosts three fundamental types of states, being extended, localized, and critical. Among various quantum phases formed by their interplay, the most nontrivial is the tripartite phase, wherein all three types of states coexist in the spectrum and are separated by anomalous mobility edges. Here, we report the realization of this exotic phase in a quasi-periodically driven orbital optical lattice with ultracold atoms. The optical lattice with a quasiperiodic Floquet modulation which couples the $s$ and $p$ orbitals is realized in experiment and shown to host the tripartite phase based on exact theory. We establish a novel two-stage protocol to achieve a controlled preparation and detection of the tripartite phase, with all three types of quantum states being observed. The characteristic exponents of these states are determined from expansion dynamics, showing their distinct universal transport properties. Our study paves the way for a comprehensive exploration of the unconventional critical phenomena and localization physics with ultracold atoms.

DOI
10.21203/rs.3.rs-9327513/v1

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Abstract The disordered quantum world hosts three fundamental types of states, being extended, localized, and critical. Among various quantum phases formed by their interplay, the most nontrivial is the tripartite phase, wherein all three types of states coexist in the spectrum and are separated by anomalous mobility edges. Here, we report the realization of this exotic phase in a quasi-periodically driven orbital optical lattice with ultracold atoms. The optical lattice with a quasiperiodic Floquet modulation which couples the sss and ppp orbitals is realized in experiment and shown to host the tripartite phase based on exact theory. We establish a novel two-stage protocol to achieve a controlled preparation and detection of the tripartite phase, with all three types of quantum states being observed. The characteristic exponents of these states are determined from expansion dynamics, showing their distinct universal transport properties. Our study paves the way for a comprehensive exploration of the unconventional critical phenomena and localization physics with ultracold atoms.
The disordered quantum world hosts three fundamental types of states: extended, localized, and critical, of which the critical states are confined to fine-tuned critical points or mobility edges in randomly disordered systems. The tripartite phase, with all three types of states coexisting over finite spectral windows, represents a hallmark distinction between quasiperiodic and truly random systems in the localization physics. Here, we report the realization of this exotic phase in a quasi-periodically driven orbital optical lattice with ultracold atoms. The optical lattice with a quasiperiodic Floquet modulation coupling s and p orbitals is realized in experiment and shown to host the tripartite phase from exact theory. We develop a two-stage protocol to precisely prepare and detect the three types of quantum states. The characteristic exponents of these states are determined from expansion dynamics, showing their distinct universal transport properties. Our study marks a significant advancement in exploring unconventional critical phenomena and localization physics with ultracold atoms.
Conventionally a mobility edge (ME) marks a critical energy that separates two different transport zones where all states are extended and localized, respectively. Here we propose a quasiperiodic spin-orbit coupled lattice model with experimental feasibility to realize a quantum phase with three coexisting energy-dependent zones, i.e., the extended, critical, and localized zones, and uncover the underlying generic mechanism for the occurrence of this quantum phase. Accordingly, this phase exhibits types of MEs which separate the extended states from critical ones and the localized states from critical ones, respectively. We introduce the diagnostic quantities to characterize and distinguish the different zones and show that the predicted phase can be detected by measuring the fractal dimension or conductivities. The experimental realization is also proposed and studied. This work extends the concept of ME and enriches the quantum phases in disordered systems, which sheds light on searching for localization and critical phenomena with transport and thermoelectric effects.
Extended, localized, and critical states form the three elementary classes of eigenstates underlying Anderson localization. While phases involving extended and localized states are widely studied, experimental realization of critical phases, particularly those where critical states coexist with others, remains a challenge. Here, we report to realize five distinct localization phases: extended, localized, critical, extended-localized mixed, and critical-localized mixed, in a one-dimensional Floquet lattice with quasiperiodic hopping amplitudes and on-site potentials. These phases are identified through characteristic spatiotemporal dynamics, supported by mutually consistent experimental observations and theoretical analysis. Our findings establish a unified experimental framework for systematic studies of localization transitions, mobility edges, and criticality in driven quasiperiodic systems.
Anderson localization has evolved far beyond the conventional dichotomy between extended and localized states. Modern localization theory predicts a complete transport hierarchy comprising extended, critical, and localized phases together with all coexistence phases among them, forming a seven-phase Anderson-localization landscape. Despite its fundamental importance, this hierarchy has never been experimentally realized within a single system. Here we realize the complete seven-phase Anderson-localization landscape in a one-dimensional Floquet photonic lattice. By engineering quasiperiodic hopping profiles containing inhomogeneously distributed hopping zeros, we generate critical states and enable their coexistence with extended and localized sectors. The resulting transport regimes are directly resolved through their distinct spatiotemporal dynamics, including ballistic expansion, confined critical oscillations, and persistent localization. We observe all seven phases, including the elusive triply coexisting extended-critical-localized phase, and experimentally track the phase transitions connecting them. Our results establish the first complete experimental map of the Anderson-localization landscape and provide a unified platform for investigating mobility edges, multifractality, and programmable coherent transport.
Abstract Anderson localization physics features three fundamental types of eigenstates: extended, localized, and critical. Confirming the presence of critical states necessitates either advancing the analysis to the thermodynamic limit or identifying a universal mechanism which can rigorously determine these states. Here we report the unambiguous experimental realization of critical states, governed by a rigorous mechanism for exact quantum critical states, and further observe a generalized mechanism that quasiperiodic zeros in hopping couplings protect the critical states. We implement a programmable quasiperiodic mosaic model with tunable couplings and on-site potentials through a multiple spin qubit system. By measuring the time-evolving observables, we identify the coexisting delocalized dynamics and incommensurately distributed zeros in the couplings, which are the defining features of the critical states. We map the localized-to-critical phase transition and demonstrate that critical states persist until quasiperiodic zeros are removed by strong long-range couplings, highlighting a novel generalized mechanism discovered in this experiment and shown with rigorous theory. Finally, we resolve the energy-dependent transition between localized and critical states, revealing the presence of anomalous mobility edges.
The critical phases, being delocalized but nonergodic, are fundamental phases different from both the many-body localization and ergodic extended quantum phases, and have so far not been realized in experiment. Here we propose an incommensurate topological insulating model of AIII symmetry class to realize such critical phases through an optical Raman lattice scheme, which possesses a one-dimensional (1D) spin-orbit coupling and an incommensurate Zeeman potential. We show the existence of both noninteracting and many-body critical phases, which can coexist with the topological phase, and show that the critical-localization transition coincides with the topological phase boundary in noninteracting regime. The dynamical detection of the critical phases is proposed and studied in detail based on the available experimental techniques. Finally, we demonstrate how the proposed critical phases can be achieved within the current ultracold atom experiments. This work paves the way to observe the novel critical phases.