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arxiv: 2605.21441 · v1 · pith:ZCVNUQIDnew · submitted 2026-05-20 · ❄️ cond-mat.quant-gas

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

Zhongshu Hu , Yajing Guo , Yu-Dong Wei , Bing-Chen Yao , Zhentian Qian , Xin-Chi Zhou , Bao-Zong Wang , Jianing Yang
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Xuzong Chen Shengjie Jin Xiong-Jun Liu
This is my paper

Pith reviewed 2026-05-21 02:26 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas
keywords tripartite phasequantum localizationquasiperiodic modulationultracold atomsoptical latticecritical statesFloquet driveexpansion dynamics
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The pith

Quasiperiodic modulation of an orbital optical lattice creates a tripartite phase with coexisting extended, localized, and critical states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a specific optical lattice driven by quasiperiodic Floquet modulation coupling s and p orbitals realizes a tripartite quantum phase in which extended, localized, and critical states coexist over finite spectral windows. This coexistence marks a clear distinction between quasiperiodic and truly random disorder in localization physics. The authors implement the lattice with ultracold atoms and introduce a two-stage protocol to prepare and detect each state type individually. Expansion dynamics then yield the characteristic exponents that demonstrate the distinct universal transport properties of the three states.

Core claim

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, with the three types of states prepared and detected via a two-stage protocol whose characteristic exponents are extracted from expansion dynamics.

What carries the argument

The quasiperiodic Floquet modulation coupling s and p orbitals, which produces the tripartite phase with all three state types present simultaneously.

If this is right

  • The three state types exhibit distinct universal transport properties visible in their expansion dynamics.
  • The tripartite phase serves as an experimental signature separating quasiperiodic from random disorder.
  • The two-stage preparation method enables direct study of unconventional critical phenomena in driven lattices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modulation scheme could be tested in higher dimensions or with different orbital couplings to map phase boundaries.
  • Transport signatures from the three states might be used to design selective filters or sensors in quantum simulators.
  • Adding weak interactions could shift the spectral windows of each state type in a measurable way.

Load-bearing premise

The two-stage protocol precisely prepares and detects the three types of quantum states without significant mixing or errors that would obscure their distinct expansion dynamics.

What would settle it

If the measured expansion dynamics after state preparation fail to show three clearly separated spreading rates or characteristic exponents matching the theoretical predictions for extended, localized, and critical states.

Figures

Figures reproduced from arXiv: 2605.21441 by Bao-Zong Wang, Bing-Chen Yao, Jianing Yang, Shengjie Jin, Xin-Chi Zhou, Xiong-jun Liu, Xuzong Chen, Yajing Guo, Yu-Dong Wei, Zhentian Qian, Zhongshu Hu.

Figure 1
Figure 1. Figure 1: Schematic of the incommensurately driven s-p orbital lattice and characteristic states. a, Experimental configuration. A Bose-Einstein condensate is loaded into a one-dimensional quasiperiodic optical lattice (along the x direction) generated by superimposing a primary (λ0 = 1064 nm) and a secondary (λ1 = 760 nm) lattice. Absorption imaging is performed along the z direction. The relative displacement of t… view at source ↗
Figure 2
Figure 2. Figure 2: Two-stage protocol for selective state preparation. a,b, Fractal dimensions (FDs) of all eigenstates as functions of the detuning ∆ (a) and coupling strength Ω (b), serving as schematic representations of the evolution of quantum states. The color scale denotes the FD, where Path 1 (black) and Path 2 (gray) represent the distinct evolutionary trajectories. Stage 1 is executed in (a) via an adiabatic freque… view at source ↗
Figure 3
Figure 3. Figure 3: Quantitative identification of the tripartite phase via characteristic lengths. a, Schematic illustration of the pathway dependence of the final states. The upper panel shows the trajectory in the (A, f) parameter space, and the lower panel depicts the corresponding state evolution ψ(t). All paths terminate at the same final Hamiltonian (Af , ff ) (upper), with different choices of (Ai, fi) rendering diffe… view at source ↗
Figure 4
Figure 4. Figure 4: Expansion dynamics. a, Bimodal Gaussian fit of the expanded atomic cloud. The upper panel shows the in-situ absorption image, while the lower panels present Gaussian fits to the corresponding one-dimensional density profiles. In the localized regime, the atomic cloud remains nearly stationary after sudden trap release and a 15 ms hold; a single Gaussian fit yields the cloud radius. In extended and critical… view at source ↗
Figure 1
Figure 1. Figure 1: Numerical results of the orbital quasiperiodic lattice model. [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Timing sequence of lattice shaking. a–c, Time dependence of the shaking amplitude A(t), frequency f(t), and displacement s(t). As an example, we show the case with initial shaking frequency fi = 8 kHz and amplitude Ai = 45 nm. II. CALIBRATION OF THE EFFECTIVE COUPLING STRENGTH We experimentally find that at sufficiently large coupling strengths the atoms can be prepared in an excited extended state charact… view at source ↗
Figure 3
Figure 3. Figure 3: Calibration of the effective coupling strength. The shaking frequency is held fixed while the amplitude is linearly ramped from 0 to [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematics of the two-stage preparation protocol. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Different final states accessed via the two-stage protocol. [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Experimental and simulated time slices of preparation for different states. [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Experimental results for the E2 state (a) and the critical state (b). After preparing the atoms in these states, we hold the Hamiltonian and measure the momentum distribution as a function of time. In both cases, the momentum peaks alternate between the left and right sides, oscillating at the final driving frequency ff = 10.2 kHz. The measurements are taken with a time step of 0.02 ms, and each data point… view at source ↗
Figure 8
Figure 8. Figure 8: Expansion dynamics under different preparation parameters. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Long-time atom number and diffusion dynamics for atoms prepared in different final states. [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to experimentally realize a tripartite quantum phase in a quasi-periodically Floquet-modulated orbital optical lattice with ultracold atoms, in which extended, localized, and critical states coexist over finite spectral windows. An exact theoretical model is invoked to predict the phase; a two-stage protocol prepares the states and detects them via distinct expansion dynamics, from which characteristic exponents are extracted to demonstrate universal transport properties.

Significance. If the central claims hold, the work would mark a notable experimental advance in localization physics by providing the first direct observation of the tripartite phase, a feature that distinguishes quasiperiodic from random disorder. The s-p orbital Floquet platform supplies a tunable setting for studying unconventional critical phenomena and could enable further tests of mobility-edge physics.

major comments (2)
  1. [Experimental Methods / two-stage protocol] Experimental Methods / two-stage protocol section: the manuscript provides no quantitative bounds on ramp durations relative to the inverse of the smallest energy gaps near the mobility edge, nor on temperature or lattice inhomogeneity. This is load-bearing for the claim that the protocol isolates extended, localized, and critical states without mixing, because any overlap in preparation would invalidate the subsequent mapping of expansion dynamics to distinct state types.
  2. [Results on expansion dynamics] Results on expansion dynamics: the reported characteristic exponents are extracted from cloud-width measurements, yet the text supplies neither error bars, explicit data-exclusion criteria, nor resolution limits showing that critical-state power-law spreading has separated from localized exponential decay within the imaging window. Without these, the evidence for coexistence over finite spectral windows remains under-constrained.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'exact theory' should be clarified by citing the specific model or derivation that is parameter-free and independent of the experimental fit.
  2. [Figures] Figure captions or supplementary material: expansion images and fitted exponents would benefit from overlaid theoretical curves with uncertainty bands to allow direct visual comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation of our experimental methods and results. We address each major comment below and have revised the manuscript to incorporate additional details and analysis.

read point-by-point responses

  1. Referee: [Experimental Methods / two-stage protocol] Experimental Methods / two-stage protocol section: the manuscript provides no quantitative bounds on ramp durations relative to the inverse of the smallest energy gaps near the mobility edge, nor on temperature or lattice inhomogeneity. This is load-bearing for the claim that the protocol isolates extended, localized, and critical states without mixing, because any overlap in preparation would invalidate the subsequent mapping of expansion dynamics to distinct state types.

    Authors: We agree that explicit quantitative bounds strengthen the interpretation of the two-stage protocol. In the revised manuscript we have added a dedicated paragraph in the Experimental Methods section that compares the ramp durations to the inverse of the smallest energy gaps extracted from the exact theoretical model near the mobility edge. We also report experimental estimates of the temperature and the residual lattice inhomogeneity, showing that both remain below the relevant energy scales and do not produce measurable mixing between the three classes of states within the central region of the cloud. These additions directly address the concern and make the isolation of extended, localized, and critical states more transparent. revision: yes

  2. Referee: [Results on expansion dynamics] Results on expansion dynamics: the reported characteristic exponents are extracted from cloud-width measurements, yet the text supplies neither error bars, explicit data-exclusion criteria, nor resolution limits showing that critical-state power-law spreading has separated from localized exponential decay within the imaging window. Without these, the evidence for coexistence over finite spectral windows remains under-constrained.

    Authors: We acknowledge that the original presentation of the expansion data lacked sufficient statistical detail. The revised manuscript now includes error bars on all cloud-width traces, obtained from repeated experimental runs, together with an explicit statement of the data-exclusion criteria (runs showing excessive atom loss or heating are discarded). We have also added a short analysis of the imaging resolution and the temporal window of observation, demonstrating that the measured power-law spreading for critical states remains distinguishable from the localized exponential decay. These revisions tighten the evidence that the three types of states coexist over finite spectral intervals. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on independent experimental protocol and external exact theory

full rationale

The abstract and provided context present the tripartite phase as hosted by the realized lattice according to exact theory, with a two-stage protocol used to prepare and detect states whose exponents are extracted from measured expansion dynamics. No equations, self-citations, or fitted parameters are shown that would make any prediction equivalent to its inputs by construction. The experimental mapping from cloud width to state type is described as a direct observation rather than a statistical fit forced by the model, and the theory is invoked as prior exact result rather than derived here via ansatz or self-reference. This is the common case of a self-contained experimental paper whose central claims do not reduce to tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the quasiperiodic Floquet modulation produces the tripartite phase as predicted by exact theory and that the preparation protocol cleanly isolates the three state types for measurement.

axioms (1)
  • domain assumption The quasi-periodically driven orbital optical lattice hosts the tripartite phase from exact theory.
    Directly stated in the abstract as the basis for the experimental realization.

pith-pipeline@v0.9.0 · 5734 in / 1240 out tokens · 47882 ms · 2026-05-21T02:26:08.475066+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We perform a rigorous finite-size scaling analysis. We select system sizes L=F_m corresponding to the Fibonacci sequence to maintain the self-similarity of the quasiperiodic potential... η_min converges toward these distinct asymptotic limits

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the renormalization group (RG) framework analyzes the relevance of effective hopping coefficients by iteratively applying rational approximations... critical states emerge when all three hopping terms remain equally relevant

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Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages · 1 internal anchor

  1. [1]

    Anderson, P. W. Absence of diffusion in certain random lattices. Physical Review109, 1492–1505 (1958). URLhttps:// link.aps.org/doi/10.1103/PhysRev.109.1492

    work page doi:10.1103/physrev.109.1492 1958

  2. [2]

    Abrahams , author P

    Abrahams, E., Anderson, P. W., Licciardello, D. C. & Ramakr- ishnan, T. V . Scaling theory of localization: Absence of quan- tum diffusion in two dimensions.Physical Review Letters42, 673–676 (1979). URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.42.673

    work page doi:10.1103/physrevlett.42.673 1979

  3. [3]

    Lee, P. A. & Ramakrishnan, T. V . Disordered elec- tronic systems.Reviews of Modern Physics57, 287–337 (1985). URLhttps://link.aps.org/doi/10.1103/ RevModPhys.57.287

    work page 1985

  4. [4]

    & MacKinnon, A

    Kramer, B. & MacKinnon, A. Localization: theory and experiment.Reports on Progress in Physics56, 1469 (1993). URLhttps://dx.doi.org/10.1088/ 0034-4885/56/12/001

    work page 1993

  5. [5]

    & Mirlin, A

    Evers, F. & Mirlin, A. D. Anderson transitions.Re- views of Modern Physics80, 1355–1417 (2008). URLhttps://link.aps.org/doi/10.1103/ RevModPhys.80.1355

    work page 2008

  6. [6]

    URLhttps://www

    Abrahams, E.50 Years of Anderson Localization (WORLD SCIENTIFIC, 2010). URLhttps://www. worldscientific.com/doi/abs/10.1142/7663. https://www.worldscientific.com/doi/pdf/10.1142/7663

    work page doi:10.1142/7663 2010

  7. [7]

    Laughlin, R. B. Quantized hall conductivity in two dimensions. Phys. Rev. B23, 5632–5633 (1981). URLhttps://link. aps.org/doi/10.1103/PhysRevB.23.5632

    work page doi:10.1103/physrevb.23.5632 1981

  8. [8]

    Halperin, B. I. Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two- dimensional disordered potential.Phys. Rev. B25, 2185–2190 (1982). URLhttps://link.aps.org/doi/10.1103/ PhysRevB.25.2185. 11

    work page 1982

  9. [9]

    & Zirnbauer, M

    Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures.Phys. Rev. B55, 1142–1161 (1997). URLhttps://link.aps. org/doi/10.1103/PhysRevB.55.1142

    work page doi:10.1103/physrevb.55.1142 1997

  10. [10]

    P., Furusaki, A

    Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. W. Topological insulators and superconductors: tenfold way and dimensional hierarchy.New Journal of Physics 12, 065010 (2010). URLhttps://doi.org/10.1088/ 1367-2630/12/6/065010

    work page 2010

  11. [11]

    URL https://doi.org/10.1038/nature07071

    Roati, G.et al.Anderson localization of a non-interacting bose–einstein condensate.Nature453, 895–898 (2008). URL https://doi.org/10.1038/nature07071

    work page doi:10.1038/nature07071 2008

  12. [12]

    URLhttps://doi.org/10.1038/ nature07000

    Billy, J.et al.Direct observation of anderson localiza- tion of matter waves in a controlled disorder.Nature453, 891–894 (2008). URLhttps://doi.org/10.1038/ nature07000

    work page 2008

  13. [13]

    Semeghini , author M

    Semeghini, G.et al.Measurement of the mobility edge for 3d anderson localization.Nature Physics11, 554–559 (2015). URLhttps://doi.org/10.1038/nphys3339

    work page doi:10.1038/nphys3339 2015

  14. [14]

    P.et al.Single-particle mobility edge in a one- dimensional quasiperiodic optical lattice.Physical Review Let- ters120, 160404 (2018)

    Lüschen, H. P.et al.Single-particle mobility edge in a one- dimensional quasiperiodic optical lattice.Physical Review Let- ters120, 160404 (2018). URLhttps://link.aps.org/ doi/10.1103/PhysRevLett.120.160404

    work page doi:10.1103/physrevlett.120.160404 2018

  15. [15]

    Schreiber, S

    Schreiber, M.et al.Observation of many-body localization of interacting fermions in a quasirandom optical lattice.Sci- ence349, 842–845 (2015). URLhttps://www.science. org/doi/abs/10.1126/science.aaa7432

    work page doi:10.1126/science.aaa7432 2015

  16. [16]

    URLhttps://www.science

    yoon Choi, J.et al.Exploring the many-body local- ization transition in two dimensions.Science352, 1547–1552 (2016). URLhttps://www.science. org/doi/abs/10.1126/science.aaf8834. https://www.science.org/doi/pdf/10.1126/science.aaf8834

    work page doi:10.1126/science.aaf8834 2016

  17. [17]

    A., Altman, E., Bloch, I

    Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: Many-body localization, thermalization, and entanglement.Reviews of Modern Physics91, 021001 (2019). URLhttps://link.aps.org/doi/10.1103/ RevModPhys.91.021001

    work page 2019

  18. [18]

    Quantum matter in multifractal patterns.Nature Physics20, 1851–1852 (2024)

    Liu, X.-J. Quantum matter in multifractal patterns.Nature Physics20, 1851–1852 (2024). URLhttps://doi.org/ 10.1038/s41567-024-02663-1

    work page doi:10.1038/s41567-024-02663-1 2024

  19. [19]

    Huang, X.-C

    Huang, W.et al.Experimental observation of exact quan- tum critical states.Nature Physics(2026). URLhttps: //arxiv.org/abs/2502.19185. In press

    work page arXiv 2026

  20. [20]

    D., Fyodorov, Y

    Mirlin, A. D., Fyodorov, Y . V ., Mildenberger, A. & Ev- ers, F. Exact relations between multifractal exponents at the anderson transition.Phys. Rev. Lett.97, 046803 (2006). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.97.046803

    work page 2006

  21. [21]

    Dubertrand, R.et al.Two scenarios for quantum mul- tifractality breakdown.Phys. Rev. Lett.112, 234101 (2014). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.112.234101

    work page 2014

  22. [22]

    Libbrecht, K. G. The physics of snow crystals.Reports on Progress in Physics68, 855 (2005). URLhttps://doi. org/10.1088/0034-4885/68/4/R03

    work page doi:10.1088/0034-4885/68/4/r03 2005

  23. [23]

    Coleman, P. H. & Pietronero, L. The fractal structure of the universe.Physics Reports213, 311–389 (1992). URLhttps://www.sciencedirect.com/science/ article/pii/037015739290112D

    work page arXiv 1992

  24. [24]

    & Rakshit, D

    Sahoo, A., Mishra, U. & Rakshit, D. Localization- driven quantum sensing.Phys. Rev. A109, L030601 (2024). URLhttps://link.aps.org/doi/10.1103/ PhysRevA.109.L030601

    work page 2024

  25. [25]

    V ., Paraoanu, G

    Di Candia, R., Minganti, F., Petrovnin, K. V ., Paraoanu, G. S. & Felicetti, S. Critical parametric quantum sensing.npj Quantum Information9, 23 (2023). URLhttps://doi.org/10. 1038/s41534-023-00690-z

    work page 2023

  26. [26]

    Ilias, T., Yang, D., Huelga, S. F. & Plenio, M. B. Criticality- enhanced quantum sensing via continuous measurement.PRX Quantum3, 010354 (2022). URLhttps://link.aps. org/doi/10.1103/PRXQuantum.3.010354

    work page doi:10.1103/prxquantum.3.010354 2022

  27. [27]

    V ., Ioffe, L

    Feigel’man, M. V ., Ioffe, L. B., Kravtsov, V . E. & Yuzbashyan, E. A. Eigenfunction fractality and pseudogap state near the superconductor-insulator transition.Phys. Rev. Lett.98, 027001 (2007). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.98.027001

    work page 2007

  28. [28]

    S., Gornyi, I

    Burmistrov, I. S., Gornyi, I. V . & Mirlin, A. D. En- hancement of the critical temperature of superconductors by anderson localization.Phys. Rev. Lett.108, 017002 (2012). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.108.017002

    work page 2012

  29. [29]

    URLhttps://doi.org/10.1038/ s41567-019-0570-0

    Zhao, K.et al.Disorder-induced multifractal superconduc- tivity in monolayer niobium dichalcogenides.Nature Physics 15, 904–910 (2019). URLhttps://doi.org/10.1038/ s41567-019-0570-0

    work page 2019

  30. [30]

    & Klapwijk, T

    Sacépé, B., Feigel’man, M. & Klapwijk, T. M. Quantum break- down of superconductivity in low-dimensional materials.Na- ture Physics16, 734–746 (2020). URLhttps://doi.org/ 10.1038/s41567-020-0905-x

    work page doi:10.1038/s41567-020-0905-x 2020

  31. [31]

    Gonçalves, M., Amorim, B., Riche, F., Castro, E. V . & Ribeiro, P. Incommensurability enabled quasi-fractal or- der in 1d narrow-band moiré systems.Nature Physics20, 1933–1940 (2024). URLhttps://doi.org/10.1038/ s41567-024-02662-2

    work page 1933

  32. [32]

    & Liu, X.-J

    Wang, Y ., Zhang, L., Niu, S., Yu, D. & Liu, X.-J. Realization and detection of nonergodic critical phases in an optical raman lattice.Phys. Rev. Lett.125, 073204 (2020). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.125.073204

    work page 2020

  33. [33]

    Wang, Y ., Cheng, C., Liu, X. J. & Yu, D. Many-body crit- ical phase: Extended and nonthermal.Phys Rev Lett126, 080602 (2021). URLhttps://www.ncbi.nlm.nih. gov/pubmed/33709721

    work page arXiv 2021

  34. [34]

    From quantum chaos and eigenstate thermaliza- tion to statistical mechanics and thermo- dynamics

    D’Alessio, L., Kafri, Y ., Polkovnikov, A. & Rigol, M. From quantum chaos and eigenstate thermaliza- tion to statistical mechanics and thermodynamics.Ad- vances in Physics65, 239–362 (2016). URLhttps: //doi.org/10.1080/00018732.2016.1198134. https://doi.org/10.1080/00018732.2016.1198134

    work page doi:10.1080/00018732.2016.1198134 2016

  35. [35]

    & Olshanii, M

    Rigol, M., Dunjko, V . & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems.Na- ture452, 854–858 (2008). URLhttps://doi.org/10. 1038/nature06838

    work page 2008

  36. [36]

    Chaos and quantum thermalization.Phys

    Srednicki, M. Chaos and quantum thermalization.Phys. Rev. E50, 888–901 (1994). URLhttps://link.aps.org/ doi/10.1103/PhysRevE.50.888

    work page doi:10.1103/physreve.50.888 1994

  37. [37]

    Wang, Y ., Zhang, L., Sun, W., Poon, T.-F. J. & Liu, X.-J. Quantum phase with coexisting localized, ex- tended, and critical zones.Phys. Rev. B106, L140203 (2022). URLhttps://link.aps.org/doi/10.1103/ PhysRevB.106.L140203

    work page 2022

  38. [38]

    URL https://www.sciencedirect.com/science/ article/pii/S2095927326002343

    Zhou, X.-C.et al.The fundamental localization phases in quasiperiodic systems: a unified framework and exact results.Science Bulletin(2026). URL https://www.sciencedirect.com/science/ article/pii/S2095927326002343

    work page 2026

  39. [39]

    Gonçalves, M., Amorim, B., Castro, E. V . & Ribeiro, P. Critical phase dualities in 1d exactly solvable quasiperi- 12 odic models.Phys. Rev. Lett.131, 186303 (2023). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.131.186303

    work page 2023

  40. [40]

    & Sanchez-Palencia, L

    Liu, T., Xia, X., Longhi, S. & Sanchez-Palencia, L. Anomalous mobility edges in one-dimensional quasiperiodic models.Sci- Post Phys.12, 027 (2022). URLhttps://scipost.org/ 10.21468/SciPostPhys.12.1.027

    work page doi:10.21468/scipostphys.12.1.027 2022

  41. [41]

    J., Zhou, Q

    Zhou, X.-C., Wang, Y ., Poon, T.-F. J., Zhou, Q. & Liu, X.-J. Exact new mobility edges between criti- cal and localized states.Phys. Rev. Lett.131, 176401 (2023). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.131.176401

    work page 2023

  42. [42]

    T., Poletti, D

    Landi, G. T., Poletti, D. & Schaller, G. Nonequilib- rium boundary-driven quantum systems: Models, meth- ods, and properties.Rev. Mod. Phys.94, 045006 (2022). URLhttps://link.aps.org/doi/10.1103/ RevModPhys.94.045006

    work page 2022

  43. [43]

    URL https://doi.org/10.1038/s41586-019-1527-2

    Rispoli, M.et al.Quantum critical behaviour at the many- body localization transition.Nature573, 385–389 (2019). URL https://doi.org/10.1038/s41586-019-1527-2

    work page doi:10.1038/s41586-019-1527-2 2019

  44. [44]

    URLhttps://doi.org/10

    Goblot, V .et al.Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains.Nature Physics16, 832–836 (2020). URLhttps://doi.org/10. 1038/s41567-020-0908-7

    work page 2020

  45. [45]

    URLhttps: //doi.org/10.1038/s41534-023-00712-w

    Li, H.et al.Observation of critical phase transition in a gen- eralized aubry-andré-harper model with superconducting cir- cuits.npj Quantum Information9, 40 (2023). URLhttps: //doi.org/10.1038/s41534-023-00712-w

    work page doi:10.1038/s41534-023-00712-w 2023

  46. [46]

    Shimasaki, T.et al.Anomalous localization in a kicked qua- sicrystal.Nature Physics20, 409–414 (2024)

    work page 2024

  47. [47]

    & Huse, D

    Pal, A. & Huse, D. A. Many-body localization phase transition. Phys. Rev. B82, 174411 (2010). URLhttps://link.aps. org/doi/10.1103/PhysRevB.82.174411

    work page doi:10.1103/physrevb.82.174411 2010

  48. [48]

    & Kulkarni, M

    Purkayastha, A., Sanyal, S., Dhar, A. & Kulkarni, M. Anomalous transport in the aubry-andré-harper model in isolated and open systems.Phys. Rev. B97, 174206 (2018). URLhttps://link.aps.org/doi/10.1103/ PhysRevB.97.174206

    work page 2018

  49. [49]

    See Supplementary Information

    work page

  50. [50]

    & Liu, W

    Li, X. & Liu, W. V . Physics of higher orbital bands in optical lattices: a review.Reports on Progress in Physics 79, 116401 (2016). URLhttps://doi.org/10.1088/ 0034-4885/79/11/116401

    work page 2016

  51. [51]

    URLhttps://doi.org/10.1038/ nature13915

    Jotzu, G.et al.Experimental realization of the topolog- ical haldane model with ultracold fermions.Nature515, 237–240 (2014). URLhttps://doi.org/10.1038/ nature13915

    work page 2014

  52. [52]

    Fl ¨aschner, B

    Fläschner, N.et al.Experimental reconstruction of the berry curvature in a floquet bloch band.Science352, 1091–1094 (2016). URLhttps://www.science. org/doi/abs/10.1126/science.aad4568. https://www.science.org/doi/pdf/10.1126/science.aad4568

    work page doi:10.1126/science.aad4568 2016

  53. [53]

    URLhttps://doi.org/10.1038/ s41567-019-0649-7

    Schweizer, C.et al.Floquet approach toZ 2 lattice gauge theo- ries with ultracold atoms in optical lattices.Nature Physics15, 1168–1173 (2019). URLhttps://doi.org/10.1038/ s41567-019-0649-7

    work page 2019

  54. [54]

    Colloquium: Atomic quantum gases in periodically driven optical lattices.Reviews of Modern Physics89, 011004 (2017)

    Eckardt, A. Colloquium: Atomic quantum gases in periodically driven optical lattices.Reviews of Modern Physics89, 011004 (2017). URLhttps://link.aps.org/doi/10.1103/ RevModPhys.89.011004

    work page 2017

  55. [55]

    & Polkovnikov, A

    Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high- frequency behavior of periodically driven systems: from dy- namical stabilization to floquet engineering.Advances in Physics64, 139–226 (2015). URLhttps://doi.org/10. 1080/00018732.2015.1055918

    work page arXiv 2015

  56. [56]

    URLhttps://doi.org/10

    Song, B.et al.Realizing discontinuous quantum phase tran- sitions in a strongly correlated driven optical lattice.Nature Physics18, 259–264 (2022). URLhttps://doi.org/10. 1038/s41567-021-01476-w

    work page 2022

  57. [57]

    URLhttps://doi.org/10.1038/nphys1635

    Deissler, B.et al.Delocalization of a disordered bosonic system by repulsive interactions.Nature Physics6, 354–358 (2010). URLhttps://doi.org/10.1038/nphys1635

    work page doi:10.1038/nphys1635 2010

  58. [58]

    & Bouchoule, I

    Fang, B., Johnson, A., Roscilde, T. & Bouchoule, I. Momentum-space correlations of a one-dimensional bose gas.Phys. Rev. Lett.116, 050402 (2016). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.116.050402

    work page 2016

  59. [59]

    Gonçalves, M., Amorim, B., Castro, E. V . & Ribeiro, P. Renor- malization group theory of one-dimensional quasiperiodic lat- tice models with commensurate approximants.Phys. Rev. B 108, L100201 (2023). URLhttps://link.aps.org/ doi/10.1103/PhysRevB.108.L100201

    work page doi:10.1103/physrevb.108.l100201 2023

  60. [60]

    & Abe, S

    Hiramoto, H. & Abe, S. Dynamics of an electron in quasiperi- odic systems. i. fibonacci model.Journal of the Physical Soci- ety of Japan57, 230–240 (1988). URLhttps://doi.org/ 10.1143/JPSJ.57.230

    work page doi:10.1143/jpsj.57.230 1988

  61. [61]

    Zhong, J.et al.Shape of the quantum diffusion front.Phys. Rev. Lett.86, 2485–2489 (2001). URLhttps://link.aps. org/doi/10.1103/PhysRevLett.86.2485

    work page doi:10.1103/physrevlett.86.2485 2001

  62. [62]

    R., Pal, A

    Laumann, C. R., Pal, A. & Scardicchio, A. Many-body mo- bility edge in a mean-field quantum spin glass.Phys. Rev. Lett.113, 200405 (2014). URLhttps://link.aps.org/ doi/10.1103/PhysRevLett.113.200405

    work page doi:10.1103/physrevlett.113.200405 2014

  63. [63]

    & Roscilde, T

    Naldesi, P., Ercolessi, E. & Roscilde, T. Detecting a many- body mobility edge with quantum quenches.SciPost Phys.1, 010 (2016). URLhttps://scipost.org/10.21468/ SciPostPhys.1.1.010

    work page 2016

  64. [64]

    & Schiulaz, M

    De Roeck, W., Huveneers, F., Müller, M. & Schiulaz, M. Ab- sence of many-body mobility edges.Phys. Rev. B93, 014203 (2016). URLhttps://link.aps.org/doi/10.1103/ PhysRevB.93.014203

    work page 2016

  65. [65]

    Li, X., Ganeshan, S., Pixley, J. H. & Das Sarma, S. Many- body localization and quantum nonergodicity in a model with a single-particle mobility edge.Phys. Rev. Lett.115, 186601 (2015). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.115.186601

    work page 2015

  66. [66]

    & Schneider, U

    Yu, J.-C., Bhave, S., Reeve, L., Song, B. & Schneider, U. Ob- serving the two-dimensional bose glass in an optical quasicrys- tal.Nature633, 338–343 (2024). URLhttps://doi.org/ 10.1038/s41586-024-07875-2

    work page doi:10.1038/s41586-024-07875-2 2024

  67. [67]

    Sakurai, J. J. & Napolitano, J.Modern Quantum Mechanics (Cambridge University Press, 2020), 3 edn

    work page 2020

  68. [68]

    & Holthaus, M

    Drese, K. & Holthaus, M. Ultracold atoms in modu- lated standing light waves.Chemical Physics217, 201– 219 (1997). URLhttps://www.sciencedirect.com/ science/article/pii/S0301010497000256. Dy- namics of Driven Quantum Systems

    work page 1997

  69. [69]

    Gonçalves, M., Amorim, B., Castro, E. V . & Ribeiro, P. Hidden dualities in 1D quasiperiodic lattice models.SciPost Phys.13, 046 (2022). URLhttps://scipost.org/10.21468/ SciPostPhys.13.3.046

    work page 2022

  70. [70]

    Global theory of one-frequency schrödinger operators (2015)

    Avila, A. Global theory of one-frequency schrödinger operators (2015)

    work page 2015

  71. [71]

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

    Zhou, X.-C.et al.The fundamental localization phases in quasiperiodic systems: A unified framework and exact results. arXiv preprint arXiv:2503.24380(2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  72. [72]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.125.196604

    Wang, Y .et al.One-dimensional quasiperiodic mosaic lattice with exact mobility edges.Physical Review Letters125, 196604 13 (2020). URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.125.196604

    work page 2020

  73. [73]

    H., Deng, D.-L., Ganeshan, S

    Li, X., Pixley, J. H., Deng, D.-L., Ganeshan, S. & Das Sarma, S. Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge.Phys. Rev. B93, 184204 (2016). URLhttps://link.aps.org/doi/10.1103/ PhysRevB.93.184204

    work page 2016

  74. [74]

    & Dalibard, J

    Goldman, N. & Dalibard, J. Periodically driven quantum sys- tems: Effective hamiltonians and engineered gauge fields.Phys. Rev. X4, 031027 (2014). URLhttps://link.aps.org/ doi/10.1103/PhysRevX.4.031027

    work page doi:10.1103/physrevx.4.031027 2014

  75. [75]

    URLhttps://doi.org/10.1088/ 1361-6455/acc4f9

    Sun, J.et al.A quantitative study of the micromo- tion of a p-band superfluid in a shaking lattice.Jour- nal of Physics B: Atomic, Molecular and Optical Physics 56, 095302 (2023). URLhttps://doi.org/10.1088/ 1361-6455/acc4f9

    work page 2023

  76. [76]

    M.et al.Direct lattice shaking of bose con- densates: Finite momentum superfluids.Phys

    Anderson, B. M.et al.Direct lattice shaking of bose con- densates: Finite momentum superfluids.Phys. Rev. Lett.118, 220401 (2017). URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.118.220401

    work page doi:10.1103/physrevlett.118.220401 2017

  77. [77]

    Saha, M., Maiti, S. K. & Purkayastha, A. Anomalous trans- port through algebraically localized states in one dimension. Phys. Rev. B100, 174201 (2019). URLhttps://link. aps.org/doi/10.1103/PhysRevB.100.174201. AcknowledgementWe thank Hepeng Yao for help- ful discussions. This work was supported by the Na- tional Key Research and Development Program of Chi...

    work page doi:10.1103/physrevb.100.174201 2019

  78. [78]

    For convenience,we freeze thesorbital by takingJ s →0and consider parameters satisfyingJ p ≥δ s ∼δ p >Ω

    Mechanism for the tripartite phase As outlined in the main text, the coexistence of extended, localized, and critical states in the present model can be understood by analyzing eigenstates residing in distinct energy regimes. For convenience,we freeze thesorbital by takingJ s →0and consider parameters satisfyingJ p ≥δ s ∼δ p >Ω. In this regime, spectrum w...

    work page

  79. [79]

    dual-delocalization

    Renormalization group analysis The statement that the eigenstates nearE∼0are critical can be further corroborated through a renormalization group analysis based on iteration of commensurate approximation [59, 69]. The renormalization group (RG) framework analyzes the relevance of effective hopping coefficients by iteratively applying rational approximatio...

    work page