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Distinct reentrant transitions in a quasi-periodic Raman lattice

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Tuning spin-orbit coupling in a quasiperiodic Raman lattice drives a reentrant transition between mixed phases and makes the system enter the purely critical phase twice.

desk verdict Solid numerical phase diagram with a credible mobility-edge-shift mechanism for reentrant delocalization, but the headline reentrant criticality transition is a perturbative slice, not an established phase. read the letter →

arxiv 2607.17104 v1 pith:ECIHTSAJ submitted 2026-07-19 cond-mat.dis-nn

Xingbo Wei , Zhentian Xie , Tong Liu , Gao Xianlong , Yunbo Zhang This is my paper · ORCID
classification cond-mat.dis-nn
keywords quasiperiodiclatticespin-orbitcouplingreentranttransitionmobilityedgescriticalphasefractaldimensionAndersonlocalizationRaman
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that a single one-dimensional lattice can cycle through qualitatively different quantum-state coexistences as one control parameter is varied. Specifically, in a spin-orbit-coupled Raman lattice with a quasiperiodic Zeeman potential, increasing the spin-orbit coupling sends the system from a mixed phase without anomalous mobility edges (M1) through a mixed phase with them (M2) and back to M1. At a very weak uniform Zeeman field the same model enters the fully critical phase twice, which the authors call a reentrant criticality transition and present as the missing counterpart to reentrant delocalization/localization transitions. It also claims a separate reentrant delocalization transition driven by a uniform Zeeman potential, whose mechanism is a mobility-edge shift splitting the localized region. If correct, this completes a framework of reentrant phenomena across extended, critical, and localized states and gives a concrete experimental platform in cold atoms.

What carries the argument

The central object is the spin-orbit-coupled tight-binding Hamiltonian: nearest-neighbor spin-conserved hopping t, spin-flip hopping Δ, and a Zeeman potential V cos(2παj+φ)+W, where α is an approximant to the inverse golden ratio. At Δ=0 it reduces to two independent quasiperiodic chains, and for W=0 and V/t=2 the spectrum is purely critical. The argument works through single-particle eigenstates categorized by inverse participation ratio, normalized participation ratio, and fractal dimension D, with the composite ratio η separating pure from mixed phases. Conventional mobility edges separate extended from localized states, while two classes of anomalous mobility edges—aME1 and aME2—separate

What would settle it

Take the same parameters as Fig. 4(a) (V/t=2, W/t=0.01) and compute the fractal dimension scaling for larger Fibonacci system sizes, such as L=4181 and L=6765, at Δ/t=0.4 and Δ/t=1.0. If ⟨D⟩ drifts toward 0 or 1 with system size, or if raising W to 0.02 removes either of the fully critical intervals, the reentrant criticality transition is not a stable thermodynamic phase.

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Extended reading notes

Core claim

The central claim is that a one-dimensional spinful lattice with a quasiperiodic Zeeman potential and spin-orbit coupling contains reentrant phase sequences as a single parameter changes. On the W–Δ phase diagram at V/t=2, the mixed phase M1 (coexisting localized and extended states, no anomalous mobility edges) is split into two regions by M2 (a mixed phase hosting anomalous mobility edges), so increasing Δ sends the system M1→M2→M1. At weak uniform field W/t=0.01 the spectrum enters the purely critical phase twice, which the authors name the reentrant criticality transition: at Δ/t=0 the critical phase is inherited from the decoupled model, while in 0.5<Δ/t<1.6 it survives the competition

Load-bearing premise

The load-bearing premise is that W/t=0.01 is a faithful stand-in for the W=0 purely critical phase in the thermodynamic limit; the paper's own supplemental section 'The effect of W on the purely critical phase' shows the purely critical region shrinks rapidly once W grows, so the double entry into the critical phase in Fig. 4(a) is established only at this single small value.

Editorial extensions

If this is right

  • When Δ is increased at V/t=2 and W/t=1, the lattice passes through M1→M2→M1, meaning a single control knob can select whether anomalous mobility edges are present or absent.
  • At W/t=0.01 the spectrum enters the purely critical phase twice, establishing a reentrant criticality transition that parallels reentrant localization/delocalization transitions.
  • The uniform Zeeman field W splits the localized region, so the lattice can delocalize, relocalize, and delocalize again in a way that does not rely on hybridization.
  • Weak random disorder and weak interactions leave the mobility edges and phase boundaries intact, so a cold-atom Raman lattice can observe the signatures via wave-packet mean-square displacement and survival probability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The two-entry critical phase is demonstrated only at W/t=0.01; the more robust statement is likely the generalized, energy-level-resolved reentrant criticality that the paper mentions in the supplement, and a finite-size scaling study across W=0.01 to 0.1 would tell whether the pure-C double entry survives as a true phase or only as a perturbative signature.
  • Because the mechanism is a mobility edge that shifts out of the spectrum, any quasiperiodic one-dimensional system with an adjustable uniform offset—not just Raman lattices—should host a similar reentrant delocalization, which is testable in photonic waveguide arrays.
  • The model's chiral symmetry and explicit matrix structure may allow exact equations for the anomalous mobility edges aME1 and aME2, turning the numerical phase boundaries into analytic curves and predicting precisely where the system re-enters the critical phase.
  • The proposed dynamical detection protocol could be turned into a quantitative mobility-edge locator: the crossover values of Δ where wave-packet growth changes should coincide with the mobility-edge positions, giving an experimental readout of the edge trajectories.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies a one-dimensional tight-binding model with spin-orbit coupling and a Zeeman potential composed of a quasiperiodic and a uniform part. Using exact diagonalization on Fibonacci-size lattices, the authors compute inverse participation ratios, normalized participation ratios, and fractal dimensions to construct W–Δ phase diagrams for V/t=2 and V/t=1.4. They report (i) a reentrant transition between two mixed phases M1→M2→M1 as Δ is tuned at fixed finite W, with M2 distinguished by the presence of anomalous mobility edges separating critical from noncritical states; (ii) a 'reentrant criticality transition' at W/t=0.01 in which the system enters the purely critical phase twice; and (iii) a reentrant delocalization transition driven by the uniform Zeeman field, attributed to a mobility-edge shift that splits the localized region. The paper includes robustness checks against random disorder and weak interactions, a multifractal analysis, an experimental protocol, and open data.

Significance. If established, the results would broaden the reentrant-phenomena framework to include critical states and would identify a mobility-edge-shift mechanism distinct from hybridization. The manuscript is careful in using multiple diagnostics (IPR/NPR, fractal dimension, β_min, composite participation ratio) and makes the numerical data openly available; the disorder and interaction robustness checks, together with the proposed wave-packet experiment, strengthen the paper's practical relevance. However, the central novel claim—the reentrant criticality transition—rests on a single perturbative value of W and lacks a controlled thermodynamic-limit analysis, so the significance is currently conditional on additional scaling evidence.

major comments (2)
  1. [§4 (Reentrant transitions), Fig. 4(a), Fig. S5(a)] The headline reentrant criticality transition is shown only at W/t=0.01. One of the two C intervals is the Δ/t=0 boundary, where the Hamiltonian reduces to two AA chains at their critical point and is critical for any W; this is a parameter-space boundary, not a re-entry. The other C interval (0.5<Δ/t<1.6) is, by the authors' own Supplement, destroyed for W/t≳0.04 ('survives only for sufficiently weak W and is rapidly replaced by mixed phases as W increases'). No scaling of the C and M2 interval widths with W→0 and L→∞ is provided. Since at W=0 the entire Δ/t<2 region is critical, the two-entry structure is a finite-W effect; whether it survives in the thermodynamic limit is unverified. The claim that this 'completes the basic framework of reentrant phenomena' therefore exceeds the present evidence. The manuscript should either supply the missing scaling analysis or explicitly restrict t
  2. [§3 (Phase diagrams) and §4 (Conventional and anomalous mobility edges), Figs. 1–3] The boundaries between M1 and M2, and the locations of conventional and anomalous mobility edges, are determined without a quantitative criterion or error estimates. In Figs. 2(b)–(d) the dotted lines are placed by inspection of ⟨D⟩ at a single system size (L=2584); in Fig. 1 the phase boundaries are quoted as sharp lines. Because the reentrant M1→M2→M1 sequence is defined by the presence/absence of anomalous mobility edges, the phase assignment should be based on a finite-size crossing analysis with confidence intervals (e.g., extrapolation of ⟨D⟩ or IPR across L), not on single-size visual inspection. This is particularly important near the W/t=0.01 slice where the C phase is claimed to survive.
minor comments (4)
  1. [References (main text and list)] The reference list contains duplicate numbers: [1], [5], [7], [8] appear multiple times with different papers. This makes citations ambiguous and must be corrected by renumbering.
  2. [Abstract and Conclusions] The phrases 'completes the basic framework of reentrant phenomena' and 'all phenomena can be realized experimentally' overstate the evidence. I suggest softening to 'extends the framework' and restricting the experimental claim to the reentrant delocalization signal for which a protocol is provided.
  3. [Fig. 2 and Fig. 4(c), mobility-edge markers] The dotted lines marking mobility edges would be reproducible if the caption or text gave the precise criterion used to place them (e.g., a threshold in ⟨D⟩ or a size-scaling crossing). At present the locations appear to be chosen by eye.
  4. [Fig. S8, DMRG robustness check] The DMRG interaction check uses L=80 with 800 kept states. Please state explicitly in the text that this is a finite-size robustness check and indicate how the phase boundary would be affected by larger L.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: phase diagrams are independent numerical output; self-citations are minor and not load-bearing.

full rationale

The paper's central results—the M1→M2→M1 reentrance, the reentrant criticality transition, and the W-driven reentrant delocalization—are obtained by direct numerical diagonalization of an explicit model Hamiltonian, with phases assigned via standard IPR/NPR/fractal-dimension diagnostics and finite-size scaling. No parameter is fitted to a subset of the data and then renamed a prediction; the claimed transitions are read from spectra across parameter space. The W=0 limiting phases are imported from external results (AA model [35]; non-Abelian AA critical phase [1,8]), and the pure-phase/no-mobility-edge criterion comes from the external unified framework of Ref. [4], so the baselines are not supplied by the present authors' own prior work. The self-citations that do appear (Ref. [23] for a generic density-of-states reentrance mechanism, Ref. [3] for a shift-invert solver, Ref. [9] for the term 'anomalous mobility edges') are supporting or methodological, not load-bearing for the new phase boundaries. The reentrant-criticality classification uses the same fractal-dimension definition that defines the critical phase; this is an operational definition rather than a reduction of the conclusion to its input. The limitation noted in the paper's own supplement—that the purely critical phase at finite W 'survives only for sufficiently weak W and is rapidly replaced by mixed phases as W increases' (Fig. S5a), and that one C entry is the Δ=0 AA critical line—is a robustness/correctness concern about the thermodynamic-limit stability of the claimed reentrant criticality, not a circularity of the derivation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard single-particle quasiperiodic localization diagnostics and external phase-diagram results. The only hand-picked parameter that directly affects the headline claim is W/t=0.01; no new physical entities are introduced.

free parameters (1)
  • W/t (uniform Zeeman strength) = 0.01 (hand-picked)
    Chosen as sufficiently small to preserve the purely critical phase; Supp. Fig. S5 shows the pure C phase is destroyed for larger W, so the reentrant criticality claim is tied to this specific perturbative value.
assumptions (5)
  • domain assumption The single-particle, noninteracting Hamiltonian Eq. (1) is the object of study; interactions are only added as perturbative robustness checks.
    All central phase diagrams are for single-particle eigenstates; DMRG in the supplement only checks weak interaction stability.
  • domain assumption The W=0 phase boundaries and the Aubry-André transition at V/t=2 from Refs. [1,8,35] are correct.
    Used as the baseline around which the reentrant criticality and reentrant delocalization claims are built.
  • domain assumption Phase classification via ⟨D⟩ and β_min finite-size scaling correctly identifies extended (D→1), localized (D→0), and critical (0<D<1) states in the thermodynamic limit.
    All pure/mixed phase labels reduce to this scaling assumption; no exact proof is given for W≠0.
  • domain assumption The pure-phase criterion of Ref. [4] (chiral symmetry, purely quasiperiodic on-site term, uniform/quasiperiodic hopping) is valid and is used to argue that W≠0 generically induces mobility edges.
    Invoked in the supplement to explain why W≠0 produces mixed phases and why a pure E phase can reappear when mobility edges fall outside the spectrum.
  • domain assumption Fibonacci approximants with L=2584 and α=1597/2584 faithfully represent the irrational quasiperiodic limit.
    All numerics are at finite L; robustness checks for other α are shown in Supp. Fig. S6 but no thermodynamic-limit proof is provided.

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Pith. "Pith review of Distinct reentrant transitions in a quasi-periodic Raman lattice." pith.science (2026). https://pith.science/paper/ECIHTSAJ

@misc{pith2026260717104,
  author       = {Pith},
  title        = {Pith review of: Distinct reentrant transitions in a quasi-periodic Raman lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECIHTSAJ}},
  note         = {Machine review of arXiv:2607.17104}
}
abstract

We investigate a one-dimensional lattice with spin-orbit coupling (SOC) and a Zeeman potential containing uniform and quasiperiodic components. By tuning SOC, anomalous mobility edges emerge that separate critical from non-critical states, yielding a reentrant transition between two mixed phases, M$_1$$\to$M$_2$$\to$M$_1$, where M$_1$ (M$_2$) lacks (hosts) anomalous mobility edges. A new \emph{reentrant criticality transition}, defined as multiple entries into the critical phase, is identified. As a counterpart to reentrant delocalization/localization transitions, it completes the basic framework of reentrant phenomena across extended, localized, and critical states. The uniform Zeeman potential drives a reentrant delocalization transition, arising from the splitting of the localized region induced by the shift of mobility edges. This reveals a distinct pathway for reentrant phenomena beyond hybridization mechanisms.

Figures

Figures reproduced from arXiv: 2607.17104 by the authors.

Figure 1
Figure 1. Phase diagrams in the [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (a) Spectrum as a function of ∆, along the white [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. (a) IPR as a function of ∆ for V /t = 2 and W/t = 0.01. (b)–(d) ⟨D⟩ at various energy levels for ∆/t = 0.0, 0.4, and 0.8, respectively. Other parameters are the same as those in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: (a)–(d) Spectra as a function of W/t for ∆/t = 0.8, 0.7, 0.6 and 0.5, sliced from [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

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Reference graph

Works this paper leans on

59 extracted references · 1 canonical work pages

  1. [48]

    E. Guan, G. Wang, X.-W. Guan, and X. Cai, Reentrant localization and mobility edges in a spinful Aubry-Andr´ e- Harper model with a non-Abelian potential, Phys. Rev. A 108, 033305 (2023)

    2023

  2. [1]

    P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev.109, 1492 (1958)

    1958

  3. [2]

    Semeghini, M

    G. Semeghini, M. Landini, P. Castilho, S. Roy, G. Spag- nolli, A. Trenkwalder, M. Fattori, M. Inguscio, and G. Modugno, Measurement of the mobility edge for 3D Anderson localization, Nature Physics11, 554 (2015)

    2015

  4. [3]

    Pasek, G

    M. Pasek, G. Orso, and D. Delande, Anderson Localiza- tion of Ultracold Atoms: Where is the Mobility Edge?, Phys. Rev. Lett.118, 170403 (2017)

    2017

  5. [4]

    Longhi, Dephasing-Induced Mobility Edges in Qua- sicrystals, Phys

    S. Longhi, Dephasing-Induced Mobility Edges in Qua- sicrystals, Phys. Rev. Lett.132, 236301 (2024)

    2024

  6. [5]

    Ganeshan, J

    S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension, Phys. Rev. Lett.114, 146601 (2015)

    2015

  7. [6]

    Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges, Phys. Rev. Lett.125, 196604 (2020)

    2020

  8. [7]

    Biddle and S

    J. Biddle and S. Das Sarma, Predicted Mobility Edges in One-Dimensional Incommensurate Optical Lattices: An Exactly Solvable Model of Anderson Localization, Phys. Rev. Lett.104, 070601 (2010)

    2010

Show all 59 references
  1. [8]

    H. Yao, A. Khoudli, L. Bresque, and L. Sanchez- Palencia, Critical Behavior and Fractality in Shallow One- Dimensional Quasiperiodic Potentials, Phys. Rev. Lett. 123, 070405 (2019)

    2019

  2. [9]

    T. Liu, X. Xia, S. Longhi, and L. Sanchez-Palencia, Anomalous mobility edges in one-dimensional quasiperi- odic models, SciPost Phys.12, 027 (2022)

    2022

  3. [10]

    Z. Wang, Y. Zhang, L. Wang, and S. Chen, Engineer- ing mobility in quasiperiodic lattices with exact mobility edges, Phys. Rev. B108, 174202 (2023)

    2023

  4. [11]

    X.-C. Zhou, Y. Wang, T.-F. J. Poon, Q. Zhou, and X.- J. Liu, Exact New Mobility Edges between Critical and Localized States, Phys. Rev. Lett.131, 176401 (2023)

    2023

  5. [12]

    Wang and Q

    Y. Wang and Q. Zhou, Exact new mobility edges (2025), arXiv:2501.17523 [math.DS]

    2025 arXiv

  6. [13]

    Mott, Electrons in disordered structures, Advances in Physics16, 49 (1967)

    N. Mott, Electrons in disordered structures, Advances in Physics16, 49 (1967)

    1967

  7. [15]

    F. A. An, K. Padavi´ c, E. J. Meier, S. Hegde, S. Ganeshan, J. H. Pixley, S. Vishveshwara, and B. Gadway, Interactions and Mobility Edges: Observing the Generalized Aubry- Andr´ e Model, Phys. Rev. Lett.126, 040603 (2021)

    2021

  8. [16]

    Wang, J.-H

    Y. Wang, J.-H. Zhang, Y. Li, J. Wu, W. Liu, F. Mei, Y. Hu, L. Xiao, J. Ma, C. Chin, and S. Jia, Observation of Interaction-Induced Mobility Edge in an Atomic Aubry- Andr´ e Wire, Phys. Rev. Lett.129, 103401 (2022)

    2022

  9. [17]

    Chang, J.-H

    Y.-J. Chang, J.-H. Zhang, Y.-H. Lu, Y.-Y. Yang, F. Mei, J. Ma, S. Jia, and X.-M. Jin, Observation of Photonic Mo- bility Edge Phases, Phys. Rev. Lett.134, 053601 (2025)

    2025

  10. [18]

    T. Xiao, D. Xie, Z. Dong, T. Chen, W. Yi, and B. Yan, Observation of topological phase with critical localiza- tion in a quasi-periodic lattice, Science bulletin66, 2175 (2021)

    2021

  11. [20]

    S. Roy, T. Mishra, B. Tanatar, and S. Basu, Reentrant Localization Transition in a Quasiperiodic Chain, Phys. 6 Rev. Lett.126, 106803 (2021)

    2021

  12. [21]

    Padhan, M

    A. Padhan, M. K. Giri, S. Mondal, and T. Mishra, Emergence of multiple localization transitions in a one- dimensional quasiperiodic lattice, Phys. Rev. B105, L220201 (2022)

    2022

  13. [22]

    ˇStrkalj, E

    A. ˇStrkalj, E. V. H. Doggen, I. V. Gornyi, and O. Zilber- berg, Many-body localization in the interpolating Aubry- Andr´ e-Fibonacci model, Phys. Rev. Res.3, 033257 (2021)

    2021

  14. [23]

    Wei, Reentrant localization transition induced by a composite potential, Phys

    X. Wei, Reentrant localization transition induced by a composite potential, Phys. Rev. B111, 134204 (2025)

    2025

  15. [24]

    Z.-S. Xu, J. Gao, A. Iovan, I. M. Khaymovich, V. Zwiller, and A. W. Elshaari, Observation of reentrant metal- insulator transition in a random-dimer disordered SSH lat- tice, npj Nanophotonics1, 8 (2024)

    2024

  16. [25]

    Vaidya, C

    S. Vaidya, C. J¨ org, K. Linn, M. Goh, and M. C. Rechtsman, Reentrant delocalization transition in one- dimensional photonic quasicrystals, Phys. Rev. Res.5, 033170 (2023)

    2023

  17. [26]

    Q. Guo, C. Cheng, Z.-H. Sun, Z. Song, H. Li, Z. Wang, W. Ren, H. Dong, D. Zheng, Y.-R. Zhang,et al., Obser- vation of energy-resolved many-body localization, Nature Physics17, 234 (2021)

    2021

  18. [27]

    Shi, Z.-H

    Y.-H. Shi, Z.-H. Sun, Y.-Y. Wang, Z.-A. Wang, Y.-R. Zhang, W.-G. Ma, H.-T. Liu, K. Zhao, J.-C. Song, G.-H. Liang,et al., Probing spin hydrodynamics on a supercon- ducting quantum simulator, Nature Communications15, 7573 (2024)

    2024

  19. [28]

    X. Li, H. Xu, J. Wang, L.-Z. Tang, D.-W. Zhang, C. Yang, T. Su, C. Wang, Z. Mi, W. Sun, X. Liang, M. Chen, C. Li, Y. Zhang, K. Linghu, J. Han, W. Liu, Y. Feng, P. Liu, G. Xue, J. Zhang, Y. Jin, S.-L. Zhu, H. Yu, S. P. Zhao, and Q.-K. Xue, Mapping the topology- localization pha...

    2024

  20. [30]

    Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science354, 83 (2016)

    2016

  21. [31]

    B. Song, C. He, S. Niu, L. Zhang, Z. Ren, X.-J. Liu, and G.-B. Jo, Observation of nodal-line semimetal with ultracold fermions in an optical lattice, Nature Physics 15, 911 (2019)

    2019

  22. [32]

    Liu, Z.-X

    X.-J. Liu, Z.-X. Liu, and M. Cheng, Manipulating Topo- logical Edge Spins in a One-Dimensional Optical Lattice, Phys. Rev. Lett.110, 076401 (2013)

    2013

  23. [33]

    Kohmoto, Metal-Insulator Transition and Scaling for Incommensurate Systems, Phys

    M. Kohmoto, Metal-Insulator Transition and Scaling for Incommensurate Systems, Phys. Rev. Lett.51, 1198 (1983)

    1983

  24. [34]

    J. Zhao, Y. Zhao, J.-G. Wang, Y. Li, and X.-D. Bai, Phase transition of a non-Abelian quasiperiodic mosaic lattice model withp-wave superfluidity, Phys. Rev. B108, 054204 (2023)

    2023

  25. [35]

    Aubry and G

    S. Aubry and G. Andr´ e, Analyticity breaking and An- derson localization in incommensurate lattices, Ann. Isr. Phys. Soc3, 18 (1980)

    1980

  26. [37]

    See the Supplemental Material for details on (i) Matrix form of the Hamiltonian, (ii) In the limiting cases, (iii) De- struction of the re-emerged extended states, (iv) The tran- sition in the case ofW/t= 0, (v) Spectrum for ∆/t= 1 and typical states (vi) ∆ induced transition ...

  27. [38]

    Yahyavi, B

    M. Yahyavi, B. Het´ enyi, and B. Tanatar, Generalized Aubry-Andr´ e-Harper model with modulated hopping and p-wave pairing, Phys. Rev. B100, 064202 (2019)

    2019

  28. [39]

    Q.-B. Zeng, S. Chen, and R. L¨ u, Generalized Aubry- Andr´ e-Harper model withp-wave superconducting pair- ing, Phys. Rev. B94, 125408 (2016)

    2016

  29. [40]

    X. Li, X. Li, and S. Das Sarma, Mobility edges in one-dimensional bichromatic incommensurate potentials, Phys. Rev. B96, 085119 (2017)

    2017

  30. [41]

    Ghosh, M

    R. Ghosh, M. Sarkar, and I. M. Khaymovich, Reentrant localization induced by short-range hopping in the frac- tal Rosenzweig-Porter model, Phys. Rev. B111, L220102 (2025)

    2025

  31. [42]

    D. J. Luitz, F. Alet, and N. Laflorencie, Universal Behav- ior beyond Multifractality in Quantum Many-Body Sys- tems, Phys. Rev. Lett.112, 057203 (2014)

    2014

  32. [43]

    De Tomasi and I

    G. De Tomasi and I. M. Khaymovich, Multifractality Meets Entanglement: Relation for Nonergodic Extended States, Phys. Rev. Lett.124, 200602 (2020)

    2020

  33. [44]

    Ji and Z

    R. Ji and Z. Xu, Fibonacci-modulation-induced multiple topological Anderson insulators, Communications Physics 8, 336 (2025)

    2025

  34. [45]

    Zhang, L.-Z

    G.-Q. Zhang, L.-Z. Tang, L. Quezada, S.-H. Dong, and D.-W. Zhang, Reentrant topological phases and spin den- sity wave induced by 1D moir´ e potentials, Communica- tions Physics8, 275 (2025)

    2025

  35. [46]

    Li and S

    X. Li and S. Das Sarma, Mobility edge and intermediate phase in one-dimensional incommensurate lattice poten- tials, Phys. Rev. B101, 064203 (2020)

    2020

  36. [47]

    R. Qi, J. Cao, and X.-P. Jiang, Multiple localization tran- sitions and novel quantum phases induced by a staggered on-site potential, Phys. Rev. B107, 224201 (2023)

    2023

  37. [49]

    Lv, T.-C

    T. Lv, T.-C. Yi, L. Li, G. Sun, and W.-L. You, Quantum criticality and universality in thep-wave-paired Aubry- Andr´ e-Harper model, Phys. Rev. A105, 013315 (2022)

    2022

  38. [50]

    T. L. G. X. Xingbo Wei, Zhentian Xie and Y. Zhang, Distinct reentrant transitions in a quasi-periodic Raman lattice 10.5281/zenodo.17082777 (2025)

    2025 doi

  39. [53]

    Tabanelli, C

    H. Tabanelli, C. Castelnovo, and A. ˇStrkalj, Reentrant lo- calization transitions and anomalous spectral properties in off-diagonal quasiperiodic systems, Phys. Rev. B110, 184208 (2024). 7 Supplementary Materials: Distinct reentrant transitions in a quasi-periodic Raman latti...

    2024

  40. [55]

    Chiral symmetry is present

  41. [56]

    The on-site matrixM j is purely quasiperiodic (i.e., no uniform part,W= 0)

  42. [57]

    generalized reentrant critical- ity transition

    The hopping matrix Π j (hereT) is either uniform or purely quasiperiodic. In our model, whenW̸= 0,M j is no longer purely quasiperiodic, thereby violating this condition. Conse- quently, although chiral symmetry is preserved, the uni- form termWis expected to induce mobility e...

  43. [58]

    Wang, X.-J

    J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phase dia- gram of a non-Abelian Aubry-Andr´ e-Harper model with p-wave superfluidity, Phys. Rev. B93, 104504 (2016)

    2016

  44. [59]

    Pietracaprina, N

    F. Pietracaprina, N. Mac´ e, D. J. Luitz, and F. Alet, Shift- invert diagonalization of large many-body localizing spin chains, SciPost Physics5, 045 (2018)

    2018

  45. [60]

    X. Wei, C. Cheng, G. Xianlong, and R. Mondaini, Inves- tigating many-body mobility edges in isolated quantum systems, Phys. Rev. B99, 165137 (2019)

    2019

  46. [61]

    Zhou, B.-C

    X.-C. Zhou, B.-C. Yao, Y. Wang, Y. Wang, Y. Wei, Q. Zhou, and X.-J. Liu, The fundamental localization phases in quasiperiodic systems: A unified framework and exact results (2025), arXiv:2503.24380 [cond-mat.dis-nn]

    2025 arXiv

  47. [62]

    Goblot, A

    V. Goblot, A. ˇStrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lema ˆ ıtre, L. Le Gratiet, A. Harouri, I. Sagnes, S. Ravets,et al., Emergence of criticality through a cas- cade of delocalization transitions in quasiperiodic chains, Nature Physics16, 832 (2020)

    2020

  48. [63]

    Tabanelli, C

    H. Tabanelli, C. Castelnovo, and A. ˇStrkalj, Reentrant lo- calization transitions and anomalous spectral properties in off-diagonal quasiperiodic systems, Phys. Rev. B110, 184208 (2024)

    2024

  49. [64]

    H. P. L¨ uschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bor- dia, X. Li, S. Das Sarma, and I. Bloch, Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Opti- cal Lattice, Phys. Rev. Lett.120, 160404 (2018)

    2018

  50. [65]

    Y. Wang, L. Zhang, S. Niu, D. Yu, and X.-J. Liu, Re- alization and Detection of Nonergodic Critical Phases in an Optical Raman Lattice, Phys. Rev. Lett.125, 073204 (2020)

    2020

  51. [100]

    The color indicates the value of ln(I)

    (c) and (d) Spectra as a function ofV /tforW/t= 3.L= 610,α= 377/610, ∆/t= 0.5, andϕ= 0. The color indicates the value of ln(I). Red dotted lines mark the transition points. ˆH ′ = 0 ˆ∆ − ˆ∆ 0 .(13) We now consider the first-order perturbation theory to study the effect of ˆH ′...

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Reviewed August 1, 2026 · model on record in the stance chip above.