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Investigation of reentrant localization transition in one-dimensional quasi-periodic lattice with long-range hopping

T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Adding long-range hopping to a quasi-periodic chain can make reentrant localization appear, not just suppress it.

desk verdict Reentrant localization finding is solid and new; the four-universality-class claim overreaches the data. read the letter →

arxiv 2412.13518 v1 pith:TVTF5F2K submitted 2024-12-18 cond-mat.dis-nn

Pei-Jie Chang , Qi-Bo Zeng , Jinghui Pi , Dong Ruan , Gui-Lu Long This is my paper · ORCID
classification cond-mat.dis-nn
keywords reentrantlocalizationquasi-periodiclatticelong-rangehoppingSu-Schrieffer-Heegermodelmobilityedgefinite-sizescalinguniversalityclassinverseparticipationratio
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

Reentrant localization is the sequence in which stronger disorder first localizes a state, then lets it become extended again, and then localizes it once more. The paper asks whether this counterintuitive sequence survives when the chain also contains long-range (next-nearest-neighbor) hopping, which earlier work suggested only weakens reentrant behavior. It argues that, with the right hopping parameters, long-range hopping actually induces reentrant localization, for both staggered and uniform disorder, and that the four localization transitions in each case belong to distinct universality classes. If correct, this widens the class of one-dimensional quasi-periodic systems in which mobility edges and reentrant phases can be engineered.

What carries the argument

The central machinery is the dimerized SSH chain (a one-dimensional tight-binding chain with alternating nearest-neighbor hoppings) extended by next-nearest-neighbor hopping terms $J_{33}$ and $J_3$, viewed as a two-leg ladder. The quasi-periodic potential is applied with opposite signs on the two sublattices (staggered disorder) or equal signs (uniform disorder). Localization is diagnosed with the inverse participation ratio (IPR) and normalized participation ratio (NPR), combined into $\eta=\log_{10}(\mathrm{IPR}\times\mathrm{NPR})$ to mark critical regions, and with the $R[L,L']$ function whose size-crossing gives the critical disorder strength and the exponent ratio $\gamma/\nu$. The correlation-length exponent $\nu$ is then fixed by collapsing $\sigma^2=L^{\gamma/\nu-1}G(\varepsilon L^{1/\nu})$ for different system sizes. The proposed mechanism is that each ladder leg retains the dimerization-disorder competition that drives reentrant localization, while the inter-leg hopping $J_{11}$ and $J_3$ tunes that competition, which is why long-range hopping can either suppress or create the effect.

What would settle it

A transfer-matrix calculation of the localization length at the four claimed critical points, with bootstrap error bars on the correlation-length exponent, would settle the universality-class claim: if the four exponents overlap within uncertainty, the claim fails. For the reentrant phenomenon itself, a waveguide experiment with engineered next-nearest-neighbor hopping could look for the localized-extended-localized sequence in band-edge states as the disorder amplitude is swept.

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Extended reading notes

Core claim

The central discovery is that long-range hopping does not simply compete with dimerization to erase reentrant localization; under specific conditions it is the ingredient that makes reentrant localization appear. For staggered disorder $\lambda_A=-\lambda_B$ with $J_{11}=J_3=0.1$ and $J_{33}/J_1=2.7$, the phase diagram in $\lambda$ shows two intervals, $0.98<\lambda<2.018$ and $2.599<\lambda<3.6$, in which localized and extended states coexist, separated by a fully localized band; eigenstates in the middle of the spectrum re-extend as $\lambda$ grows. For uniform disorder $\lambda_A=\lambda_B$ with $J_{33}/J_1=0.7$ and $J_{11}=J_3=0.1$, eigenstates near the band edges undergo localized-extended-localized reentrance in a narrow window near $\lambda\simeq 2.3$-$2.4$. The paper interprets the long-range chain as a two-leg ladder: reentrant localization reflects competition between dimerization and disorder within each leg, with inter-leg hopping acting as a tunable perturbation. Finite-size scaling of the $R$-function yields four critical points in each disorder class whose exponents $\nu$ and ratios $\gamma/\nu$, $\beta/\nu$ satisfy the hyperscaling relation $2\beta/\nu+\gamma/\nu=1$, and the paper concludes the four transitions belong to distinct universality classes.

Load-bearing premise

The conclusion that the four transitions are genuinely different in character rests on matching numerical curves at different system sizes, and the key fitted number is quoted without an error bar.

Editorial extensions

If this is right

  • In the staggered-disorder case, long-range hopping removes the earlier coincidence of universality classes: the second and third localization transitions now have different correlation-length exponents, so each transition is its own universality class.
  • In the uniform-disorder case, reentrant localization appears where it was previously thought absent, with the re-extended states sitting at the highest and lowest eigenenergies rather than near the band center.
  • Tuning the inter-leg hopping strength $J_{11}=J_3$ controls the phenomenon: for staggered disorder reentrant localization disappears above $J_{11}=J_3\simeq0.207$, while for uniform disorder it appears only for nonzero inter-leg hopping and vanishes above about $J_{11}=J_3\simeq0.35$.
  • The four critical points in each disorder class satisfy the hyperscaling relation $2\beta/\nu+\gamma/\nu=1$, so the extracted exponents are internally consistent and describe genuine transitions.

Reading between the lines

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

  • Editorially, the two-leg ladder picture predicts that at $J_{11}=J_3=0$ the spectrum is just the union of two decoupled chains; checking whether the two observed pairs of mobility edges coincide with the individual legs' mobility edges would cleanly separate a superposition effect from a genuine long-range-hopping effect.
  • An implication the paper leaves implicit: the reentrant window location (band center for staggered disorder, band edges for uniform disorder) could serve as a spectroscopic fingerprint of the disorder type in experimental realizations.
  • A testable extension is to compute the dynamics: a wave packet launched at a band-edge state should show non-monotonic spreading as $\lambda$ is swept through the reentrant window, which would make the effect visible in time-dependent cold-atom or waveguide experiments.
  • The distinct-universality-class claim would be strengthened by an independent transfer-matrix computation of localization lengths, since the paper's exponents come from eigenstate windows selected after inspecting the data.
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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

4 major / 7 minor

Summary. The manuscript studies a one-dimensional dimerized (SSH) chain with next-nearest-neighbor hopping and a quasi-periodic potential, under either staggered (λA = −λB) or uniform (λA = λB) disorder. Using exact diagonalization up to L = 35422, it computes IPR/NPR-based diagnostics and η phase diagrams. It reports that with J11 = J3 = 0.1 and J33/J1 = 2.7 (staggered case) the system shows two critical regions, and with J33/J1 = 0.7 (uniform case) the band-edge states show localized–extended–localized reentrant behavior. It also extracts critical exponents (λc, γ/ν, β/ν, ν) at four transitions in each case and concludes that the transitions belong to distinct universality classes.

Significance. The reentrant localization observation is potentially interesting because it extends reentrant behavior to a model with next-nearest-neighbor hopping and to uniform disorder, where previous work (e.g., Refs. [47,65]) suggested it is absent or requires large detuning. The finite-size checks in the insets of Figs. 4(b) and 11 support the persistence of the reentrant feature. However, the universality-class conclusion is not supported by the current analysis: the ν values are obtained by a visually adjusted scaling collapse without quoted uncertainties, in post-hoc eigenstate windows, and the hyperscaling check is not independent. The paper's main numerical evidence for reentrant localization is credible, but the stronger claim about distinct universality classes needs substantial additional analysis.

major comments (4)
  1. [Sec. III, Eq. (10) and Fig. 7] The correlation-length exponent ν is determined by 'adjusting' ν to achieve 'optimal overlap' of the scaling curves σ²L^{1−γ/ν} versus εL^{1/ν}, with no collapse metric, no goodness-of-fit, and no uncertainty estimate. The reported values (ν1 = 0.9, ν2 = 1.1, ν3 = 1.8, ν4 = 0.9) are separated by amounts comparable to typical scaling-collapse uncertainties for this method, so the claim that the four transitions belong to distinct universality classes is not supported. Please provide error bars, for example via bootstrap or by reporting the sensitivity of ν to the range of system sizes and to the eigenstate window.
  2. [Sec. IV, Fig. 13] The same issue applies to the uniform-disorder case, where the ν values (ν1 = 0.814, ν2 = 1.10, ν3 = 0.938, ν4 = 0.680) are even closer together. In addition, the eigenstate windows used for the average NPR are selected post hoc (m/L ∈ [0.167, 0.177] for the first transition and m/L ∈ [0, 0.1] for the others), with no independent justification. This can bias both λc and ν. The analysis should demonstrate that the extracted exponents are stable under reasonable variations of these windows.
  3. [Sec. III, Eq. (12) and Fig. 8] The hyperscaling relation 2β/ν + γ/ν = 1 is presented as a confirmation, but both β/ν and γ/ν are obtained by least-squares fits to the same NPR-derived data, so the check is not independent. Moreover, the quoted uncertainties (e.g., γ1/ν1 = 0.5735 ± 8e−05) are only fit standard errors and do not include systematic uncertainties from finite-size corrections or eigenstate-window choices; they should not be used to argue for the precision of the exponents. Please either obtain one of the exponents from an independent quantity or explicitly state that Eq. (12) is a consistency condition rather than an independent verification.
  4. [Sec. II, Eqs. (8)–(10)] The finite-size scaling analysis assumes a single correlation-length exponent ν for each transition, but the identified 'critical regions' are intervals where localized and extended states coexist (single-particle mobility edges), not simple critical points. The R-function crossing method is designed for a transition at a single λc. The manuscript should justify its applicability to these mobility-edge transitions or test it with a direct scaling collapse that includes the energy dependence explicitly.
minor comments (7)
  1. [Figs. 2, 3, 9, 15, 16 captions] The captions state that the color represents log10(η), but η is already defined as log10(IPR × NPR) in Eq. (11), so the label appears to be a double logarithm. Please clarify whether the color bar shows η or log10(IPR × NPR).
  2. [Sec. III, Fig. 4(b) inset] The finite-size check is shown only for the average NPR; the text says this rules out finite-size effects for the reentrant localization feature, but the corresponding IPR behavior for the second critical region is not shown for all sizes. Please include IPR or another order parameter in the finite-size analysis.
  3. [Sec. I] The term 'long-range hopping' is used for next-nearest-neighbor hopping. This is consistent with some prior literature, but since it can be confused with power-law or algebraically decaying hopping, a brief clarification in the introduction would help.
  4. [Eq. (5)] The definition of NPR(m) appears to have a formatting issue with parentheses; please ensure the equation reads NPR(m) = ( L ∑_i |φ_i^{(m)}|^4 )^{-1}.
  5. [Sec. V, first paragraph] The text refers to 'Fig 3(c)', but Fig. 3 has only panels (a) and (b); the intended reference is likely Fig. 3(b). Similarly, 'Fig 9(c)' is correct.
  6. [Sec. V, paragraph after Fig. 15] The phrase 'In a word that long-range hopping introduces additional pairs of mobility edges' is grammatically awkward; please rephrase.
  7. [Sec. VI] The phrase 'each characterized by unique critical exponents' is ambiguous; 'distinct critical exponents' would be clearer, since 'unique' might be read as 'each having exactly one exponent', which is not the intended meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the reentrant-localization and critical-exponent claims are direct numerical observations; the statistical weaknesses noted by the skeptic do not reduce any result to its inputs.

full rationale

The paper is an exact-diagonalization study, and its central claims—reentrant localization under staggered and uniform disorder with long-range hopping—are read off directly from IPR/NPR phase diagrams, eigenstate-resolved colormaps, spatial distributions, and system-size scaling checks. No parameter is fitted to a subset of the data and then used to predict a closely related quantity: the critical exponents are extracted from the same transitions they characterize, which is a statistical fragility (notably the absence of quoted uncertainties on nu and the post-hoc choice of eigenstate windows) but not a circularity. The hyperscaling check in Eq. (12) is an internal consistency check of the scaling ansatz in Eq. (10) rather than an independent confirmation, but it is not load-bearing for the paper's main conclusions. The only self-citation (Ref. [43] by co-author Zeng) appears in the introduction as an example of mobility edges in mosaic lattices and is not load-bearing. The comparison with Ref. [72] is an external benchmark of prior work, not a self-citation chain. Accordingly, the numerical derivation is self-contained, and no step reduces by construction to its inputs; the circularity score is therefore 0.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central numerical results depend on the model parameters J11, J3, and J33, all chosen by hand, and on the eigenstate averaging windows used in the scaling analysis. No new entities are introduced. The main axioms are the standard finite-size scaling assumptions for the R-function and the ladder mapping, which is heuristic rather than derived.

free parameters (5)
  • J11/J1 = 0.1
    Chosen by hand as a representative small inter-leg hopping; the reentrant localization result is demonstrated at this value.
  • J3/J1 = 0.1
    Chosen by hand together with J11; maintained equal to J11 throughout the main analysis.
  • J33/J1 for staggered case = 2.7
    Selected from the phase diagram as the value where the reentrant localization reappears clearly; the result is shown for this specific value.
  • J33/J1 for uniform case = 0.7
    Selected from the phase diagram in Fig. 9(c) as the value where the spike structure is visible.
  • eigenstate averaging windows = e.g., m/L in [0.238,0.243] for first critical point
    Chosen post hoc from colormaps to track the localized-to-extended transition at each lambda_c; affects the extracted exponents.
assumptions (4)
  • domain assumption The finite-size scaling ansatz of Eq. 10, with a single correlation length exponent nu, applies to each transition.
    Invoked when computing nu via scaling collapse; not proved for this model with mobility edges and multiple transitions.
  • standard math The R-function defined in Eq. 9 gives an unbiased estimate of lambda_c and gamma/nu.
    Method taken from Refs. [72,73]; the curves are assumed to cross at a single common point for different sizes.
  • ad hoc to paper The system can be described as a two-leg ladder in the presence of long-range hopping.
    Used in Sec. III and V to interpret reentrant localization as competition between dimers and disorder in each leg; no rigorous mapping is provided.
  • domain assumption The hyperscaling relation 2*beta/nu + gamma/nu = 1 holds.
    Used to validate the exponents; standard for continuous transitions but not derived for this quasiperiodic model.

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Cite this review

Pith. "Pith review of Investigation of reentrant localization transition in one-dimensional quasi-periodic lattice with long-range hopping." pith.science (2026). https://pith.science/paper/TVTF5F2K

@misc{pith2026241213518,
  author       = {Pith},
  title        = {Pith review of: Investigation of reentrant localization transition in one-dimensional quasi-periodic lattice with long-range hopping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TVTF5F2K}},
  note         = {Machine review of arXiv:2412.13518}
}
read the original abstract

Reentrant localization has recently been observed in systems with quasi-periodic nearest-neighbor hopping, where the interplay between dimerized hopping and staggered disorder is identified as the driving mechanism. However, the robustness of reentrant localization in the presence of long-range hopping remains an open question. In this work, we investigate the phenomenon of reentrant localization in systems incorporating long-range hopping. Our results reveal that long-range hopping induces reentrant localization regardless of whether the disorder is staggered or uniform. We demonstrate that long-range hopping does not inherently disrupt localization; instead, under specific conditions, it facilitates the emergence of reentrant localization. Furthermore, by analyzing critical exponents, we show that the inclusion of long-range hopping modifies the critical behavior, leading to transitions that belong to distinct universality classes.

Figures

Figures reproduced from arXiv: 2412.13518 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic representation of the modulated SSH [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) and (b) show the average IPR and NPR [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a)–(e) show the distributions of IPR (red) and NPR (blue) as functions of the eigenstate index for [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a)–(d) depict the values of the function [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The critical exponent ratio (a) [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The average IPR and NPR over eigenstates with [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (a) The IPR and (b) NPR associated with [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (a)–(e) show the distributions of IPR (red) and NPR (blue) as functions of the eigenstate index for [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. (a)–(d) depict the values of the function [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The critical exponent ratio (a) [PITH_FULL_IMAGE:figures/full_fig_p009_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The [PITH_FULL_IMAGE:figures/full_fig_p010_16.png]

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Reentrant localization transition in a dimerized quasiperiodic dipolar chain

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    A dimerized quasiperiodic chain of dipolar emitters exhibits a reentrant localization transition that survives all-to-all coupling for a specifically chosen incommensurate period.

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Reviewed August 11, 2026 · model on record in the stance chip above.