Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Driven dynamics of localization phase transition in the Aubry-Andr\'{e} model with initial gapless extended states

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Linearly driving the Aubry-André model across its localization transition from a gapless extended state still obeys Kibble-Zurek and finite-time scaling, with newly derived two-regime scaling functions for the inverse participation ratio…

desk verdict A solid, workmanlike FTS extension to extended initial states in the AA model, with a new D~R^2 law; the main caveat is an imported sufficiency criterion that the numerics support but don't prove. read the letter →

arxiv 2509.06358 v1 pith:MCUCSXJN submitted 2025-09-08 cond-mat.dis-nn cond-mat.stat-mech

Xin-Yu Wang , Wen-Jing Yu , Yue-Mei Sun , Liang-Jun Zhai This is my paper · ORCID
classification cond-mat.dis-nncond-mat.stat-mech
keywords Aubry-AndrémodellocalizationtransitionKibble-Zurekmechanismfinite-timescalinggaplessinitialstateinverseparticipationratiodrivenquantumdynamicscriticality
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 asks whether the Kibble-Zurek mechanism, whose adiabatic-impulse picture normally assumes a gapped ground state, can still describe a localization phase transition when the ramp begins in a gapless extended state. It argues yes for the Aubry-André model because the extended phase's dynamical exponent $z' = 2$ is smaller than $r = z + 1/\nu = 3.37$, the criterion imported from the finite-time-scaling literature. On that basis it derives and numerically validates scaling functions for the inverse participation ratio $\mathcal{I}$ and the deviation from instantaneous ground-state energy $\mathcal{D}$: each takes one form for slow ramps and another for fast ramps, and both differ from the known results for localized initial states. The payoff is a set of explicit power laws in system size $L$ and driving rate $R$ that cold-atom experiments can test directly.

What carries the argument

The load-bearing machinery is the finite-time scaling (FTS) ansatz combined with the applicability criterion $z' < r$, where $r = z + 1/\nu$. Here $z'$ is the dynamical exponent of the gapless extended phase ($z' = 2$), and $z = 2.37$, $\nu = 1$ are the critical exponents at the localization transition, so $r = 3.37$; the criterion says that excitations generated while the state is still extended do not dominate the universal critical dynamics. The scaling ansatz treats the driving rate $R$ as a scaling field, asserting that observables at the critical point are homogeneous functions of $R L^r$ and $\varepsilon R^{-1/(r\nu)}$, with prefactors that change between the small- and large-$R$ limits. The numerical verification is a sequence of data collapses: rescaled $\mathcal{I}$ and $\mathcal{D}$ curves for different $L$ and $R$ fall onto single master curves, which is the evidence that the ansatz, not just the individual power laws, is correct.

What would settle it

Take the same linear ramp but engineer the extended phase so its dynamical exponent satisfies $z' \geq r$ (for instance by adding long-range hopping that changes the low-energy dispersion); the criterion predicts the data collapse of $\mathcal{I}$ and $\mathcal{D}$ should fail, so a clean collapse in that regime would refute the explanation. Within the Aubry-André model itself, measuring $\mathcal{D}(\varepsilon=0)$ at fixed small $R$ and fitting its $L$ and $R$ exponents would also settle the small-$R$ ansatz: any systematic departure from $\mathcal{D} \propto R^2 L^{2r-z}$ at the smallest rates would falsify Eq. (12).

Watch

Extended reading notes

Core claim

The central claim is that the driven dynamics from an extended initial state across the Aubry-André localization transition is governed by the same universal scaling as KZM, even though the initial state is gapless, provided $z' < r$. For the inverse participation ratio the scaling functions are $\mathcal{I} = L^{-s/\nu} f_1(R L^r, \varepsilon R^{-1/(r\nu)})$ for small $R$ and $\mathcal{I} = L^{-1} R^{(s-\nu)/(r\nu)} f_2(R L^r, \varepsilon R^{-1/(r\nu)})$ for large $R$; for the energy deviation they are $\mathcal{D} = R^2 L^{2r-z} g_1(R L^r, \varepsilon R^{-1/(r\nu)})$ and $\mathcal{D} = R^{z/r} g_2(R L^r, \varepsilon R^{-1/(r\nu)})$. The large-$R$ inverse participation ratio behaves like an extended state ($\mathcal{I} \propto L^{-1}$), while the small-$R$ energy deviation grows as $R^2$, a behavior the paper reports for the first time. The paper validates these forms through data collapse over system sizes from $L = 8$ to $987$ and driving rates spanning many decades.

Load-bearing premise

The load-bearing premise is that a gapless starting phase with fast-enough low-energy dynamics (dynamical exponent below $r = z + 1/\nu$) produces excitations that never matter for the universal scaling at the transition; the paper checks only that the final curves collapse consistently with this, without directly measuring whether the adiabatic and impulse stages really separate.

Editorial extensions

If this is right

  • At the critical point $\varepsilon = 0$, the inverse participation ratio crosses over from the static finite-size form $\mathcal{I} \propto L^{-s/\nu}$ at slow driving to a rate-dominated extended-like form $\mathcal{I} \propto L^{-1} R^{(s-\nu)/(r\nu)}$ at fast driving, so either regime can be used to extract $s$, $\nu$, and $r$.
  • The energy deviation $\mathcal{D}$ at $\varepsilon = 0$ obeys $\mathcal{D} \propto R^2 L^{2r-z}$ for slow ramps and $\mathcal{D} \propto R^{z/r}$ for fast ramps; the $R^2$ law is a new diagnostic for extended-initial-state ramps.
  • Because the scaling functions hold across $L = 8$ to $987$ and a wide range of $R$, experiments at moderate system sizes can use the collapse to extrapolate to the thermodynamic limit.
  • The difference between these forms and the localized-initial-state formulas (for example $\mathcal{I} \propto R^{s/(r\nu)}$) means the initial-state character is visible in the final scaling, not washed out by the critical dynamics.
  • In any model satisfying the criterion $z' < r$, the same FTS-based scaling description should apply, so the present Aubry-André results are a template for other quasiperiodic or disordered localization transitions.

Reading between the lines

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

  • A stress test the paper does not perform is to tune the extended phase until $z' \geq r$; the criterion predicts the data collapse should fail, so a model with long-range hopping would directly probe whether the collapse is caused by the criterion or by the broader applicability of FTS.
  • The small-$R$ $\mathcal{D} \propto R^2$ law looks like the leading even term in a power-series expansion of an observable that must be positive for both signs of $R$; if so, it should appear in other drives of finite-size gapped single-particle states, and its presence alone may not be unique to the gapless-start criterion.
  • Because the paper works at the single-particle level, an immediate extension is to test the same scaling functions in interacting or non-Hermitian Aubry-André models; deviations from the predicted collapse would show where many-body effects or non-Hermitian terms enter the universal scaling.
  • Experimentally, the $R^2$ law for $\mathcal{D}$ could be checked by measuring the overlap with the instantaneous ground state during slow ramps in an ultracold-atom realization, giving a time-resolved test of the scaling ansatz rather than only a final-state measurement.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the driven dynamics of the localization-delocalization transition in the one-dimensional Aubry-André (AA) model, with the initial state chosen as a gapless extended state rather than the localized states used in earlier work. The authors linearly ramp the quasiperiodic potential strength across the critical point and compute the inverse participation ratio (I) and the deviation from the instantaneous ground-state energy (D). They claim that the driven dynamics starting from the extended state satisfies the criterion z' < r, where z' is the dynamical exponent of the extended phase and r = z + 1/ν is the combination of critical exponents that controls finite-time scaling (FTS). They propose two-branch scaling forms for I and D, valid in the small- and large-driving-rate regimes, and validate these forms by data collapse in Figs. 3, 5, 6 and 7. The exponents appearing in the scaling functions are not fitted to the data but are combinations of known critical exponents (s = 0.333, ν = 1, z = 2.37, r = 3.37). The paper reports, for the first time, a quadratic dependence D ∝ R² in the small-R regime.

Significance. If the central claim is correct, the paper extends the applicability of Kibble-Zurek theory and finite-time scaling to localization transitions starting from gapless extended states, a regime previously thought to violate the adiabatic-impulse hypothesis. The scaling forms for I and D provide concrete, testable predictions that could be checked in ultracold-atom realizations of the AA model. A notable strength is that the scaling exponents are not free parameters fitted to the collapses; they are derived from independent static critical exponents, which makes the validation nontrivial. Another strength is the systematic numerical verification across system sizes L = 8 to 987 and a wide range of driving rates. The work thus goes beyond a purely empirical scaling analysis and offers an experimentally accessible framework.

major comments (3)
  1. [Sec. III A, Eq. (3)] The central claim that the driven extended-state dynamics satisfies the criterion for KZM/FTS applicability rests entirely on the imported inequality z' < r from Ref. [97], but the paper does not establish that this criterion is sufficient for the AA model. The criterion was derived for Dirac-like gapless systems (linear dispersion), whereas the AA extended phase has z' = 2, indicating a different low-energy structure. The authors check the inequality and then observe data collapse, but the collapse is a necessary rather than a sufficient test: initial-phase excitations might contribute with the same power-law forms over the simulated ranges and still produce a collapse while the universal critical-region interpretation is wrong. A direct test is needed, for example by varying the starting point λ₀ within the extended phase (changing the initial-state gap) or by comparing with a case where z' > r to see whether the scaling forms break down. Without such a test, the paper has not demonstrated the sufficiency of the criterion for the AA model; it has only shown consistency with an assumed sufficiency.
  2. [Sec. III B, Eq. (7)] The two-branch scaling ansatz for I is introduced as a reconciliation of the small-R and large-R asymptotic behaviors, but it is not derived from the FTS framework. In particular, it is unclear whether the two branches are asymptotic limits of a single homogeneous scaling function or represent genuinely different scaling forms. The paper also does not provide a quantitative crossover condition between the small-R and large-R regimes; the boundary between them is identified by eye in Fig. 3(a). A derivation or at least a more explicit statement of how the crossover scale behaves (e.g., R_c ∝ L^{-r}) would make the scaling claim stronger and more testable.
  3. [Sec. III C, Eq. (12)] The small-R scaling D ∝ R² L^{2r-z} is justified by a parity argument: 'D must remain positive regardless of whether R is positive or negative, so the expansion should contain only even powers of R.' However, the driving protocol always has R > 0 by construction (λ increases linearly with time), so positivity does not force the odd terms to vanish. If the authors intend to invoke adiabatic perturbation theory for a finite system, they should state the explicit condition under which the quadratic term dominates, especially because the initial state is in a gapless phase and the adiabatic theorem does not directly apply. As written, the derivation of this scaling law is not convincing, although the numerical data are consistent with it.
minor comments (4)
  1. [Fig. 7 caption] The caption of Fig. 7 refers to rescaling according to Eq. (7) and Eq. (8), but the data shown are for the dynamic deviation D and should be rescaled according to Eq. (12) and Eq. (13), respectively. This typo should be corrected.
  2. [Sec. III A] The phrase 'the system satisfies the following precondition z' < r' is stated without explanation of the physical mechanism behind the criterion. A reader unfamiliar with Ref. [97] is left with no intuition for why an inequality between two dynamical exponents guarantees that initial-phase excitations are irrelevant. At least a qualitative description of the mechanism would improve the readability.
  3. [Figs. 3, 5, 6, 7] The numerical data are averaged over only 10 samples of the phase φ, and no error bars or standard deviations are shown. Given that the scaling collapses are the central evidence, a quantitative measure of the collapse quality (e.g., residuals or confidence intervals for the power-law fits) would strengthen the claims. At present, the collapses are assessed by eye.
  4. [Sec. III C, after Eq. (12)] The text states that 'for sufficiently large L, Eq. (12) can be simplified to D = R^{z/r} g₃(...)', but Eq. (12) has two branches. It is not explained why the small-R branch becomes negligible in the large-L limit; the crossover between the two branches should be quantified, otherwise the simplification appears to select one branch arbitrarily.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: scaling forms are FTS-based with fixed independent exponents, and the KZM/FTS applicability criterion is imported from an external work rather than inferred from the data being predicted.

full rationale

The paper's central claims are the two-branch finite-time scaling (FTS) forms for I and D, Eqs. (7) and (12). These are not obtained by fitting the collapse; the exponents entering them (s = 0.333, nu = 1, z = 2.37, r = 3.37) come from known AA critical properties and from independent references, and the collapse is then used as a consistency check. The applicability criterion z' < r in Eq. (3) is explicitly imported from Ref. [97], which is not authored by the present group, so the load-bearing sufficiency condition is external rather than a self-citation loop. The measured value z' = 2 enters only into this criterion check, not into the scaling functions of I and D. The small-R D proportional-to-R^2 behavior is first read off the data and then rationalized by parity and dimensional arguments, but the paper presents it as an observed scaling law with an FTS-based explanation rather than as a parameter-free prediction, so this is not a fitted input disguised as a prediction. Self-citations such as [25] and [107] provide known critical exponents and prior localized-initial-state results used for comparison; they do not by themselves force the extended-initial-state scaling forms. The epsilon-dependence collapses at fixed R L^r and R L^{-1/(r nu)} provide additional constraints beyond the asymptotic power-law fits, so the validation is not tautological. Consequently, no step in the derivation chain reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities or ad hoc parameters. The central claim rests on the imported z' < r criterion from Ref. [97], on standard critical exponents, and on scaling ansatze that are validated by numerical data collapse. The only fitted quantity is z', which is an input to the applicability check rather than to the scaling functions.

free parameters (1)
  • z' (dynamical exponent of extended state) = 2
    Obtained from power-law fitting of the energy gap versus system size in Fig. 1. Used to evaluate the criterion z' < r. Not part of the scaling functions themselves, but an input to the central applicability claim.
assumptions (6)
  • domain assumption Sufficiency of the z' < r criterion for KZM/FTS applicability to gapless initial states (Ref. [97])
    The paper relies on this criterion to argue that driven dynamics from extended states can be described by FTS. It is not proved in this paper; it is imported from prior work.
  • domain assumption Static scaling form I = L^{-s/ν} f(ε L^{1/ν}) with s = 0.333 and ν = 1 (Ref. [25])
    The static finite-size scaling of the inverse participation ratio and the critical exponents are taken from earlier literature and used to construct the dynamic scaling forms.
  • domain assumption Critical exponents ν = 1, z = 2.37, and r = 3.37 for the AA model
    These values are inputs from previous numerical studies and are used to form the predicted power-law exponents in the driving-rate scaling.
  • ad hoc to paper The scaling ansatze in Eqs. (7) and (12) are the correct homogeneous forms for the driven dynamics
    The paper proposes these scaling forms based on FTS intuition and asymptotic limits; they are not derived from a first-principles equation. Their validation is the data collapse.
  • standard math The Fibonacci approximation of γ with periodic boundary conditions is a valid representation of the quasiperiodic potential
    This is a standard numerical technique for simulating the AA model with periodic boundary conditions.
  • domain assumption Single-particle Schrödinger evolution is sufficient
    The Hamiltonian is quadratic (non-interacting hard-core bosons), and the dynamics reduces to single-particle evolution. This is standard for the non-interacting AA model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Driven dynamics of localization phase transition in the Aubry-Andr\'{e} model with initial gapless extended states." pith.science (2026). https://pith.science/paper/MCUCSXJN

@misc{pith2026250906358,
  author       = {Pith},
  title        = {Pith review of: Driven dynamics of localization phase transition in the Aubry-Andr\'e model with initial gapless extended states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCUCSXJN}},
  note         = {Machine review of arXiv:2509.06358}
}
abstract

Recently, the driven dynamics of localization phase transitions have garnered growing interest. However, studies so far have mainly considered initial localized states, whose driven dynamics follow the Kibble-Zurek mechanism (KZM). In this study, we investigate the driven dynamics of the localization phase transition in the Aubry-Andr\'e (AA) model starting from a gapless extended state, which violates the adiabatic-impulse scenario of KZM. By linearly driving the quasiperiodic potential strength across the critical point, we numerically simulate the driven dynamics and analyze the scaling behavior of both the inverse participation ratio ($\mathcal{I}$) and the dynamic deviation from the instantaneous ground state energy $(\mathcal{D})$. We demonstrate that the driven dynamics starting from initially extended states satisfies the criterion for the applicability of KZM and its extension, finite-time scaling (FTS). The scaling functions governing the driven dynamics of both $\mathcal{I}$ and $\mathcal{D}$ have been derived based on FTS and numerically validated. We found that the scaling functions exhibit significant differences at large $R$ and small $R$, and also differ considerably from the scaling functions when the initial state is localized, highlighting the crucial role of initial state behavior. The established scaling laws remain robust across a wide range of system sizes and driving rates, providing testable predictions for experimental realizations.

Figures

Figures reproduced from arXiv: 2509.06358 by the authors.

Figure 1
Figure 1. FIG. 1. Energy gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Static [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The snapshots of states during a slow ramp taken at [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Curves of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Curves of [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Universal Driven Critical Dynamics near the Boundary

    cond-mat.stat-mech 2025-09 conditional novelty 6.0 of 10

    Driven critical dynamics near boundaries of the Ising model obey boundary finite-time scaling in most cases, but cooling through the ordinary boundary universality class produces an anomalous logarithmic dependence on...

Reference graph

Works this paper leans on

116 extracted references · 62 canonical work pages · cited by 1 Pith paper

  1. [97]

    Zeng, Y.-K

    Z. Zeng, Y.-K. Yu, Z.-X. Li, Z.-X. Li, and S. Yin, Finite-time scaling beyond the Kibble-Zurek prerequi- site in Dirac systems, Nature Communications16, 6181 (2025)

    work page 2025

  2. [1]

    P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev.109, 1492 (1958)

    1958

  3. [2]

    Abrahams, P.W

    E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett.42, 673-676 (1979)

    1979

  4. [3]

    P. A. Lee and T. V. Ramakrishnan, Disordered elec- tronic systems, Rev. Mod. Phys.57, 287 (1985)

    1985

  5. [4]

    Kramer, A

    B. Kramer, A. MacKinnon, Localization: theory and experiment, Rep. Prog. Phys.56, 1469 (1993)

    1993

  6. [5]

    Schwartz, G

    T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Transport and Anderson localization in disordered two- dimensional photonic lattices, Nature446, 7131 (2007)

    2007

  7. [6]

    Evers, A.D

    F. Evers, A.D. Mirlin, Anderson transitions, Rev. Mod. Phys.,80, 1355-1417 (2008). 7

    2008

  8. [7]

    Segev, Y

    M. Segev, Y. Silberberg, and D. N. Christodoulides, An- derson localization of light, Nature Photonics7, 197 (2013)

    2013

Show all 116 references
  1. [8]

    D. S. Wiersma, Disordered photonics, Nature Photonics 7, 188 (2013)

    2013

  2. [9]

    Yamilov, S

    A. Yamilov, S. E. Skipetrov, T. W. Hughes, M. Minkov, Z. Yu, and H. Cao, Anderson localization of electromag- netic waves in three dimensions, Nat. Phys.19, 1308 (2023)

    2023

  3. [10]

    Aubry and G

    S. Aubry and G. Andr´ e, Analyticity breaking and ander- son localization in incommensurate lattices, Ann. Israel Phys. Soc.3, 133 (1980)

    1980

  4. [11]

    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

  5. [12]

    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

  6. [13]

    Schirmann, S

    J. Schirmann, S. Franca, F. Flicker, and A. G. Grushin, Physical Properties of an Aperiodic Monotile with Graphene-like Features, Chirality, and Zero Modes, Phys. Rev. Lett.132, 086402 (2024)

    2024

  7. [14]

    Gon¸ calves, B

    M. Gon¸ calves, B. Amorim, E. V. Castro, and P. Ribeiro, Critical Phase Dualities in 1D Exactly Solvable Quasiperiodic Models, Phys. Rev. Lett.131, 186303 (2024)

    2024

  8. [15]

    ˇSuntajs, T

    J. ˇSuntajs, T. c. v. Prosen, and L. Vidmar, Localization challenges quantum chaos in the finite two-dimensional Anderson model, Phys. Rev. B107, 064205 (2023)

    2023

  9. [16]

    Agrawal, S

    U. Agrawal, S. Gopalakrishnan, and R. Vasseur, Uni- versality and quantum criticality in quasiperiodic spin chains, Nat. Commun.11, 2225 (2020)

    2020

  10. [17]

    Agrawal, S

    U. Agrawal, S. Gopalakrishnan, and R. Vasseur, Quan- tum Criticality in the 2D Quasiperiodic Potts Model, Phys. Rev. Lett.125, 265702 (2020)

    2020

  11. [18]

    Goblot, A

    V. Goblot, A. ˇStrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaˆ ıtre, L. L. Gratiet, A. Harouri, I. Sagnes, S. Ravets, A. Amo, J. Bloch, and O. Zilberberg, Emer- gence of criticality through a cascade of delocalization transitions in quasiperiodic chains, Nat. Phys.16, 832 (2020)

    2020

  12. [19]

    S. Roy, S. Chattopadhyay, T. Mishra, and S. Basu, Crit- ical analysis of the reentrant localization transition in a one-dimensional dimerized quasiperiodic lattice, Phys. Rev. B105, 214203 (2022)

    2022

  13. [20]

    Agrawal, R

    U. Agrawal, R. Vasseur, and S. Gopalakrishnan, Quasiperiodic many-body localization transition in di- mensiond>1, Phys. Rev. B106, 094206 (2022)

    2022

  14. [21]

    Zhou, Y.-J

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

    2023

  15. [22]

    Smith, J

    A. Smith, J. Knolle, R. Moessner, D. L. Kovrizhin, Ab- sence of ergodicity without quenched disorder: From quantum disentangled liquids to many-body localiza- tion, Phys. Rev. Lett.119, 176601 (2017)

    2017

  16. [23]

    Biddle, D

    J. Biddle, D. J. Priour, B. Wang, and S. Das Sarma, Lo- calization in one-dimensional lattices with non-nearest- neighbor hopping: Generalized Anderson and Aubry- Andr´ e models, Phys. Rev. B83, 075105 (2011)

    2011

  17. [24]

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological States and Adiabatic Pumping in Quasicrystals, Phys. Rev. Lett.109, 106402 (2012)

    2012

  18. [25]

    Bu, L.-J

    X. Bu, L.-J. Zhai, and S. Yin, Quantum criticality in the disordered Aubry-Andr´ e model, Phys. Rev. B106, 214208 (2022)

    2022

  19. [26]

    Lv, Y.-B

    T. Lv, Y.-B. Liu, T.-C. Yi, L. Li, M. Liu, and W.-L. You, Exploring unconventional quantum criticality in thep- wave-paired Aubry-Andr´ e-Harper model, Phys. Rev. B 106, 144205 (2022)

    2022

  20. [27]

    Sahoo, A

    A. Sahoo, A. Saha, and D. Rakshit, Stark localization near Aubry-Andr´ e criticality, Phys. Rev. B111, 024205 (2025)

    2025

  21. [28]

    C. Y. Guo, Multiple intermediate phases in the inter- polating Aubry-Andr´ e-Fibonacci model, Phys. Rev. B 109, 174203 (2024)

    2024

  22. [29]

    Yi, Y.-Y

    T.-C. Yi, Y.-Y. Fang, W. Chen, W.-L. You, and Y. Zhang, Unveiling quantum criticality of disordered Aubry-Andr´ e-Harper models via typical fidelity suscep- tibility, Phys. Rev. A112, 023308 (2025)

    2025

  23. [30]

    S. Saha, S. K. Maiti, and S. Karmakar, Multiple mobil- ity edges in a 1D Aubry chain with Hubbard interaction in presence of electric field: Controlled electron trans- port, Physica E83, 358 (2016)

    2016

  24. [31]

    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

  25. [32]

    Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, One-dimensional quasiperiodic mo- saic lattice with exact mobility edges, Phys. Rev. Lett. 125, 196604 (2020)

    2020

  26. [33]

    H.-T. Hu, Y. Chen, X. Lin, A.-M. Guo, Z. Lin, and M. Gong, Exact mobility edges in quasiperiodic net- work models with slowly varying potentials, Phys. Rev. B112, 054201 (2025)

    2025

  27. [34]

    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

  28. [35]

    Gao, Ivan M

    J. Gao, Ivan M. Khaymovich, X.-W. Wang, Z.-S. Xu, A. Iovan, G. Krishna, J. Jieensi, A. Cataldo, A. V. Bal- atsky, V. Zwiller, and A. W. Elshaari, Probing multi- mobility edges in quasiperiodic mosaic lattices, Sci. Bull.09, 030 (2024)

    2024

  29. [36]

    Banerjee, S

    S. Banerjee, S. R. Padhi, T. Mishra, Emergence of distinct exact mobility edges in a quasiperiodic chain, arXiv:2503.19834 (2025)

    2025 arXiv

  30. [37]

    S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Many-body localization in a quasiperiodic system, Phys. Rev. B87, 134202 (2013)

    2013

  31. [38]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice, Science 349, 842 (2015)

    2015

  32. [39]

    Mastropietro, Localization of Interacting Fermions in the Aubry-Andr´ e Model, Phys

    V. Mastropietro, Localization of Interacting Fermions in the Aubry-Andr´ e Model, Phys. Rev. Lett.115, 180401 (2015)

    2015

  33. [40]

    Khemani, D

    V. Khemani, D. N. Sheng, and D. A. Huse, Two univer- sality classes for the many-body localization transition, Phys. Rev. Lett.119, 075702 (2017)

    2017

  34. [41]

    Zhang and H

    S.-X. Zhang and H. Yao, Universal Properties of Many- Body Localization Transitions in Quasiperiodic Sys- tems, Phys. Rev. Lett.121, 206601 (2018)

    2018

  35. [42]

    Rispoli, A

    M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. L´ eonard, and M. Greiner, Quantum critical behaviour 8 at the many-body localization transition, Nature573, 385 (2019)

    2019

  36. [43]

    Weiner, F

    F. Weiner, F. Evers, and S. Bera, Slow dynamics and strong finite-size effects in many-body localization with random and quasiperiodic potentials, Phys. Rev. B100, 104204 (2019)

    2019

  37. [44]

    S. Xu, X. Li, Y.-T. Hsu, B. Swingle, and S. Das Sarma, Buttery effect in interacting Aubry-Andr´ e model: Ther- malization, slow scrambling, and many-body localiza- tion, Phys. Rev. Research1, 032039 (2019)

    2019

  38. [45]

    L.-J. Zhai, S. Yin, and G.-Y. Huang, Many-body local- ization in a non-hermitian quasiperiodic system, Phys. Rev. B102, 064206 (2020)

    2020

  39. [46]

    Cookmeyer, J

    T. Cookmeyer, J. Motruk, and J. E. Moore, Critical properties of the ground-state localization- delocalization transition in the many-particle Aubry- Andr´ e model, Phys. Rev. B101, 174203 (2020)

    2020

  40. [47]

    ˇStrkalj, E

    A. ˇStrkalj, E. V. H. Doggen, I. V. Gornyi, and O. Zil- berberg, Many-body localization in the interpolating Aubry-Andr´ e-Fibonacci model, Phys. Rev. Research3, 033257 (2021)

    2021

  41. [48]

    Tang, G.-Q

    L.-Z. Tang, G.-Q. Zhang, L.-F. Zhang, D.-W. Zhang, Localization and topological transitions in non- Hermitian quasiperiodic lattices, Phys. Rev. A,103, 033325 (2021)

    2021

  42. [49]

    ˇStrkalj, E.V.H

    A. ˇStrkalj, E.V.H. Doggen, C. Castelnovo, Coexis- tence of localization and transport in many-body two- dimensional Aubry-Andr´ e models, Phys. Rev. B,106, 184209 (2022)

    2022

  43. [50]

    Prasad and A

    Y. Prasad and A. Garg, Single-particle excitations across the localization and many-body localization tran- sition in quasiperiodic systems, Phys. Rev. B109, 094204 (2024)

    2024

  44. [51]

    Huang, D

    K. Huang, D. Vu, S. Das Sarma, X. Li, Interaction- enhanced many-body localization in a generalized Aubry-Andr´ e model, Phys. Rev. Research,6, L022054 (2024)

    2024

  45. [52]

    Chakrabarty, S

    A. Chakrabarty, S. Banerjee, T. Mishra, and S. Datta, Emergence of non-trivial phases in interacting non- Hermitian quasiperiodic chains with power-law hopping, arXiv:2508.14724 (2025)

    2025 arXiv

  46. [53]

    Q.-B. Zeng, S. Chen, R. L¨ u, Anderson localization in the non-Hermitian Aubry-Andr´ e-Harper model with physi- cal gain and loss, Phys. Rev. A,95, 062118 (2017)

    2017

  47. [54]

    Jiang, L.-J

    H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen, Interplay of non-hermitian skin effects and Anderson lo- calization in nonreciprocal quasiperiodic lattices, Phys. Rev. B100, 054301 (2019)

    2019

  48. [55]

    Longhi, Metal-insulator phase transition in a non- Hermitian Aubry-Andr´ e-Harper model, Phys

    S. Longhi, Metal-insulator phase transition in a non- Hermitian Aubry-Andr´ e-Harper model, Phys. Rev. B 100, 125157 (2019)

    2019

  49. [56]

    Longhi, Non-Hermitian maryland model, Phys

    S. Longhi, Non-Hermitian maryland model, Phys. Rev. B103, 224206 (2021)

    2021

  50. [57]

    Y. Liu, Y. Wang, X.-J. Liu, Q. Zhou, and S. Chen, Exact mobility edges, PT-symmetry breaking, and skin effect in one-dimensional non-hermitian quasicrystals, Phys. Rev. B103, 014203 (2021)

    2021

  51. [58]

    Guo, C.-H

    C.-X. Guo, C.-H. Liu, X.-M. Zhao, Y. Liu, and S. Chen, Exact solution of non-Hermitian systems with general- ized boundary conditions: Size-dependent boundary ef- fect and fragility of the skin effect, Phys. Rev. Lett.127, 116801 (2021)

    2021

  52. [59]

    Zhai, G.-Y

    L.-J. Zhai, G.-Y. Huang, and S. Yin, Cascade of the delocalization transition in a non-Hermitian interpolat- ing Aubry-Andr´ e-Fibonacci chain, Phys. Rev. B104, 014202 (2021)

    2021

  53. [60]

    Lin, T.-Y Li, L

    Q. Lin, T.-Y Li, L. Xiao, K. -K Wang, W. Yi, P. Xue, Topological Phase Transitions and Mobility Edges in Non-Hermitian Quasicrystals, Phys. Rev. Lett.129, 113601 (2022)

    2022

  54. [61]

    A. P. Acharya and S. Datta, Localization transitions in a non-Hermitian quasiperiodic lattice, Phys. Rev. B 109, 024203 (2024)

    2024

  55. [62]

    C.-X. Guo, L. H. Su, Y. L. Wang, L. Li, J. Z. Wang, X. H Ruan, Y. J. Du, D. N. Zheng, S. Chen, and H. P Hu, Scale-tailored localization and its observation in non- Hermitian electrical circuits, Nat. Commun.15, 9120 (2024)

    2024

  56. [63]

    C.-C. Zeng, Z. Cai, G.-H. Wang, G. Sun, Fidelity and criticality in the nonreciprocal Aubry-Andr´ e-Harper model, Europhys. Lett.,149, 38001 (2025)

    2025

  57. [64]

    Sun, X.-Y

    Y.-M. Sun, X.-Y. Wang and L.-J. Zhai, Hybrid scal- ing properties of the localization transition in a non- Hermitian disordered Aubry-Andr´ e model, Phys. Rev. B110, 054202 (2024)

    2024

  58. [65]

    Chen, G.-Q

    R.-J. Chen, G.-Q. Zhang, Z. Li, D.-W. Zhang, Mobil- ity rings in a non-Hermitian non-Abelian quasiperiodic lattice, Phys. Rev. A,112, 013320 (2025)

    2025

  59. [66]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nature Phys.8, 277 (2012)

    2012

  60. [67]

    Mukherjee, A

    S. Mukherjee, A. Spracklen, M. Valiente, E. Anders- son, P. ¨Ohberg, N. Goldman, and R. R. Thomson, Ex- perimental observation of anomalous topological edge modes in a slowly driven photonic lattice, Nature Com- mun.8, 13918 (2017)

    2017

  61. [68]

    Gross and I

    C. Gross and I. Bloch, Quantum simulations with ultra- cold atoms in optical lattices, Science357, 995 (2017)

    2017

  62. [69]

    Ebadi, T

    S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Se- meghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuleti´ c, M. D. Lukin, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227-232 (2021)

    2021

  63. [70]

    L. Xiao, D. -K. Qu, K. -K. Wang, H. -W Li, J.-Y Dai, B. D´ ora , M. Heyl, R. Moessner, W. Yi, and P. Xue, Non- Hermitian Kibble-Zurek Mechanism with Tunable Com- plexity in Single-Photon Interferometry, PRX QUAN- TUM2, 020313 (2021)

    2021

  64. [71]

    Bedow, E

    J. Bedow, E. Mascot, and D. K. Morr, Emergence and manipulation of non-equilibrium Yu-Shiba-Rusinov states. Commun. Phys.5, 281 (2022)

    2022

  65. [72]

    Perfetto, F

    G. Perfetto, F. Carollo, M. Magoni, I. Lesanovsky, Designing nonequilibrium states of quantum mat- ter through stochastic resetting, Phys. Rev. B104, L180302 (2021)

    2021

  66. [73]

    Mankowsky, M

    R. Mankowsky, M. F¨ orst, A. Cavalleri, Non-equilibrium control of complex solids by nonlinear phononics, Rep. Prog. Phys.79, 064503 (2016)

    2016

  67. [74]

    Popkov, R

    V. Popkov, R. Livi, Manipulating energy and spin cur- rents in non-equilibrium systems of interacting qubits, New J. Phys.15, 023030 (2013)

    2013

  68. [75]

    Giudici, M.D

    G. Giudici, M.D. Lukin, and H. Pichler, Dynamical Preparation of Quantum Spin Liquids in Rydberg Atom Arrays, Phys. Rev. Lett.129, 090401 (2022)

    2022

  69. [76]

    Cheng, C

    Y. Cheng, C. Li, and H. Zhai, Variational approach to quantum spin liquid in a Rydberg atom simulator, New J. Phys.,25, 033010 (2023). 9

    2023

  70. [77]

    Keesling, A

    A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pich- ler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg sim- ulator, Nat...

    2019

  71. [78]

    Dupont and J

    M. Dupont and J. E. Moore, Quantum criticality using a superconducting quantum processor, Phys. Rev. B106, L041109 (2022)

    2022

  72. [79]

    Reichhardt, A

    C.J.O. Reichhardt, A. del Campo, C. Reichhardt, Kibble-Zurek mechanism for nonequilibrium phase tran- sitions in driven systems with quenched disorder, Com- mun. Phys.5, 173 (2022)

    2022

  73. [80]

    A. D. King, J. Raymond, T. Lanting, R. Harris, A. Zucca, F. Altomare, A. J. Berkley, K. Boothby, S. Ejtemaee, C. Enderud, E. Hoskinson, S. Huang, E. Ladizinsky, A. J. R. MacDonald, G. Marsden, R. Molavi, T. Oh, G. Poulin-Lamarre, M. Reis, C. Rich, Y. Sato, N. Tsai, M. Volkmann...

    2023

  74. [81]

    Soto Garcia and N

    J. Soto Garcia and N. Chepiga, Resolving chiral transi- tions in one-dimensional Rydberg arrays with quantum Kibble-Zurek mechanism and finite-time scaling, Phys. Rev. B110, 125113 (2024)

    2024

  75. [82]

    T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen.9, 1387 (1976)

    1976

  76. [83]

    W. H. Zurek, Cosmological experiments in superfluid helium?, Nature317, 505 (1985)

    1985

  77. [84]

    W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett.95, 105701 (2005)

    2005

  78. [85]

    Dziarmaga, Dynamics of a quantum phase transition: exact solution of the quantum Ising model, Phys

    J. Dziarmaga, Dynamics of a quantum phase transition: exact solution of the quantum Ising model, Phys. Rev. Lett.95, 245701 (2005)

    2005

  79. [86]

    Dziarmaga, Dynamics of a quantum phase transition and relaxation to a steady state, Adv

    J. Dziarmaga, Dynamics of a quantum phase transition and relaxation to a steady state, Adv. Phys.59, 1063 (2010)

    2010

  80. [87]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Ven- galattore, Colloquium: nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011)

    2011

  81. [88]

    Yin, G.-Y

    S. Yin, G.-Y. Huang, C.-Y. Lo, and P. Chen, Kibble- Zurek scaling in the Yang-Lee edge singularity, Phys. Rev. Lett.118, 065701 (2017)

    2017

  82. [89]

    Zhai, H.-Y

    L.-J. Zhai, H.-Y. Wang, and S. Yin, Hybridized Kibble- Zurek scaling in the driven critical dynamics across an overlapping critical region, Phys. Rev. B97, 134108 (2018)

    2018

  83. [90]

    Shu, S.-K

    Y.-R. Shu, S.-K. Jian, A. W. Sandvik, and S. Yin, Equilibration of topological defects near the deconfined quantum multicritical point, Nat. Commun.16, 3402 (2025)

    2025

  84. [91]

    S. Deng, G. Ortiz, and L. Viola, Dynamical non-ergodic scaling in continuous finite-order quantum phase tran- sitions, Europhys. Lett.84, 67008 (2008)

    2008

  85. [92]

    Polkovnikov and V

    A. Polkovnikov and V. Gritsev, Breakdown of the adi- abatic limit in low-dimensional gapless systems, Nat. Phys.4, 477 (2008)

    2008

  86. [93]

    Pellegrini, S

    F. Pellegrini, S. Montangero, G. E. Santoro, and R. Fazio, Adiabatic quenches through an extended quan- tum critical region, Phys. Rev. B77, 140404 (2008)

    2008

  87. [94]

    S. Gong, F. Zhong, X. Huang, and S. Fan, Finite-time scaling via linear driving, New J. Phys.12, 043036 (2010)

    2010

  88. [95]

    Huang, S

    Y. Huang, S. Yin, B. Feng, and F. Zhong, Kibble-Zurek mechanism and finite-time scaling, Phys. Rev. B90, 134108 (2014)

    2014

  89. [96]

    B. Feng, S. Yin, and F. Zhong, Theory of driven nonequilibrium critical phenomena, Phys. Rev. B94, 144103 (2016)

    2016

  90. [98]

    Morales-Molina, E

    L. Morales-Molina, E. Doerner, C. Danieli, and S. Flach, Resonant extended states in driven quasiperiodic lat- tices: Aubry-Andre localization by design, Phys. Rev. A90, 043630 (2014)

    2014

  91. [99]

    Bairey, G

    E. Bairey, G. Refael, and N. H. Lindner, Driving in- duced many-body localization, Phys. Rev. B96, 020201 (2017)

    2017

  92. [100]

    Zhou, Floquet engineering of topological localization transitions and mobility edges in one-dimensional non- Hermitian quasicrystals, Phys

    L. Zhou, Floquet engineering of topological localization transitions and mobility edges in one-dimensional non- Hermitian quasicrystals, Phys. Rev. Research3, 033184 (2021)

    2021

  93. [101]

    Tiwari, D

    V. Tiwari, D. S. Bhakuni, and A. Sharma, Dynami- cal localization and slow dynamics in quasiperiodically driven quantum systems, Phys. Rev. B109, L161104 (2024)

    2024

  94. [102]

    C. Yang, Y. C. Wang, P. Wang, X. L. Gao, and S. Chen, Dynamical signature of localization-delocalization tran- sition in a one-dimensional incommensurate lattice, Phys. Rev. B95, 184201 (2017)

    2017

  95. [103]

    Z. Xu, H. Huangfu, Y. Zhang, S. Chen, Dynamical ob- servation of mobility edges in one-dimensional incom- mensurate optical lattices, New J. Phys.22, 013036 (2020)

    2020

  96. [104]

    Z. H. Xu and S. Chen, Dynamical evolution in a one- dimensional incommensurate lattice with PT symmetry, Phys. Rev. A103, 043325 (2021)

    2021

  97. [105]

    Cheng, S

    J.-Q. Cheng, S. Yin, and D.-X. Yao, Dynamical localiza- tion transition in the non-Hermitian lattice gauge the- ory, Commun. Phys.7, 58 (2024)

    2024

  98. [106]

    Sinha, M

    A. Sinha, M. M. Rams, and J. Dziarmaga, Kibble- zurek mechanism with a single particle: Dynamics of the localization-delocalization transition in the Aubry- Andr´ e model, Phys. Rev. B99, 094203 (2019)

    2019

  99. [107]

    Zhai, G.-Y

    L.-J. Zhai, G.-Y. Huang, and S. Yin, Nonequilibrium dynamics of the localization-delocalization transition in the non-Hermitian Aubry-Andr´ e model, Phys. Rev. B 106, 014204 (2022)

    2022

  100. [108]

    Zhai, L.-L

    L.-J. Zhai, L.-L. Hou, Q. Gao, and H.-Y. Wang, Kibble- Zurek scaling of the dynamical localization-skin effect phase transition in a non-Hermitian quasi-periodic sys- tem under the open boundary condition, Front. Phys. 10, 1098551 (2022)

    2022

  101. [109]

    Bu, L.-J

    X. Bu, L.-J. Zhai, and S. Yin, Kibble-Zurek scaling in one-dimensional localization transitions, Phys. Rev. A 108, 023312 (2023)

    2023

  102. [110]

    Liang, L.-Z

    E.-W. Liang, L.-Z. Tang, and D.-W. Zhang, Quan- tum criticality and Kibble-Zurek scaling in the Aubry- Andr´ e-Stark model, Phys. Rev. B110, 024207 (2024)

    2024

  103. [111]

    Sun, X.-Y

    Y.-M. Sun, X.-Y. Wang, and L.-J. Zhai, Non- equilibrium dynamics of localization phase transition in the non-Hermitian disorder-Aubry-Andr´ e model, Phys. 10 Rev. A112, 022204 (2025)

    2025

  104. [112]

    Wei, Fidelity susceptibility in one-dimensional disordered lattice models, Phys

    B.-B. Wei, Fidelity susceptibility in one-dimensional disordered lattice models, Phys. Rev. A99, 042117 (2019)

    2019

  105. [113]

    Bauer, T

    J. Bauer, T. M. Chang, and J. L. Skinner, Correlation length and inverse-participation-ratio exponents and multifractal structure for anderson localization, Phys. Rev. B42, 8121 (1990)

    1990

  106. [114]

    Y. V. Fyodorov and A. D. Mirlin, Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems, Phys. Rev. Lett.69, 1093 (1992)

    1992

  107. [115]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno and M. In- guscio, Anderson localization of a non-interacting Bose- Einstein condensate, Nature453, 895-898 (2008)

    2008

  108. [116]

    Billy, V

    J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Cl´ ement, L. Sanchez-Palencia, P. Bouyer, A. Aspect, Direct observation of Anderson localization of matter waves in a controlled disorder, Nature453, 891-894(2008)

    2008

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.