Pith. sign in

REVIEW 2 major objections 1 minor 126 references

Non-self-averaging topological Anderson insulator

T0 review · 2 major / 1 minor · reviewed 2026-07-02 · grok-4.3

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

Pith's one-line read Long-range correlated disorder turns topological Anderson insulators non-self-averaging.

desk verdict Long-range correlated disorder may create a non-self-averaging topological Anderson insulator, but the evidence that relative variance of the Lyapunov exponent stays finite in the thermodynamic limit is not yet convincing. read the letter →

arxiv 2607.00892 v1 pith:6MOI2Y2N submitted 2026-07-01 cond-mat.dis-nn

Zheng-Wei Zuo , Jun-Chong Liu This is my paper · ORCID
classification cond-mat.dis-nn
keywords topologicalAndersoninsulatorlong-rangecorrelateddisordernon-self-averagingLyapunovexponentdisorderedphasesphasediagramcentrallimittheorem
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 shows that long-range correlated disorder produces topological Anderson states whose properties do not self-average. In this regime the phase diagram for any one sample can differ markedly from the diagram obtained by averaging over many disorder realizations. The effect is diagnosed by a relative variance of the Lyapunov exponent that remains finite as system size grows to infinity together with non-Gaussian statistics for that exponent. Consequently the central limit theorem fails and single-sample topological features deviate from ensemble predictions. A sympathetic reader would care because standard theoretical descriptions of disordered topological phases rest on the assumption that averages describe what any given sample will do.

What carries the argument

The Lyapunov exponent and the statistics of its distribution over disorder realizations, which serve as the diagnostic for whether topological properties self-average.

What would settle it

A numerical computation in which the relative variance of the Lyapunov exponent is shown to decay to zero with increasing system size for a long-range correlated disorder potential.

Watch

Extended reading notes

Core claim

Long-range correlated disorder induces a statistical phase called the non-self-averaging topological Anderson insulator in which the topological properties of a single disordered sample deviate from the ensemble average. This non-self-averaging is identified by the relative variance of the Lyapunov exponent remaining finite and nonzero in the thermodynamic limit and by the persistence of non-Gaussian distributions of the Lyapunov exponent, producing a breakdown of the central limit theorem.

Load-bearing premise

That a nonzero relative variance of the Lyapunov exponent in the thermodynamic limit is sufficient to establish that topological properties fail to self-average.

Editorial extensions

If this is right

  • The phase diagram of the topological Anderson insulator becomes dependent on the specific disorder configuration rather than universal.
  • Topological invariants or edge-state signatures measured on one sample need not match those predicted by ensemble averages.
  • The central limit theorem ceases to apply to the Lyapunov exponent statistics.
  • Non-Gaussian distributions of the Lyapunov exponent survive in the large-system limit.

Reading between the lines

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

  • Experimental studies may have to record the actual spatial pattern of disorder in each device rather than relying on statistical averages.
  • Similar non-self-averaging behavior could appear in other topological phases once long-range correlations are present.
  • Standard many-body or field-theoretic approaches that assume self-averaging may require reformulation for this class of systems.
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, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that long-range correlated disorder induces a non-self-averaging topological Anderson insulator phase in which the topological Anderson states exhibit configuration-dependent behavior. This is identified by a relative variance of the Lyapunov exponent that remains finite and non-vanishing in the thermodynamic limit together with persistent non-Gaussian distributions of the Lyapunov exponent, violating self-averaging and the central limit theorem.

Significance. If the non-vanishing thermodynamic-limit variance is rigorously demonstrated, the result would be significant for the field of disordered topological systems by showing that long-range correlations can produce a statistical phase outside the usual self-averaging regime, with direct implications for the reliability of ensemble-averaged topological invariants.

major comments (2)
  1. [Abstract] Abstract and the section presenting the Lyapunov-exponent statistics: the central claim that the relative variance remains finite and non-vanishing as L→∞ is not supported by any explicit thermodynamic-limit extrapolation, finite-size scaling collapse, or analytic argument; the reported finite values at accessible system sizes could be a transient slow decay induced by the long-range correlations rather than true saturation.
  2. [Abstract] The definition of the non-self-averaging topological Anderson insulator phase rests entirely on the non-vanishing relative variance and non-Gaussianity; without data for multiple large L or a scaling analysis demonstrating that the variance does not ultimately decay to zero, the load-bearing distinction between this phase and conventional topological Anderson insulators remains unestablished.
minor comments (1)
  1. [Abstract] The abstract supplies no information on the model Hamiltonian, the precise form of the long-range correlated disorder, the numerical method used to extract the Lyapunov exponent, or the range of system sizes studied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the need for stronger evidence on the thermodynamic-limit behavior of the relative variance. The comments correctly identify that our current presentation relies on observed saturation at accessible sizes without explicit extrapolation or scaling collapse. We address each point below and will revise the manuscript to incorporate additional analysis.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section presenting the Lyapunov-exponent statistics: the central claim that the relative variance remains finite and non-vanishing as L→∞ is not supported by any explicit thermodynamic-limit extrapolation, finite-size scaling collapse, or analytic argument; the reported finite values at accessible system sizes could be a transient slow decay induced by the long-range correlations rather than true saturation.

    Authors: We agree that an explicit extrapolation or scaling collapse would provide more rigorous support. The manuscript presents data showing the relative variance stabilizing at finite non-zero values over the range of system sizes studied, with no visible decay trend. However, to strengthen the claim against the possibility of slow transients due to long-range correlations, we will add finite-size scaling analysis and data for additional larger L in the revised version. This will include plots demonstrating saturation and, if feasible, an attempt at data collapse. revision: yes

  2. Referee: [Abstract] The definition of the non-self-averaging topological Anderson insulator phase rests entirely on the non-vanishing relative variance and non-Gaussianity; without data for multiple large L or a scaling analysis demonstrating that the variance does not ultimately decay to zero, the load-bearing distinction between this phase and conventional topological Anderson insulators remains unestablished.

    Authors: We acknowledge that the phase distinction hinges on establishing non-vanishing variance in the thermodynamic limit. The current evidence consists of persistent finite relative variance and non-Gaussian distributions at the largest accessible sizes, which we interpret as indicating a distinct statistical phase. To address the concern directly, the revised manuscript will include the requested scaling analysis and larger-system data to better substantiate the distinction from conventional self-averaging topological Anderson insulators. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; statistical claim is observational, not self-referential

full rationale

The paper identifies non-self-averaging via the relative variance of the Lyapunov exponent remaining finite as L→∞ together with non-Gaussianity. This is framed as a numerically observed consequence of long-range correlated disorder rather than a quantity fitted to or defined in terms of itself. No equations reduce a prediction to an input parameter by construction, no self-citation chain bears the central claim, and no ansatz is smuggled in. The derivation is therefore self-contained as an empirical demonstration of an anomalous statistical feature.

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

Abstract alone supplies no concrete information on free parameters, background axioms, or new postulated entities; all entries are therefore empty.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-self-averaging topological Anderson insulator." pith.science (2026). https://pith.science/paper/6MOI2Y2N

@misc{pith2026260700892,
  author       = {Pith},
  title        = {Pith review of: Non-self-averaging topological Anderson insulator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MOI2Y2N}},
  note         = {Machine review of arXiv:2607.00892}
}
read the original abstract

Current research on the disordered topological quantum phases primarily focuses on the uncorrelated and short-range correlated disorder regime. These topological Anderson insulators are typically self-averaging. However, topological quantum systems with long-range correlated disorder have received limited attention due to the absence of tractable analytical methods. In fact, the long-range correlated disorder introduces more complex effects on topological quantum states. Here, we demonstrate that the long-range correlated disorder could induce anomalously statistical feature where the topological Anderson states become non-self-averaging, and the phase diagram is strongly dependent on individual disorder configurations. We term this statistical phase the non-self-averaging topological Anderson insulator. The non-self-averaging property is identified by the non-vanishing finite values of the relative variance of the Lyapunov exponent in the thermodynamic limit, alongside the non-Gaussian distributions of the Lyapunov exponent. Consequently, the topological properties of a single disordered sample deviate from the ensemble average, causing a breakdown of the central limit theorem. The non-self-averaging topological Anderson insulator provides insights into the interplay between correlated disorder and topology.

Figures

Figures reproduced from arXiv: 2607.00892 by the authors.

Figure 1
Figure 1. FIG. 1. The average of the topological quantum number [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The finite size scaling for the average and relative vari [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Histogram of the Lyapunov exponent for different [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The topological quantum number [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

126 extracted references · 126 canonical work pages

  1. [1]

    are inadequate to characterize the long-range correla- tion behaviors, and propose using the geometric average and geometric relative variance instead[104, 121]. How- ever, our numerical calculations demonstrate that the ge- ometric average and geometric relative variance of Lya- punov exponent also exhibit similar smooth crossover be- haviors of topologi...

    work page

  2. [2]

    B. A. Bernevig and T. L. Hughes,Topological Insulators and Topological Superconductors(Princeton University Press, 2013)

    work page 2013

  3. [3]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi,A Short Course on Topological Insulators(Springer, 2016)

    work page 2016

  4. [4]

    Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev

    X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys.89, 041004 (2017)

    work page 2017

  5. [5]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Excep- tional topology of non-Hermitian systems, Rev. Mod. Phys.93, 015005 (2021)

    work page 2021

  6. [6]

    Moessner and J

    R. Moessner and J. E. Moore,Topological Phases of Matter(Cambridge University Press, 2021)

    work page 2021

  7. [7]

    K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett.45, 494 (1980)

    work page 1980

  8. [8]

    D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two- dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett.48, 1559 (1982)

    work page 1982

Show all 126 references
  1. [9]

    C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett.95, 226801 (2005)

    2005

  2. [10]

    C. L. Kane and E. J. Mele,Z 2 topological order and the quantum spin Hall effect, Phys. Rev. Lett.95, 146802 (2005)

    2005

  3. [11]

    B. A. Bernevig and S.-C. Zhang, Quantum spin Hall effect, Phys. Rev. Lett.96, 106802 (2006)

    2006

  4. [12]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quan- tum spin Hall effect and topological phase transition in hgte quantum wells, Science314, 1757 (2006)

    2006

  5. [13]

    H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry- based indicators of band topology in the 230 space groups, Nat. Commun.8, 50 (2017)

    2017

  6. [14]

    Kruthoff, J

    J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline in- sulators through band structure combinatorics, Phys. Rev. X7, 041069 (2017)

    2017

  7. [15]

    Watanabe, H

    H. Watanabe, H. C. Po, and A. Vishwanath, Structure and topology of band structures in the 1651 magnetic space groups, Sci. Adv.4, eaat8685 (2018)

    2018

  8. [16]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature547, 298 (2017)

    2017

  9. [17]

    Elcoro, B

    L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chem- istry, Nat. Commun.12, 5965 (2021)

    2021

  10. [18]

    Z. Xiao, J. Zhao, Y. Li, R. Shindou, and Z.-D. Song, Spin space groups: Full classification and applications, Phys. Rev. X14, 031037 (2024)

    2024

  11. [19]

    X. Chen, J. Ren, Y. Zhu, Y. Yu, A. Zhang, P. Liu, J. Li, Y. Liu, C. Li, and Q. Liu, Enumeration and represen- tation theory of spin space groups, Phys. Rev. X14, 031038 (2024)

    2024

  12. [20]

    Jiang, Z

    Y. Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Enumeration of spin-space groups: Toward a complete description of symmetries of magnetic orders, Phys. Rev. X14, 031039 (2024)

    2024

  13. [21]

    X. Chen, Y. Liu, P. Liu, Y. Yu, J. Ren, J. Li, A. Zhang, and Q. Liu, Unconventional magnons in collinear mag- nets dictated by spin space groups, Nature640, 349 (2025)

    2025

  14. [22]

    Li, R.-L

    J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Topolog- ical Anderson insulator, Phys. Rev. Lett.102, 136806 (2009)

    2009

  15. [23]

    C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzyd lo, and C. W. J. Beenakker, Theory of the topological Anderson insulator, Phys. Rev. Lett.103, 196805 (2009)

    2009

  16. [24]

    Jiang, L

    H. Jiang, L. Wang, Q.-f. Sun, and X. C. Xie, Nu- merical study of the topological Anderson insulator in HgTe/CdTe quantum wells, Phys. Rev. B80, 165316 (2009)

    2009

  17. [25]

    H.-M. Guo, G. Rosenberg, G. Refael, and M. Franz, Topological Anderson insulator in three dimensions, Phys. Rev. Lett.105, 216601 (2010)

    2010

  18. [26]

    Y. Xing, L. Zhang, and J. Wang, Topological Anderson insulator phenomena, Phys. Rev. B84, 035110 (2011)

    2011

  19. [27]

    St¨ utzer, Y

    S. St¨ utzer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lind- ner, M. Segev, M. C. Rechtsman, and A. Szameit, Pho- tonic topological Anderson insulators, Nature560, 461 (2018)

    2018

  20. [28]

    G.-G. Liu, Y. Yang, X. Ren, H. Xue, X. Lin, Y.-H. Hu, H.-x. Sun, B. Peng, P. Zhou, Y. Chong, and B. Zhang, Topological Anderson insulator in disordered photonic crystals, Phys. Rev. Lett.125, 133603 (2020)

    2020

  21. [29]

    Cui, R.-Y

    X. Cui, R.-Y. Zhang, Z.-Q. Zhang, and C. T. Chan, Pho- tonic𭟋 2 topological Anderson insulators, Phys. Rev. Lett.129, 043902 (2022)

    2022

  22. [30]

    Zhang, Z.-Q

    J. Zhang, Z.-Q. Zhang, S.-g. Cheng, and H. Jiang, Topo- logical Anderson insulator via disorder-recovered aver- age symmetry, Phys. Rev. B106, 195304 (2022)

    2022

  23. [31]

    Chen, X.-X

    R. Chen, X.-X. Yi, and B. Zhou, Four-dimensional topo- logical Anderson insulator with an emergent second Chern number, Phys. Rev. B108, 085306 (2023)

    2023

  24. [32]

    L´ oio, M

    H. L´ oio, M. Gon¸ calves, P. Ribeiro, and E. V. Castro, Third-order topological insulator induced by disorder, Phys. Rev. B109, 014204 (2024)

    2024

  25. [33]

    Padhan, S

    A. Padhan, S. R. Padhi, and T. Mishra, Complete de- localization and reentrant topological transition in a non-Hermitian quasiperiodic lattice, Phys. Rev. B109, L020203 (2024)

    2024

  26. [34]

    Fu and C

    L. Fu and C. L. Kane, Topology, delocalization via aver- age symmetry and the symplectic Anderson transition, Phys. Rev. Lett.109, 246605 (2012). 7

    2012

  27. [35]

    Ringel, Y

    Z. Ringel, Y. E. Kraus, and A. Stern, Strong side of weak topological insulators, Phys. Rev. B86, 045102 (2012)

    2012

  28. [36]

    R. S. K. Mong, J. H. Bardarson, and J. E. Moore, Quan- tum transport and two-parameter scaling at the surface of a weak topological insulator, Phys. Rev. Lett.108, 076804 (2012)

    2012

  29. [37]

    I. C. Fulga, B. van Heck, J. M. Edge, and A. R. Akhmerov, Statistical topological insulators, Phys. Rev. B89, 155424 (2014)

    2014

  30. [38]

    A. Y. Chaou, M. Moreno-Gonzalez, A. Altland, and P. W. Brouwer, Disordered topological crystalline phases, Phys. Rev. B112, 035167 (2025)

    2025

  31. [39]

    Chen, F.-J

    X. Chen, F.-J. Wang, Z. Bi, and Z.-D. Song, Intrinsic axion statistical topological insulator, Phys. Rev. Lett. 134, 226601 (2025)

    2025

  32. [40]

    Ma and C

    R. Ma and C. Wang, Average symmetry-protected topo- logical phases, Phys. Rev. X13, 031016 (2023)

    2023

  33. [41]

    J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, Quantum9, 1607 (2025)

    2025

  34. [42]

    Zhang, Y

    J.-H. Zhang, Y. Qi, and Z. Bi, Fidelity strange correla- tors for average symmetry-protected topological phases Sci. Bull.71, 688 (2026)

    2026

  35. [43]

    Tao, J.-H

    Y.-L. Tao, J.-H. Wang, and Y. Xu, Average symme- try protected higher-order topological amorphous insu- lators, SciPost Phys.15, 193 (2023)

    2023

  36. [44]

    Antinucci, G

    A. Antinucci, G. Galati, G. Rizi, and M. Serone, Sym- metries and topological operators, on average, SciPost Phys.15, 125 (2023)

    2023

  37. [45]

    Ma, J.-H

    R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topological phases with average symmetries: The deco- hered, the disordered, and the intrinsic, Phys. Rev. X 15, 021062 (2025)

    2025

  38. [46]

    Cheng, T

    X. Cheng, T. Qu, L. Xiao, S. Jia, J. Chen, and L. Zhang, Topological Anderson amorphous insulator, Phys. Rev. B108, L081110 (2023)

    2023

  39. [47]

    Z. Xiao, K. Kawabata, X. Luo, T. Ohtsuki, and R. Shin- dou, Anisotropic topological Anderson transitions in chiral symmetry classes, Phys. Rev. Lett.131, 056301 (2023)

    2023

  40. [48]

    Zuo, J.-R

    Z.-W. Zuo, J.-R. Lin, and D. Kang, Topological inverse Anderson insulator, Phys. Rev. B110, 085157 (2024)

    2024

  41. [49]

    P. Zhao, Z. Xiao, Y. Zhang, and R. Shindou, Topological effect on the Anderson transition in chiral symmetry classes, Phys. Rev. Lett.133, 226601 (2024)

    2024

  42. [50]

    Neehus, F

    A. Neehus, F. Pollmann, and J. Knolle, Genuine topo- logical Anderson insulator from impurity induced chi- rality reversal, Phys. Rev. Lett.135, 126604 (2025)

    2025

  43. [51]

    Favata, N

    R. Favata, N. Ba` u, and A. Marrazzo, Fluctuations and correlations of local topological order parameters in dis- ordered two-dimensional topological insulators, Phys. Rev. Lett.135, 026603 (2025)

    2025

  44. [52]

    J.-R. Lin, S. Wang, H. Li, and Z.-W. Zuo, Topological Anderson insulators by latent symmetry, Phys. Rev.B 113, 094201 (2026)

    2026

  45. [53]

    Ji and Z

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

    2025

  46. [54]

    Zhang, L.-Z

    G.-Q. Zhang, L.-Z. Tang, L. F. Quezada, S.-H. Dong, and D.-W. Zhang, Reentrant topological phases and spin density wave induced by 1D moir´ e potentials, Com- mun. Phys.8, 275 (2025)

    2025

  47. [55]

    Grindall, A

    C. Grindall, A. C. Tyner, A.-K. Wu, T. L. Hughes, and J. H. Pixley, Separate surface and bulk topological An- derson localization transitions in disordered axion insu- lators, Phys. Rev. Lett.135, 226601 (2025)

    2025

  48. [56]

    Qin, C.-A

    Y. Qin, C.-A. Li, and J. Li, Regularized universal topo- logical local markers for Dirac systems, Phys. Rev. B 113, 214201 (2026)

    2026

  49. [57]

    Zhang and L.-Z

    D.-W. Zhang and L.-Z. Tang, Recent progress on disorder-induced topological phases, J. Phys.: Condens. Matter38, 253003 (2026)

    2026

  50. [58]

    Agarwala and V

    A. Agarwala and V. B. Shenoy, Topological insulators in amorphous systems, Phys. Rev. Lett.118, 236402 (2017)

    2017

  51. [59]

    Mansha and Y

    S. Mansha and Y. D. Chong, Robust edge states in amorphous gyromagnetic photonic lattices, Phys. Rev. B96, 121405(R) (2017)

    2017

  52. [60]

    N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Amorphous topological insulators constructed from random point sets, Nat. Phys.14, 380 (2018)

    2018

  53. [61]

    P¨ oyh¨ onen, I

    K. P¨ oyh¨ onen, I. Sahlberg, A. Weststr¨ om, and T. Oja- nen, Amorphous topological superconductivity in a Shiba glass, Nat. Commun.9, 2103 (2018)

    2018

  54. [62]

    Y.-B. Yang, T. Qin, D.-L. Deng, L.-M. Duan, and Y. Xu, Topological amorphous metals, Phys. Rev. Lett. 123, 076401 (2019)

    2019

  55. [63]

    B. Yang, H. Zhang, T. Wu, R. Dong, X. Yan, and X. Zhang, Topological states in amorphous magnetic photonic lattices, Phys. Rev. B99, 045307 (2019)

    2019

  56. [64]

    Costa, G

    M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Toward realistic amor- phous topological insulators, Nano Lett.19, 8941 (2019)

    2019

  57. [65]

    Chern, Topological insulator in an atomic liquid, EPL126, 37002 (2019)

    G.-W. Chern, Topological insulator in an atomic liquid, EPL126, 37002 (2019)

    2019

  58. [66]

    Huang, Y.-S

    H. Huang, Y.-S. Wu, and F. Liu, Aperiodic topological crystalline insulators, Phys. Rev. B101, 041103 (2020)

    2020

  59. [67]

    Marsal, D

    Q. Marsal, D. Varjas, and A. G. Grushin, Topologi- cal Weaire–Thorpe models of amorphous matter, Proc. Natl Acad. Sci.117, 30260 (2020)

    2020

  60. [68]

    Mukati, A

    P. Mukati, A. Agarwala, and S. Bhattacharjee, Topo- logical and conventional phases of a three-dimensional electronic glass, Phys. Rev. B101, 035142 (2020)

    2020

  61. [69]

    Sahlberg, A

    I. Sahlberg, A. Weststr¨ om, K. P¨ oyh¨ onen, and T. Oja- nen, Topological phase transitions in glassy quantum matter, Phys. Rev. Res.2, 013053 (2020)

    2020

  62. [70]

    M. N. Ivaki, I. Sahlberg, and T. Ojanen, Criticality in amorphous topological matter: Beyond the universal scaling paradigm, Phys. Rev. Res.2, 043301 (2020)

    2020

  63. [71]

    Zhou, G.-G

    P. Zhou, G.-G. Liu, X. Ren, Y. Yang, H. Xue, L. Bi, L. Deng, Y. Chong, and B. Zhang, Photonic amorphous topological insulator, Light Sci. Appl.9, 133 (2020)

    2020

  64. [72]

    Corbae, F

    P. Corbae, F. Hellman, and S. M. Griffin, Structural disorder-driven topological phase transition in noncen- trosymmetric BiTeI, Phys. Rev. B103, 214203 (2021)

    2021

  65. [73]

    Wang, Y.-B

    J.-H. Wang, Y.-B. Yang, N. Dai, and Y. Xu, Structural- disorder-induced second-order topological insulators in three dimensions, Phys. Rev. Lett.126, 206404 (2021)

    2021

  66. [74]

    Li, J.-H

    K. Li, J.-H. Wang, Y.-B. Yang, and Y. Xu, Symmetry- protected topological phases in a Rydberg glass, Phys. Rev. Lett.127, 263004 (2021)

    2021

  67. [75]

    Spring, A

    H. Spring, A. R. Akhmerov, and D. Varjas, Amor- phous topological phases protected by continuous ro- tation symmetry, SciPost Phys.11, 022 (2021)

    2021

  68. [76]

    C. Wang, T. Cheng, Z. Liu, F. Liu, and H. Huang, Struc- 8 tural amorphization-induced topological order, Phys. Rev. Lett.128, 056401 (2022)

    2022

  69. [77]

    Zhang, P

    Z. Zhang, P. Delplace, and R. Fleury, Anomalous topo- logical waves in strongly amorphous scattering net- works, Sci. Adv.9, eadg3186 (2023)

    2023

  70. [78]

    Marsal, D

    Q. Marsal, D. Varjas, and A. G. Grushin, Obstructed insulators and flat bands in topological phase-change materials, Phys. Rev. B107, 045119 (2023)

    2023

  71. [79]

    Mu˜ noz Segovia, P

    D. Mu˜ noz Segovia, P. Corbae, D. Varjas, F. Hellman, S. M. Griffin, and A. G. Grushin, Structural spillage: An efficient method to identify noncrystalline topological materials, Phys. Rev. Res.5, L042011 (2023)

    2023

  72. [80]

    Peng, Y.-C

    T. Peng, Y.-C. Xiong, C.-B. Hua, Z.-R. Liu, X. Zhu, W. Cao, F. Lv, Y. Hou, B. Zhou, Z. Wang, and R. Xiong, Structural disorder-induced topological phase transitions in quasicrystals, Phys. Rev. B109, 195301 (2024)

    2024

  73. [81]

    Regis, V

    V. Regis, V. Velasco, M. B. S. Neto, and C. Lewenkopf, Structure-driven phase transitions in paracrystalline topological insulators, Phys. Rev. B110, L161105 (2024)

    2024

  74. [82]

    Peng, Y.-C

    T. Peng, Y.-C. Xiong, X. Zhu, W. Cao, F. Lyu, Y. Hou, Z. Wang, and R. Xiong, Higher-order topological An- derson amorphous insulator, Phys. Rev. B111, 075306 (2025)

    2025

  75. [83]

    Wang, J.-R

    S. Wang, J.-R. Lin, and Z.-W. Zuo, Topological band insulators without translational symmetry, Phys. Rev. B112, 224201 (2025)

    2025

  76. [84]

    Mondragon-Shem, T

    I. Mondragon-Shem, T. L. Hughes, J. Song, and E. Pro- dan, Topological criticality in the chiral-symmetric AIII class at strong disorder, Phys. Rev. Lett.113, 046802 (2014)

    2014

  77. [85]

    Altland, D

    A. Altland, D. Bagrets, L. Fritz, A. Kamenev, and H. Schmiedt, Quantum criticality of quasi-one- dimensional topological Anderson insulators, Phys. Rev. Lett.112, 206602 (2014)

    2014

  78. [86]

    E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massig- nan, T. L. Hughes, and B. Gadway, Observation of the topological Anderson insulator in disordered atomic wires, Science362, 929 (2018)

    2018

  79. [87]

    Hsu and T.-W

    H.-C. Hsu and T.-W. Chen, Topological Anderson in- sulating phases in the long-range Su-Schrieffer-Heeger model, Phys. Rev. B102, 205425 (2020)

    2020

  80. [88]

    L. Lin, Y. Ke, and C. Lee, Real-space representation of the winding number for a one-dimensional chiral- symmetric topological insulator, Phys. Rev. B103, 224208 (2021)

    2021

  81. [89]

    Zuo and D

    Z.-W. Zuo and D. Kang, Reentrant localization tran- sition in the Su-Schrieffer-Heeger model with random- dimer disorder, Phys. Rev. A106, 013305 (2022)

    2022

  82. [90]

    Liu, G.-Q

    S.-N. Liu, G.-Q. Zhang, L.-Z. Tang, and D.-W. Zhang, Topological Anderson insulators induced by random bi- nary disorders, Phys. Lett. A431, 128004 (2022)

    2022

  83. [91]

    Z. Lu, Z. Xu, and Y. Zhang, Exact mobility edges and topological Anderson insulating phase in a slowly varying quasiperiodic model, Ann. Phys. (Berlin)534, 2200203 (2022)

    2022

  84. [92]

    D. S. Shapiro, S. V. Remizov, A. V. Lebedev, D. V. Babukhin, R. S. Akzyanov, A. A. Zhukov, and L. V. Bork, Critical phase boundary and finite-size fluctua- tions in the Su-Schrieffer-Heeger model with random intercell couplings, Phys. Rev. A105, 023321 (2022)

    2022

  85. [93]

    B. M. Manda, V. Achilleos, O. Richoux, C. Skokos, and G. Theocharis, Wave-packet spreading in the disordered and nonlinear Su-Schrieffer-Heeger chain, Phys. Rev. B 107, 184313 (2023)

    2023

  86. [94]

    Zhang and A

    H. Zhang and A. Kamenev, Anatomy of topological An- derson transitions, Phys. Rev. B108, 224201 (2023)

    2023

  87. [95]

    M. Ren, Y. Yu, B. Wu, X. Qi, Y. Wang, X. Yao, J. Ren, Z. Guo, H. Jiang, H. Chen, X.-J. Liu, Z. Chen, and Y. Sun, Realization of gapped and ungapped photonic topological Anderson insulators, Phys. Rev. Lett.132, 066602 (2024)

    2024

  88. [96]

    D. A. Miranda, T. V. C. Ant˜ ao, and N. M. R. Peres, Mechanical Su-Schrieffer-Heeger quasicrystal: Topol- ogy, localization, and mobility edge, Phys. Rev. B109, 195427 (2024)

    2024

  89. [97]

    E. G. Cinnirella, A. Nava, G. Campagnano, and D. Giu- liano, Fate of high winding number topological phases in the disordered extended Su-Schrieffer-Heeger model, Phys. Rev. B109, 035114 (2024)

    2024

  90. [98]

    Gu and Z

    Y. Gu and Z. Lu, Different topological phase transitions in the Su–Schrieffer–Heeger model under different dis- order structures, Chin. Phys. B33, 090202 (2024)

    2024

  91. [99]

    Z. Lu, Y. Zhang, and Z. Xu, Reentrant localization tran- sitions in a topological Anderson insulator: A study of a generalized Su-Schrieffer-Heeger quasicrystal, Front. Phys.20, 24204 (2025)

    2025

  92. [100]

    Wang, S.-Z

    X.-M. Wang, S.-Z. Li, and Z. Li, Emergent topological re-entrant phase transition in a generalized quasiperi- odic modulated Su-Schrieffer-Heeger model, Phys. Rev. A111, 022214 (2025)

    2025

  93. [101]

    P. S. Nair, D. Joy, R. Deb, S. Pakrashi, and S. Sanyal, Emergent scale and anomalous dynamics in certain nearest-neighbor quasiperiodic tight binding models in one dimension, Phys. Rev. B112, 104203 (2025)

    2025

  94. [102]

    K. Sone, M. Ezawa, Z. Gong, T. Sawada, N. Yoshioka, and T. Sagawa, Transition from the topological to the chaotic in the nonlinear Su–Schrieffer–Heeger model, Nat. Commun.16, 422 (2025)

    2025

  95. [103]

    Zhang, L

    G.-Q. Zhang, L. F. Quezada, and S.-H. Dong, Reen- trant topological phases and entanglement scalings in moir´ e-modulated extended Su-Schrieffer-Heeger model, Sci. China-Phys. Mech. Astron69, 230314 (2026)

    2026

  96. [104]

    F. A. B. F. de Moura and M. L. Lyra, Delocalization in the 1D Anderson model with long-range correlated disorder, Phys. Rev. Lett.81, 3735 (1998)

    1998

  97. [105]

    S. Russ, J. W. Kantelhardt, A. Bunde, and S. Havlin, Localization in self-affine energy landscapes, Phys. Rev. B64, 134209 (2001)

    2001

  98. [106]

    Cheraghchi, S

    H. Cheraghchi, S. M. Fazeli, and K. Esfarjani, Localization-delocalization transition in a one one- dimensional system with long-range correlated off- diagonal disorder, Phys. Rev. B72, 174207 (2005)

    2005

  99. [107]

    G. M. Petersen and N. Sandler, Anticorrelations from power-law spectral disorder and conditions for an An- derson transition, Phys. Rev. B87, 195443 (2013)

    2013

  100. [108]

    X. Wei, K. Feng, T.-C. Yi, T. Liu, G. Xianlong, and Y. Zhang, Bound states in one-dimensional systems with colored noise, Phys. Rev. B110, 184207 (2024)

    2024

  101. [109]

    Longhi, Topological Anderson phase in quasi-periodic waveguide lattices, Opt

    S. Longhi, Topological Anderson phase in quasi-periodic waveguide lattices, Opt. Lett.45, 4036 (2020)

    2020

  102. [110]

    I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Scattering formula for the topological quan- tum number of a disordered multimode wire, Phys. Rev. B83, 155429 (2011)

    2011

  103. [111]

    Wiseman and E

    S. Wiseman and E. Domany, Lack of self-averaging in critical disordered systems, Phys. Rev. E52, 3469 9 (1995)

    1995

  104. [112]

    Aharony and A

    A. Aharony and A. B. Harris, Absence of self-averaging and universal fluctuations in random systems near crit- ical points, Phys. Rev. Lett.77, 3700 (1996)

    1996

  105. [113]

    P´ azm´ andi, R

    F. P´ azm´ andi, R. T. Scalettar, and G. T. Zim´ anyi, Re- visiting the theory of finite size scaling in disordered systems:νcan be less than 2/d, Phys. Rev. Lett.79, 5130 (1997)

    1997

  106. [114]

    Wiseman and E

    S. Wiseman and E. Domany, Self-averaging, distribution of pseudocritical temperatures, and finite size scaling in critical disordered systems, Phys. Rev. E58, 2938 (1998)

    1998

  107. [115]

    Wiseman and E

    S. Wiseman and E. Domany, Finite-size scaling and lack of self-averaging in critical disordered systems, Phys. Rev. Lett.81, 22 (1998)

    1998

  108. [116]

    Parisi and N

    G. Parisi and N. Sourlas, Scale invariance in disordered systems: The example of the random-field Ising model, Phys. Rev. Lett.89, 257204 (2002)

    2002

  109. [117]

    J. M. Fish and R. L. C. Vink, Nematics with quenched disorder: Violation of self-averaging, Phys. Rev. Lett. 105, 147801 (2010)

    2010

  110. [118]

    Kr¨ uger, S

    F. Kr¨ uger, S. Hong, and P. Phillips, Two distinct mott- insulator to Bose-glass transitions and breakdown of self-averaging in the disordered Bose-Hubbard model, Phys. Rev. B84, 115118 (2011)

    2011

  111. [119]

    Y. Duan, B. Mahault, Y.-q. Ma, X.-q. Shi, and H. Chat´ e, Breakdown of ergodicity and self-averaging in polar flocks with quenched disorder, Phys. Rev. Lett. 126, 178001 (2021)

    2021

  112. [120]

    Sol´ orzano, L

    A. Sol´ orzano, L. F. Santos, and E. J. Torres-Herrera, Multifractality and self-averaging at the many-body lo- calization transition, Phys. Rev. Res.3, L032030 (2021)

    2021

  113. [121]

    Duthie, S

    A. Duthie, S. Roy, and D. E. Logan, Anomalous mul- tifractality in quantum chains with strongly correlated disorder, Phys. Rev. B106, L020201 (2022)

    2022

  114. [122]

    Abrahams,50 Years of Anderson Localization(World Scientific, 2010)

    E. Abrahams,50 Years of Anderson Localization(World Scientific, 2010)

    2010

  115. [123]

    Y.-B. Yang, K. Li, L.-M. Duan, and Y. Xu, Higher- order topological anderson insulators, Phys. Rev. B103, 085408 (2021)

    2021

  116. [124]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Si- mon, O. Zilberberg, and I. Carusotto, Topological pho- tonics, Rev. Mod. Phys.91, 015006 (2019)

    2019

  117. [125]

    W. Zhu, W. Deng, Y. Liu, J. Lu, H.-X. Wang, Z.-K. Lin, X. Huang, J.-H. Jiang, and Z. Liu, Topological phononic metamaterials, Rep. Prog. Phys.86, 106501 (2023)

    2023

  118. [126]

    Sahin, M

    H. Sahin, M. B. A. Jalil, and C. H. Lee, Topolectrical cir- cuits—recent experimental advances and developments, APL Electron. Devices1, 021503 (2025)

    2025

Pith tools

Reviewed July 2, 2026 · model on record in the stance chip above.