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arxiv: 2606.02476 · v1 · pith:PYBSQ6R4new · submitted 2026-06-01 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn

Leveraging structural disorder to enhance topological phases

Laura Gomez Paz , Peru d'Ornellas , Adolfo G Grushin This is my paper

Pith reviewed 2026-06-28 13:02 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.dis-nn
keywords topological insulatorstructural disordertwo dimensionsthree dimensionsspectral localizerZ2 invarianttime-reversal symmetry
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The pith

Structural disorder that keeps sites apart sustains topological phases in 2D up to strong disorder.

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

The paper examines whether structural disorder alone, without added on-site potentials, can create or preserve topological insulator phases. In two dimensions, disorder that penalizes atoms from sitting too close to one another enhances the phase and keeps it stable even when disorder is strong. The same kind of disorder destroys the phase in three dimensions. Because strong disorder can scramble any global spin reference frame, conventional spin-based markers fail, so the authors instead use a marker built directly from the time-reversal operator.

Core claim

Structural disorder with a minimum-distance penalty between sites enhances and sustains the Z2 topological phase in two-dimensional lattice models up to strong disorder strengths, while generic structural disorder is detrimental in three dimensions; the spectral localizer, defined from the time-reversal symmetry operator, correctly diagnoses the invariant even when disorder scrambles the spin frame.

What carries the argument

The spectral localizer, a local marker constructed directly from the time-reversal symmetry operator rather than a spin projection, which remains well-defined when disorder eliminates a global spin reference frame.

If this is right

  • In two dimensions the topological phase can survive stronger structural disorder than models without the distance penalty predict.
  • Spin-Chern and spin-Bott markers become unreliable once disorder scrambles the global spin frame, while the spectral localizer remains usable.
  • Three-dimensional topological phases are generically more fragile to structural disorder than their two-dimensional counterparts.
  • Metamaterial or solid-state samples can be engineered with controlled minimum inter-site distances to stabilize two-dimensional topology.

Where Pith is reading between the lines

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

  • Materials with built-in repulsive interactions between neighboring sites could be designed to favor two-dimensional topological order.
  • The same distance-penalty idea might be tested in artificial lattices or photonic systems where structural disorder is easier to control.
  • The spectral localizer could be applied to other symmetry-protected phases whose conventional markers rely on a fixed basis that disorder can scramble.

Load-bearing premise

The chosen lattice models with purely structural disorder represent realistic situations and the spectral localizer still identifies the correct Z2 invariant when spins are scrambled.

What would settle it

A direct numerical check showing that the topological phase collapses at weaker disorder than predicted when the minimum-distance penalty is enforced in two dimensions, or that the spectral localizer disagrees with an independent invariant calculation in a scrambled-spin system.

Figures

Figures reproduced from arXiv: 2606.02476 by Adolfo G Grushin, Laura Gomez Paz, Peru d'Ornellas.

Figure 1
Figure 1. Figure 1: FIG. 1. A comparison between different types of disorder. (a) In a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Failure of the spin-Bott index in the presence of spin-frame [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Two examples of progressively disordering a system starting [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Phase diagram of the localizer index for the two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Phase diagram of the localizer index for the three-dimensional [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

On-site disorder can be leveraged to induce a transition from a trivial to a topological insulator. However it is unclear if structural disorder in the absence of on-site disorder can aid a similar transition and, if so, which kind of structural disorder is more favourable. We numerically show that structural disorder can enhance and sustain a topological phase up to strong disorder in two dimensions provided that one penalises atomic sites from being close to one another. However, we find this effect is absent in three dimensions, where structural disorder appears generically detrimental to the phase. In our calculations we include disorder that can scramble the global spin-reference frame, an overlooked type of disorder expected to exist in strongly disordered solids. This disorder fatally scrambles the information necessary for the spin-Bott and the spin-Chern marker to correctly diagnose a topological phase. By using the spectral localizer, a local marker directly defined using the time-reversal symmetry operator rather than a spin-projection, we show how one can circumvent this limitation, providing a basis-indifferent theory for calculating Z2 invariants. Our work showcases that not all structural disorders are equally beneficial to topology, and highlights guiding principles to enhance and detect topological phases in both solid-state and metamaterial realisations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper claims that structural disorder (specifically, penalizing close atomic sites) can enhance and sustain a 2D topological phase up to strong disorder, while the same effect is absent in 3D where structural disorder is generically detrimental. The authors introduce spin-scrambling disorder that invalidates spin-Bott and spin-Chern markers and instead employ the spectral localizer (defined from the time-reversal operator) to diagnose the Z2 invariant in a basis-independent manner, supported by numerical simulations.

Significance. If the numerical evidence is robust and the localizer is shown to be faithful, the result would demonstrate that specific forms of structural disorder can stabilize topology in 2D (but not 3D) and would supply a practical, spin-basis-independent diagnostic for Z2 phases in disordered systems relevant to both solid-state and metamaterial realizations.

major comments (2)
  1. [Section describing the spectral localizer and its application to disordered systems] The central claim that the spectral localizer correctly identifies the Z2 invariant when disorder scrambles the global spin reference frame rests on an unbenchmarked assumption. No explicit validation against exactly solvable limits (clean TI with artificial per-site spin rotations, or models where Z2 is independently known) is described, leaving open whether the localizer remains faithful once the spin frame is lost.
  2. [Numerical methods and results sections] The numerical evidence for 2D enhancement versus 3D detriment is presented without visible details on lattice sizes, number of disorder realizations, convergence criteria, or the precise implementation of the 'penalize close sites' structural disorder (e.g., the functional form of the penalty or how it is sampled). These omissions make it impossible to assess whether the reported phase stability is robust or an artifact of finite-size or sampling effects.
minor comments (1)
  1. [Abstract] The abstract states that 'structural disorder appears generically detrimental' in 3D but does not quantify what 'generically' means or contrast it explicitly with the 2D penalization protocol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight areas where additional detail and validation will strengthen the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Section describing the spectral localizer and its application to disordered systems] The central claim that the spectral localizer correctly identifies the Z2 invariant when disorder scrambles the global spin reference frame rests on an unbenchmarked assumption. No explicit validation against exactly solvable limits (clean TI with artificial per-site spin rotations, or models where Z2 is independently known) is described, leaving open whether the localizer remains faithful once the spin frame is lost.

    Authors: We agree that explicit benchmarks against solvable limits would strengthen the claim. While the spectral localizer is defined directly from the time-reversal operator (rather than any spin projection) and is therefore basis-independent by construction, we will add in the revised manuscript explicit validations using clean topological insulators subjected to artificial per-site spin rotations as well as comparisons against models with independently known Z2 invariants. These additions will confirm faithfulness under spin-frame scrambling. revision: yes

  2. Referee: [Numerical methods and results sections] The numerical evidence for 2D enhancement versus 3D detriment is presented without visible details on lattice sizes, number of disorder realizations, convergence criteria, or the precise implementation of the 'penalize close sites' structural disorder (e.g., the functional form of the penalty or how it is sampled). These omissions make it impossible to assess whether the reported phase stability is robust or an artifact of finite-size or sampling effects.

    Authors: We agree that these methodological details are essential for assessing robustness. In the revised manuscript we will explicitly report the lattice sizes employed, the number of disorder realizations averaged, the convergence criteria used, and the precise implementation of the structural disorder, including the functional form of the site-separation penalty and the sampling procedure. revision: yes

Circularity Check

0 steps flagged

Numerical results on structural disorder and Z2 invariants are direct simulation outputs with no reduction to inputs by construction

full rationale

The paper reports numerical lattice simulations showing that certain structural disorder (penalizing close sites) sustains 2D topological phases to strong disorder while being detrimental in 3D. The spectral localizer is introduced as a marker defined directly from the time-reversal operator to handle spin-scrambling disorder where spin-Bott and spin-Chern fail. No equations, definitions, or self-citations reduce the reported phase diagrams or invariant values to fitted parameters or tautological quantities. The chain consists of model construction, disorder generation, and direct computation of the localizer; it does not match any enumerated circularity pattern. The work is self-contained against external benchmarks in the sense required by the guidelines.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claims rest on the unstated assumption that the chosen numerical models faithfully capture structural disorder physics.

pith-pipeline@v0.9.1-grok · 5752 in / 1285 out tokens · 33185 ms · 2026-06-28T13:02:09.627896+00:00 · methodology

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Reference graph

Works this paper leans on

156 extracted references · 10 canonical work pages

  1. [1]

    4 (a), we find a familiar story [1, 2, 4, 5, 54, 95, 115]

    Two-dimensional quantum-spin Hall In the case of Anderson disorder, shown in Fig. 4 (a), we find a familiar story [1, 2, 4, 5, 54, 95, 115]. Here, provided that we start with M approximately ∈[−5.4,−4] , the system can be driven into a topological phase by increasing the Anderson disorder coefficientW. In contrast, in the case of structural disorder we fi...

  2. [2]

    5 (a), is 8 in good agreement with previous studies of disordered three- dimensional Z2 topological insulators [3, 91, 115, 143]

    Three-dimensionalZ 2 topological insulator In the three-dimensional case with on-site disorder, the phase diagram obtained from the localizer, shown in Fig. 5 (a), is 8 in good agreement with previous studies of disordered three- dimensional Z2 topological insulators [3, 91, 115, 143]. Sim- ilar to the two-dimensional case, we find that within a range of ...

  3. [3]

    Li, R.-L

    J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Physical Review Letters102, 136806 (2009)

    2009

  4. [4]

    C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzydło, and C. W. J. Beenakker, Physical Review Letters103, 196805 (2009)

    2009

  5. [5]

    H.-M. Guo, G. Rosenberg, G. Refael, and M. Franz, Phys. Rev. Lett.105, 216601 (2010)

    2010

  6. [6]

    Prodan, Phys

    E. Prodan, Phys. Rev. B83, 195119 (2011)

    2011

  7. [7]

    Yamakage, K

    A. Yamakage, K. Nomura, K.-I. Imura, and Y . Kuramoto, Jour- nal of Physics: Conference Series400, 042070 (2012)

    2012

  8. [8]

    J. Song, H. Liu, H. Jiang, Q.-f. Sun, and X. C. Xie, Phys. Rev. B85, 195125 (2012)

    2012

  9. [9]

    Prodan, J

    E. Prodan, J. Phys. A44, 113001 (2011)

    2011

  10. [10]

    Ringel, Y

    Z. Ringel, Y . E. Kraus, and A. Stern, Phys. Rev. B86, 045102 (2012)

    2012

  11. [11]

    S.-H. Lv, J. Song, and Y .-X. Li, Journal of Applied Physics 114, 183710 (2013)

    2013

  12. [12]

    I. C. Fulga, B. van Heck, J. M. Edge, and A. R. Akhmerov, Physical Review B89, 155424 (2014)

    2014

  13. [13]

    X. Ni, H. Huang, and F. Liu, Phys. Rev. B101, 125114 (2020)

    2020

  14. [14]

    J. a. S. Silva, E. V . Castro, R. Mondaini, M. A. H. V ozmediano, and M. P. L´opez-Sancho, Phys. Rev. B109, 125145 (2024)

    2024

  15. [15]

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

    2025

  16. [16]

    Agarwala and V

    A. Agarwala and V . B. Shenoy, Physical Review Letters118, 236402 (2017)

    2017

  17. [17]

    Mansha and Y

    S. Mansha and Y . D. Chong, Physical Review B96, 121405 (2017)

    2017

  18. [18]

    Xiao and S

    M. Xiao and S. Fan, Physical Review B96, 100202 (2017)

    2017

  19. [19]

    N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Nature Physics14, 380 (2018)

    2018

  20. [20]

    Bourne and E

    C. Bourne and E. Prodan, Journal of Physics A: Mathematical and Theoretical51, 235202 (2018)

    2018

  21. [21]

    P¨oyh¨onen, I

    K. P¨oyh¨onen, I. Sahlberg, A. Weststr¨om, and T. Ojanen, Nature Communications9, 2103 (2018)

    2018

  22. [22]

    E. L. Minarelli, K. P ¨oyh¨onen, G. A. R. van Dalum, T. Ojanen, and L. Fritz, Physical Review B99, 165413 (2019)

    2019

  23. [23]

    Chern, Europhysics Letters126, 37002 (2019)

    G.-W. Chern, Europhysics Letters126, 37002 (2019)

    2019

  24. [24]

    Mano and T

    T. Mano and T. Ohtsuki, Journal of the Physical Society of Japan88, 123704 (2019)

    2019

  25. [25]

    Costa, G

    M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Nano Letters19, 8941 (2019)

    2019

  26. [26]

    Marsal, D

    Q. Marsal, D. Varjas, and A. G. Grushin, Proceedings of the National Academy of Sciences117, 30260 (2020)

    2020

  27. [27]

    Mukati, A

    P. Mukati, A. Agarwala, and S. Bhattacharjee, Physical Review B101, 035142 (2020)

    2020

  28. [28]

    Sahlberg, A

    I. Sahlberg, A. Weststr¨om, K. P¨oyh¨onen, and T. Ojanen, Physi- cal Review Research2, 013053 (2020)

    2020

  29. [29]

    M. N. Ivaki, I. Sahlberg, and T. Ojanen, Physical Review Re- search2, 043301 (2020)

    2020

  30. [30]

    Agarwala, V

    A. Agarwala, V . Juriˇci´c, and B. Roy, Physical Review Research 2, 012067 (2020)

    2020

  31. [31]

    A. G. Grushin, inLow-Temperature Thermal and Vibrational Properties of Disordered Solids, edited by M. A. Ramos (World Scientific, 2022) Chap. 11

    2022

  32. [32]

    Huang and F

    H. Huang and F. Liu, Research 2020, 10.34133/2020/7832610 (2020), https://spj.science.org/doi/pdf/10.34133/2020/7832610

    work page doi:10.34133/2020/7832610 2020

  33. [33]

    Wang, Y .-B

    J.-H. Wang, Y .-B. Yang, N. Dai, and Y . Xu, Physical Review Letters126, 206404 (2021)

    2021

  34. [34]

    Focassio, G

    B. Focassio, G. R. Schleder, M. Costa, A. Fazzio, and C. Lewenkopf, 2D Materials8, 025032 (2021)

    2021

  35. [35]

    Focassio, G

    B. Focassio, G. R. Schleder, F. C. d. Lima, C. Lewenkopf, and A. Fazzio, Physical Review B104, 214206 (2021)

    2021

  36. [36]

    N. P. Mitchell, A. M. Turner, and W. T. M. Irvine, Physical Review E104, 025007 (2021)

    2021

  37. [37]

    Spring, A

    H. Spring, A. Akhmerov, and D. Varjas, SciPost Physics11, 022 (2021)

    2021

  38. [38]

    C. Wang, T. Cheng, Z. Liu, F. Liu, and H. Huang, Physical Review Letters128, 056401 (2022)

    2022

  39. [39]

    Ma and H

    J. Ma and H. Huang, Physical Review B106, 195150 (2022)

    2022

  40. [40]

    Peng, C.-B

    T. Peng, C.-B. Hua, R. Chen, Z.-R. Liu, H.-M. Huang, and B. Zhou, Phys. Rev. B106, 125310 (2022)

    2022

  41. [41]

    A. J. Ur´ıa-´Alvarez, D. Molpeceres-Mingo, and J. J. Palacios, Phys. Rev. B105, 155128 (2022)

    2022

  42. [42]

    Guzm´an, D

    M. Guzm´an, D. Bartolo, and D. Carpentier, SciPost Physics12, 038 (2022)

    2022

  43. [43]

    Mu ˜noz Segovia, P

    D. Mu ˜noz Segovia, P. Corbae, D. Varjas, F. Hellman, S. M. Griffin, and A. G. Grushin, Phys. Rev. Res.5, L042011 (2023)

    2023

  44. [44]

    Cassella, P

    G. Cassella, P. d’Ornellas, T. Hodson, W. M. H. Natori, and J. Knolle, Nature Communications14, 10.1038/s41467-023- 42105-9 (2023)

    work page doi:10.1038/s41467-023- 2023

  45. [45]

    A. G. Grushin and C. Repellin, Phys. Rev. Lett.130, 186702 (2023). 10

    2023

  46. [46]

    Cheng, T

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

    2023

  47. [47]

    Zhang, P

    Z. Zhang, P. Delplace, and R. Fleury, Science Advances9, 10.1126/sciadv.adg3186 (2023)

    work page doi:10.1126/sciadv.adg3186 2023

  48. [48]

    Manna, S

    S. Manna, S. K. Das, and B. Roy, Phys. Rev. B109, 174512 (2024)

    2024

  49. [49]

    D. Li, C. Wang, and H. Huang, SciPost Phys.17, 086 (2024)

    2024

  50. [50]

    Marsal, D

    Q. Marsal, D. Varjas, and A. G. Grushin, Phys. Rev. B107, 045119 (2023)

    2023

  51. [51]

    A. J. Ur´ıa-´Alvarez and J. J. Palacios, Phys. Rev. Res.7, 043263 (2025)

    2025

  52. [52]

    Y . E. Kraus, Y . Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett.109, 106402 (2012)

    2012

  53. [53]

    D. T. Tran, A. Dauphin, N. Goldman, and P. Gaspard, Phys. Rev. B91, 085125 (2015)

    2015

  54. [54]

    M. A. Bandres, M. C. Rechtsman, and M. Segev, Phys. Rev. X 6, 011016 (2016)

    2016

  55. [55]

    Fuchs and J

    J.-N. Fuchs and J. Vidal, Phys. Rev. B94, 205437 (2016)

    2016

  56. [56]

    Huang and F

    H. Huang and F. Liu, Phys. Rev. B98, 125130 (2018)

    2018

  57. [57]

    Fuchs, R

    J.-N. Fuchs, R. Mosseri, and J. Vidal, Phys. Rev. B98, 145 (2018)

    2018

  58. [58]

    T. A. Loring, Journal of Mathematical Physics60, 081903 (2019)

    2019

  59. [59]

    Varjas, A

    D. Varjas, A. Lau, K. P¨oyh¨onen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, Phys. Rev. Lett.123, 196401 (2019)

    2019

  60. [60]

    Huang and F

    H. Huang and F. Liu, Phys. Rev. B100, 085119 (2019)

    2019

  61. [61]

    Chen, D.-H

    R. Chen, D.-H. Xu, and B. Zhou, Phys. Rev. B100, 115311 (2019)

    2019

  62. [62]

    He, L.-R

    A.-L. He, L.-R. Ding, Y . Zhou, Y .-F. Wang, and C.-D. Gong, Phys. Rev. B100, 214109 (2019)

    2019

  63. [63]

    C. W. Duncan, S. Manna, and A. E. B. Nielsen, Phys. Rev. B 101, 115413 (2020)

    2020

  64. [64]

    Fan and H

    J. Fan and H. Huang, Frontiers of Physics17, 13203 (2021)

    2021

  65. [65]

    Zilberberg, Opt

    O. Zilberberg, Opt. Mater. Express11, 1143 (2021)

    2021

  66. [66]

    D. V . Else, S.-J. Huang, A. Prem, and A. Gromov, Phys. Rev. X11, 041051 (2021)

    2021

  67. [67]

    Hua, Z.-R

    C.-B. Hua, Z.-R. Liu, T. Peng, R. Chen, D.-H. Xu, and B. Zhou, Phys. Rev. B104, 155304 (2021)

    2021

  68. [68]

    J. Jeon, M. J. Park, and S. Lee, Phys. Rev. B105, 045146 (2022)

    2022

  69. [69]

    Schirmann, S

    J. Schirmann, S. Franca, F. Flicker, and A. G. Grushin, Phys. Rev. Lett.132, 086402 (2024)

    2024

  70. [70]

    Roche Carrasco, J

    H. Roche Carrasco, J. Schirmann, A. Mordret, and A. G. Grushin, Phys. Rev. Lett.135, 236603 (2025)

    2025

  71. [71]

    Caiger, F

    W. Caiger, F. Flicker, and M. ´Angel S´anchez-Mart´ınez, Fractal topology of majorana bound states in superconducting qua- sicrystals (2026), arXiv:2602.02796 [cond-mat.mes-hall]

    arXiv 2026

  72. [72]

    Brzezi´nska, A

    M. Brzezi´nska, A. M. Cook, and T. Neupert, Phys. Rev. B98, 205116 (2018)

    2018

  73. [73]

    Manna and B

    S. Manna and B. Roy, Communications Physics6, 10.1038/s42005-023-01130-2 (2023)

    work page doi:10.1038/s42005-023-01130-2 2023

  74. [74]

    D. M. Urwyler, P. M. Lenggenhager, I. Boettcher, R. Thomale, T. Neupert, and T. c. v. Bzduˇsek, Phys. Rev. Lett.129, 246402 (2022)

    2022

  75. [75]

    Liu, C.-B

    Z.-R. Liu, C.-B. Hua, T. Peng, and B. Zhou, Phys. Rev. B105, 245301 (2022)

    2022

  76. [76]

    P. M. Lenggenhager, A. Stegmaier, L. K. Upreti, T. Hofmann, T. Helbig, A. V ollhardt, M. Greiter, C. H. Lee, S. Imhof, H. Brand, T. Kießling, I. Boettcher, T. Neupert, R. Thomale, and T. Bzduˇsek, Nature Communications13, 4373 (2022)

    2022

  77. [77]

    Kitaev, Ann

    A. Kitaev, Ann. Phys.321, 2 (2006), january Special Issue

    2006

  78. [78]

    Bianco and R

    R. Bianco and R. Resta, Phys. Rev. B84, 241106 (2011)

    2011

  79. [79]

    Marrazzo and R

    A. Marrazzo and R. Resta, Phys. Rev. B95, 121114 (2017), publisher: American Physical Society

    2017

  80. [80]

    T. A. Loring and M. B. Hastings, EPL92, 67004 (2010)

    2010

Showing first 80 references.