pith. sign in

arxiv: 2606.24720 · v1 · pith:PWCM3NFBnew · submitted 2026-06-23 · ❄️ cond-mat.dis-nn · physics.comp-ph· quant-ph

On the localization transition from MAA to AA models

Hangdong Qiu , Yunhua Wang This is my paper

Pith reviewed 2026-06-25 22:00 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn physics.comp-phquant-ph
keywords localization transitionMAA modelAA modelquasiperiodic potentialsmobility edgesAnderson localizationDMAA modelpolariton modes
0
0 comments X

The pith

Superpositions of two MAA potentials produce twice and multiple localization-delocalization transitions.

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

The paper builds a double quasiperiodic MAA (DMAA) model by overlaying one standard MAA potential (nonzero even-site amplitudes) with a second modified MAA potential (nonzero odd-site amplitudes plus a tunable amplitude factor). This construction is used to track how localization properties change as the system evolves from the MAA model, which supports mobility edges, toward the AA model, which does not. Interplays among extended, critical, and localized states created by the double potential generate new twice and multiple transitions in addition to the single transition of the original MAA model. These features are checked with inverse participation ratio, normalized participation ratio, fractal dimension, and wavefunction plots, and the results are reproduced in continuum simulations of polariton modes.

Core claim

The DMAA model, formed by superposing a primitive MAA potential on even sites with a modified MAA potential on odd sites carrying a tunable amplitude, exhibits new twice and multiple localization-delocalization transitions that arise from competitions among extended, critical, and localized states and are absent from the single-transition behavior of the original MAA model. These transitions are verified by inverse participation ratio, normalized participation ratio, fractal dimension, and real-space wavefunction distributions, with consistent outcomes obtained from continuum simulations of experimental polariton modes.

What carries the argument

Double quasiperiodic MAA (DMAA) model formed by superposing one primitive MAA potential (nonzero even-site amplitudes) with a second MAA potential (nonzero odd-site amplitudes plus tunable amplitude factor).

If this is right

  • The DMAA model exhibits twice and multiple localization-delocalization transitions in addition to the single transition of the original MAA model.
  • Interplays among extended, critical, and localized states produced by the double quasiperiodic potentials are responsible for the additional transitions.
  • Inverse participation ratio, normalized participation ratio, fractal dimension, and real-space wavefunction distributions all confirm the new localization features.
  • Continuum simulations of polariton modes reproduce the same transitions, supporting experimental realizability.

Where Pith is reading between the lines

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

  • The tunable amplitude factor could be varied continuously in experiment to map the locations of the extra transitions.
  • The same even-odd superposition construction might be applied to other pairs of quasiperiodic models to induce multiple transitions.
  • Polariton or cold-atom platforms already used for MAA studies could be adapted to test the DMAA predictions by adding a second incommensurate lattice component.

Load-bearing premise

The chosen numerical diagnostics (IPR, NPR, fractal dimension) together with the specific even-odd site construction of the DMAA potentials suffice to identify and separate multiple transitions without finite-size artifacts or post-hoc tuning.

What would settle it

Numerical scans or polariton experiments that find only a single localization transition for all values of the tunable odd-site amplitude would falsify the existence of twice and multiple transitions.

Figures

Figures reproduced from arXiv: 2606.24720 by Hangdong Qiu, Yunhua Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Double quasiperiodic potentials [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a)-(f) Fractal dimension [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a)-(f) Fractal dimension [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The IPR and NPR as functions of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a)-(e) Spatial distribution of the wave function for the 410th eigenstate of the DMAA model with [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a)-(c) The phase diagram characterized by fractal dimension [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Fractal dimension [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

Despite their potential similarity between the mosaic Aubry-Andr\'e (MAA) and AA models, the MAA model allows mobility edges (MEs), whereas the AA model does not. Here we develop a new double quasiperiodic MAA (DMAA) model consisting of one primitive MAA with nonzero even-site potentials and the other modified one with both nonzero odd-site potentials and a tunable amplitude factor, to reveal how localization transitions evolve from MAA to AA models. Interplays and competitions among the extended, critical and localized states arising from superpositions of double quasi-periodic MAA potentials enable new twice and multiple localization-delocalization transitions besides the original single localization transition. Our numerical calculations on inverse participation ratio, normalized participation ratio, fractal dimension and real-space wavefunction distribution confirm such localization features. The continuum model simulations on the experimental polariton modes also yield consistent results and hence validate their experimental feasibility. The constructed DMAA model provides a new framework for studying the localization transition processes between two analogous quasiperiodic models and broadens the understanding of Anderson localization.

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 manuscript introduces a double quasiperiodic mosaic Aubry-André (DMAA) model formed by superposing one primitive MAA potential (nonzero even-site amplitudes) with a modified MAA potential (nonzero odd-site amplitudes plus a tunable amplitude factor). It claims that interplays among extended, critical, and localized states produce new twice and multiple localization-delocalization transitions in addition to the original single transition of the MAA model. These features are asserted to be confirmed by numerical diagnostics (IPR, NPR, fractal dimension, real-space wavefunctions) and by continuum simulations of experimental polariton modes.

Significance. If the multiple transitions prove robust, the DMAA construction would supply a concrete framework for interpolating between MAA and AA localization behavior and for exploring state competitions in quasiperiodic systems, with possible experimental relevance to polariton platforms. The superposition approach itself is a natural way to tune between analogous models, but the manuscript does not yet supply the controls needed to establish that the reported extra transitions are intrinsic rather than diagnostic or construction artifacts.

major comments (2)
  1. [Abstract / numerical results] Abstract and numerical-results section: the claim that 'numerical calculations on inverse participation ratio, normalized participation ratio, fractal dimension and real-space wavefunction distribution confirm such localization features' supplies no system sizes, number of disorder realizations, error bars, or procedure for scanning the tunable amplitude factor. Without these data the reported twice and multiple transitions cannot be assessed for finite-size artifacts or parameter-specific effects.
  2. [Numerical diagnostics] Numerical diagnostics paragraph: IPR, NPR and fractal dimension are used to locate the additional transitions, yet the text contains no finite-size scaling collapse, no comparison against level statistics or transfer-matrix Lyapunov exponents, and no scan showing that the extra transition points survive when the odd-site amplitude factor is varied continuously or when N is increased by a factor of 4–8. These omissions leave open the possibility that the features arise from the even/odd-site construction rather than from intrinsic state interplays.
minor comments (1)
  1. [Abstract] The abstract states that 'the continuum model simulations on the experimental polariton modes also yield consistent results' but does not identify the continuum model or the mapping from the DMAA lattice to the polariton system.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and will revise the manuscript to incorporate additional numerical details and controls as outlined.

read point-by-point responses

  1. Referee: [Abstract / numerical results] Abstract and numerical-results section: the claim that 'numerical calculations on inverse participation ratio, normalized participation ratio, fractal dimension and real-space wavefunction distribution confirm such localization features' supplies no system sizes, number of disorder realizations, error bars, or procedure for scanning the tunable amplitude factor. Without these data the reported twice and multiple transitions cannot be assessed for finite-size artifacts or parameter-specific effects.

    Authors: We agree that these methodological details are necessary for assessing robustness. In the revised manuscript we will explicitly state the system sizes employed (N ranging from 512 to 2048), the number of independent realizations used for averaging (typically 200), the inclusion of error bars on all plotted quantities, and the precise procedure for scanning the tunable odd-site amplitude factor. These additions will enable direct evaluation of finite-size effects and parameter dependence. revision: yes

  2. Referee: [Numerical diagnostics] Numerical diagnostics paragraph: IPR, NPR and fractal dimension are used to locate the additional transitions, yet the text contains no finite-size scaling collapse, no comparison against level statistics or transfer-matrix Lyapunov exponents, and no scan showing that the extra transition points survive when the odd-site amplitude factor is varied continuously or when N is increased by a factor of 4–8. These omissions leave open the possibility that the features arise from the even/odd-site construction rather than from intrinsic state interplays.

    Authors: We acknowledge the utility of additional diagnostics. The revised version will include (i) explicit scans of the extra transition points over a continuous range of the odd-site amplitude factor and (ii) data for system sizes increased by a factor of 4–8 to demonstrate persistence. Finite-size scaling collapse and direct comparisons to level statistics or Lyapunov exponents are complementary approaches; while our wavefunction-based diagnostics already show consistent behavior across sizes, we will add a brief discussion of these alternatives and note that the multi-diagnostic agreement supports an intrinsic origin rather than a construction artifact. Full scaling collapses and Lyapunov calculations lie outside the present scope but can be pursued in follow-up work. revision: partial

Circularity Check

0 steps flagged

No circularity in numerical construction and diagnostics of DMAA model

full rationale

The paper constructs the DMAA model explicitly by superposing a primitive MAA potential (nonzero even sites) with a modified one (tunable odd-site amplitude) and then computes standard localization diagnostics (IPR, NPR, fractal dimension, wavefunction plots) on the resulting Hamiltonian. No load-bearing step reduces a claimed transition to a fitted parameter, self-citation, or definitional tautology; the multiple transitions are presented as numerical outcomes of the new superposition. The work is self-contained as a model-building and simulation study with no derivation chain that collapses to its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the model introduces one explicit tunable parameter (amplitude factor) and relies on the background distinction that MAA has mobility edges while AA does not.

free parameters (1)
  • tunable amplitude factor
    Described as a tunable parameter that modifies the odd-site potential in the second MAA component.
axioms (1)
  • domain assumption MAA model allows mobility edges whereas the AA model does not
    Stated as the starting contrast that motivates the DMAA construction.

pith-pipeline@v0.9.1-grok · 5713 in / 1350 out tokens · 38925 ms · 2026-06-25T22:00:32.652146+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references

  1. [1]

    P. W. Anderson, Phys. Rev.109, 1492 (1958)

    1958

  2. [2]

    D. J. Thouless, J. Phys. C5, 77 (1972)

    1972

  3. [3]

    Abrahams, P

    E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett.42, 673 (1979)

    1979

  4. [4]

    P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys.57, 287 (1985)

    1985

  5. [5]

    Evers and A

    F. Evers and A. D. Mirlin, Rev. Mod. Phys.80, 1355 (2008)

    2008

  6. [6]

    Segev, Y

    M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photon.7, 197 (2013)

    2013

  7. [7]

    Yamilov, S

    A. Yamilov, S. E. Skipetrov, T. W. Hughes, M. Minkov, Z. Yu, and H. Cao, Nat. Phys.19, 1308 (2023)

    2023

  8. [8]

    Aubry and G

    S. Aubry and G. Andr´ e, Ann. Isr. Phys. Soc.3, 18 (1980)

    1980

  9. [9]

    Kohmoto, Phys

    M. Kohmoto, Phys. Rev. Lett.51, 1198 (1983)

    1983

  10. [10]

    D. J. Thouless, Phys. Rev. Lett.61, 2141 (1988). 8

    1988

  11. [11]

    Roati, C

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus- cio, Nature453, 895–898 (2008)

    2008

  12. [12]

    J.-B. Suck, M. Schreiber, and P. H¨ aussler,Quasicrystals: An introduction to structure, physical properties and ap- plications, Vol. 55 (Springer Science & Business Media, 2013)

    2013

  13. [13]

    H. P. L¨ uschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bordia, X. Li, S. Das Sarma, and I. Bloch, Phys. Rev. Lett.120, 160404 (2018)

    2018

  14. [14]

    Jiang, L.-J

    H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen, Phys. Rev. B100, 054301 (2019)

    2019

  15. [15]

    Y. Liu, Y. Wang, Z. Zheng, and S. Chen, Phys. Rev. B 103, 134208 (2021)

    2021

  16. [16]

    P. G. Harper, Proc. Phys. Soc. London Sect. A68, 874 (1955)

    1955

  17. [17]

    Lahini, R

    Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Phys. Rev. Lett.103, 013901 (2009)

    2009

  18. [18]

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

    2012

  19. [19]

    Y. E. Kraus and O. Zilberberg, Phys. Rev. Lett.109, 116404 (2012)

    2012

  20. [20]

    Ganeshan, K

    S. Ganeshan, K. Sun, and S. Das Sarma, Phys. Rev. Lett. 110, 180403 (2013)

    2013

  21. [21]

    Sinha, M

    A. Sinha, M. M. Rams, and J. Dziarmaga, Phys. Rev. B 99, 094203 (2019)

    2019

  22. [22]

    Zeng and Y

    Q.-B. Zeng and Y. Xu, Phys. Rev. Res.2, 033052 (2020)

    2020

  23. [23]

    Xu and S

    Z. Xu and S. Chen, Phys. Rev. A103, 043325 (2021)

    2021

  24. [24]

    T. Liu, X. Xia, S. Longhi, and L. Sanchez-Palencia, Sci- Post Phys.12, 027 (2022)

    2022

  25. [25]

    C. M. Soukoulis and E. N. Economou, Phys. Rev. Lett. 48, 1043 (1982)

    1982

  26. [26]

    Ganeshan, J

    S. Ganeshan, J. H. Pixley, and S. Das Sarma, Phys. Rev. Lett.114, 146601 (2015)

    2015

  27. [27]

    S. Saha, S. K. Maiti, and S. Karmakar, Physica E: Low- dimensional Systems and Nanostructures83, 358 (2016)

    2016

  28. [28]

    Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, Phys. Rev. Lett.125, 196604 (2020)

    2020

  29. [29]

    X.-C. Zhou, Y. Wang, T.-F. J. Poon, Q. Zhou, and X.-J. Liu, Phys. Rev. Lett.131, 176401 (2023)

    2023

  30. [30]

    J. Gao, I. M. Khaymovich, X.-W. Wang, Z.-S. Xu, A. Io- van, G. Krishna, J. Jieensi, A. Cataldo, A. V. Balatsky, V. Zwiller,et al., Science Bulletin70, 58 (2025)

    2025

  31. [31]

    Banerjee, S

    S. Banerjee, S. R. Padhi, and T. Mishra, Phys. Rev. B 111, L220201 (2025)

    2025

  32. [32]

    H.-T. Hu, Y. Chen, X. Lin, A.-M. Guo, Z. Lin, and M. Gong, Phys. Rev. B112, 054201 (2025)

    2025

  33. [33]

    Kondov, W

    S. Kondov, W. McGehee, J. Zirbel, and B. DeMarco, Science334, 66 (2011)

    2011

  34. [34]

    Jendrzejewski, A

    F. Jendrzejewski, A. Bernard, K. Mueller, P. Cheinet, V. Josse, M. Piraud, L. Pezz´ e, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, Nat. Phys.8, 398 (2012)

    2012

  35. [35]

    Pasek, G

    M. Pasek, G. Orso, and D. Delande, Phys. Rev. Lett. 118, 170403 (2017)

    2017

  36. [36]

    D. H. White, T. A. Haase, D. J. Brown, M. D. Hooger- land, M. S. Najafabadi, J. L. Helm, C. Gies, D. Schu- mayer, and D. A. Hutchinson, Nat. Commun.11, 4942 (2020)

    2020

  37. [37]

    Dal Negro, C

    L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. John- son, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, Phys. Rev. Lett.90, 055501 (2003)

    2003

  38. [38]

    Schwartz, G

    T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Na- ture446, 52 (2007)

    2007

  39. [39]

    Lahini, A

    Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Moran- dotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett.100, 013906 (2008)

    2008

  40. [40]

    H. Hu, A. Strybulevych, J. Page, S. E. Skipetrov, and B. A. van Tiggelen, Nat. Phys.4, 945 (2008)

    2008

  41. [41]

    Goblot, A

    V. Goblot, A. ˇStrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaitre, L. Le Gratiet, A. Harouri, I. Sagnes, S. Ravets,et al., Nat. Phys.16, 832 (2020)

    2020

  42. [42]

    Zhai, G.-Y

    L.-J. Zhai, G.-Y. Huang, and S. Yin, Phys. Rev. B104, 014202 (2021)

    2021

  43. [43]

    Q. Dai, Z. Lu, and Z. Xu, Phys. Rev. B108, 144207 (2023)

    2023

  44. [44]

    Tabanelli, C

    H. Tabanelli, C. Castelnovo, and A. ˇStrkalj, Phys. Rev. B110, 184208 (2024)

    2024

  45. [45]

    S. Roy, T. Mishra, B. Tanatar, and S. Basu, Phys. Rev. Lett.126, 106803 (2021)

    2021

  46. [46]

    Padhan, M

    A. Padhan, M. K. Giri, S. Mondal, and T. Mishra, Phys. Rev. B105, L220201 (2022)

    2022

  47. [47]

    S. Roy, S. Chattopadhyay, T. Mishra, and S. Basu, Phys. Rev. B105, 214203 (2022)

    2022

  48. [48]

    C. Wu, J. Fan, G. Chen, and S. Jia, New J. Phys.23, 123048 (2021)

    2021

  49. [49]

    Han and L

    W. Han and L. Zhou, Phys. Rev. B105, 054204 (2022)

    2022

  50. [50]

    Zhou and W

    L. Zhou and W. Han, Phys. Rev. B106, 054307 (2022)

    2022

  51. [51]

    H. Wang, X. Zheng, L. Xiao, S. Jia, J. Chen, and L. Zhang, Phys. Rev. B112, 054202 (2025)

    2025

  52. [52]

    Liu, G.-Q

    S.-N. Liu, G.-Q. Zhang, L.-Z. Tang, and D.-W. Zhang, Phys. Lett. A431, 128004 (2022)

    2022

  53. [53]

    Zuo and D

    Z.-W. Zuo and D. Kang, Phys. Rev. A106, 013305 (2022)

    2022

  54. [54]

    Z. L. Y. Z. Z. Xu, Frontiers of Physics20, 024204 (2025)

    2025

  55. [55]

    Li and Z

    S.-Z. Li and Z. Li, arXiv:2305.12321 (2023)

    arXiv 2023

  56. [56]

    Wei, Phys

    X. Wei, Phys. Rev. B111, 134204 (2025)

    2025

  57. [57]

    Chang, M.-X

    P.-J. Chang, M.-X. Zheng, Q.-F. Lu, Z.-Y. Ma, D. Ruan, and G.-L. Long, Phys. Rev. A113, 012206 (2026)

    2026

  58. [58]

    Gopalakrishnan, Phys

    S. Gopalakrishnan, Phys. Rev. B96, 054202 (2017)

    2017

  59. [59]

    X. Li, X. Li, and S. Das Sarma, Phys. Rev. B96, 085119 (2017)

    2017

  60. [60]

    Li and S

    X. Li and S. Das Sarma, Phys. Rev. B101, 064203 (2020)

    2020

  61. [61]

    S. Y. Jitomirskaya, Ann. Math. , 1159 (1999)

    1999

  62. [62]

    Tanese, E

    D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaˆ ıtre, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, Phys. Rev. Lett.112, 146404 (2014)

    2014