pith. machine review for the scientific record. sign in

arxiv: 2604.06400 · v1 · submitted 2026-04-07 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall

Recognition: 2 theorem links

· Lean Theorem

Disorder averaging in random lattice models with periodic boundary conditions: Application to models with uncorrelated and correlated disorder

Bal\'azs Het\'enyi , Lu\'is Miguel Martelo , Andr\'as L\'aszl\'offy
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:55 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hall
keywords disorder averagingmodern theory of polarizationperiodic boundary conditionsAnderson localizationcorrelated disorderBinder cumulantgeometric phasedelocalization indicator
0
0 comments X

The pith

Disorder averaging applied to the geometric phase formulation of polarization yields its variance, higher moments, and a delocalization indicator in periodic systems.

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

The paper shows how to perform disorder averaging inside the modern theory of polarization, which expresses polarization as a geometric phase rather than a direct expectation value. This makes it possible to compute the variance of polarization, its higher-order moments, and the excess kurtosis under periodic boundary conditions. The same framework produces a delocalization diagnostic based on how energy eigenvalues degenerate when boundary conditions are twisted. The methods are demonstrated on a fully Anderson-localized chain, where polarization fluctuations vanish, and on the de Moura-Lyra model with tunable power-law correlated disorder, where pairwise degeneracies appear near half filling for strong correlations.

Core claim

Within the geometric-phase expression for polarization, disorder averaging can be carried out to obtain the variance of the polarization, its higher moments, and the Binder cumulant, while the degeneracy pattern of eigenstates under twisted boundary conditions supplies a direct indicator of localization or delocalization.

What carries the argument

Disorder averaging performed on the geometric (Berry) phase that defines polarization, together with the counting of boundary-condition-induced degeneracies.

If this is right

  • Polarization statistics become computable for any disordered lattice model that can be diagonalized under periodic boundaries.
  • The excess kurtosis of the polarization distribution can be tracked as a function of disorder strength or correlation parameter.
  • Pairwise degeneracies near band center for correlation exponent alpha greater than 2 signal the region previously conjectured to host a mobility edge.
  • The delocalization indicator can be evaluated without computing localization lengths directly.

Where Pith is reading between the lines

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

  • The same averaging procedure could be applied to other geometric quantities, such as Berry curvatures or Chern numbers, in disordered settings.
  • Extension to two or three dimensions would allow statistical characterization of polarization in higher-dimensional disordered insulators.
  • The degeneracy diagnostic might be compared directly with participation-ratio or inverse-participation-ratio measures on the same finite samples.

Load-bearing premise

The modern theory of polarization extends directly to disordered systems under periodic boundary conditions in a way that permits straightforward disorder averaging without additional uncontrolled approximations.

What would settle it

In the one-dimensional Anderson model, the computed polarization variance must remain zero for any disorder strength; any nonzero value obtained by the averaging procedure would contradict the claim.

Figures

Figures reproduced from arXiv: 2604.06400 by Andr\'as L\'aszl\'offy, Bal\'azs Het\'enyi, Lu\'is Miguel Martelo.

Figure 1
Figure 1. Figure 1: FIG. 1. Size scaling exponent ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Geometric Binder cumulant ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Geometric Binder cumulant ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Contour plot of the size scaling exponent, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Geometric Binder cumulant, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Contour plot of the size scaling exponent, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Upper panel: analysis of the energy spectrum as a [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Contour plot of the degeneracy indicator for the de [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Contour plot of the geometric Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Phase diagram of the de Moura-Lyra model. [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

Periodic boundary conditions are not always used in the study of disordered systems, but it can be advantageous to apply them to mimick thermodynamically large systems. In this case, polarization and its cumulants can not be obtained directly, but through the tools of the modern theory of polarization. This theory casts the polarization in crystalline systems as a geometric phase, rather than an operator expectation value. We develop disorder averaging techniques within the context of this theory which can calculate the variance of the polarization, its higher order moments, and the excess kurtosis (or Binder cumulant). We also derive an indicator of delocalization based on the degeneracy as a function of boundary conditions. We apply the computational techniques to two model systems. To test localization, we use a one-dimensional disordered model which is fully Anderson localized. Our calculations verify this. We also apply our techniques to the one dimensional de Moura-Lyra model, developed to study power law correlated (controlled by a parameter, $\alpha$) disorder. While this model is a pathological one, our method is validated. We also point out the significance of pairwise degeneracies found in the parameter range, $\alpha>2$ and near the band center (or near half filling), where the model was conjectured to exhibit a mobility edge.

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

1 major / 3 minor

Summary. The manuscript develops disorder-averaging techniques for the polarization and its cumulants (variance, higher moments, excess kurtosis/Binder cumulant) in random lattice models with periodic boundary conditions by applying the modern theory of polarization (Berry/Zak phase) to each disorder realization independently before averaging. It also derives a delocalization indicator based on polarization degeneracy as a function of boundary conditions. The methods are tested on a 1D Anderson-localized model (confirming strong localization) and the 1D de Moura-Lyra model with power-law correlated disorder (validating the approach while noting pairwise degeneracies for α>2 near band center).

Significance. If the central construction holds, the work supplies a concrete, per-sample computational route to polarization statistics under PBC that avoids additional uncontrolled approximations in the averaging step itself. This is useful for numerical studies of localization and mobility edges in disordered systems, and the degeneracy diagnostic provides an independent observable. The explicit validation on the Anderson case and the de Moura-Lyra exercise (including the noted degeneracies) constitute reproducible checks that strengthen the claim.

major comments (1)
  1. [de Moura-Lyra section] § on de Moura-Lyra application: the observation of pairwise degeneracies for α>2 near half-filling is presented as a separate diagnostic, but the text should explicitly state whether these degeneracies are excluded from the disorder average or handled by a symmetry factor; otherwise the variance and Binder cumulant formulas could be affected by overcounting.
minor comments (3)
  1. [Abstract] Abstract: 'mimick' should be 'mimic'; the phrase 'excess kurtosis (or Binder cumulant)' should be clarified because the Binder cumulant is a normalized fourth-moment ratio, not identical to excess kurtosis.
  2. [Methods] The manuscript would benefit from an explicit algorithmic pseudocode or flowchart for the per-realization Zak-phase computation followed by the disorder-average step, especially for readers implementing the method.
  3. [Figures] Figure captions and axis labels should include the precise definition of the polarization (e.g., in units of e a / 2π) and the number of disorder realizations used for each data point.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of its significance, and recommendation for minor revision. We address the single major comment below and will incorporate the requested clarification.

read point-by-point responses

  1. Referee: [de Moura-Lyra section] § on de Moura-Lyra application: the observation of pairwise degeneracies for α>2 near half-filling is presented as a separate diagnostic, but the text should explicitly state whether these degeneracies are excluded from the disorder average or handled by a symmetry factor; otherwise the variance and Binder cumulant formulas could be affected by overcounting.

    Authors: We agree that an explicit statement is needed to avoid any ambiguity. In our implementation, the disorder average is performed over independent realizations sampled from the ensemble; each realization contributes to the polarization statistics (including variance and Binder cumulant) exactly once according to its natural occurrence, without exclusion or an additional symmetry factor. The pairwise degeneracies are instead used solely as the separate delocalization diagnostic described in the text. To make this handling unambiguous, we will revise the de Moura-Lyra section to state explicitly that (i) no realizations are excluded on the basis of degeneracy and (ii) no symmetry factor is applied in the cumulant formulas. This clarification does not change the numerical results or the formulas already presented. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central construction applies the pre-existing modern theory of polarization (geometric phase via Berry/Zak) independently to each disorder realization, then computes standard statistical moments (variance, higher cumulants, Binder cumulant) and a separate degeneracy-based delocalization diagnostic on the resulting per-sample values. These steps follow directly from the definitions of cumulants and boundary-condition degeneracy without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations that would make the outputs equivalent to the inputs by construction. Validation on the Anderson model and de Moura-Lyra model serves as external checks rather than internal forcing. The derivation chain remains self-contained and independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of the modern theory of polarization to disordered periodic systems and on the representativeness of the two chosen 1D models for testing localization and correlation effects.

axioms (1)
  • domain assumption Modern theory of polarization applies to disordered systems with periodic boundary conditions for computing cumulants via disorder averaging
    Invoked as the foundation for developing the averaging techniques described in the abstract.

pith-pipeline@v0.9.0 · 5551 in / 1180 out tokens · 36644 ms · 2026-05-10T17:55:55.887515+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

75 extracted references · 2 canonical work pages

  1. [1]

    A verages were taken within each segment

    The reduced energy range was divided into one hundred equal segments. A verages were taken within each segment. The Binder cumulants are smaller at the band edges, and they reach their maxima around the band center. In the interval 0 < W < 1 the system is known to be localized [18], but this is not easy to establish numerically. The GBC is often greater t...

    2048

  2. [2]

    The degeneracy indicator is shown in Fig. 9. Again, there is a definite region, α > 2, around a filling den- sity of ρ ≈ 0.5, where a degeneracy is seen. Both the GBC and the degeneracy indicator can be considered as gauges of localization, reaching their maximum values at maximal delocalization. We summarize our results by proposing a phase diagram for t...

    2017

  3. [3]

    Kramer and A

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

    1993

  4. [4]

    Kohn, ”Theory of the Insulating State” Phys

    W. Kohn, ”Theory of the Insulating State” Phys. Rev. 133 A171 (1963)

    1963

  5. [5]

    In this particular sentence, however, we are re- ferring to the localization of the center of charge of a crystalline system

    The term many-body localization, nowadays, refers to the ergodicity breaking that occurs in certain many-body systems. In this particular sentence, however, we are re- ferring to the localization of the center of charge of a crystalline system

  6. [6]

    R. D. King-Smith and D. Vanderbilt, ”Theory of po- larization of crystalline solids” Phys. Rev. B 47 R1651 (1993)

    1993

  7. [7]

    Resta, ”Macroscopic polarization in crystalline di- electrics”Rev

    R. Resta, ”Macroscopic polarization in crystalline di- electrics”Rev. Mod. Phys. 66 899 (1994)

    1994

  8. [8]

    Vanderbilt, Berry Phases in Electronic Structure Theory, Cambridge University Press, Cambridge, U.K

    D. Vanderbilt, Berry Phases in Electronic Structure Theory, Cambridge University Press, Cambridge, U.K. (2018)

    2018

  9. [9]

    Resta, ”Manifestations of Berry’s phase in molecules and condensed matter” J

    R. Resta, ”Manifestations of Berry’s phase in molecules and condensed matter” J. Phys.: Cond. Mat. 12 R107 (2000)

    2000

  10. [10]

    Spaldin, ”A beginner’s guide to the modern theory of polarization”, J

    N. Spaldin, ”A beginner’s guide to the modern theory of polarization”, J. Sol. Stat. Chem. 195 2 (2012)

    2012

  11. [11]

    Resta, ”Quantum Mechanical Position Operator in Extended Systems” Phys

    R. Resta, ”Quantum Mechanical Position Operator in Extended Systems” Phys. Rev. Lett. 80 1800 (1998)

    1998

  12. [12]

    Zak, ”Berry’s phase for energy bands in solids” Phys

    J. Zak, ”Berry’s phase for energy bands in solids” Phys. Rev. Lett. 62 2747 (1989)

    1989

  13. [13]

    M. V. Berry, ”’Quantal phase factors accompanying adi- abatic changes” Proc. Roy. Soc. London A392 45 (1984)

    1984

  14. [14]

    Resta and S

    R. Resta and S. Sorella, ”Electron Localization in the Insulating State” Phys. Rev. Lett. 82 370 (1999)

    1999

  15. [15]

    Souza, T

    I. Souza, T. Wilkens, and R. M. Martin, ”Polarization and localization in insulators: Generating function ap- proach” Phys. Rev. B 62 1666 (2000)

    2000

  16. [16]

    Langedijk, B

    A. Langedijk, B. van Tiggelen, and D. S. Wiersma, ”Fifty years of Anderson localization” Physics Today 62 24 (2009)

    2009

  17. [17]

    Evers and A

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

    2008

  18. [18]

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

    1985

  19. [19]

    P. W. Anderson, ”Absence of Diffusion in Certain Ran- dom Lattices” Phys. Rev. 109 1492 (1958)

    1958

  20. [20]

    Abrahams, P

    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 (1979)

    1979

  21. [21]

    Cheng, H

    M. Cheng, H. Chen, and J. Chen ”Regulating Anderson localization with structural defect disorder” . Chin. Phys. B 33 107201 (2024). 12

    2024

  22. [22]

    Bhattacharjee, P

    S. Bhattacharjee, P. Sierant, M. Dudińsky, J. Wehr, J. Zakrzewski, and M.Lewenstein, ”Anderson localiza- tion induced by structural disorder”, Phys. Rev. B 111 L180202 (2025)

    2025

  23. [23]

    T. Li, Y. Peng, Y. Wang, and H. Hu, ”Anderson tran- sition and mobility edges on hyperbolic lattices with randomly connected boundaries”, Comm. Phys. 7 371 (2024)

    2024

  24. [24]

    Mondal, S

    M. Mondal, S. K. Maiti, ”Bias driven circular current in a ring nanojunction: Critical role of environmental interaction” J. Appl. Phys. 134 174302 (2025)

    2025

  25. [25]

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

    2013

  26. [26]

    J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions , Lecture Notes on Physica, vol. 919, Springer International Publishing, (2016)

    2016

  27. [27]

    Li, R.-L

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

    2009

  28. [28]

    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. B 80, 165316 (2009)

    2009

  29. [29]

    Prodan, ”Disordered topological insulators: a non- commutative geometry perspective” J

    E. Prodan, ”Disordered topological insulators: a non- commutative geometry perspective” J. Phys. A: Math. Theor. 44 113001 (2011)

    2011

  30. [30]

    Nomura, ”Disorder-induced quantum phase transitions in three-dimensional topological insu- lators and superconductors”, Phys

    S.Ryu and K. Nomura, ”Disorder-induced quantum phase transitions in three-dimensional topological insu- lators and superconductors”, Phys. Rev. B 85 155138 (2012)

    2012

  31. [31]

    B. Wu, J. Song, J. Zhou, H. Jiang, ”Disorder Effects in Topological States: Brief review of the recent develop- ments”, Chin. Phys. B 25 117311 (2016)

    2016

  32. [32]

    C. P. Orth, T. Sekera, C. Bruder, and T. L Schmidt, ”The topological Anderson insulator phase in the Kane-Mele model”, Sci. Rep. 6 24007 (2016)

    2016

  33. [33]

    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”, Science 362 929 (2018)

    2018

  34. [34]

    O. A. Awoga and J. Cayao, ”Identifying trivial and Majo- rana zero-energy modes using the Majorana polarization” Phys. Rev. B 110 165404 (2024)

    2024

  35. [35]

    Ahmed, Y

    E. Ahmed, Y. Tanaka, J. Cayao, ”Anomalous proxim- ity effect under Andreev and Majorana bound states”, arxiv.org:2505.03726

    work page arXiv

  36. [36]

    Kerala Varma and S

    V. Kerala Varma and S. Pilati, ”Kohn’s localisation in disordered fermionic systems with and without interac- tions” Phys. Rev. B 92 134207 (2015)

    2015

  37. [37]

    G. L. Bendazzoli, S. Evangelisti, A. Monari, and R. Resta, ”Kohn’s localization in the insulating state: one- dimensional lattices, crystalline versus disordered”, J. Chem. Phys. 133 064703 (2010)

    2010

  38. [38]

    Hetényi, S

    B. Hetényi, S. Parlak, and M. Yahyavi, ”Scaling and renormalization in the modern theory of polarization: Application to disordered systems” Phys. Rev. B 104 214207 (2021)

    2021

  39. [39]

    D. M. Basko, I. L. Aleiner, and B. L. Altschuler, ”Metal– insulator transition in a weakly interacting many-electron system with localized single-particle states”, Ann. Phys. 321 1126 (2006)

    2006

  40. [40]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, ”Many-Body Localiza- tion and Thermalization in Quantum Statistical Mechan- ics”, Ann. Rev. Cond. Mat. Phys. 6 15 (2015)

    2015

  41. [41]

    X. Li, J. H. Pixley, D.-L. Deng, S. Ganeshan, and S. Das Sarma,”Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge”, Phys. Rev. B 93 184204 (2016)

    2016

  42. [42]

    Ryu and Y

    S. Ryu and Y. Hatsugai, ”Entanglement entropy and the Berry phase in the solid state”, Phys. Rev. B 73 245115 (2006)

    2006

  43. [43]

    W. N. Faugno and T. Ozawa, ”Characterizing the many- body localization crossover as a metal-insulator transi- tion: Localization length from polarization and quantum metric”, Phys. Rev. B 113 014306 (2026)

    2026

  44. [44]

    Simonov and A

    A. Simonov and A. L. Goodwin, ”Designing disorder into crystalline materials” Nat. Rev. Chem. 4 , 657 (2020)

    2020

  45. [45]

    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. B 110 184207 (2024)

    2024

  46. [46]

    A. L. Goodwin, ”Structural Complexity and Correlated Disorder in Materials Chemistry” arxiv:2509.09171

    work page arXiv

  47. [47]

    D. H. Dunlap, P.-L. Wu and P. W. Phillips, ”Absence of Localization in a Random Dimer Model”, Phys. Rev. Lett. 65 88 (1990)

    1990

  48. [48]

    E. G. Meekel and A. L. Goodwin, ”Correlated disorder in metal–organic frameworks”, CrystEngComm 23 2915 (2021)

    2021

  49. [49]

    V. D. Neverov, A. E. Lukyanov, A. V. Krasavin, A. Vagov, M. D. Croitoru, ”Correlated disorder as a way towards robust superconductivity” Comm. Phys. 5 177 (2022)

    2022

  50. [50]

    V. D. Neverov, A. E. Lukyanov, A. V. Krasavin, A. Vagov, M. D. Croitoru, ”Spatial correlations in disor- der: Impact on the superconducting critical tempera- ture” Phys. Rev. B 111 184514 (2025)

    2025

  51. [51]

    C.-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, F. Sciortino, M. Simons, and H. E. Stanley, ”Long-range correlations in nucleotide sequences”, Nature 356 168 (1992)

    1992

  52. [52]

    Y. Li, W. Zhou, Y. Zhou, H. F. Cheng, B. Lee, X. Hu, E. W. Roth, V. P. Dravid, S. C. Glotzer, and C. A. Mirkin, ”Cocrystals combining order and correlated disorder via colloidal crystal engineering with DNA”, Sci. Adv. 11 eadu4919 (2025)

    2025

  53. [53]

    X. Liu, D. Lyu, C. Merlet, M. J. A. Leesmith, X. Hua, Z. Xu, C. P. Grey, and A. C. Forse, ”Structural disorder determines capacitance in nanoporous carbons”, Science 384 321 (2024)

    2024

  54. [54]

    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

  55. [55]

    Domínguez-Adame, V

    F. Domínguez-Adame, V. A. Malyshev, F. A. B. F. de Moura, and M. L. Lyra, ”Bloch-like Oscillations in a One- Dimensional Lattice with Long-Range Correlated Disor- der”, Phys. Rev. Lett. 91 197402 (2003)

    2003

  56. [56]

    Delocalization in the 1D An- derson Model with Long-Range Correlated Disorder

    J.W. Kantelhardt, S. Russ, A. Bunde, S. Havlin, and I.Webman, Comment on “Delocalization in the 1D An- derson Model with Long-Range Correlated Disorder” Phys. Rev. Lett. 84, 198 (2000)

    2000

  57. [57]

    Delo- calization in the 1D Anderson Model with Long-Range Correlated Disorder

    F. A. B. F. de Moura and M. L. Lyra, Reply to “Delo- calization in the 1D Anderson Model with Long-Range Correlated Disorder”, Phys. Rev. Lett. 84 199 (2000)

    2000

  58. [58]

    Bunde, S

    A. Bunde, S. Havlin, J.W. Kantelhardt, S. Russ, and I. Webman, ”Localization in a Highly Correlated Potential Landscape”, J. Mol. Liq. 86 151 (2000). 13

    2000

  59. [59]

    S. Russ, J. W. Kantelhardt,A. Bunde,and S. Havlin, ”Lo- calization in self-affine energy landscapes” Phys. Rev. B 64 134209 (2001)

    2001

  60. [60]

    Shima, T

    H. Shima, T. Nomura, and T. Nakayama ”Localization- delocalization transition in one-dimensional electron sys- tems with long-range correlated disorder” Phys. Rev. B 70 075116 (2004)

    2004

  61. [61]

    Kaya, ”Localization-delocalization transition in chains with long-range correlated disorder”, Eur

    T. Kaya, ”Localization-delocalization transition in chains with long-range correlated disorder”, Eur. Phys. J. B 55 49 (2007)

    2007

  62. [62]

    G. M. Petersen and N. Sandler, ”Anticorrelations from power-law spectral disorder and conditions for an Ander- son transition”, Phys. Rev. B 87 195443 (2013)

    2013

  63. [63]

    J. P. Santos Pires, N. A. Khan, J. M. Viana Parente Lopes, and J. M. B. Lopes dos Santos, ”Global delocal- ization transition in the de Moura-Lyra model”, Phys. Rev. B 99 205248 (2019)

    2019

  64. [64]

    Izrailev, A.A

    F.M. Izrailev, A.A. Krokhin, N.M. Makarov ”Anomalous localization in low-dimensional systems with correlated disorder”, Phys. Rep. 512 125 (2012)

    2012

  65. [65]

    Duthie, S

    A. Duthie, S. Roy, and D. E. Logan, ”Anomalous mul- tifractality in quantum chains with strongly correlated disorder” Phys. Rev. B 106 L020201 (2022)

    2022

  66. [66]

    N. A. Khan, S. Muhammad, and M. Sajid, ”Single pa- rameter scaling in the correlated Anderson model” Phys- ica E 139 115150 (2022)

    2022

  67. [67]

    N. A. Khan, M. Jan, and M. Shah, ”Conductance dis- tributions in correlated disordered metals”, Ann. Phys. 455 169387 (2023)

    2023

  68. [68]

    Hetényi and B

    B. Hetényi and B. Dóra, ”Quantum phase transitions from analysis of the polarization amplitude”, Phys. Rev. B 99 085126 (2019)

    2019

  69. [69]

    Hetényi and S

    B. Hetényi and S. Cengiz, ”Geometric cumulants asso- ciated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator tran- sitions in crystalline electronic systems”, Phys. Rev. B 106 195151 (2022)

    2022

  70. [70]

    Hetényi, ”Scaling of the bulk polarization in extended and localized phases of a quasiperiodic model”, Phys

    B. Hetényi, ”Scaling of the bulk polarization in extended and localized phases of a quasiperiodic model”, Phys. Rev. B 110 125124 (2024)

    2024

  71. [71]

    Binder, ”Finite size scaling analysis of ising model block distribution functions” Z

    K. Binder, ”Finite size scaling analysis of ising model block distribution functions” Z. Phys. B 43 119 (1981)

    1981

  72. [72]

    Binder, ”Critical Properties from Monte Carlo Coarse Graining and Renormalization” Phys

    K. Binder, ”Critical Properties from Monte Carlo Coarse Graining and Renormalization” Phys. Rev. Lett. 47 693 (1981)

    1981

  73. [73]

    M. E. Fisher, in Critical Phenomena , Proc. 51st Enrico Fermi Summer School, Varena, edited by M. S. Green (Academic Press, N.Y.) 1972

    1972

  74. [74]

    M. E. Fisher and M. N. Barber, ”Scaling Theory for Fi- nite Size Effects in the Critical Region” Phys. Rev. Lett. 28 1516 (1972)

    1972

  75. [75]

    Baeriswyl, ”Variational scheme for the Mott transi- tion” Found

    D. Baeriswyl, ”Variational scheme for the Mott transi- tion” Found. Phys. 30 2033 (2000)

    2033