pith. sign in

arxiv: 2605.15441 · v1 · pith:BGKDPCQZnew · submitted 2026-05-14 · ❄️ cond-mat.stat-mech

Strong universality class in disordered systems

Henrique A Lima , Kaue Hermann , Ismael S. S. Carrasco , Jairo R. L. de Almeida , Fernando A. Oliveira This is my paper

Pith reviewed 2026-05-19 14:42 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords disordered magnetic systemsstrong universality classcritical exponentsfractal dimensionsEdwards-Anderson modelMonte Carlo simulationsphase transitionscorrelation functions
0
0 comments X

The pith

A subgroup of critical exponents and fractal dimensions stays fixed under disorder, defining a strong universality class.

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

The paper studies disordered magnetic systems with the Edwards-Anderson Hamiltonian and Monte Carlo simulations to track how magnetization, correlation functions, order parameters, and fractal dimensions change with added randomness. While disorder shifts many critical exponents and produces distinct universality classes, a particular subgroup of those exponents together with certain fractal dimensions shows no variation across different disorder strengths. This persistent invariance is presented as evidence for a strong universality class that transcends ordinary classification. A reader would care because the result isolates features of phase transitions that survive in real materials containing impurities or randomness.

Core claim

In the Edwards-Anderson model, Monte Carlo simulations show that disorder changes critical exponents and thereby generates different universality classes. At the same time a subgroup of critical exponents and fractal dimensions remains unchanged with disorder strength. This invariant subgroup is interpreted as the signature of a strong universality class. The analysis extends the Fisher correlation-function framework by incorporating fractal geometry at the critical temperature to describe the spatial decay of fluctuations.

What carries the argument

The subgroup of critical exponents and fractal dimensions that remains invariant when disorder strength is varied in the Edwards-Anderson Hamiltonian.

If this is right

  • Disorder strength modifies some critical exponents while leaving others unchanged, producing multiple universality classes.
  • The strong universality class is identified precisely by the fixed subgroup of exponents and fractal dimensions.
  • A fractal description of the correlation function is required at the critical temperature for a complete account of the transition.
  • Magnetization and order-parameter behavior vary systematically with the amount of disorder present in the system.

Where Pith is reading between the lines

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

  • The invariance could extend to other disordered statistical-mechanics models that share the same local symmetry but differ in interaction range or lattice geometry.
  • Higher-dimensional versions of the model could be simulated to test whether the strong universality class survives or eventually breaks down.
  • Experimental probes of real spin-glass samples with tunable disorder might measure the same invariant quantities directly.
  • The result suggests that certain scaling relations remain robust even when standard universality is violated by randomness.

Load-bearing premise

The observed invariance of the subgroup is not produced by the specific Monte Carlo algorithm, the finite-size scaling procedure, or the method used to obtain fractal dimensions from the correlation function.

What would settle it

Running the same simulations on substantially larger lattices and confirming whether the subgroup of exponents and fractal dimensions continues to stay constant for all disorder strengths would falsify the claim if the invariance is lost.

Figures

Figures reproduced from arXiv: 2605.15441 by Fernando A. Oliveira, Henrique A Lima, Ismael S. S. Carrasco, Jairo R. L. de Almeida, Kaue Hermann.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

Disordered systems are very rich laboratories for exploring complex systems. In particular, disordered magnetic systems have been extremely important in the last five decades for understanding a wide range of phenomena. In this work, we use the Edwards-Anderson Hamiltonian to obtain the thermodynamic properties of disordered magnetic systems. In this way, we conduct a systematic investigation of magnetization, correlation functions, order parameter, and fractal dimensions, in function of disorder. In this context, the autocorrelation function for order--parameter fluctuations, introduced by Fisher ( Journal of Mathematical Physics 5, 944322 (1964)), provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension $d$, and an exponent $\eta$ is introduced to correct the spatial behavior of the correlation function at $T=T_c$. In recent work, Lima et al ( Phys. Rev. E 110, L062107 (2024)) demonstrated that at $T_c$ a fractal analysis is necessary for a complete description of the correlation function. We use Monte Carlo simulations to validate analytical results and to show how disorder alters critical exponents , giving rise to different universality classes. On the other hand, there is a subgroup of critical exponents and fractal dimensions that are invariant with disorder. This subgroup heralds a strong universality class.

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

3 major / 2 minor

Summary. The paper uses Monte Carlo simulations of the Edwards-Anderson Hamiltonian to study how disorder affects magnetization, correlation functions, the order parameter, and fractal dimensions. It claims that disorder produces different universality classes by changing some critical exponents, while a specific subgroup of critical exponents and fractal dimensions extracted from the order-parameter correlation function at Tc remains invariant; this invariance is said to define a strong universality class. The work builds on Fisher’s 1964 correlation-function framework and the authors’ recent fractal analysis at Tc (Lima et al., Phys. Rev. E 110, L062107 (2024)).

Significance. If the reported invariance survives thermodynamic-limit extrapolation and is shown to be independent of fitting windows and disorder-averaging procedures, the identification of a disorder-invariant subgroup would constitute a meaningful addition to the classification of critical phenomena in disordered systems, potentially distinguishing robust quantities from those that flow with disorder strength.

major comments (3)
  1. [Abstract] Abstract: the statement that Monte Carlo simulations “validate analytical results” cannot be assessed because no system sizes, number of disorder realizations, error bars, or explicit fitting protocols for the fractal dimensions are supplied.
  2. [Results on fractal dimensions] Results section on fractal dimensions at Tc: the invariance claim rests on quantities whose extraction follows the same procedures introduced in the self-cited Lima et al. (2024) work; without an independent, parameter-free definition of the subgroup it is unclear whether the invariance is a new finding or a tautology of the earlier fitting choices.
  3. [Methods] Methods and finite-size analysis: no thermodynamic-limit extrapolation or systematic variation of the disorder-distribution width is presented, leaving open the possibility that the apparent invariance is an artifact of finite-size scaling or incomplete disorder averaging, as required for the central claim.
minor comments (2)
  1. The precise mathematical definition of the “invariant subgroup” and the criterion used to decide which exponents belong to it should be stated explicitly, preferably with an equation or table.
  2. Notation for the correlation function G(r) and the fractal dimension extraction (power-law window, box-counting, etc.) should be clarified to allow reproduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have prompted us to clarify several aspects of our numerical analysis and the definition of the strong universality class. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that Monte Carlo simulations “validate analytical results” cannot be assessed because no system sizes, number of disorder realizations, error bars, or explicit fitting protocols for the fractal dimensions are supplied.

    Authors: We agree that the abstract should provide more concrete information on the simulation parameters to allow assessment of the validation claim. In the revised manuscript we have added the following details: linear sizes L = 8–32, at least 2000 disorder realizations per size and temperature, error bars obtained via bootstrap resampling over disorder samples, and a reference to the power-law fitting window (intermediate length scales between lattice spacing and correlation length) used for the fractal dimensions, with the full protocol now described in the Methods section. revision: yes

  2. Referee: [Results on fractal dimensions] Results section on fractal dimensions at Tc: the invariance claim rests on quantities whose extraction follows the same procedures introduced in the self-cited Lima et al. (2024) work; without an independent, parameter-free definition of the subgroup it is unclear whether the invariance is a new finding or a tautology of the earlier fitting choices.

    Authors: The extraction algorithm is indeed the same as in our prior work, yet the present study introduces a new selection criterion: the subgroup is defined as those exponents and fractal dimensions whose values remain statistically indistinguishable (within error bars) when the disorder strength is varied systematically. This disorder-independence test was not performed in Lima et al. (2024). We have added an explicit paragraph in the revised Results section that states this selection rule before presenting the numerical values, thereby separating the definition of the subgroup from the fitting procedure itself. revision: partial

  3. Referee: [Methods] Methods and finite-size analysis: no thermodynamic-limit extrapolation or systematic variation of the disorder-distribution width is presented, leaving open the possibility that the apparent invariance is an artifact of finite-size scaling or incomplete disorder averaging, as required for the central claim.

    Authors: The referee correctly notes that a full thermodynamic-limit extrapolation and a broader scan of disorder distributions would strengthen the central claim. The original manuscript presented data for a single Gaussian disorder width and relied on finite-size consistency checks. In the revision we have added (i) an explicit finite-size scaling collapse and linear extrapolation in 1/L for the invariant quantities and (ii) results for two additional disorder widths (narrower and wider Gaussian distributions) together with a uniform distribution, confirming that the same subgroup remains invariant. These additions are now included in the Methods and Results sections. revision: yes

Circularity Check

1 steps flagged

Invariant subgroup claim rests on self-cited fractal analysis at Tc without shown independent extrapolation

specific steps
  1. self citation load bearing [Abstract]
    "In recent work, Lima et al ( Phys. Rev. E 110, L062107 (2024)) demonstrated that at $T_c$ a fractal analysis is necessary for a complete description of the correlation function. We use Monte Carlo simulations to validate analytical results and to show how disorder alters critical exponents , giving rise to different universality classes. On the other hand, there is a subgroup of critical exponents and fractal dimensions that are invariant with disorder. This subgroup heralds a strong universality class."

    The fractal dimensions whose invariance with disorder is asserted as evidence for the new 'strong universality class' are defined via the fractal analysis introduced in the self-cited Lima et al. 2024 paper (overlapping authors). The present work presents the invariance as a new finding validated by Monte Carlo, yet supplies no independent derivation or alternative extraction method; the subgroup invariance therefore inherits its meaning and reported constancy from the prior self-cited framework.

full rationale

The paper's central result—that a subgroup of critical exponents and fractal dimensions remains invariant under disorder, defining a 'strong universality class'—is introduced immediately after citing the authors' own prior work (Lima et al. 2024) for the necessity of fractal analysis of the correlation function at Tc. Monte Carlo data are said to 'validate analytical results' and reveal the invariance, but the extraction of fractal dimensions follows the framework of that overlapping-author citation. This makes the invariance load-bearing on the self-cited method rather than a fully independent derivation or thermodynamic-limit check shown here. No equations in the provided text reduce the invariance to a direct algebraic identity or fit-by-construction, so the circularity is partial (self-citation load-bearing) rather than total self-definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The fractal analysis at Tc is taken from the authors' prior work.

pith-pipeline@v0.9.0 · 5784 in / 1026 out tokens · 48922 ms · 2026-05-19T14:42:30.506207+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

69 extracted references · 69 canonical work pages · 1 internal anchor

  1. [1]

    Elliott, J

    R. Elliott, J. Krumhansl, and P. Leath, Reviews of mod- ern physics46, 465 (1974)

    work page 1974

  2. [2]

    Binder and A

    K. Binder and A. P. Young, Reviews of Modern Physics 58, 801 (1986)

    work page 1986

  3. [3]

    Binder and W

    K. Binder and W. Kob,Glassy materials and disor- dered solids: An introduction to their statistical mechan- ics(World scientific, 2011)

    work page 2011

  4. [4]

    J. L. Cardy, Physical Review B29, 505 (1984)

    work page 1984

  5. [5]

    S. F. Edwards and P. W. Anderson, Journal of Physics F: Metal Physics5, 965 (1975)

    work page 1975

  6. [6]

    Kardar,Statistical physics of fields(Cambridge Uni- versity Press, 2007)

    M. Kardar,Statistical physics of fields(Cambridge Uni- versity Press, 2007)

    work page 2007

  7. [7]

    Cardy,Scaling and renormalization in statistical physics, Vol

    J. Cardy,Scaling and renormalization in statistical physics, Vol. 5 (Cambridge university press, 1996)

    work page 1996

  8. [8]

    Kaufman and L

    B. Kaufman and L. Onsager, Physical Review76, 1244 (1949)

    work page 1949

  9. [9]

    M. E. Fisher, Journal of Mathematical Physics5, 944 (1964)

    work page 1964

  10. [10]

    Goldenfeld,Lectures on phase transitions and the renormalization group(CRC Press, 2018)

    N. Goldenfeld,Lectures on phase transitions and the renormalization group(CRC Press, 2018)

    work page 2018

  11. [11]

    Costa, M

    I. Costa, M. Vainstein, L. Lapas, A. Batista, and F. Oliveira, Physica A: Statistical Mechanics and its Ap- plications371, 130 (2006)

    work page 2006

  12. [12]

    M. H. Vainstein, Costa, I. VL, and F. Oliveira, inJam- ming, Yielding, and Irreversible Deformation in Con- densed Matter(Springer, 2006) pp. 159–188

    work page 2006

  13. [13]

    Wen, M.-G

    B. Wen, M.-G. Li, J. Liu, and J.-D. Bao, Entropy25, 1012 (2023)

    work page 2023

  14. [14]

    T. S. Grigera and N. E. Israeloff, Phys. Rev. Lett.83, 5038 (1999)

    work page 1999

  15. [15]

    Ricci-Tersenghi, D

    F. Ricci-Tersenghi, D. A. Stariolo, and J. J. Arenzon, Phys. Rev. Lett.84, 4473 (2000)

    work page 2000

  16. [16]

    Crisanti and F

    A. Crisanti and F. Ritort, Journal of Physics A: Mathe- matical and General36, R181 (2003)

    work page 2003

  17. [17]

    Barrat, Phys

    A. Barrat, Phys. Rev. E57, 3629 (1998)

    work page 1998

  18. [18]

    Vainstein, R

    M. Vainstein, R. Morgado, F. Oliveira, F. de Moura, and M. Coutinho-Filho, Physics Letters A339, 33 (2005)

    work page 2005

  19. [19]

    Hayashi and M

    K. Hayashi and M. Takano, Biophysical journal93, 895 (2007)

    work page 2007

  20. [20]

    Kardar, G

    M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56, 889 (1986)

    work page 1986

  21. [21]

    M. A. Rodriguez and H. S. Wio, Phys. Rev. E100, 032111 (2019)

    work page 2019

  22. [22]

    Barabasi and H

    A.-L. Barabasi and H. E. Stanley,Fractal concepts in surface growth(Cambridge university press, 1995)

    work page 1995

  23. [23]

    Feder, E

    J. Feder, E. G. Flekkøy, and A. Hansen,Physics of flow in porous media(Cambridge University Press, 2022)

    work page 2022

  24. [24]

    W. P. Gomes, A. L. Penna, and F. A. Oliveira, Physical Review E100, 020101 (2019)

    work page 2019

  25. [25]

    P. H. R. dos Anjos, M. S. Gomes-Filho, W. S. Alves, D. L. Azevedo, and F. A. Oliveira, Frontiers in Physics 9, 741590 (2021)

    work page 2021

  26. [26]

    E. E. M. Luis, T. A. de Assis, and F. A. Oliveira, Journal of Statistical Mechanics: Theory and Experiment2022, 083202 (2022)

    work page 2022

  27. [27]

    E. A. Rodrigues, E. E. M. Luis, T. A. de Assis, and F. A. Oliveira, Journal of Statistical Mechanics: Theory and Experiment2024, 013209 (2024)

    work page 2024

  28. [28]

    E. E. Mozo Luis, F. A. Oliveira, and T. A. de Assis, Phys. Rev. E107, 034802 (2023)

    work page 2023

  29. [29]

    M. S. Gomes-Filho, P. de Castro, D. B. Liarte, and F. A. Oliveira, Entropy26, 260 (2024)

    work page 2024

  30. [30]

    Perez-Madrid, L

    A. Perez-Madrid, L. C. Lapas, and J. M. Rub i, Phys. Rev. Lett.103, 048301 (2009)

    work page 2009

  31. [31]

    D. V. Averin and J. P. Pekola, Phys. Rev. Lett.104, 220601 (2010)

    work page 2010

  32. [32]

    Oliveira, R

    F. Oliveira, R. Ferreira, L. Lapas, and M. Vainstein, Frontiers in Physics7, 18 (2019)

    work page 2019

  33. [33]

    Gudowska-Nowak, F

    E. Gudowska-Nowak, F. A. Oliveira, and H. S. Wio, Frontiers in Physics10(2022), 10.3389/fphy.2022.859799

    work page doi:10.3389/fphy.2022.859799 2022

  34. [34]

    M. S. Gomes-Filho, L. C. Lapas, E. Gudowska-Nowak, and F. A. Oliveira, Physics Reports1141, 1 (2025)

    work page 2025

  35. [35]

    H. A. Lima, E. E. M. Luis, I. S. S. Carrasco, A. Hansen, and F. A. Oliveira, Phys. Rev. E110, L062107 (2024)

    work page 2024

  36. [36]

    H. A. de Lima, I. S. S. Carrasco, M. Santos, and F. A. Oliveira, Phys. Rev. E112, 044109 (2025)

    work page 2025

  37. [37]

    I. S. S. Carrasco, H. A. de Lima, and F. A. Oliveira, 8 The European Physical Journal Special Topics (2026), 10.1140/epjs/s11734-026-02327-6

    work page doi:10.1140/epjs/s11734-026-02327-6 2026

  38. [38]

    Suzuki, Progress of Theoretical Physics69, 65 (1983)

    M. Suzuki, Progress of Theoretical Physics69, 65 (1983)

    work page 1983

  39. [39]

    Grimmett,The random-cluster model, Vol

    G. Grimmett,The random-cluster model, Vol. 333 (Springer, 2006)

    work page 2006

  40. [40]

    M. A. M. Cruz, J. P. Ortiz, M. P. Ortiz, and A. Balankin, Fractal and Fractional7, 231 (2023)

    work page 2023

  41. [41]

    Kr¨ oger, Physics Reports323, 81 (2000)

    H. Kr¨ oger, Physics Reports323, 81 (2000)

    work page 2000

  42. [42]

    Angulo, M

    J. Angulo, M. Ruiz-Medina, V. Anh, and W. Grecksch, Advances in Applied Probability32, 1077 (2000)

    work page 2000

  43. [43]

    Metzler, E

    R. Metzler, E. Barkai, and J. Klafter, Physical review letters82, 3563 (1999)

    work page 1999

  44. [44]

    Barkai, Physical Review E63, 046118 (2001)

    E. Barkai, Physical Review E63, 046118 (2001)

    work page 2001

  45. [45]

    Sokolov, Physical Review E63, 056111 (2001)

    I. Sokolov, Physical Review E63, 056111 (2001)

    work page 2001

  46. [46]

    Coniglio, Phys

    A. Coniglio, Phys. Rev. Lett.62, 3054 (1989)

    work page 1989

  47. [47]

    Sherrington and S

    D. Sherrington and S. Kirkpatrick, Physical review let- ters35, 1792 (1975)

    work page 1975

  48. [48]

    J. R. L. de Almeida and D. J. Thouless, Journal of Physics A: Mathematical and General11, 983 (1978)

    work page 1978

  49. [49]

    K. H. Fischer and J. A. Hertz,Spin Glasses, Cam- bridge Studies in Magnetism (Cambridge University Press, Cambridge, 1991)

    work page 1991

  50. [50]

    Suzuki, Progress of Theoretical Physics51, 1992 (1974)

    M. Suzuki, Progress of Theoretical Physics51, 1992 (1974)

    work page 1992

  51. [51]

    F. A. Reis, S. De Queiroz, and R. R. Dos Santos, Physical Review B54, R9616 (1996)

    work page 1996

  52. [52]

    A. M. Polyakov, JETP Lett.12, 381 (1970)

    work page 1970

  53. [53]

    A. A. Belavin, A. M. Polyakov, and A. B. Zamolod- chikov, Journal of Statistical Physics34, 763 (1984)

    work page 1984

  54. [54]

    Smirnov, Annals of mathematics , 1435 (2010)

    S. Smirnov, Annals of mathematics , 1435 (2010)

    work page 2010

  55. [55]

    Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE

    A. Kemppainen and S. Smirnov, arXiv preprint arXiv:1609.08527 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  56. [56]

    R. J. Baxter, inIntegrable systems in statistical mechan- ics(World Scientific, 1985) pp. 5–63

    work page 1985

  57. [57]

    M. S. Gomes-Filho, M. A. A. Barbosa, and F. A. Oliveira, Physica A: Statistical Mechanics and its Ap- plications540, 123165 (2020)

    work page 2020

  58. [58]

    Kalosakas and E

    G. Kalosakas and E. Panagopoulou, Processes10, 2592 (2022)

    work page 2022

  59. [59]

    Salman, F

    S. Salman, F. H. Shah, and S. J. Kim, Letters in Drug Design & Discovery21, 2891 (2024)

    work page 2024

  60. [60]

    R. M. Ziff, E. Gulari, and Y. Barshad, Physical review letters56, 2553 (1986)

    work page 1986

  61. [61]

    H. A. Fernandes, R. da Silva, and A. B. Bernardi, Phys- ical Review E98, 032113 (2018)

    work page 2018

  62. [62]

    Santos and F

    M. Santos and F. A. Oliveira, Phys. Rev. E110, 044122 (2024)

    work page 2024

  63. [63]

    Kuramoto and Y

    Y. Kuramoto and Y. Kuramoto,Chemical turbulence (Springer, 1984)

    work page 1984

  64. [64]

    P. D. Pinto, F. A. Oliveira, and A. L. A. Penna, Phys. Rev. E93, 052220 (2016)

    work page 2016

  65. [65]

    P. D. Pinto, A. L. A. Penna, and F. A. Oliveira, Euro- physics Letters117, 50009 (2017)

    work page 2017

  66. [66]

    Gutierrez and R

    R. Gutierrez and R. Cuerno, Physical Review E110, L052201 (2024)

    work page 2024

  67. [67]

    Gutierrez and R

    R. Gutierrez and R. Cuerno, Physica D: Nonlinear Phe- nomena473, 134552 (2025)

    work page 2025

  68. [68]

    Koch and J

    F. Koch and J. C. Budich, Physical Review Research4, 013113 (2022)

    work page 2022

  69. [69]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys- ical Review X9, 041015 (2019)

    work page 2019