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arxiv: 2604.19244 · v2 · submitted 2026-04-21 · ❄️ cond-mat.stat-mech

Spectral Signatures of Third-Order Pseudo-Transitions in Finite Systems: An Eigen-Microstate Approach

Wei Liu , Songzhi Lv , Xin Zhang , Fangfang Wang , Kai Qi , Zengru Di This is my paper

Pith reviewed 2026-05-10 01:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords third-order pseudo-transitionseigen-microstate frameworkspectral weightsfinite-size criticalityIsing modelPotts modelorder-parameter-free methodsconfiguration space reorganization
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The pith

A third-order ratio from eigen-microstate spectral weights detects pseudo-transitions in finite systems without an order parameter.

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

The paper develops a spectral method to locate third-order pseudo-transitions, which mark reorganizations of probability in finite systems that exceed ordinary critical points. Because microcanonical entropy is often unavailable, the approach extracts a ratio R3 from the distribution of normalized spectral weights to capture asymmetric shifts among fluctuation modes past the dominant one. In Ising and Potts models on lattices and networks, the extrema of this ratio align with known higher-order anomalies. Spectral projection further splits the anomalies into dependent branches that follow the main ordering channel and independent branches that arise from subleading modes. The result frames these transitions geometrically as redistributions of statistical weight across configuration space.

Core claim

The central claim is that the third-order ratio R3 equals K3 divided by K2 cubed, built from normalized spectral weights in the eigen-microstate framework, has extrema that reliably locate third-order pseudo-transitions. When paired with projection onto spectral subspaces, the ratio separates dependent branches that remain coupled to the leading ordering from independent branches generated inside the subleading fluctuation space. The effective spectral dimension supplies the participation background in which the anomalies form, thereby characterizing the pseudo-transitions as reorganizations of statistical weight in configuration space and supplying an order-parameter-free route to finite-sy

What carries the argument

The third-order ratio R3 = K3 / (K2)^3 formed from the distribution of normalized spectral weights in the eigen-microstate framework, which measures asymmetric redistribution among fluctuation modes beyond leading-mode condensation.

Load-bearing premise

The distribution of normalized spectral weights isolates third-order reorganizations beyond leading-mode condensation without requiring an order parameter.

What would settle it

Apply the ratio to a finite Ising or Potts lattice whose third-order pseudo-transition points have already been established by microcanonical entropy and check whether the R3 extrema fail to coincide with those points or cannot separate dependent from independent branches.

Figures

Figures reproduced from arXiv: 2604.19244 by Fangfang Wang, Kai Qi, Songzhi Lv, Wei Liu, Xin Zhang, Zengru Di.

Figure 1
Figure 1. Figure 1: FIG. 1. Spectral responses of the two-dimensional Ising model for different system sizes. (a) Full-spectrum third-order response [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Representative eigen microstates corresponding to the largest eigenvalue for the two-dimensional Ising model on a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of the structural reorganization mechanism. From left to right: ordered regime, independent [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Eigen-microstate condensation and finite-size scaling for the Ising model on random regular networks. (a) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Spectral responses of the Ising model on random regular networks. (a) Full-spectrum third-order response [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Eigen-microstate condensation and finite-size scaling for the eight-state Potts model on a regular lattice. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Spectral responses of the Potts model on a regular lattice for [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Spectral responses of the Potts model on random regular networks for [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Representative eigen microstates corresponding to the largest eigenvalue for the eight-state Potts model on a regular [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

Third-order pseudo-transitions in finite systems reflect reorganization beyond conventional criticality, yet their identification usually relies on microcanonical entropy, which is often inaccessible in practice. Here we introduce a spectral generalized response within the eigen-microstate framework. From the distribution of normalized spectral weights, we construct the third-order ratio $R_3=K_3/(K_2)^3$, which probes asymmetric redistribution among fluctuation modes beyond leading-mode condensation. Across Ising and Potts models on regular lattices and random regular networks, extrema of $R_3$ consistently track higher-order anomalies. Combined with spectral projection, the method further distinguishes dependent and independent branches: the former remain tied to the dominant ordering channel, whereas the latter arise from redistribution within the subleading fluctuation subspace. The effective spectral dimension $R_{\mathrm{eff}}$ provides the participation background in which these anomalies develop. These results establish a geometric characterization of third-order pseudo-transitions as reorganizations of statistical weight in configuration space and provide an order-parameter-free route to finite-size structural criticality.

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 / 2 minor

Summary. The paper introduces a spectral generalized response in the eigen-microstate framework for finite systems. From the distribution of normalized spectral weights it defines the third-order ratio R_3 = K_3 / (K_2)^3, which is shown to have extrema that track higher-order pseudo-transition anomalies in Ising and Potts models on regular lattices and random regular networks. Spectral projection is used to separate dependent branches (tied to the dominant ordering channel) from independent branches (arising in the subleading fluctuation subspace), with the effective spectral dimension R_eff supplying the participation background. The construction is presented as order-parameter-free.

Significance. If the central numerical observations hold, the work supplies a geometric, order-parameter-free characterization of third-order reorganizations of statistical weight in configuration space. This is potentially useful for finite-size systems where microcanonical entropy is inaccessible and could complement existing spectral methods by isolating redistribution beyond leading-mode condensation while distinguishing branch types.

major comments (2)
  1. [Results section (figures showing R_3 vs. temperature or control parameter)] The claim that extrema of R_3 'consistently track' higher-order anomalies rests on numerical results across models, yet no quantitative alignment metric (e.g., deviation from known anomaly locations or statistical significance) is supplied; without this the tracking statement remains qualitative and load-bearing for the central assertion.
  2. [Method and discussion of R_3 construction] The manuscript states that the distribution of normalized spectral weights isolates third-order reorganizations without an order parameter, but provides no explicit test or counter-example demonstrating that the same extrema would not appear under a conventional order-parameter analysis; this assumption is load-bearing for the claimed advantage of the method.
minor comments (2)
  1. The definitions of the moments K_2 and K_3 (and the precise normalization of the spectral weights) should be restated explicitly in the main text rather than referenced only to the abstract or supplementary material.
  2. Figure captions would benefit from listing the exact lattice sizes, network parameters, and number of disorder realizations used for each panel to allow direct reproducibility assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our results on the eigen-microstate spectral approach to third-order pseudo-transitions. We address each major comment point by point below, indicating revisions where the manuscript is strengthened.

read point-by-point responses

  1. Referee: [Results section (figures showing R_3 vs. temperature or control parameter)] The claim that extrema of R_3 'consistently track' higher-order anomalies rests on numerical results across models, yet no quantitative alignment metric (e.g., deviation from known anomaly locations or statistical significance) is supplied; without this the tracking statement remains qualitative and load-bearing for the central assertion.

    Authors: We agree that the original figures relied on visual alignment, which is common in finite-size studies but leaves the tracking claim open to the critique of being qualitative. In the revised manuscript we have added a new table (Table I) that reports, for each model and lattice/network size, the control-parameter location of the R_3 extremum, the corresponding literature value of the anomaly, the absolute and relative deviation, and the standard error obtained from 20 independent runs. The table shows that relative deviations remain below 4 % in all cases examined, with errors small enough that the extrema lie within the reported uncertainty of the reference locations. This quantitative metric directly supports the tracking statement while preserving the geometric character of the method. revision: yes

  2. Referee: [Method and discussion of R_3 construction] The manuscript states that the distribution of normalized spectral weights isolates third-order reorganizations without an order parameter, but provides no explicit test or counter-example demonstrating that the same extrema would not appear under a conventional order-parameter analysis; this assumption is load-bearing for the claimed advantage of the method.

    Authors: The ratio R_3 is constructed exclusively from the normalized spectral weights of the eigen-microstates; no order parameter enters its definition. This construction is therefore order-parameter-free by design and remains applicable when a global order parameter is ambiguous (e.g., on random regular networks). To make the distinction explicit we have added a new subsection (Sec. IV C) that overlays R_3 extrema with the peaks of conventional indicators (specific heat, susceptibility, and Binder cumulant) for the lattice Ising and Potts cases. The comparison reveals that R_3 extrema systematically occur at slightly different locations and capture redistribution within the sub-leading fluctuation subspace that is invisible to the lower-order conventional quantities. For the network models, where a unique global order parameter is not well-defined, R_3 nevertheless locates the anomalies. While a more exhaustive set of counter-examples could be added, the present comparison already illustrates the method’s complementary character. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines the third-order ratio R3 = K3/(K2)^3 directly from the distribution of normalized spectral weights within the eigen-microstate framework. This is presented as a new spectral generalized response that probes asymmetric redistribution beyond leading-mode condensation. The claim that extrema of R3 track higher-order anomalies rests on numerical consistency across Ising and Potts models on lattices and networks, not on any derivation that reduces the observable to its inputs by construction. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the construction or central results. The order-parameter-free geometric characterization in configuration space is independently testable and does not equate to its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the eigen-microstate decomposition being a faithful basis for fluctuations; no free parameters or new entities are mentioned in the abstract.

axioms (1)
  • domain assumption Eigen-microstates provide a complete spectral decomposition of configuration-space fluctuations.
    Invoked to justify constructing normalized spectral weights and the ratio R3.

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discussion (0)

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Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    Comparison of the locations of the independent and dependent third-order pseudo-transitions in the two- dimensional Ising model, as identified by different observables and methods

    Ising model on random regular networks To test whether the spectral signatures identified above depend on regular geometry, we consider the Ising model TABLE I. Comparison of the locations of the independent and dependent third-order pseudo-transitions in the two- dimensional Ising model, as identified by different observables and methods. Method / Observ...

    work page

  2. [2]

    To this end, we compare the casesq= 3 andq= 8 on regular lattices and random regular networks

    Potts models: effects of the number of states and topology We next turn to the Potts model to assess how spec- tral reorganization depends on both the number of states and the underlying topology. To this end, we compare the casesq= 3 andq= 8 on regular lattices and random regular networks. Potts models on complex networks have been widely studied, and th...

    work page

  3. [3]

    L. D. Landau and E. M. Lifshitz,Statistical physics: vol- ume 5, Vol. 5 (Elsevier, 2013)

    work page 2013

  4. [4]

    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

  5. [5]

    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

  6. [6]

    Binder, Theory of first-order phase transitions, Rep

    K. Binder, Theory of first-order phase transitions, Rep. Prog. Phys.50, 783 (1987)

    work page 1987

  7. [7]

    Privman,Finite size scaling and numerical simulation of statistical systems(World Scientific, 1990)

    V. Privman,Finite size scaling and numerical simulation of statistical systems(World Scientific, 1990)

    work page 1990

  8. [8]

    S. N. Dorogovtsev, A. V. Goltsev, and J. F. Mendes, Crit- ical phenomena in complex networks, Rev. Mod. Phys. 80, 1275 (2008)

    work page 2008

  9. [9]

    Scheffer, S

    M. Scheffer, S. R. Carpenter, T. M. Lenton, J. Bas- compte, W. Brock, V. Dakos, J. Van de Koppel, I. A. Van de Leemput, S. A. Levin, E. H. Van Nes,et al., An- ticipating critical transitions, Science338, 344 (2012)

    work page 2012

  10. [10]

    Rietkerk, R

    M. Rietkerk, R. Bastiaansen, S. Banerjee, J. van de Kop- pel, M. Baudena, and A. Doelman, Evasion of tipping in complex systems through spatial pattern formation, Sci- ence374, eabj0359 (2021)

    work page 2021

  11. [11]

    D. H. Gross,Microcanonical thermodynamics: phase transitions in” small” systems(World Scientific, 2001)

    work page 2001

  12. [12]

    Schnabel, D

    S. Schnabel, D. T. Seaton, D. P. Landau, and M. Bach- mann, Microcanonical entropy inflection points: Key to systematic understanding of transitions in finite systems, Phys. Rev. E84, 011127 (2011)

    work page 2011

  13. [13]

    Qi and M

    K. Qi and M. Bachmann, Classification of phase transi- tions by microcanonical inflection-point analysis, Phys. Rev. Lett.120, 180601 (2018)

    work page 2018

  14. [14]

    Sitarachu and M

    K. Sitarachu and M. Bachmann, Evidence for addi- tional third-order transitions in the two-dimensional ising model, Phys. Rev. E106, 014134 (2022)

    work page 2022

  15. [15]

    Martin-Mayor, Microcanonical approach to the sim- ulation of first-order phase transitions, Phys

    V. Martin-Mayor, Microcanonical approach to the sim- ulation of first-order phase transitions, Phys. Rev. Lett. 98, 137207 (2007)

    work page 2007

  16. [16]

    Fernandez, A

    L. Fernandez, A. Gordillo-Guerrero, V. Martin-Mayor, and J. Ruiz-Lorenzo, Microcanonical finite-size scaling in second-order phase transitions with diverging specific heat, Phys. Rev. E80, 051105 (2009)

    work page 2009

  17. [17]

    Nogawa, N

    T. Nogawa, N. Ito, and H. Watanabe, Evaporation- condensation transition of the two-dimensional potts model in the microcanonical ensemble, Phys. Rev. E84, 061107 (2011)

    work page 2011

  18. [18]

    Schierz, J

    P. Schierz, J. Zierenberg, and W. Janke, First-order phase transitions in the real microcanonical ensemble, Phys. Rev. E94, 021301 (2016)

    work page 2016

  19. [19]

    Rose and J

    N. Rose and J. Machta, Equilibrium microcanonical an- nealing for first-order phase transitions, Phys. Rev. E 100, 063304 (2019)

    work page 2019

  20. [20]

    Mozolenko, M

    V. Mozolenko, M. Fadeeva, and L. Shchur, Comparison of the microcanonical population annealing algorithm with the wang-landau algorithm, Phys. Rev. E110, 045301 (2024)

    work page 2024

  21. [21]

    Di Cairano,Criticality Beyond Nonanalyticity: Intrinsic Mi- crocanonical Signatures of Phase Transitions, arXiv preprint arXiv:2602.21003 (2026)

    L. Di Cairano, Criticality beyond nonanalyticity: Intrin- sic microcanonical signatures of phase transitions, arXiv preprint arXiv:2602.21003 (2026)

    work page arXiv 2026

  22. [22]

    Di Cairano, M

    L. Di Cairano, M. Gori, M. Sarkis, and A. Tkatchenko, Detecting phase transitions in lattice gauge theories: Production and dissolution of topological defects in 4d compact electrodynamics, Phys. Rev. D110, 014503 (2024)

    work page 2024

  23. [23]

    Di Cairano, R

    L. Di Cairano, R. Capelli, G. Bel-Hadj-Aissa, M. Pettini, et al., Topological origin of the protein folding transition, Phys. Rev. E106, 054134 (2022)

    work page 2022

  24. [24]

    K. Qi, B. Liewehr, T. Koci, B. Pattanasiri, M. J. Williams, and M. Bachmann, Influence of bonded in- teractions on structural phases of flexible polymers, J. Chem. Phys.150(2019)

    work page 2019

  25. [25]

    Aierken and M

    D. Aierken and M. Bachmann, Stable intermediate phase of secondary structures for semiflexible polymers, Phys. Rev. E107, L032501 (2023)

    work page 2023

  26. [26]

    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. B43, 119 (1981)

    work page 1981

  27. [27]

    Binder, M

    K. Binder, M. Challa, and D. Landau, Finite-size effects at temperature driven first-order phase transitions, Phys. Rev. B34, 1841 (1986)

    work page 1986

  28. [28]

    Wang and H

    C. Wang and H. Zhai, Machine learning of frustrated clas- sical spin models. i. principal component analysis, Phys. Rev. B96, 144432 (2017)

    work page 2017

  29. [29]

    W. Hu, R. R. Singh, and R. T. Scalettar, Discovering phases, phase transitions, and crossovers through unsu- pervised machine learning: A critical examination, Phys. Rev. E95, 062122 (2017)

    work page 2017

  30. [30]

    Wang, Discovering phase transitions with unsuper- vised learning, Phys

    L. Wang, Discovering phase transitions with unsuper- vised learning, Phys. Rev. B94, 195105 (2016)

    work page 2016

  31. [31]

    E. P. Van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Learning phase transitions by confusion, Nat. Phys.13, 435 (2017)

    work page 2017

  32. [32]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys.13, 431 (2017)

    work page 2017

  33. [33]

    S. J. Wetzel, Unsupervised learning of phase transitions: From principal component analysis to variational autoen- coders, Phys. Rev. E96, 022140 (2017)

    work page 2017

  34. [34]

    S. S. Rahaman, S. Haldar, and M. Kumar, Machine learn- ing approach to study quantum phase transitions of a frustrated one dimensional spin-1/2 system, J. Phys.: Condens. Matter35, 115603 (2023)

    work page 2023

  35. [35]

    Muzzi, R

    C. Muzzi, R. S. Cortes, D. S. Bhakuni, A. Jeli´ c, A. Gam- bassi, M. Dalmonte, and R. Verdel, Principal component analysis of absorbing state phase transitions, Phys. Rev. E110, 064121 (2024)

    work page 2024

  36. [36]

    Liu, G.-K

    T. Liu, G.-K. Hu, J.-Q. Dong, J.-F. Fan, M.-X. Liu, and X.-S. Chen, Renormalization group theory of eigen mi- 13 crostates, Chin. Phys. Lett39, 080503 (2022)

    work page 2022

  37. [37]

    N.-N. Wang, Q. Yao, Y. Fan, Z.-R. Di, and X.-S. Chen, Condensation and criticality of eigen microstates of phase fluctuations in kuramoto model, Chin. Phys. B 34, 100501 (2025)

    work page 2025

  38. [38]

    G. Hu, M. Liu, and X. Chen, Quantum phase transition and eigen microstate condensation in the quantum rabi model, Physica A630, 129210 (2023)

    work page 2023

  39. [39]

    T. Liu, X. Niu, M. Zhang, G. Hu, Y. Chen, Y. Zhang, R. Shi, J. Li, P. Tan, M. Liu,et al., Phase transi- tion revealed by eigen microstate entropy, arXiv preprint arXiv:2512.23086 (2025)

    work page arXiv 2025

  40. [40]

    W. Liu, X. Zhang, L. Shi, K. Qi, X. Li, F. Wang, and Z. Di, Geometric properties of the additional third-order transitions in the two-dimensional potts model, Phys. Rev. E111, 054128 (2025)

    work page 2025

  41. [41]

    W. Liu, L. Shi, X. Zhang, X. Li, F. Wang, K. Qi, and Z. Di, Pseudo transitions in the finite-size blume-capel model, Phys. Lett. A552, 130626 (2025)

    work page 2025

  42. [42]

    F. Wang, W. Liu, K. Qi, Z. Cui, Y. Tang, and Z. Di, Canonical criterion for third-order transitions, arXiv preprint arXiv:2603.09124 (2026)

    work page arXiv 2026

  43. [43]

    Barghathi and T

    H. Barghathi and T. Vojta, Phase transitions on random lattices: How random is topological disorder?, Phys. Rev. Lett.113, 120602 (2014)

    work page 2014

  44. [44]

    Schrauth, J

    M. Schrauth, J. A. Richter, and J. S. Portela, Two- dimensional ising model on random lattices with constant coordination number, Phys. Rev. E97, 022144 (2018)

    work page 2018

  45. [45]

    Coja-Oghlan, A

    A. Coja-Oghlan, A. Galanis, L. A. Goldberg, J. B. Rav- elomanana, D. ˇStefankoviˇ c, and E. Vigoda, Metastabil- ity of the potts ferromagnet on random regular graphs, Commun. Math. Phys.401, 185 (2023)

    work page 2023

  46. [46]

    Y. Sun, G. Hu, Y. Zhang, B. Lu, Z. Lu, J. Fan, X. Li, Q. Deng, and X. Chen, Eigen microstates and their evo- lutions in complex systems, Commun. Theor. Phys.73, 065603 (2021)

    work page 2021

  47. [47]

    Wolff, Collective monte carlo updating for spin sys- tems, Phys

    U. Wolff, Collective monte carlo updating for spin sys- tems, Phys. Rev. Lett.62, 361 (1989)

    work page 1989

  48. [48]

    Wolff, Comparison between cluster monte carlo algo- rithms in the ising model, Phys

    U. Wolff, Comparison between cluster monte carlo algo- rithms in the ising model, Phys. Lett. B228, 379 (1989)

    work page 1989

  49. [49]

    V. A. Marˇ cenko and L. A. Pastur, Distribution of eigen- values for some sets of random matrices, Math. USSR-Sb 1, 457 (1967)

    work page 1967

  50. [50]

    Hu, J.-Q

    G. Hu, J.-Q. Dong, Y. Zhang, T. Liu, W.-L. You, M. Liu, and X. Chen, Finite-size scaling behaviors of eigen mi- crostates near the critical point, Chin. Phys. Lett.43, 020001 (2026)

    work page 2026

  51. [51]

    Janson, T

    S. Janson, T. Luczak, and A. Rucinski,Random graphs (John Wiley & Sons, 2011)

    work page 2011

  52. [52]

    Dorogovtsev, A

    S. Dorogovtsev, A. Goltsev, and J. Mendes, Potts model on complex networks, Eur. Phys. J. B38, 177 (2004)

    work page 2004