pith. sign in

arxiv: 2507.05961 · v2 · pith:KROALRFEnew · submitted 2025-07-08 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· physics.soc-ph

Chladni states in Ising Spin Lattices

Giulio Iannelli , Pablo Villegas This is my paper

Pith reviewed 2026-05-21 23:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnphysics.soc-ph
keywords Chladni statesIsing spin latticesLaplacian eigenmodesmetastable configurationstopological mode decompositionspin glassesnon-ergodic relaxationinteraction networks
0
0 comments X

The pith

Binarizing the eigenmodes of the interaction Laplacian yields Chladni states that organize metastable configurations in Ising spin systems.

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

The paper introduces Chladni states as spin configurations obtained by binarizing the eigenmodes of the interaction Laplacian. These states organize the long-lived metastable configurations that Ising systems reach under non-ergodic relaxation at low temperatures. The approach covers ferromagnets, frustrated antiferromagnets, and spin glasses by providing a geometric description of frozen patterns. It introduces Topological Mode Decomposition as a compact method to monitor and reconstruct these configurations.

Core claim

Chladni states obtained by binarizing the eigenmodes of the interaction Laplacian organize the metastable configurations reached by Ising systems under non-ergodic relaxation. The resulting Topological Mode Decomposition provides a compact way to monitor and reconstruct frozen spin configurations in ferromagnets, frustrated antiferromagnets, and spin glasses.

What carries the argument

Chladni states formed by binarizing the eigenmodes of the interaction Laplacian, which organize and reconstruct the dominant metastable spin patterns.

Load-bearing premise

Binarized Laplacian eigenmodes of the interaction graph are sufficient to capture and reconstruct the dominant long-lived metastable spin configurations without additional fitting or selection rules.

What would settle it

An Ising system on a specific interaction network whose observed long-lived metastable states cannot be reconstructed from the binarized eigenmodes of its Laplacian would falsify the claim.

Figures

Figures reproduced from arXiv: 2507.05961 by Giulio Iannelli, Pablo Villegas.

Figure 1
Figure 1. Figure 1: (a) Chladni states for selected fractional indices i/N, using N = 214. From left to right: i = (2, 2 8 , 2 12 , 2 · 2 12 , 3 · 2 12). (b) Energy of Chladni states (Ising Hamiltonian evaluated on binarized eigenvector) versus normalized index, for several system sizes (see legend, N = L 2 ). The curve col￾lapse across sizes, with an inflection point at half the system size, separating the positive from the … view at source ↗
Figure 2
Figure 2. Figure 2: (c) shows the reconstructability index, R (i.e., the overlap with the simulated configuration, site by site) for a triangular antiferromagnetic lattice initialized at se￾lected Chladni states at T = 0 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: Chladni states for a 2D square lattice with side [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Temporal evolution and snapshots of the composition with weights of Chladni states for the square lattice of size [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Temporal evolution and snapshots of the composition by element-wise product of Chladni states for the square [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Chladni states for a 2D triangular lattice with side [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the energy corresponding to each normalized Chladni state as a function of the normalized growing eigenvectors for a 2D triangular lattice, being independent of the system size. Note that the energy curve does not exhibit symmetry about the origin, as a consequence of the lack of an antiferromagnetic phase in this case. Again, Chladni states with negative energy are stable over time at T = 0. 0.0 0.5… view at source ↗
Figure 6
Figure 6. Figure 6: Temporal evolution and snapshots of the composition with weights of Chladni states for the triangular lattice of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Chladni states for a 2D hexagonal lattice of size [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: shows the energy corresponding to each normalized Chladni state as a function of the normalized growing eigenvectors. The energy curve exhibits a characteristic double S-shape, independent of system size, and is symmetric about the origin, as expected due to the existence of an antiferromagnetic phase. We have performed extensive Monte Carlo simulations of selected Chaldni states at T = 0 to monitor their … view at source ↗
Figure 9
Figure 9. Figure 9: Temporal evolution and snapshots of the composition with weights of Chladni states for the hexagonal lattice of size [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Chladni states for a 2D Supercrystal structure with side [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the energy corresponding to each normalized Chladni state as a function of the normalized growing eigenvectors. The energy curve exhibits symmetry about the origin, as it is able to sustain an antiferromagnetic phase. Note that the vibrational spectrum shows new symmetries and characteristic scales: a large-scale effective square lattice that starts to combine with the microscopic scale, giving rise… view at source ↗
Figure 12
Figure 12. Figure 12: shows the energy corresponding to each normalized Chladni state as a function of the normalized growing eigenvectors for a 2D triangular antiferromagnetic lattice, being independent of the system size. Note that the energy curve does not exhibit symmetry about the origin (it is a mirroring of the curve shown before), as a consequence of the lack of an antiferromagnetic phase in this case. Again, Chladni s… view at source ↗
Figure 13
Figure 13. Figure 13: Temporal evolution and snapshots of the composition with weights of Chladni states for the antiferromagnetic [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: shows the energy corresponding to each normalized Chladni state as a function of the normalized growing eigenvectors for a 2D hexagonal lattice on the spin-glass phase (p = 0.2, with p > pc ≃ 0.064), being independent of the system size. Note that the energy curve becomes linear and exhibits trivial symmetry about the origin. Again, Chladni states with negative energy are stable over time at T = 0. 0.0 0.… view at source ↗
Figure 15
Figure 15. Figure 15: Three-dimensional reconstructability phase diagram using the top three eigenmodes, [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Original image. Square 2D AntiTri 2D Cubic 3D [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Reconstructed image using the first 100 (left column), [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Reconstruction of a MNIST digit using the first [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
read the original abstract

Low-temperature spin dynamics can become trapped in long-lived patterns shaped by the geometry of the interaction network. Here we introduce Chladni states: spin configurations obtained by binarizing the eigenmodes of the interaction Laplacian. These graph-spectral patterns organize the metastable configurations reached by Ising systems under non-ergodic relaxation. The resulting Topological Mode Decomposition provides a compact way to monitor and reconstruct frozen spin configurations in ferromagnets, frustrated antiferromagnets, and spin glasses.

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 / 3 minor

Summary. The paper introduces Chladni states, defined as spin configurations obtained by binarizing the eigenmodes of the interaction Laplacian L = D - A on Ising lattices. It claims that these states organize the metastable configurations reached under non-ergodic relaxation, and introduces a Topological Mode Decomposition to monitor and reconstruct frozen states in ferromagnets, frustrated antiferromagnets, and spin glasses. The construction is illustrated on small grids and random graphs with visual comparisons.

Significance. If the central claim is substantiated, the work would provide a graph-spectral, largely parameter-free route to identifying dominant long-lived patterns in Ising systems, linking Laplacian eigenvectors directly to dynamical basins. This could complement existing approaches to metastability and offer a compact decomposition for monitoring relaxation in complex spin networks.

major comments (3)
  1. §4 (numerical illustrations): the manuscript shows visual agreement between binarized low-lying eigenmodes and metastable states for small systems, but supplies no quantitative metric (e.g., average Hamming distance or energy ranking relative to actual long-time Glauber trajectories) across an ensemble of realizations; this is load-bearing for the claim that the states 'organize' the metastable configurations rather than merely resembling them in selected cases.
  2. Eq. (3) and surrounding text: the binarization (sign or zero-thresholding) of eigenmodes is presented as directly yielding the relevant patterns, yet the selection of which modes (or linear combinations) to retain is not derived from the spectrum or dynamics; no argument rules out the necessity of higher modes or post-hoc rules for the claimed generality across ferromagnets, frustrated AF, and spin glasses.
  3. §5 (spin-glass examples): the reconstruction of frozen configurations is asserted to be compact, but the manuscript provides only illustrative snapshots without systematic comparison to independently sampled metastable states (e.g., via parallel tempering or basin-hopping), leaving the link between Laplacian spectrum and actual non-ergodic attractors illustrative rather than established.
minor comments (3)
  1. The term 'Topological Mode Decomposition' is used in the abstract but first defined only in the main text; an early definition or acronym introduction would improve readability.
  2. Figure captions should specify the precise temperature, coupling distribution, and relaxation protocol used for each panel to allow direct reproduction.
  3. Notation for the adjacency matrix A (signed couplings) is introduced without an explicit statement of how negative bonds are handled in the degree matrix D; a short clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address each of the major comments below and outline the revisions we will make to improve the manuscript.

read point-by-point responses

  1. Referee: §4 (numerical illustrations): the manuscript shows visual agreement between binarized low-lying eigenmodes and metastable states for small systems, but supplies no quantitative metric (e.g., average Hamming distance or energy ranking relative to actual long-time Glauber trajectories) across an ensemble of realizations; this is load-bearing for the claim that the states 'organize' the metastable configurations rather than merely resembling them in selected cases.

    Authors: We agree that quantitative metrics would strengthen the evidence. In the revised manuscript, we will add ensemble-averaged Hamming distances and energy comparisons between the binarized eigenmodes and metastable states obtained from long Glauber dynamics simulations across multiple random realizations. This will provide statistical support for the organizing role of Chladni states. revision: yes

  2. Referee: Eq. (3) and surrounding text: the binarization (sign or zero-thresholding) of eigenmodes is presented as directly yielding the relevant patterns, yet the selection of which modes (or linear combinations) to retain is not derived from the spectrum or dynamics; no argument rules out the necessity of higher modes or post-hoc rules for the claimed generality across ferromagnets, frustrated AF, and spin glasses.

    Authors: The choice of low-lying modes is guided by their association with the slowest relaxation modes in the system, as they correspond to the smallest eigenvalues of the Laplacian. We will revise the text around Eq. (3) to provide a clearer motivation based on the variational characterization of the Laplacian eigenvalues and their relation to the energy barriers in the Ising model. While a complete derivation from first principles is beyond the current scope, we will discuss the empirical evidence for the sufficiency of low modes and acknowledge that higher modes may be relevant in certain regimes, particularly in spin glasses. revision: partial

  3. Referee: §5 (spin-glass examples): the reconstruction of frozen configurations is asserted to be compact, but the manuscript provides only illustrative snapshots without systematic comparison to independently sampled metastable states (e.g., via parallel tempering or basin-hopping), leaving the link between Laplacian spectrum and actual non-ergodic attractors illustrative rather than established.

    Authors: We accept that the current examples are illustrative. We will enhance §5 by including quantitative comparisons, such as the fraction of variance explained or reconstruction fidelity, against metastable states sampled via parallel tempering. This will better substantiate the compactness and relevance of the Topological Mode Decomposition for spin glasses. revision: yes

Circularity Check

0 steps flagged

No significant circularity: Chladni states introduced as a direct definition applied to standard Laplacian spectra without reduction to fitted inputs or self-citation chains.

full rationale

The manuscript defines Chladni states explicitly as the result of binarizing eigenmodes of the interaction Laplacian L = D - A and then asserts that these patterns organize metastable Ising configurations under non-ergodic dynamics. This construction is presented as a new descriptive tool rather than a derivation whose outputs are forced by its own inputs. No equations are exhibited that fit parameters on a data subset and then relabel the same quantities as predictions, nor is a uniqueness theorem imported from prior self-work to forbid alternatives. The link to dynamics is supported by illustrative examples on small systems; the central claim therefore remains self-contained against external benchmarks such as direct Glauber dynamics simulations and does not reduce to a tautology by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that the interaction Laplacian eigenmodes, once binarized, directly correspond to physically realized metastable states; this introduces one new entity and relies on standard spectral graph theory without additional free parameters or ad-hoc axioms stated in the abstract.

axioms (1)
  • domain assumption The interaction network of the Ising model can be represented by a Laplacian whose eigenmodes encode the relevant geometric constraints on spin configurations.
    Invoked to justify using Laplacian eigenmodes as the basis for Chladni states.
invented entities (1)
  • Chladni states no independent evidence
    purpose: Binarized eigenmodes of the interaction Laplacian that label metastable spin patterns.
    Newly defined object whose correspondence to physical metastability is asserted but not independently evidenced in the abstract.

pith-pipeline@v0.9.0 · 5601 in / 1271 out tokens · 27407 ms · 2026-05-21T23:53:55.943288+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.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    By applying the sign function to these eigenvectors (i.e., binarizing them), we obtain discrete spin configurations that we refer to as Chladni states... low-index modes correspond to metastable configurations at T = 0

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the eigenvalues λi and the eigenvectors |λi⟩ of L act as the discrete analogue of Fourier modes, capturing the intrinsic topological or geometric features of the underlying lattice

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

56 extracted references · 56 canonical work pages

  1. [1]

    E. F. F. Chladni, Die Akustik (Breitkopf und Härtel, Leipzig, 1802) Wellcome Library

    work page

  2. [2]

    S. R. Jain and R. Samajdar, Rev. Mod. Phys.89, 045005 (2017)

    work page 2017

  3. [3]

    Hakonen, and R

    Q.Zhou, V.Sariola, K.Latifi, V.Liimatainen, S.Tuukka- nen, P. Hakonen, and R. H. A. Ras, Nat. Commun.7, 12764 (2016)

    work page 2016

  4. [4]

    Kopitca, K

    A. Kopitca, K. Latifi, and Q. Zhou, Sci. Adv.7, eabi7716 (2021)

    work page 2021

  5. [5]

    Latifi, H

    K. Latifi, H. Wijaya, and Q. Zhou, Phys. Rev. Lett.122, 184301 (2019)

    work page 2019

  6. [6]

    Agrawal, F

    R. Agrawal, F. Corberi, F. Insalata, and S. Puri, Phys. Rev. E 105, 034131 (2022)

    work page 2022

  7. [7]

    Olejarz, P

    J. Olejarz, P. L. Krapivsky, and S. Redner, Phys. Rev. Lett. 109, 195702 (2012)

    work page 2012

  8. [8]

    Bray, Adv

    A. Bray, Adv. Phys.43, 357 (1994)

    work page 1994

  9. [9]

    E. N. M. Cirillo and J. L. Lebowitz, J. Stat. Phys.90, 211 (1998)

    work page 1998

  10. [10]

    Nattermann, Theory of the random field ising model, in Spin Glasses and Random Fields, pp

    T. Nattermann, Theory of the random field ising model, in Spin Glasses and Random Fields, pp. 277–298

    work page

  11. [11]

    Barros, P

    K. Barros, P. L. Krapivsky, and S. Redner, Phys. Rev. E 80, 040101 (2009)

    work page 2009

  12. [12]

    Spirin, P

    V. Spirin, P. Krapivsky, and S. Redner, Phys. Rev. E65, 016119 (2001)

    work page 2001

  13. [13]

    Ramirez, Nature421, 483 (2003)

    A. Ramirez, Nature421, 483 (2003)

    work page 2003

  14. [14]

    P. W. Anderson, Phys. Rev.102, 1008 (1956)

    work page 1956

  15. [15]

    G. H. Wannier, Phys. Rev.79, 357 (1950)

    work page 1950

  16. [16]

    Saito and K

    Y. Saito and K. Igeta, J. Phys. Soc. Jpn.53, 3060 (1984)

    work page 1984

  17. [17]

    S. H. Skjærvø, C. H. Marrows, R. L. Stamps, and L. J. Heyderman, Nat. Rev. Phys.2, 13 (2020)

    work page 2020

  18. [18]

    Wang, X.-J

    K. Wang, X.-J. Liu, L.-M. Tu, J.-J. Zhang, V. N. Glad- ilin, and J.-Y. Ge, Phys. Rev. B111, 224418 (2025)

    work page 2025

  19. [20]

    Kirkpatrick, Phys

    S. Kirkpatrick, Phys. Rev. B16, 4630 (1977)

    work page 1977

  20. [21]

    Hubert and R

    A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures, 1st ed. (Springer, Berlin, 1998)

    work page 1998

  21. [22]

    A. J. Bray and M. A. Moore, Phys. Rev. Lett.58, 57 (1987)

    work page 1987

  22. [23]

    Parisi, Phys

    G. Parisi, Phys. Rev. Lett.43, 1754 (1979)

    work page 1979

  23. [24]

    Castellani and A

    T. Castellani and A. Cavagna, J. Stat. Mech. Theory Exp. 2005, P05012 (2005)

    work page 2005

  24. [25]

    Jonason, E

    K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud, and P. Nordblad, Phys. Rev. Lett.81, 3243 (1998)

    work page 1998

  25. [27]

    C. Fan, M. Shen, Z. Nussinov, C. Wang, D. Li, and S. Zhang, Nat. Commun.14, 725 (2023)

    work page 2023

  26. [28]

    Mézard, Indian J

    M. Mézard, Indian J. Phys.98, 3757 (2024)

    work page 2024

  27. [29]

    Newman and D

    C. Newman and D. Stein, arXiv preprint cond- mat/0411426 (2004)

    work page arXiv 2004

  28. [30]

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

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

    work page 2007

  29. [31]

    J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. New- man, The theory of critical phenomena: an introduction to the renormalization group(Oxford University Press, Oxford, 1992)

    work page 1992

  30. [32]

    D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena, 3rd ed. (World Scientific, Singapore, 2005)

    work page 2005

  31. [33]

    Simonov and A

    A. Simonov and A. L. Goodwin, Nat. Rev. Chem.4, 657 (2020)

    work page 2020

  32. [34]

    M. Hehn, K. Ounadjela, J.-P. Bucher, F. Rousseaux, D. Decanini, B. Bartenlian, and C. Chappert, Science 272, 1782 (1996)

    work page 1996

  33. [35]

    Nahaset al., Nature 577, 47 (2020)

    Y. Nahaset al., Nature 577, 47 (2020)

    work page 2020

  34. [36]

    Ramesh and D

    R. Ramesh and D. G. Schlom, Nat. Rev. Mater.4, 257 (2019)

    work page 2019

  35. [38]

    Marro and R

    J. Marro and R. Dickman,Nonequilibrium Phase Tran- sitions in Lattice Models(Cambridge University Press, Cambridge, 1999)

    work page 1999

  36. [39]

    Burioni and D

    R. Burioni and D. Cassi, Phys. Rev. Lett. 76, 1091 (1996)

    work page 1996

  37. [40]

    Villegas, T

    P. Villegas, T. Gili, G. Caldarelli, and A. Gabrielli, Nat. Phys. 19, 445 (2023)

    work page 2023

  38. [41]

    Villegas, Phys

    P. Villegas, Phys. Rev. E111, L042301 (2025)

    work page 2025

  39. [42]

    See Supplemental Material at [] for further technical de- tails, extended results, and additional simulations sup- porting the findings in the main text

    work page

  40. [43]

    The updating rule when ∆E = 0 is key for this point

    That is, making no flip when∆E = 0. The updating rule when ∆E = 0 is key for this point. Hence, the Ising- Glauber dynamics smooth domain walls, making some patterns unstable

    work page

  41. [44]

    Godrèche and J

    C. Godrèche and J. M. Luck, J. Phys.: Condens. Matter 17, S2573 (2005)

    work page 2005

  42. [45]

    Drisko, T

    J. Drisko, T. Marsh, and J. Cumings, Nat. Commun.8, 14009 (2017)

    work page 2017

  43. [46]

    A. P. Ramirez, Nature421, 483 (2003)

    work page 2003

  44. [47]

    Shokef, A

    Y. Shokef, A. Souslov, and T. C. Lubensky, Proc. Natl. Acad. Sci. U.S.A.108, 11804 (2011)

    work page 2011

  45. [48]

    Yamada, S

    Y. Yamada, S. Miyashita, T. Horiguchi, M. Kang, and O. Nagai, Physica A217, 125 (1995)

    work page 1995

  46. [49]

    Kawamura and S

    H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn.53, 4138 (1984)

    work page 1984

  47. [50]

    Chung, M

    K. Chung, M. Cheon, and I. Chang, Int. J. Mod. Phys. C 17, 591 (2006)

    work page 2006

  48. [51]

    S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5, 965 (1975)

    work page 1975

  49. [52]

    Mézard, G

    M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Phys. Rev. Lett.52, 1156 (1984)

    work page 1984

  50. [53]

    Belkin and P

    M. Belkin and P. Niyogi, Neural Comput. 15, 1373 (2003)

    work page 2003

  51. [54]

    F. Xin, L. Falsi, Y. Gelkop, D. Pierangeli, G. Zhang, F. Bo, F. Fusella, A. J. Agranat, and E. DelRe, Phys. Rev. Lett. 132, 066603 (2024). Chladni states in Ising Spin Lattices Giulio Iannelli1, 2,∗and Pablo Villegas1, 3,† 1‘Enrico Fermi’ Research Center (CREF), Via Panisperna 89A, 00184 - Rome, Italy 2Dipartimento di Fisica, Università degli Studi di Pal...

    work page 2024

  52. [55]

    Iannelli, P

    G. Iannelli, P. Villegas, T. Gili, and A. Gabrielli, arXiv preprint arXiv:2504.00144 (2025)

    work page arXiv 2025

  53. [56]

    D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena , 3rd ed. (World Scientific, Singapore, 2005)

    work page 2005

  54. [57]

    J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. Newman,The theory of critical phenomena: an introduction to the renormalization group (Oxford University Press, Oxford, 1992)

    work page 1992

  55. [58]

    Liu and J

    Y. Liu and J. Shen, Taiwanese J. Math.19, 505 (2015)

    work page 2015

  56. [59]

    Falsi, P

    L. Falsi, P. Villegas, T. Gili, A. J. Agranat, and E. DelRe, arXiv , 2406.14646 (2024)

    work page arXiv 2024