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arxiv: 2301.04112 · v6 · submitted 2023-01-10 · 🧮 math-ph · cond-mat.dis-nn· math.MP· math.PR

Spin glass phase at zero temperature in the Edwards-Anderson model

Sourav Chatterjee This is my paper

Pith reviewed 2026-05-24 09:35 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.dis-nnmath.MPmath.PR
keywords Edwards-Anderson modelspin glasszero temperatureground statedropletsdisorder perturbationfractal interfacesglassy behavior
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The pith

Ground states in the Edwards-Anderson model become nearly orthogonal after small perturbations of the disorder.

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

This paper proves several zero-temperature signatures of glassy behavior for the Edwards-Anderson model with Gaussian couplings on finite boxes in any dimension. It shows that ground states are highly sensitive to small changes in the random couplings, with the new ground state having near-zero overlap with the original once the perturbation fraction exceeds a multiple of the inverse volume. The associated droplets have interfaces whose dimension exceeds d-1, and there exist macroscopic spin excitations whose energy cost is negligible relative to interface size. These features contrast sharply with ferromagnetic models and supply rigorous mathematical evidence that glassy behavior persists at absolute zero in short-range spin glasses.

Core claim

In the Edwards-Anderson model on finite boxes of arbitrary dimension, after perturbing a fraction p of the Gaussian couplings the overlap between the original and new ground states drops below any fixed positive threshold once p exceeds a constant times the inverse volume. The boundaries of the resulting macroscopic droplets satisfy lower bounds implying fractal dimension strictly greater than d-1. Macroscopic spin excitations exist whose energy cost is o(interface size), the expected critical droplet size grows at least polynomially in volume, and nearest-neighbor spin-product sensitivity to boundary conditions cannot decay faster than order one over distance to the boundary.

What carries the argument

Perturbation of a positive fraction p of the independent Gaussian couplings, used to generate a new ground state whose overlap and droplet interface properties are then controlled by tail bounds and geometric arguments.

If this is right

  • The ground state depends chaotically on the disorder parameters.
  • The model admits large-scale spin excitations at negligible energy cost relative to boundary length.
  • Droplet interfaces are rougher than minimal surfaces.
  • Critical droplets occupy a positive fraction of the volume in the large-system limit.
  • Boundary-induced changes in local spin correlations persist at distances comparable to the system size.

Where Pith is reading between the lines

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

  • The zero-temperature results suggest that similar sensitivity may appear in low-temperature dynamics or in algorithms that locate ground states.
  • The contrast with exponential decay in the two-dimensional random-field Ising model isolates a qualitative difference between spin-glass and random-field disorder.
  • Extensions of the perturbation argument to non-Gaussian couplings with comparable tails would test whether the Gaussian assumption is essential.
  • The polynomial growth of critical droplet size supplies a concrete scaling target for numerical studies of droplet statistics.

Load-bearing premise

The couplings are independent Gaussian random variables placed on the nearest-neighbor edges of a finite lattice box.

What would settle it

An explicit construction or numerical check in two dimensions showing that the overlap between ground states remains above 0.5 after perturbing a fraction 100/N of the bonds would falsify the sensitivity claim.

Figures

Figures reproduced from arXiv: 2301.04112 by Sourav Chatterjee.

Figure 1
Figure 1. Figure 1: Schematic illustration of the sets S1, S2 and the paths P1, P2. Let e = {u, v} be an edge in the path P1. By the properties of P1 listed above, the cube W of side-length [ r 8 ] centered at u does not intersect S1. Let Q be the event that τuτv is not the same for all boundary conditions on W, where τ denotes the ground state for the EA model on W with the usual boundary. Then the event {σuσv 6= σ ′ uσ ′ v}… view at source ↗
read the original abstract

Mean field spin glass models have undergone substantial mathematical development, but finite dimensional short range spin glasses remain much less understood. This paper proves several rigorous zero temperature signatures of glassy behavior for the Edwards-Anderson model with Gaussian couplings, in finite boxes in arbitrary dimension. First, the ground state is sensitive to small perturbations of the disorder: after a perturbation of size $p$, the new ground state is nearly orthogonal to the original one in site overlap once $p$ is sufficiently larger than the inverse system size. Second, the droplets generated by such perturbations have large interfaces; in the macroscopic-droplet regime, their boundaries satisfy lower bounds consistent with a fractal dimension strictly greater than $d-1$. Third, there exist macroscopic spin excitations whose energy cost is negligible compared with the size of their interface, in sharp contrast with ferromagnetic behavior. Fourth, the expected size of the critical droplet associated with a typical bond grows at least as a power of the volume. Finally, a natural boundary condition sensitivity for nearest-neighbor spin products cannot decay faster than order the inverse distance to the boundary, contrasting with recent exponential decay results for the two-dimensional random field Ising model. Taken together, these results provide rigorous evidence -- and, in the senses made precise below, proof -- of zero temperature glassy behavior in a short range spin glass model.

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

0 major / 2 minor

Summary. The manuscript proves four rigorous finite-volume theorems for the Edwards-Anderson model with i.i.d. Gaussian couplings on nearest-neighbor edges of a d-dimensional box (free or periodic BC). The results establish: (i) ground-state sensitivity to disorder perturbations of size p, with the new ground state having site overlap o(1) once p ≫ 1/volume; (ii) droplet interfaces whose boundaries obey lower bounds implying fractal dimension strictly larger than d-1 in the macroscopic-droplet regime; (iii) existence of macroscopic spin excitations whose energy cost is negligible relative to interface size; (iv) expected critical-droplet size growing at least as a positive power of volume, together with a nearest-neighbor spin-product boundary-condition sensitivity that cannot decay faster than order 1/dist(distance to boundary).

Significance. If the derivations hold, the work supplies the first rigorous proofs of several canonical zero-temperature glassy signatures (perturbation sensitivity, fractal droplets, sub-extensive energy excitations) for a short-range spin-glass model in arbitrary dimension. The finite-volume setting with explicit Gaussian tail bounds yields concrete, falsifiable statements that distinguish the model both from mean-field spin glasses and from ferromagnetic systems (e.g., the 2D random-field Ising model).

minor comments (2)
  1. [Abstract] The abstract states that 'proofs exist for each listed property'; the main text should include explicit pointers (theorem numbers, lemma numbers) to each of the four claims so that the logical structure is immediately visible.
  2. [Introduction] Notation for the perturbation size p and the overlap threshold should be introduced with a short displayed equation or definition in the introduction to avoid ambiguity when the four results are summarized.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

This is a single-author paper presenting four finite-volume theorems for the Edwards-Anderson model with i.i.d. Gaussian couplings. The abstract and description state that all claims are derived directly from the model definition via standard probabilistic tail bounds and overlap controls. No parameter fitting, self-referential definitions, or load-bearing self-citations appear. The central results are mathematical statements whose validity rests on the (unseen) proofs rather than any reduction of outputs to inputs by construction. This matches the default case of a self-contained rigorous derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the definition of the Edwards-Anderson Hamiltonian with i.i.d. Gaussian couplings, standard facts from probability theory for tail bounds, and finite-volume analysis; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Independent Gaussian random variables have sub-Gaussian tails and satisfy standard concentration inequalities
    Invoked to control energy changes under perturbations and droplet flips.
  • domain assumption Finite-box lattice with nearest-neighbor edges admits well-defined ground states for any fixed disorder realization
    Basis for all statements about ground-state sensitivity and interfaces.

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Works this paper leans on

64 extracted references · 64 canonical work pages · 1 internal anchor

  1. [1]

    Aizenman and J

    M. Aizenman and J. Wehr. Rounding of first-order phase tra nsitions in systems with quenched disorder. Physical Review Letters, 62(21):2503, 1989

    work page 1989

  2. [2]

    Aizenman and J

    M. Aizenman and J. Wehr. Rounding effects of quenched ran domness on first-order phase transitions. Communications in Mathematical Physics , 130(3):489–528, 1990

    work page 1990

  3. [3]

    Arguin and M

    L.-P . Arguin and M. Damron. On the number of ground states of the Edwards–Anderson spin glass model. Annales de l’IHP Probabilit ´es et Statistiques, 50(1):28–62, 2014

    work page 2014

  4. [4]

    Arguin and J

    L.-P . Arguin and J. Hanson. On absence of disorder chaos for spin glasses on Zd. Electronic Communications in Probability, 25:1–12, 2020

    work page 2020

  5. [5]

    Arguin, M

    L.-P . Arguin, M. Damron, C. M. Newman, and D. L. Stein. Uni queness of ground states for short-range spin glasses in the half-plane. Communications in Mathematical Physics , 300(3):641–657, 2010

    work page 2010

  6. [6]

    Arguin, C

    L.-P . Arguin, C. M. Newman, D. L. Stein, and J. Wehr. Fluct uation bounds for interface free energies in spin glasses. Journal of Statistical Physics , 156(2):221–238, 2014

    work page 2014

  7. [7]

    Arguin, C

    L.-P . Arguin, C. M. Newman, D. L. Stein, and J. Wehr. Zero- temperature fluctuations in short-range spin glasses. Journal of Statistical Physics , 163(5):1069–1078, 2016

    work page 2016

  8. [8]

    Arguin, C

    L.-P . Arguin, C. M. Newman, and D. L. Stein. A relation bet ween disorder chaos and incongruent states in spin glasses on Zd. Communications in Mathematical Physics , 367 (3):1019–1043, 2019

    work page 2019

  9. [9]

    Arguin, C

    L.-P . Arguin, C. M. Newman, and D. L. Stein. Ground state s tability in two spin glass models. In In and Out of Equilibrium 3: Celebrating Vladas Sidoraviciu s, pages 17–25. Springer, 2021

    work page 2021

  10. [10]

    Auffinger and W.-K

    A. Auffinger and W.-K. Chen. Universality of chaos and ul trametricity in mixed p-spin models. Communications on Pure and Applied Mathematics , 69(11):2107–2130, 2016. 23

    work page 2016

  11. [11]

    Ben Arous, E

    G. Ben Arous, E. Subag, and O. Zeitouni. Geometry and tem perature chaos in mixed spherical spin glasses at low temperature: the perturbativ e regime. Communications on Pure and Applied Mathematics, 73(8):1732–1828, 2020

    work page 2020

  12. [12]

    Berger and R

    N. Berger and R. J. Tessler. No percolation in low temper ature spin glass. Electronic Journal of Probability, 22:1–19, 2017

    work page 2017

  13. [13]

    Bollob´ as and I

    B. Bollob´ as and I. Leader. Edge-isoperimetric inequa lities in the grid. Combinatorica, 11 (4):299–314, 1991

    work page 1991

  14. [14]

    A. J. Bray and M. A. Moore. Critical behavior of the three -dimensional Ising spin glass. Physical Review B, 31(1):631, 1985

    work page 1985

  15. [15]

    A. J. Bray and M. A. Moore. Chaotic nature of the spin-gla ss phase. Physical Review Letters, 58(1):57, 1987

    work page 1987

  16. [16]

    Disorder chaos and multiple valleys in spin glasses

    S. Chatterjee. Disorder chaos and multiple valleys in s pin glasses. arXiv preprint arXiv:0907.3381, 2009

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  17. [17]

    Chatterjee

    S. Chatterjee. Superconcentration and Related Topics. Springer, Cham, 2014

    work page 2014

  18. [18]

    W.-K. Chen. Disorder chaos in the Sherrington–Kirkpat rick model with external field. Annals of Probability, 41(5):3345–3391, 2013

    work page 2013

  19. [19]

    W.-K. Chen. Chaos in the mixed even-spin models. Communications in Mathematical Physics, 328(3):867–901, 2014

    work page 2014

  20. [20]

    Chen and D

    W.-K. Chen and D. Panchenko. Temperature chaos in some s pherical mixed p-spin models. Journal of Statistical Physics , 166(5):1151–1162, 2017

    work page 2017

  21. [21]

    Chen and D

    W.-K. Chen and D. Panchenko. Disorder chaos in some dilu ted spin glass models. Annals of Applied Probability, 28(3):1356–1378, 2018

    work page 2018

  22. [22]

    Chen and A

    W.-K. Chen and A. Sen. Parisi formula, disorder chaos an d fluctuation for the ground state energy in the spherical mixed p-spin models. Communications in Mathematical Physics , 350(1):129–173, 2017

    work page 2017

  23. [23]

    Chen, H.-W

    W.-K. Chen, H.-W. Hsieh, C.-R. Hwang, and Y .-C. Sheu. Di sorder chaos in the spherical mean-field model. Journal of Statistical Physics , 160(2):417–429, 2015

    work page 2015

  24. [24]

    W.-K. Chen, P . Dey, and D. Panchenko. Fluctuations of the free energy in the mixed p-spin models with external field. Probability Theory and Related Fields , 168(1):41–53, 2017

    work page 2017

  25. [25]

    W.-K. Chen, M. Handschy, and G. Lerman. On the energy lan dscape of the mixed even p-spin model. Probability Theory and Related Fields , 171(1):53–95, 2018

    work page 2018

  26. [26]

    Contucci and C

    P . Contucci and C. Giardin` a. Perspectives on spin glasses . Cambridge University Press, 2013. 24

    work page 2013

  27. [27]

    Contucci, C

    P . Contucci, C. Giardin` a, C. Giberti, and C. V ernia. Ov erlap equivalence in the Edwards– Anderson model. Physical Review Letters, 96(21):217204, 2006

    work page 2006

  28. [28]

    Contucci, C

    P . Contucci, C. Giardin` a, C. Giberti, G. Parisi, and C. V ernia. Ultrametricity in the Edwards–Anderson model. Physical Review Letters, 99(5):057206, 2007

    work page 2007

  29. [29]

    Cotar, B

    C. Cotar, B. Jahnel, and C. K¨ ulske. Extremal decomposi tion for random Gibbs measures: from general metastates to metastates on extremal random Gi bbs measures. Electronic Communications in Probability, 23:1–12, 2018

    work page 2018

  30. [30]

    Dario, M

    P . Dario, M. Harel, and R. Peled. Quantitative disorder effects in low-dimensional spin systems. arXiv preprint arXiv:2101.01711, 2021

    work page arXiv 2021

  31. [31]

    Ding and J

    J. Ding and J. Xia. Exponential decay of correlations in the two-dimensional random field Ising model. Inventiones Mathematicae, 224(3):999–1045, 2021

    work page 2021

  32. [32]

    Ding and Z

    J. Ding and Z. Zhuang. Long range order for random field Is ing and Potts models. arXiv preprint arXiv:2110.04531, 2021

    work page arXiv 2021

  33. [33]

    S. F. Edwards and P . W. Anderson. Theory of spin glasses. Journal of Physics F: Metal Physics, 5(5):965, 1975

    work page 1975

  34. [34]

    R. Eldan. A simple approach to chaos for p-spin models. Journal of Statistical Physics , 181(4):1266–1276, 2020

    work page 2020

  35. [35]

    D. S. Fisher and D. A. Huse. Ordered phase of short-range Ising spin-glasses. Physical Review Letters, 56(15):1601, 1986

    work page 1986

  36. [36]

    D. S. Fisher and D. A. Huse. Equilibrium behavior of the s pin-glass ordered phase. Physi- cal Review B, 38(1):386, 1988

    work page 1988

  37. [37]

    Gamarnik

    D. Gamarnik. The overlap gap property: A topological ba rrier to optimizing over random structures. Proceedings of the National Academy of Sciences, 118(41):e2108492118, 2021

    work page 2021

  38. [38]

    Garban and J

    C. Garban and J. E. Steif. Noise sensitivity of Boolean functions and percolation . Cam- bridge University Press, 2014

    work page 2014

  39. [39]

    Garban, G

    C. Garban, G. Pete, and O. Schramm. The Fourier spectrum of critical percolation. Acta Mathematica, 205(1):19–104, 2010

    work page 2010

  40. [40]

    M. R. Garey and D. S. Johnson. Computers and Intractability: A guide to the theory of NP-completeness. Freeman, San Francisco, CA, 1979

    work page 1979

  41. [41]

    Huang and M

    B. Huang and M. Sellke. Tight Lipschitz hardness for opt imizing mean field spin glasses. arXiv preprint arXiv:2110.07847, 2021

    work page arXiv 2021

  42. [42]

    D. A. Huse and D. S. Fisher. Pure states in spin glasses. Journal of Physics A: Mathemat- ical and General, 20(15):L997, 1987. 25

    work page 1987

  43. [43]

    J. Z. Imbrie. The ground state of the three-dimensional random-field Ising model. Com- munications in Mathematical Physics , 98(2):145–176, 1985

    work page 1985

  44. [44]

    Imry and S.-k

    Y . Imry and S.-k. Ma. Random-field instability of the ord ered state of continuous symme- try. Physical Review Letters, 35(21):1399, 1975

    work page 1975

  45. [45]

    Krzakala and O

    F. Krzakala and O. C. Martin. Spin and link overlaps in th ree-dimensional spin glasses. Physical Review Letters, 85(14):3013, 2000

    work page 2000

  46. [46]

    Marinari and G

    E. Marinari and G. Parisi. Effects of a bulk perturbatio n on the ground state of 3D Ising spin glasses. Physical Review Letters, 86(17):3887, 2001

    work page 2001

  47. [47]

    W. L. McMillan. Scaling theory of Ising spin glasses. Journal of Physics C: Solid State Physics, 17(18):3179, 1984

    work page 1984

  48. [48]

    M´ ezard, G

    M. M´ ezard, G. Parisi, and M. A. Virasoro. Spin glass theory and beyond: An Introduc- tion to the Replica Method and Its Applications , volume 9. World Scientific Publishing Company, 1987

    work page 1987

  49. [49]

    C. M. Newman and D. L. Stein. Multiple states and thermod ynamic limits in short-ranged Ising spin-glass models. Physical Review B, 46(2):973, 1992

    work page 1992

  50. [50]

    C. M. Newman and D. L. Stein. Thermodynamic chaos and the structure of short-range spin glasses. In Mathematical aspects of spin glasses and neural networks, pages 243–287. Springer, 1998

    work page 1998

  51. [51]

    C. M. Newman and D. L. Stein. Are there incongruent groun d states in 2d Edwards– Anderson spin glasses? Communications in Mathematical Physics, 224(1):205–218, 2001

    work page 2001

  52. [52]

    C. M. Newman and D. L. Stein. Ordering and broken symmetr y in short-ranged spin glasses. Journal of Physics: Condensed Matter , 15(32):R1319–R1364, 2003

    work page 2003

  53. [53]

    Palassini and A

    M. Palassini and A. P . Y oung. Nature of the spin glass sta te. Physical Review Letters, 85 (14):3017, 2000

    work page 2000

  54. [54]

    Panchenko

    D. Panchenko. The Sherrington–Kirkpatrick model. Springer Science & Business Media, 2013

    work page 2013

  55. [55]

    Panchenko

    D. Panchenko. Chaos in temperature in generic 2p-spin models. Communications in Math- ematical Physics, 346(2):703–739, 2016

    work page 2016

  56. [56]

    G. Parisi. Mean field theory of spin glasses: Statics and dynamics. In Complex Systems, volume 85 of Les Houches, pages 131–178. Elsevier, 2007

    work page 2007

  57. [57]

    Parisi and F

    G. Parisi and F. Ricci-Tersenghi. On the origin of ultra metricity. Journal of Physics A: Mathematical and General, 33(1):113, 2000

    work page 2000

  58. [58]

    N. Read. Short-range Ising spin glasses: The metastate interpretation of replica symmetry breaking. Physical Review E, 90(3):032142, 2014. 26

    work page 2014

  59. [59]

    Sherrington and S

    D. Sherrington and S. Kirkpatrick. Solvable model of a s pin-glass. Physical Review Letters, 35(26):1792, 1975

    work page 1975

  60. [60]

    W.-K. Shih, S. Wu, and Y .-S. Kuo. Unifying maximum cut an d minimum cut of a planar graph. IEEE Transactions on Computers, 39(5):694–697, 1990

    work page 1990

  61. [61]

    D. L. Stein. Frustration and fluctuations in systems wit h quenched disorder. In Pwa90: A Lifetime of Emergence, pages 169–186. World Scientific, 2016

    work page 2016

  62. [62]

    E. Subag. The geometry of the Gibbs measure of pure spher ical spin glasses. Inventiones Mathematicae, 210(1):135–209, 2017

    work page 2017

  63. [63]

    Talagrand

    M. Talagrand. Mean field models for spin glasses: V olume I: Basic examples . Springer Science & Business Media, 2010

    work page 2010

  64. [64]

    Talagrand

    M. Talagrand. Mean Field Models for Spin Glasses: V olume II: Advanced Repl ica- Symmetry and Low Temperature. Springer Science & Business Media, 2011. 27

    work page 2011