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A long-prism geometry measures the lower critical dimension of the 3D Ising spin glass at 2.49, matching the mean-field prediction of 5/2.

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2026-08-03 10:58 UTC pith:GVF52O7O

load-bearing objection Send to review, but the headline D_lc=2.49 rides on a dimensional-independence assumption the paper never tests. the 3 major comments →

arxiv 2601.07926 v2 pith:GVF52O7O submitted 2026-01-12 cond-mat.dis-nn cond-mat.stat-mechphysics.comp-ph

Low energy excitations in a long prism geometry: testing the lower critical dimension of the Ising spin glass

Massimo Bernaschi , Luis Antonio Fern\'andez , Isidoro Gonz\'alez-Adalid Pemart\'in , V\'ictor Mart\'in-Mayor , Giorgio Parisi , Federico Ricci-Tersenghi This is my paper
classification cond-mat.dis-nn cond-mat.stat-mechphysics.comp-ph
keywords Ising spin glasslower critical dimensionlong prism geometrycorrelation length scalingmultifractal spectrumdroplet modelreplica symmetry breakingHoudayer cluster algorithm
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to settle a long-standing question: at what spatial dimension does the Ising spin glass stop ordering at any positive temperature? It argues that in a long rectangular prism, correlations decay exponentially along the long direction at all temperatures, and the way the longitudinal correlation length grows with the transverse size encodes the lower critical dimension. Simulating three-dimensional prisms with transverse sizes up to 24 and effectively infinite length, the authors measure the growth exponent a3=1.34(3)(3), which yields D_lc=2.49(3)(3), matching mean-field theory's 5/2 and slightly exceeding the droplet-model value. They also measure a multifractal spectrum for correlation-function moments that disagrees with the droplet model's simplest prediction, so the prism setup appears able to discriminate between the two competing pictures of 3D spin-glass order.

Core claim

The central claim is that a long rectangular prism acts as a one-dimensional filter: even below the critical temperature, plane-to-plane spin-glass correlations decay exponentially along the long axis, with correlation length ξ(L) ∝ L^{a3}. From this power law the lower critical dimension follows as D_lc = 1 + 2/a3. Using prisms with transverse sizes up to 24, the paper obtains a3=1.34(3)(3) and D_lc=2.49(3)(3), matching the mean-field value 5/2 and slightly exceeding the droplet-model value a3=1.24(1). The higher correlation-function moments scale as C^{(n)} ~ [C^{(1)}]^{τ_n} with τ2≈1.43, τ3≈1.71, τ4≈1.92, incompatible with the droplet-model prediction and compatible with an effective one-

What carries the argument

The central mechanism is the long-prism free-energy relation ΔF ∼ L^{D-1}/M^b. Setting this free-energy cost to order one at M = ξ(L) gives ξ(L) ∝ L^{(D-1)/b} and therefore D_lc = 1 + (D-1)/a_D. A one-dimensional effective chain with coupling distribution P(J) ∼ |J|^λ near J=0 supplies the multifractal analysis: correlation lengths diverge as 1/T^{1+λ} and the spectrum is τ_n = I_{2n}/I_2. The numerical work relies on two devices: Houdayer cluster moves, which become highly effective in the prism because clusters do not percolate along the long direction, and open boundary conditions along that direction, which remove the large finite-length corrections that periodic boundaries introduce.

Load-bearing premise

The conversion from the measured longitudinal exponent a3 to the lower critical dimension assumes that the free-energy exponent b is the same in all dimensions and takes its mean-field value 3/2; the paper announces this assumption but does not prove it, so if b varies with dimension the reported D_lc=2.49 is not determined.

What would settle it

Measure the free-energy cost of opposite boundary conditions in a D=3 prism directly as a function of the longitudinal length M at fixed transverse size L, and check whether it follows ΔF ∼ L^2/M^{3/2}; if the M-exponent is not 3/2, or if repeating the measurement in D=4 gives a different b, the inferred D_lc=2.49 collapses. Alternatively, measure ξ_even/ξ_odd at larger L: if it trends to zero, the replica-symmetry-breaking expectation of soft Q^2 excitations is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If D_lc = 2.49(3)(3) is correct, three-dimensional Ising spin glasses are marginal: the glassy phase exists but the free-energy cost of low-energy excitations grows only slowly with system size.
  • The measured multifractal spectrum rules out the droplet model's simplest effective-chain prediction (λ=0), so the prism data provide a quantitative test separating droplet and replica-symmetry-breaking pictures.
  • The prism method gives a finite-temperature route to lower critical dimensions that does not rely on ground-state stiffness exponents or critical-point extrapolations, and can be applied to other systems with soft excitations.
  • Houdayer cluster moves are shown to accelerate equilibration by about three orders of magnitude in elongated 3D spin-glass prisms, making systems of 2^17 spins at T≈0.63Tc practical.
  • Because replica symmetry breaking predicts soft excitations for even operators such as Q^2 while the droplet model does not, measuring ξ_even/ξ_odd as L grows is a proposed decisive next experiment.

Where Pith is reading between the lines

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

  • If the dimension independence of b holds, the same construction in D=4 should yield a4 = 3/b = 2 and hence again D_lc=2.5; measuring a4 would provide a direct check of the paper's key connecting assumption.
  • The numerical proximity of a3=1.34 to the mean-field value 4/3 could partly reflect crossover or correction-to-scaling effects at the accessible sizes; the multifractal spectrum, being farther from the droplet prediction, may end up being the more robust discriminator.
  • The effective one-dimensional coupling exponent λ≈0.36 could be tested directly by coarse-graining the plane overlap in the prism and measuring the distribution of effective couplings, rather than only inferring λ from the spectrum.
  • The appendix's short-prism, end-to-end correlation method has a natural application beyond spin glasses, such as random-field systems where cluster moves of the Houdayer type are not available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a rectangular-prism geometry for studying systems with low-energy excitations and applies it to the 3D Ising spin glass. At T≈0.63Tc the longitudinal correlation length is reported to scale as ξ_{n=1}(L)∝L^{a3} with a3=1.34(3)[3], from which the authors infer D_lc=1+2/a3=2.49(3)[3], in agreement with the mean-field prediction 5/2 and arguably only slightly away from the droplet-model value a3=1.24(1). The paper also reports a multifractal spectrum τ_2≈1.43, τ_3≈1.71, τ_4≈1.92, incompatible with the droplet-model λ=0 prediction and consistent with λ≈0.36 in a 1D effective model. The main technical contributions are the use of open boundary conditions along the long axis, a Houdayer-cluster speedup in the prism geometry, and careful handling of sub-leading exponential corrections.

Significance. If the central inference is valid, this is a significant result: a new geometric route to the lower critical dimension, supported by large-scale simulations (transverse size up to L=24) and by detailed supplemental analysis. The numerical work is careful in several respects: thermalization is checked by multiple criteria, correlation lengths are extracted with integral estimators that control short-distance contamination, and the sub-leading correlation lengths are quantitatively bounded. The source code is made available. However, the headline value D_lc=2.49 is conditional on an unverified assumption about the dimension dependence of the exponent b in Eq. (1), and the mapping from Eq. (1) to Eq. (2) is heuristic. The significance of the paper is therefore real but contingent.

major comments (3)
  1. [Introduction, Eqs. (1)-(2)] The central conversion a3→D_lc assumes that the exponent b in ΔF ∼ L^{D-1}/M^b is independent of D. The paper states this as an 'important goal' but never returns to it; the subsequent derivation in Eq. (2) uses b as a single constant. From the D=3 measurement alone one obtains only b_3 = 2/a3 ≈ 1.49(3), and the lower critical dimension is defined by D_lc = 1 + b(D_lc). Without a measurement or derivation of b(D) for D≠3, the abstract's D_lc=2.49 is not a computed value but a conditional inference. Please either provide evidence for D-independence (e.g., repeating the prism measurement in D=2 and D=4) or reformulate the central claim so that it does not overstate what has been established.
  2. [Introduction and Eq. (2)] Eq. (2) is obtained by setting M=ξ(L) and requiring ΔF∼L^0, but Eq. (1) was introduced for the free-energy excess induced by mutually incongruent boundary conditions at the prism end planes. It is not self-evident that the same ΔF(M) controls the exponential decay of the bulk overlap correlation C^{(1)}(z,z+r) in the M→∞ limit; the Wilson-RG/effective-1D argument is only sketched. Since the entire method rests on this identification, please provide a more explicit derivation, or a controlled check on a system whose answer is known (e.g., the Heisenberg ferromagnet in the same prism geometry, where b=1 and a_D=D-1), showing that the estimators used here reproduce that known result.
  3. [Numerical results and Fig. 3 / Eq. (7)] The statistical precision of a3 is load-bearing for distinguishing the measured value from the droplet prediction a3=1.24(1). The fit ξ=BL^{a3}+c is a three-parameter form applied to five data points after excluding L=4, and the text notes that a3 drifts upward when the fitting window is moved to larger L. Please report a stability table showing the fit results (B, a3, c, χ²/DoF) for the different L-windows used to compute the systematic error given in square brackets. It is important to know whether the L≥8 or L≥12 windows produce central values consistent with 1.34(3); otherwise the 'slightly away from the droplet model' conclusion may be driven by scaling corrections rather than by the true exponent.
minor comments (5)
  1. [Abstract vs. Discussion] The abstract states that the measured D_lc is 'in agreement with expectations from both the Replica Symmetry Breaking theory and the Droplet model', while the body reports a3=1.34(3) as 'perfect agreement with MFT' but 'slightly away from the DM predictions' a3=1.24(1). With the quoted errors these differ by roughly 3σ; the summary should be harmonized with the body.
  2. [Fig. 3 caption] The caption says the T=0.7 fit is to ξ=BL^{4/3}+c, while the text describes fitting ξ=BL^{a3}+c with free a3. Please clarify whether the plotted line fixes a3=4/3 or uses the fitted a3.
  3. [SM Eq. (24)] In the displayed expression for C^{1D,(n)}(r)|_{OBC}, the notation 'tr^{2n}' is ambiguous; it should be t_{2n}^r (with the usual definition of t_{2n}). Please correct the typographical convention.
  4. [Numerical results: singularity spectrum] The statement 'the reader may check that the ansatz τ_n=1+A log n fits our results' should be replaced by a quantitative fit (best A, χ², and the range of n used). Similarly, the agreement with λ=0.36 is a post-hoc comparison; please provide an uncertainty estimate for λ or state clearly that it is not a fitted parameter.
  5. [SM title] The supplemental title says 'computing the lowest critical dimension'; the standard terminology used in the main text is 'lower critical dimension'. Please make the titles consistent.

Circularity Check

1 steps flagged

Central a3/Dlc measurement is an independent simulation result; only a post-hoc λ=0.36 multifractal match and an unverified D-independence premise introduce minor circular/derivation-gap content.

specific steps
  1. fitted input called prediction [Numerical results: the singularity spectrum (main text, after Fig. 4)]
    "Interestingly enough, {τ2, τ3, τ4}1D,λ=0.36 ≈ {1.43, 1.71, 1.92} agrees with the prism singularity spectrum, which suggests that the DM prediction λ = 0 for the 1D effective Hamiltonian needs to be improved."

    The 1D toy-model spectrum, Eq. (6), depends on the free parameter λ through τn = I2n/I2. The paper does not derive λ=0.36; it is selected so that the 1D formula reproduces the measured prism values {1.43,1.71,1.92}. The reported 'agreement' is therefore a restatement of the fit, not an independent test of the DM λ=0 hypothesis. This is an interpretive post-hoc match and does not enter the independently measured a3/Dlc.

full rationale

The central numerical result, a3 = 1.34(3)[3] and Dlc = 2.49(3)[3], comes from direct Monte Carlo measurements of the longitudinal correlation length ξn=1(L) in L=4..24 prisms, fit to ξn=1(L) = B L^{a3} + c (Fig. 3). This measurement is not constructed from MFT or Droplet-model inputs; the comparison with a3^MFT = 4/3 and a3^DM = 1.24(1) is an external benchmark. The only risky inference step is the conversion a3 -> Dlc via Eq. (2), which assumes that the exponent b in ΔF ∼ L^{D-1}/M^b is dimension-independent. The Introduction announces this as a goal ('An important goal of this work will be to confirm that b is D-independent for spin glasses as well') but the manuscript never returns to verify it. That is a derivation gap / correctness risk rather than a circular reduction: the measured a3 is not used to define b and then re-extracted as a prediction. The multifractal λ=0.36 agreement is a one-parameter fit to the measured τ spectrum and thus carries little independent discriminating power, but it is not load-bearing for the central Dlc claim. There is no self-citation chain or ansatz-smuggling driving the main result.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The ledger counts the assumptions behind the central claim. The method rests on a hypothesized free-energy scaling with a dimension-independent exponent b (Eq. 1), an effective-1D picture for the long direction, and an assumed 1D toy-model form P(J)∼|J|^λ. The numerical analysis introduces two free fit parameters (B and c) in addition to the exponent a3; λ=0.36 is selected post hoc to match the multifractal data. No new physical entities are postulated.

free parameters (5)
  • amplitude B and constant background c in ξ(L)=B L^{a3}+c = B=1.046(5), c=2.01(8) for fixed a3=4/3
    The final a3 = 1.34(3)(3) comes from a three-parameter fit (B, a3, c) to only five (or four, excluding L=6) data points; c absorbs scaling corrections.
  • longitudinal correlation length exponent a3 = 1.34(3)(3)
    Fitted from ξ(L) data at T=0.7; used to derive Dlc.
  • constant A in critical fit ξ=AL = A=0.977(2)
    Fit at T≈Tc to confirm linear growth; L=4 excluded.
  • exponent λ in 1D effective coupling distribution = 0.36 (quoted)
    Chosen so that the 1D formula τ_n(λ) matches the measured prism multifractal spectrum; not stated as a fit but evidently selected to reproduce {1.43,1.71,1.92}.
  • discard distances zdiscard = 5 (L=16), 10 (L=24)
    Hand-chosen analysis parameters to restore translation invariance under OBC.
axioms (5)
  • domain assumption Free-energy excess ΔF ∼ L^{D-1}/M^b with b independent of D (Eq. 1)
    Used to derive ξ(L) ∝ L^{(D-1)/b} and D_lc = 1+b; the paper states confirming D-independence is a goal but does not prove it.
  • domain assumption For M→∞, longitudinal correlations decay exponentially, governed by an effective 1D Hamiltonian
    Invoked in the Introduction via Wilson RG; justifies setting M=ξ(L) to obtain Eq. (2).
  • domain assumption 1D Edwards-Anderson chain with P(J) ∼ |J|^λ g(J) captures the leading scaling of prism correlations
    Used for the 1D toy model predictions (multifractal spectrum, OBC/PBC finite-M behavior); the main text acknowledges the toy model is not the full effective model (missing subleading lengths).
  • standard math Replica trick / transfer-matrix spectral decomposition (Eq. 8) applies to the overlap correlation functions
    Used to analyze subdominant correlation lengths and the short-prism R(M) expansion.
  • domain assumption Simulated prisms are in equilibrium at T=0.7
    SM shows detailed equilibration tests for L=12; for L=16 and L=24 the strict energy-block criterion is claimed but less documented in the main text.

pith-pipeline@v1.3.0-alltime-deepseek · 28253 in / 16508 out tokens · 137668 ms · 2026-08-03T10:58:22.166042+00:00 · methodology

0 comments
read the original abstract

We propose a general method for studying systems that display excitations with arbitrarily low energy in their low-temperature phase. We argue that in a rectangular right prism geometry, with longitudinal size much larger than the transverse size, correlations decay exponentially (at all temperatures) along the longitudinal dimension, but the scaling of the correlation length with the transverse size carries crucial information from which the lower critical dimension can be inferred. The method is applied in the particularly demanding context of Ising spin glasses at zero magnetic field. The lower critical dimension and the multifractal spectrum for the correlation function are computed from large-scale numerical simulations. Several technical novelties (such as the unexpectedly crucial performance of Houdayer's cluster method or the convenience of using open - rather than periodic - boundary conditions) allow us to study three-dimensional prisms with transverse dimensions up to $L=24$ and effectively infinite longitudinal dimensions down to low temperatures. The value that we find for the lower critical dimension turns out to be in agreement with expectations from both the Replica Symmetry Breaking theory and the Droplet model for spin glasses. We argue that our novel setting holds promise in clarifying which of the two competing theories more accurately describes three-dimensional spin glasses.

Figures

Figures reproduced from arXiv: 2601.07926 by Federico Ricci-Tersenghi, Giorgio Parisi, Isidoro Gonz\'alez-Adalid Pemart\'in, Luis Antonio Fern\'andez, Massimo Bernaschi, V\'ictor Mart\'in-Mayor.

Figure 1
Figure 1. Figure 1: FIG. 1. The eight largest clusters for a typical configuration [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A comparison of the correlation functions [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Ratio of correlation functions [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Scaling of the logarithm of the (normalized) corre [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Scaling of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The open boundary condition on the last [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Overlap squared evolution with and without Hou [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diagonal elements of the correlation matrix [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Estimation of the ratio of correlation length [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Integral estimator of the correlation length for a [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Correlation [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Correlation [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

93 extracted references · 11 linked inside Pith · cited by 2 Pith papers

  1. [1]

    So, comparing with Eq

    In the M → ∞ limit, C 1D,(n)(z1, z2) = e −|z1−z2|/ξn , ξn diverges when T1D,L → 0 as ξn ∼ 1/T 1+λ 1D,L. So, comparing with Eq. ( 2), ρ = a3/(1 + λ)

  2. [2]

    We have C 1D,(n) ∼ [C 1D,(1)]τn with τn < n, hence a multifractal with a singularity spectrum τn = I2n I2 , I k = ∫ ∞ 0 du uλ(1 − tanhku) . (6)

  3. [3]

    Instead, for pe- riodic boundary conditions (PBC), it holds only in the M → ∞ limit, and C 1D,PBC,(n)(z1, z2) is plagued by finite- M corrections

    With open boundary conditions (OBC), the expo- nential decay, C 1D,(n)(z1, z2) = e −|z1−z2|/ξn , holds true even for finite values of M . Instead, for pe- riodic boundary conditions (PBC), it holds only in the M → ∞ limit, and C 1D,PBC,(n)(z1, z2) is plagued by finite- M corrections. This probably ex- plains the very large M values required for the PBC pris...

  4. [4]

    [ ξn=1(L) will be used 3 FIG. 1. The eight largest clusters for a typical configuration on a lattice 8 × 8 × 512 with periodic boundary conditions at T = 0 .7 are depicted at y = 4. There is no percolation through the lattice along the Z direction. The Z scale has been reduced by a factor of 10, in order to improve visibility. 1 × 10− 2 10− 3 10− 2 10− 1 0...

  5. [5]

    J. A. Mydosh, Spin Glasses: an Experimental Introduc- tion (Taylor and Francis, London, 1993)

    1993

  6. [6]

    The MFT prediction, aMFT 3 = 4 /3 is numerically close to the one from the DM

    one gets τn = ∑ n−1 k=0 1/(2k + 1). The MFT prediction, aMFT 3 = 4 /3 is numerically close to the one from the DM. However, the MFT does not regard the 1D toy model as a true effective model, be- cause the corresponding 1D theory, which we do not dis- cuss here, is much more complex. Moreover, being yD an increasing concave function of D, the DM prediction...

  7. [7]

    National Centre for HPC, Big Data and Quan- tum Computing - HPC

    In fact, for T ≪ Tc, C (1)(z, z) turns out to be quite large and essentially M -independent, which results in a close to one-dimensional geometry for Hou- dayer’s clusters. So, flipping Houdayer cluster (at low T and M ≫ ξn=1) is a non-trivial move that has resulted in a three-order-of-magnitude speed-up for large M [ 15]. Instead, C (1)(z, z) is small for...

    2000

  8. [8]

    Parisi, Statistical Field Theory (Addison-Wesley, 1988)

    G. Parisi, Statistical Field Theory (Addison-Wesley, 1988)

    1988

  9. [9]

    Zinn-Justin, Quantum Field Theory and Critical Phe- nomena, 4th ed

    J. Zinn-Justin, Quantum Field Theory and Critical Phe- nomena, 4th ed. (Clarendon Press, Oxford, 2005)

    2005

  10. [10]

    P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev. 130, 439 (1963)

    1963

  11. [11]

    P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett. 13, 508 (1964)

    1964

  12. [12]

    Charbonneau, E

    P. Charbonneau, E. Marinari, M. M´ ezard, G. Parisi, F. Ricci-Tersenghi, G. Sicuro, and F. Zamponi, eds., Spin Glass Theory and Far Beyond (World Scientific, 2023)

    2023

  13. [13]

    Dahlberg, I

    E. Dahlberg, I. G.-A. Pemart ´ ın, E. Marinari, G. Parisi, F. Ricci-Tersenghi, V. Martin-Mayor, J. Moreno-Gordo, R. Orbach, I. Paga, J. Ruiz-Lorenzo, et al., Spin-glass dy- namics: Experiment, theory, and simulation, Rev. Mod. Phys. 97, 045005 (2025)

    2025

  14. [14]

    Baity-Jesi, E

    M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi´ on, I. Gonz´ alez-Adalid Pemart ´ ın, A. Gordillo- Guerrero, D. ´I˜ niguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu˜ noz Sudupe, D. Navarro, I. Paga, G. Parisi, S. Perez-Gaviro, F. Ricci- Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, D. Seoane, A. Tarancon, and...

    2024

  15. [15]

    This is one of the geometries employed by Brezin in his investigation of Finite Size Scaling at the critical temper- ature Tc [ 49]

  16. [16]

    Because every pair of consecutive planes contributes an excess free-energy ∼ [φ/(M − 1)]2LD−1 and there are M − 1 pairs of consecutive planes —writing M or M − 1 is irrelevant in asymptotic expansions such as Eq. ( 1)

  17. [17]

    Maiorano and G

    A. Maiorano and G. Parisi, Support for the value 5/2 for the spin glass lower critical dimension at zero magnetic field, Proc. Natl. Acad. Sci. USA 115, 5129 (2018)

    2018

  18. [18]

    Franz, G

    S. Franz, G. Parisi, and M. Virasoro, Interfaces and lower critical dimension in a spin glass model, J. Phys. (France) 4, 1657 (1994)

    1994

  19. [19]

    K. G. Wilson, The renormalization group: Critical phe- nomena and the kondo problem, Rev. Mod. Phys. 47, 773 (1975)

    1975

  20. [20]

    The probability that the order parameter differs signif- icantly in the two planes xD = n and xD = n + d is e−∆F (L,M =d)

  21. [21]

    [ 50– 59]

    See supplemental information (2025), [URL will be in- serted by publisher] for a short description of our simu- lations and equilibration checks, the computation of cor- relation lengths with periodic and open boundary condi- tions, the derivation of crucial results for the 1D Ising- Edwards-Anderson model, the discussion of finite- M ef- fects, and the be...

    2025

  22. [22]

    S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F 5, 965 (1975)

    1975

  23. [23]

    S. F. Edwards and P. W. Anderson, Theory of spin glasses. ii, J. Phys. F 6, 1927 (1976)

    1927

  24. [24]

    The importance of ge- ometry was emphasized in [ 60]

    However, the multifractal spectrum that we compute here cannot be directly compared with the results of [ 8], because of the different geometry. The importance of ge- ometry was emphasized in [ 60]

  25. [25]

    D. S. Fisher and D. A. Huse, Equilibrium behavior of the spin-glass ordered phase, Phys. Rev. B 38, 386 (1988)

    1988

  26. [26]

    A. C. Carter, A. J. Bray, and M. A. Moore, Aspect-ratio scaling and the stiffness exponent θ for ising spin glasses, Phys. Rev. Lett. 88, 077201 (2002)

    2002

  27. [27]

    Palassini and A

    M. Palassini and A. P. Young, Phys. Rev. Lett. 83, 5126 (1999)

    1999

  28. [28]

    A. K. Hartmann, Scaling of stiffness energy for 3d ±j ising spin glasses, Phys. Rev. E. 59, 84 (1999), arXiv:cond-mat/9806114

    Pith/arXiv arXiv 1999

  29. [29]

    Boettcher, Stiffness exponents for lattice spin glasse s in dimensions d = 3,

    S. Boettcher, Stiffness exponents for lattice spin glasse s in dimensions d = 3, . . . ,6, Eur. Phys. J. B 38, 83 (2004) , arXiv:cond-mat/0310698

    Pith/arXiv arXiv 2004

  30. [30]

    Billoire, L

    A. Billoire, L. A. Fernandez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, G. Parisi, F. Ricci- Tersenghi, and J. J. Ruiz-Lorenzo, Dynamic variational study of chaos: spin glasses in three dimensions, J. Stat. Mech. 2018, 033302 (2018)

    2018

  31. [31]

    Alvarez Ba˜ nos, A

    R. Alvarez Ba˜ nos, A. Cruz, L. A. Fernandez, J. M. Gil- Narvion, A. Gordillo-Guerrero, M. Guidetti, A. Maio- rano, F. Mantovani, E. Marinari, V. Mart ´ ın-Mayor, J. Monforte-Garcia, A. Mu˜ noz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes (Janus Collaboration), N...

    Pith/arXiv arXiv 2010

  32. [32]

    Baity-Jesi, R

    M. Baity-Jesi, R. A. Ba˜ nos, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. Iniguez, A. Maiorano, F. Mantovani, E. Marinari, V. Mart ´ ın- Mayor, J. Monforte-Garcia, A. Mu˜ noz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, M. Pivanti, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, ...

    Pith/arXiv arXiv 2014

  33. [33]

    Hukushima and K

    K. Hukushima and K. Nemoto, Exchange Monte Carlo method and application to spin glass simulations, J. Phys. Soc. Japan 65, 1604 (1996) , arXiv:cond- mat/9512035

    arXiv 1996

  34. [35]

    L. A. Fernandez, E. Marinari, V. Martin-Mayor, G. Parisi, and J. J. Ruiz-Lorenzo, Universal critical be- havior of the two-dimensional Ising spin glass, Phys. Rev. B 94, 024402 (2016)

    2016

  35. [36]

    Z. Zhu, A. J. Ochoa, and H. G. Katzgraber, Efficient cluster algorithm for spin glasses in any space dimension, Phys. Rev. Lett. 115, 077201 (2015) . 6

    2015

  36. [37]

    Baity-Jesi, R

    M. Baity-Jesi, R. A. Ba˜ nos, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. Iniguez, A. Maiorano, F. Mantovani, E. Marinari, V. Mart ´ ın- Mayor, J. Monforte-Garcia, A. Mu˜ noz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, M. Pivanti, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, ...

    Pith/arXiv arXiv 2013

  37. [38]

    Benzi, G

    R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, On the multifractal nature of fully developed turbulence and chaotic systems, Journal of Physics A: Mathematical and General 17, 3521 (1984)

    1984

  38. [39]

    Frisch and G

    U. Frisch and G. Parisi, On the singularity structure of fully developed turbulence, in Turbulence and predictabil- ity in geophysical fluid dynamics and climate dynamics (1983 International School of Physics ”Enrico Fermi”, Varenna), edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam, 1985)

    1983

  39. [40]

    Castellani and L

    C. Castellani and L. Peliti, Multifractal wavefunction at the localisation threshold, Journal of physics A: mathe- matical and general 19, L429 (1986)

    1986

  40. [41]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Fractal measures and their singular- ities: The characterization of strange sets, Phys. Rev. A 33, 1141 (1986)

    1986

  41. [42]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Erratum: Fractal measures and their singularities: The characterization of strange sets [phys. rev. a 33, 1141 (1986)], Phys. Rev. A 34, 1601 (1986)

    1986

  42. [43]

    Harte, Multifractals

    D. Harte, Multifractals. Theory and applications , 1st ed. (Chapman and Hall/CRC, New York, 2001)

    2001

  43. [44]

    The corresponding results for T = 1 .1019 ≈ Tc are sim- ilar: τ2 = 1 .437(6), τ3 = 1 .712(15), τ4 = 1 .90(3) for L = 16 and τ2 = 1 .42(3), τ3 = 1 .64(6), τ4 = 1 .73(12) for L = 24

  44. [45]

    Palassini and S

    M. Palassini and S. Caracciolo, Universal finite-size sca l- ing functions in the 3D Ising spin glass, Phys. Rev. Lett. 82, 5128 (1999) , arXiv:cond-mat/9904246

    Pith/arXiv arXiv 1999

  45. [46]

    H. G. Ballesteros, A. Cruz, L. A. Fernandez, V. Mart ´ ın- Mayor, J. Pech, J. J. Ruiz-Lorenzo, A. Tarancon, P. Tellez, C. L. Ullod, and C. Ungil, Critical behavior of the three-dimensional Ising spin glass, Phys. Rev. B 62, 14237 (2000) , arXiv:cond-mat/0006211

    Pith/arXiv arXiv 2000

  46. [47]

    Gunnarsson, P

    K. Gunnarsson, P. Svedlindh, P. Nordblad, L. Lundgren, H. Aruga, and A. Ito, Static scaling in a short-range Ising spin glass, Phys. Rev. B 43, 8199 (1991)

    1991

  47. [48]

    Guchhait and R

    S. Guchhait and R. Orbach, Direct dynamical evidence for the spin glass lower critical dimension, Phys. Rev. Lett. 112, 126401 (2014)

    2014

  48. [49]

    L. A. Fernandez, E. Marinari, V. Martin-Mayor, I. Paga, and J. J. Ruiz-Lorenzo, Dimensional crossover in the ag- ing dynamics of spin glasses in a film geometry, Phys. Rev. B 100, 184412 (2019)

    2019

  49. [50]

    Boettcher, Stiffness of the edwards-anderson model in all dimensions, Phys

    S. Boettcher, Stiffness of the edwards-anderson model in all dimensions, Phys. Rev. Lett. 95, 197205 (2005) , arXiv:cond-mat/0508061

    Pith/arXiv arXiv 2005

  50. [51]

    Aguilar-Janita, V

    M. Aguilar-Janita, V. Mart ´ ın-Mayor, J. Moreno-Gordo, and J. J. Ruiz-Lorenzo, Evidence of de Almeida–Thouless line below six dimensions, J. Stat. Mech. 2025, 113301 (2025)

    2025

  51. [52]

    Parisi, On the probabilistic formulation of the replica approach to spin glasses, cond-mat/9801081 (1998), preprint

    G. Parisi, On the probabilistic formulation of the replica approach to spin glasses, cond-mat/9801081 (1998), preprint

    Pith/arXiv arXiv 1998

  52. [53]

    Parisi and F

    G. Parisi and F. Ricci-Tersenghi, On the origin of ultra- metricity, J. Phys. A: Math. Gen. 33, 113 (2000)

    2000

  53. [54]

    Alvarez Ba˜ nos, A

    R. Alvarez Ba˜ nos, A. Cruz, L. A. Fernandez, J. M. Gil- Narvion, A. Gordillo-Guerrero, M. Guidetti, A. Maio- rano, F. Mantovani, E. Marinari, V. Mart ´ ın-Mayor, J. Monforte-Garcia, A. Mu˜ noz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes (Janus Collaboration), S...

    Pith/arXiv arXiv 2010

  54. [55]

    Br´ ezin, E., An investigation of finite size scaling, J. Phys. France 43, 15 (1982)

    1982

  55. [66]

    Marinari, V

    E. Marinari, V. Martin-Mayor, G. Parisi, F. Ricci- Tersenghi, and J. J. Ruiz-Lorenzo, Multiscaling in the 3d critical site-diluted Ising ferromagnet, J. Stat. Mech. 2024, 013301 (2024)

    2024

  56. [67]

    M´ ezard, G

    M. M´ ezard, G. Parisi, and M. Virasoro, Spin-Glass The- ory and Beyond (World Scientific, Singapore, 1987)

    1987

  57. [68]

    J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51, 659 (1979)

    1979

  58. [69]

    Lucibello, F

    C. Lucibello, F. Morone, and T. Rizzo, One-dimensional disordered ising models by replica and cavity methods, Phys. Rev. E 90, 012140 (2014) . 7 END MA TTER In this work, we have estimated correlation lengths with high accuracy (about 2% for our largest system; see Table I). This is quite a serious numerical challenge, because one should be confident that ...

    2014

  59. [70]

    8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0

  60. [71]

    1 L = 4 L = 6 L = 8 L = 12 L = 16 L = 24 K(r) r/ξ n=1(L) FIG

    1 0 0 . 1 L = 4 L = 6 L = 8 L = 12 L = 16 L = 24 K(r) r/ξ n=1(L) FIG. 5. Scaling of the logarithm of the (normalized) corre- lation function C (1)(r), see Eq. ( 9). Data for systems of size L = 4, 6, 8, 12, 16, and 24 at temperature T = 0.7. The black line is the best fit to K(r) = ( r/ξn=1) +d for the L = 16 data in the range 0 .05 < r/ξ n=1 < 1.1. Inset:...

  61. [72]

    To assess whether or not this scenario is realized for Ising spin glass models in the elongated prism geometry, we represent in Fig

    — because they would indicate that ˆQ produces further soft excitations in the system. To assess whether or not this scenario is realized for Ising spin glass models in the elongated prism geometry, we represent in Fig. 5 the following quantity K(r) = − log C (1)(r) C (1)(r = 0) . (9) as a function of r/ξn=1 (the values of ξn=1 are in Table I). In order t...

  62. [73]

    Low energy excitations in a long prism geome try: computing the lowest critical dimension of the Ising spin glass

    9 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 L = 24 L = 16 L = 12 L = 8 L = 6 L = 4 E(r) r/ξ n=1(L) FIG. 6. Scaling of E(r), see Eq. ( 10). Data for systems of size L = 4 , 6, 8, 12, 16, and 24 at temperature T = 0 .7. The dashed line is a fit to f (x) = Ae−W x + c for the L = 24 data with x = r/ξn=1 > 0.02, returning A = 0 .1268(19), W = 18.1(3), c = 0.8833(3), χ2/DoF...

    2026

  63. [74]

    Its main advantage is that code changes are minimal. In the beginning, we tested three variants of this scheme that differ for the value of the coupling along the auxil- iary link: i) weak, the value of the coupling is the opposite of the value of the coupling along y, so that they cancel each other; ii) strong, the value of the coupling is the same as the...

  64. [75]

    It turns out that we can choose a safe distance for the bor- ders, zdiscard, such that if we choose the matrix indices z1 < z 2 with z1 > z discard and z2 < M − 1 − zdiscard, the 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 5 10 15 20 25 30 35 40 45 PBC z = 5 z = 15 z = 25 z = 35 C (1)(z, z+ r) r FIG. 5. Rows of the matrix C (1)(z1 = z, z2 = z + r) as a func...

  65. [76]

    , M − 2rmin − 1

    as C(rmin+r) = A e−rmin/ξ [ e−r/ξ + e−(M −2rmin−r)/ξ ] , (2) with r = 0 , 1, . . . , M − 2rmin − 1. We see that introduc- ing the short-distance cutoff rmin merely amounts to a redefinition of the constant amplitude and a shorter ef- fective length of the system. Hence, we can directly use the standard formulae that we recall next. Let us define M ∗ = M − 2r...

  66. [77]

    Therefore, the leading behavior is det C(r) = A e−(Eodd 1 +Eodd 2 +Eodd 3 ) r +

    with two coincident indices are zero, because the corresponding matrix columns are proportional. Therefore, the leading behavior is det C(r) = A e−(Eodd 1 +Eodd 2 +Eodd 3 ) r + . . . (17) where A is an amplitude. An analogous argument gives us the leading behavior of the 2 × 2 minors of det C(r) det C(i,j)(r) = A(i,j)e−(Eodd 1 +Eodd 2 ) r + . . . (18) Thi...

  67. [78]

    ξn=1;k=2/ξn=1 tends to a positive constant when L grows, and

  68. [79]

    This large ratio explains a posteriori the large accuracy, of a few percent, that we reached in the computation of ξn=1

    ξn=1;k=2 ≈ ξn=1/17. This large ratio explains a posteriori the large accuracy, of a few percent, that we reached in the computation of ξn=1. As soon as rmin becomes larger than ξn=1;k=2 (and this is easy, given the smallness of ξn=1;k=2), the contribution of this correction term becomes negligible as com- pared to the main contribution. V. THE 1D ISING-ED...

  69. [80]

    Therefore, Eq

    is independent of M . Therefore, Eq. ( 24) tells us that the spin-glass correlation functions in the M → ∞ limit display a simple exponential decay C 1D,(n)(r) ⏐ ⏐ ⏐ M =∞ = e −mn r , m n = 1 ξn = − log t2n . (26) [Mind that mp+1 > m p because t2p+2 < t 2p; t0 = 1 fol- lows from the normalization condition for the probabil- ity density P 1D(J )]. We shall ...

  70. [81]

    is recovered with PBC is the subject of Sect. V B. A. The 1D-scaling limit for an infinite chain Let us now consider how the correlation lengths ξn diverge as the inverse temperature β1D goes to ∞ (or, equivalently, T1D → 0 for the temperature). We shall rather work with the masses mn = 1 /ξn and rewrite Eq. (26) in a form more convenient to discuss the sc...

  71. [82]

    (42) Let us now consider the simplest spin-glass correlation function C 1D,(1)(r) = ⟨σ0σr⟩2 C 1D,(1)(r) ⏐ ⏐ ⏐ PBC = k2 + ˜k2 + 2k˜k (1 + k˜k)2

    for PBC is nicely expressed in terms of the following two statistically independent random variables: k = Π r z=1 tanh (β1DJz−1,z) , (40) ˜k = Π M −1 z=r tanh(β1DJz,z+1) , (41) ⟨σ0σd⟩PBC = k + ˜k 1 + k˜k . (42) Let us now consider the simplest spin-glass correlation function C 1D,(1)(r) = ⟨σ0σr⟩2 C 1D,(1)(r) ⏐ ⏐ ⏐ PBC = k2 + ˜k2 + 2k˜k (1 + k˜k)2 . (43) W...

  72. [83]

    ( 56) will tell us that the image term is simply Cimage(r) = Cdirect(M − r)

    with Eq. ( 56) will tell us that the image term is simply Cimage(r) = Cdirect(M − r). In fact, the image term is completely standard when work- ing with PBC. Instead, the constant term Cconst lacks an analog in our experience with non-disordered systems. The computation of the disorder average starts with the Taylor expansion 1 (1 + k˜k)2 = ∞∑ n=0 (−1)n(n...

  73. [84]

    Only for M = 192, namely M/ξn=1 ≈ 10.3 we find mutual consistency among the different rmin and the two integral estimators ξ0,1 and ξ1,2

    will cause undesir- able drifts as rmin is varied. Only for M = 192, namely M/ξn=1 ≈ 10.3 we find mutual consistency among the different rmin and the two integral estimators ξ0,1 and ξ1,2. We also observe from Fig. 2 in the main text that our data from prism’s size 16 ×16×512 and PBC are rmin independent ( M/ξn=1 ≈ 11.5 in that case). We conclude that, give...

  74. [85]

    We shall be computing R(M ) = ⟨Q(z1 = 0)Q(z2 = M − 1)⟩ ⟨[Q(z1 = 0)] 2⟩ , (61) = C (1)(0, M − 1) C (1)(0, 0) . (62) The transfer-matrix analysis of a finite- M correlation function is more involved than its M → ∞ limit, be- cause it has contributions from both even-parity and odd- parity states (see, e.g., Ref. [ 14] for an example worked out in detail). Th...

  75. [86]

    The coefficients a1, a2 and b1 are expected to be of order one

    stand for the contribu- tions of higher excited states. The coefficients a1, a2 and b1 are expected to be of order one. Hence, un- der the hypothesis that M Eeven 1 and M Eodd 2 are large- enough (larger than two would be enough in practice), it is meaningful to truncate the expansions to the order shown in Eq. (

  76. [87]

    and Taylor-expand in the supposedly small quantity b1e−M Eeven 1 , obtaining R(M ) = a1e−M Eodd 1 + a2e−M Eodd 2 − a1b1e −M ( Eodd 1 +Eeven 1 ) + . . . . (64) Eq. ( 64) is finally ready for the data analysis that we explain next. Take a set of R(M ), obtained in prisms of varying length M . Because each R(M ) comes from a different 10 simulation, they will ...

  77. [88]

    [ 15]. As we shall show in the following, the surface tension deeply modifies the results found in the main text, in the sense that the correlations along the longi- tudinal dimension of the prism have a correlation length that grows exponentially with LD−1 (recall that L is the prism’s transverse size), rather than with a power law in L. Let us start cons...

  78. [89]

    [ 13], makes the computation straightforward): Z 1D +− Z 1D +− = 1 − tanhL−1 κFM,1D 1 + tanhL−1 κFM,1D

    (the Transfer matrix, see e.g. [ 13], makes the computation straightforward): Z 1D +− Z 1D +− = 1 − tanhL−1 κFM,1D 1 + tanhL−1 κFM,1D . (79) Our matching condition for κFM,1D simply states that Y = Z 1D +− Z 1D ++ or tanh κFM,1D = ( 1 − Y 1 + Y ) 1 L−1 . (80) We are finally ready to go to the infinitely long prism M → ∞ for which it is well known that ξFM,1...

  79. [90]

    Barma and B

    M. Barma and B. Sriram Shastry, d-dimensional Hub- bard model as a (d + 1)-dimensional classical problem, Physics Letters A 61, 15–18 (1977)

    1977

  80. [91]

    Bernaschi, I

    M. Bernaschi, I. Gonz´ alez-Adalid Pemart ´ ın, V. Mart ´ ın- Mayor, and G. Parisi, The QISG suite: High-performance codes for studying quantum Ising spin glasses, Comp. Phys. Comm. 298, 109101 (2024)

    2024

Showing first 80 references.