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REVIEW 2 major objections 6 minor 58 references

Domain Growth in Long-range Ising Models with Disorder

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Pith's one-line read Long-range interactions do not rescue power-law domain growth in disordered Ising magnets: growth stays logarithmic in time, with a reduced exponent.

desk verdict Useful first 2D map of disorder+long-range coarsening, but the headline logarithmic-growth exponent rests on a collapse with a free length and a short scaling range. read the letter →

arxiv 2507.03154 v1 pith:UBHZW64W submitted 2025-07-03 cond-mat.stat-mech cond-mat.soft

Ramgopal Agrawal , Federico Corberi , Eugenio Lippiello , Sanjay Puri This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.soft
keywords long-rangeIsingmodelrandomfieldsquencheddisorderdomaingrowthcoarseningactivateddynamicslogarithmicdynamicalfreezing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether quenched random fields still force logarithmic domain growth in an Ising ferromagnet whose interactions decay as $J(r)\sim r^{-(D+\sigma)}$, instead of the usual power-law coarsening. The authors argue that in one dimension the answer is yes for every $\sigma>0$: domains grow as $R(t)\sim(\ln t)^{\alpha(\sigma)}$, and the measured exponents $\alpha(0.5)=1.31(5)$, $\alpha(0.9)=1.17(1)$, and $\alpha(1.5)=1.26(1)$ are all below the nearest-neighbor value $\alpha=2$. In two dimensions they identify a crossover governed by $\sigma$: for weak disorder ($\Delta\le T$) and $\sigma=3$, a scaling collapse of the effective exponent gives a barrier exponent $\psi\simeq 1.23$ and confirms logarithmic growth, whereas for $\sigma<1$ long-range interactions suppress disorder effects up to $R\simeq 100$. For strong disorder ($\Delta>T$) the dynamics becomes extremely slow and shows signs of dynamical freezing. The interest is that long-range drift accelerates early growth but does not restore power-law coarsening; it only changes the logarithmic-growth exponent.

What carries the argument

The load-bearing object is the disorder-barrier scaling ansatz used to identify logarithmic growth: in the activated regime the effective dynamic exponent satisfies $z_{\mathrm{eff}}-\bar z = a[R(t)/\ell(\Delta)]^{\psi(\sigma)}$, where $\bar z$ is the pre-asymptotic power-law exponent of the pure long-range model, $\ell(\Delta)$ is a disorder-dependent crossover length adjusted to collapse data for different $\Delta$, and $\psi$ is the barrier exponent controlling energy barriers $E_B\sim\Delta R^{\psi}$. A successful collapse under this form implies $R(t)\sim(\ln t)^{1/\psi}$ and converts the upward drift of $z_{\mathrm{eff}}(t)$ into a quantitative growth exponent. The reference values of $\bar z$ come from the Bray-Rutenberg laws and the pure long-range Ising model: $\bar z=1+\sigma$ for $\sigma<1$, $\bar z=2$ for $\sigma>1$, plus the $z=4/3$ zero-temperature regime in $D=2$. The diagnostic power of the method is precisely what distinguishes logarithmic growth from a plateau or freezing: logarithmic growth gives the power-law collapse, while the strong-disorder data resist any such collapse.

What would settle it

Simulate the two-dimensional RFLRIM at $T=0.1$, $\Delta=0.1$, and $\sigma=3$ on lattices large enough that $R(t)>500$; if the scaled effective exponent $z_{\mathrm{eff}}-2$ departs from $a(R/\ell)^\psi$ with $\psi\simeq 1.23$, or if $R(t)$ instead fits a power law over two decades, the claimed asymptotic logarithmic growth is falsified.

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Extended reading notes

Core claim

In the random-field long-range Ising model, $H=-\sum_{j<i} r_{ij}^{-(D+\sigma)}s_i s_j - \sum_i h_i s_i$ with Gaussian random fields of width $\Delta$, quenched to low temperature, the asymptotic domain-growth law remains the activated logarithmic form $R(t)\sim(\ln(t/\tau))^{\alpha(\sigma)}$ rather than a power law. The paper's new quantitative finding in $D=2$ is that for $\Delta\le T$ and $\sigma=3$ the effective exponent obeys $z_{\mathrm{eff}}-\bar z = a(R/\ell(\Delta))^\psi$ with $\psi\simeq 1.23$, giving a clean data collapse and hence logarithmic growth, while the same analysis fails for $\Delta>T$, where the sharp rise of $z_{\mathrm{eff}}$ with $R$ suggests dynamical freezing. In $D=1$ the paper reaffirms, following its earlier study, that logarithmic growth holds for all $\sigma>0$ with $\alpha(\sigma)<2$, so long-range interactions reduce the efficiency of activated coarsening relative to the nearest-neighbor case. In both dimensions the equal-time correlation function collapses onto a disorder-independent, superuniversal scaling function. The paper's overall claim is that the Huse-Henley activated mechanism survives long-range drift, but with a $\sigma$-dependent growth exponent and with a strong-disorder regime that may be dynamically frozen.

Load-bearing premise

The logarithmic-growth conclusion rests on assuming the scaling form $z_{\mathrm{eff}}-\bar z = a(R/\ell(\Delta))^\psi$, with a length scale chosen freely for each disorder strength to force the data collapse, and then identifying $\alpha=1/\psi$; if that assumed form is not the true asymptotic behavior, the extracted exponents do not establish logarithmic growth.

Editorial extensions

If this is right

  • In one dimension, no matter how long-ranged the interactions are ($\sigma>0$), quenched disorder wins asymptotically: coarsening is logarithmic, though the exponent $\alpha(\sigma)$ lies below the nearest-neighbor value $2$.
  • In two dimensions, once $\sigma$ is large enough (here $\sigma=3$), the same activated logarithmic regime becomes visible on accessible scales, with barrier exponent $\psi\simeq 1.23$, so the nearest-neighbor random-field phenomenology survives as a limit.
  • Small-$\sigma$ long-range coupling protects power-law growth against weak disorder up to $R\simeq 100$ in $L=2048$ systems; whether logarithmic growth eventually takes over beyond that scale is left open.
  • For disorder stronger than temperature ($\Delta>T$) across all studied $\sigma$, growth becomes extremely slow with no scaling collapse, and the paper interprets this as dynamical freezing rather than logarithmic coarsening.
  • Superuniversality holds: the scaled correlation function is independent of $\Delta$ for all $\sigma>0$ studied, so disorder affects only the rate of growth, not the morphology of domains.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One consequence the paper leaves implicit: if the two-dimensional $\psi\simeq 1.23$ result is asymptotic, then long-range interactions make activated growth faster in exponent ($\alpha=1/1.23\simeq 0.81$) than the nearest-neighbor random-field value ($\alpha\simeq 1/1.5$), opposite to the trend reported in one dimension.
  • The failure to collapse the strong-disorder data cannot by itself distinguish true freezing from logarithmic growth with a very long crossover; a direct measurement of the disorder-induced barrier-height distribution, or simulations reaching $R(t)\gg\ell$, would decide between those readings.
  • A testable consequence of the scaling ansatz is that two-time quantities such as the autocorrelation function should age with $\ln t$ rather than $t$ scaling, using the same barrier exponent; measuring them would independently confirm the activated mechanism.
  • The crossover length $\ell(\Delta)$ is treated as a fitting parameter; a theory predicting its dependence on $\Delta$, $\sigma$, and $T$ from the barrier distribution would turn this collapse method into a predictive scheme.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper reports Monte Carlo simulations of domain growth in the random-field long-range Ising model in one and two dimensions after a deep quench. It recapitulates the authors' earlier 1D result that growth is asymptotically logarithmic for all sigma > 0 with alpha(sigma) below the nearest-neighbor value alpha = 2, and presents new 2D simulations. For Delta <= T and sigma = 0.6, 0.9 the data show disorder-independent growth up to R <= 100, while for sigma = 3 a Lippiello-type collapse of the effective exponent is used to infer logarithmic growth with barrier exponent psi ~ 1.23. For Delta > T the dynamics is extremely slow and is interpreted as possible dynamical freezing. Dynamical scaling and superuniversality of C(r,t) are also checked.

Significance. If established, these results would extend the Huse-Henley activated-dynamics scenario to long-range disordered ferromagnets and would give quantitative guidance on when long-range interactions dominate disorder during coarsening. The paper is transparent about its main limitation for sigma < 1 in 2D, where only scales up to R ~ 100 are accessible, and it uses an Ewald-type summation for the long-range interactions. However, the central quantitative claims---alpha < 2 in 1D and psi ~ 1.23 for sigma = 3 in 2D---rest on a scaling collapse with a free per-disorder crossover length and are not supported by error bars or direct fits. The 2D sigma = 3 fit is especially fragile because it lies mostly in the crossover region rather than in the asymptotic activated regime.

major comments (2)
  1. [Sections 4-5, Eq. (13), Figs. 5 and 11] The exponents alpha(sigma) and psi are extracted by plotting zeff - zbar against R/ell(Delta), where ell(Delta) is chosen separately for each disorder strength to force data collapse, and then fitting zeff - zbar = a(R/ell)^psi. Because ell is a free parameter per curve and no quantitative collapse metric or uncertainty is reported, this test has limited ability to distinguish logarithmic growth from other slowly varying growth laws over the simulated window. In addition, in the asymptotic activated regime Eq. (13) is precisely the form implied by R ~ (ln t)^alpha with alpha = 1/psi, so the fit is not an independent confirmation of logarithmic growth. I ask the authors to report direct fits of R(t) to a logarithmic law for each Delta, to give bootstrap estimates of the exponents, and to show the fitted range explicitly.
  2. [Section 5, Fig. 11] The evidence for logarithmic growth at sigma = 3 in 2D is not yet asymptotic. With zbar = 2, the collapsed data extend only to zeff - zbar ~ 2, so the entire fitted region has zeff - zbar comparable to zbar rather than much larger than it; the asymptotic activated regime is not reached. Since the 2D results are averaged over only 25 runs and no error bars or confidence intervals are displayed, the quoted psi ~ 1.23 (versus the nearest-neighbor value ~ 1.5) is not statistically secured. Please either provide additional simulation data at larger L and longer times, or present a careful bootstrap or chi-square analysis of the collapse and fit range before claiming logarithmic growth.
minor comments (6)
  1. [Eq. (4)] The Metropolis transition rate contains an explicit factor N^{-1} in addition to min(1, exp(-Delta E/T)); since time is measured in Monte Carlo steps, please clarify whether this factor is intentional or a typographical error.
  2. [Section 3, Eq. (6)] The phrase 'Except in D = 1' before Eq. (6) appears to be a slip; Eq. (6) is the exact 1D Hurwitz-zeta expression, while Ewald summation is used in D > 1.
  3. [Section 5, Fig. 10] For sigma = 0.6 the text says growth 'initially follows the BR regime, characterized by zeff ~ 1 + sigma, and gradually approaches the asymptotic value z = 4/3,' but the caption of Fig. 10 and the discussion in Section 5 label z = 4/3 as the pre-asymptotic universal law and z = 1 + sigma as the BR asymptotic law; the text and caption should be made consistent.
  4. [Sections 1 and 6] The manuscript is referred to as a 'review' in Section 1 and Section 6, but it presents original numerical results; please adjust the wording to avoid confusion.
  5. [Section 6] The limitation stated in Section 6---that for sigma < 1 in 2D whether true logarithmic growth sets in at larger length scales up to R ~ 500 remains an open question---is an important caveat and should also be reflected in the abstract, which currently states only that the 2D dynamics is 'more complex.'
  6. [Sections 3-5, Figs. 3-15] No error bars are given for R(t) or zeff(t) in any figure; given the small number of disorder realizations in 2D, a brief statement of the statistical uncertainty would help the reader judge the collapse quality.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the logarithmic-growth inference is an explicitly stated empirical hypothesis tested by data collapse, with reported failure cases; the cited self-work supplies method and prior data, not a forced conclusion.

full rationale

The paper's central quantitative claims are the 1D result alpha(sigma)<2 and the 2D sigma=3 evidence for logarithmic growth. Both rest on the scaling analysis of Sec. 4: one plots z_eff-zbar vs R/ell(Delta) with ell(Delta) chosen per disorder strength to obtain a collapse, fits Eq. (13), z_eff-zbar = a(R/ell)^psi, and sets alpha=1/psi. This is a fit of a theoretically motivated master curve, not a derivation from first principles, and it is presented as a 'working hypothesis' rather than as a consequence of the model. The inference has empirical content because the collapse can fail: for Delta>T the paper reports 'no collapse could be obtained – suggesting instead a dynamical freezing' (Sec. 5, Fig. 12), and for sigma<1 it states 'no conclusive evidence of asymptotic logarithmic growth can be established' (Sec. 5). The relation alpha=1/psi follows from the assumed R~(ln t)^alpha in the activated regime, so alpha is a reparametrization of the fitted psi rather than an independent prediction; this is a standard consistency test, not a definitional identity. The method is cited to Lippiello et al. [30,57,58], which includes the present authors, and the 1D section cites Ref. [39] as its predecessor, but the relevant figures and fits are reproduced in this paper, and no uniqueness theorem or citational authority is used to forbid alternative growth laws. The free per-disorder length ell(Delta) weakens the discriminative power of the collapse and deserves a robustness check, but that is a statistical/correctness concern, not circularity: none of the paper's equations reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central quantitative claims (logarithmic growth exponents) are extracted by fitting to an assumed scaling form with per-disorder free lengths; the model, interactions, and disorder distribution are standard and not invented. No new entities are postulated.

free parameters (3)
  • 1D growth exponent α(σ) = α(0.5)=1.31(5), α(0.9)=1.17(1), α(1.5)=1.26(1)
    Extracted by fitting the collapsed effective-exponent data with zeff − zbar = a(R/ℓ)^ψ; the claim of logarithmic growth depends on these fitted exponents.
  • 2D barrier exponent ψ for σ=3 = ψ ≈ 1.23
    Power-law fit to the collapsed data in Fig. 11; used to confirm logarithmic growth with α = 1/ψ.
  • Crossover length ℓ(Δ) = not listed numerically; one value per Δ chosen for collapse
    For each disorder strength, ℓ(Δ) is adjusted to obtain data collapse in Figs. 5, 11, and 14. This is a free parameter that enables the extraction of ψ.
assumptions (5)
  • domain assumption Dynamical scaling of the correlation function, C(r,t) = f(r/R(t)).
    Used to define R(t) and to analyze super-universality; assumed for all disorder strengths and times.
  • domain assumption In the crossover to activated growth, zeff − zbar is a power-law function of R/ℓ(Δ).
    Working hypothesis from Lippiello et al.; the logarithmic-growth conclusion depends on this scaling form.
  • domain assumption Energy barriers scale as E_B ∼ Δ R^ψ (Huse-Henley picture).
    Motivates the activated dynamics and the interpretation of ψ as a barrier exponent.
  • standard math The infinite-range periodic interactions are correctly computed by the Ewald/Hurwitz zeta summation of Refs. [35,51,52].
    The numerical evolution relies on this implementation; errors here would affect all results.
  • domain assumption The early-time growth exponents are the Bray-Rutenberg values for the pure system (z=1+σ for σ<1, z=2 for σ>1, z=4/3 for T=0 in 2D).
    Used as zbar in the scaling collapse; assumes disorder does not modify the pre-asymptotic exponent.

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Cite this review

Pith. "Pith review of Domain Growth in Long-range Ising Models with Disorder." pith.science (2026). https://pith.science/paper/UBHZW64W

@misc{pith2026250703154,
  author       = {Pith},
  title        = {Pith review of: Domain Growth in Long-range Ising Models with Disorder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UBHZW64W}},
  note         = {Machine review of arXiv:2507.03154}
}
abstract

Recent advances have highlighted the rich low-temperature kinetics of the long-range Ising model (LRIM). This study investigates domain growth in an LRIM with quenched disorder, following a deep low-temperature quench. Specifically, we consider an Ising model with interactions that decay as $J(r) \sim r^{-(D+\sigma)}$, where $D$ is the spatial dimension and $\sigma > 0$ is the power-law exponent. The quenched disorder is introduced via random pinning fields at each lattice site. For nearest-neighbor models, we expect that domain growth during activated dynamics is logarithmic in nature: $R(t) \sim (\ln t)^{\alpha}$, with growth exponent $\alpha >0$. Here, we examine how long-range interactions influence domain growth with disorder in dimensions $D = 1$ and $D = 2$. In $D = 1$, logarithmic growth is found to persist for various $\sigma > 0$. However, in $D = 2$, the dynamics is more complex due to the non-trivial interplay between extended interactions, disorder, and thermal fluctuations.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.