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arxiv: 2605.00567 · v1 · submitted 2026-05-01 · ❄️ cond-mat.stat-mech

Fixed points and crossovers for the hysteresis scaling of dynamic mean-field models

Jiapeng Yang , Fan Zhong This is my paper

Pith reviewed 2026-05-09 18:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords hysteresis scalingfirst-order phase transitionsmean-field modelsrenormalization groupuniversality classesfixed pointsdriving ratesdynamic scaling
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The pith

A single driven mean-field quartic model exhibits several distinct universality classes in hysteresis scaling, each controlled by its own renormalization-group fixed point.

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

The paper establishes that hysteresis scaling in a dynamic mean-field quartic model driven through a first-order phase transition depends on driving rate magnitude and the presence or absence of noise, producing multiple universality classes with their own universal exponents. Each class is governed by a corresponding fixed point identified through renormalization-group scaling analysis, including a previously unreported exponent for fast driving that originates in critical phenomena. A sympathetic reader cares because this supplies a systematic classification for scaling behaviors in discontinuous transitions, where such universality was previously thought to be absent or limited to a single cubic fixed point. The analysis also yields crossovers between regimes and full curve collapse, all confirmed by direct numerical integration of the model equations.

Core claim

We discover a new exponent for large driving rates arising from critical phenomena and show that, depending on the magnitude of the driving rates and on the absence or presence of noise, the same mean-field model exhibits several universality classes with definite universal scaling exponents governed by their corresponding fixed points through a systematic scaling analysis based on renormalization group theory. The theories and their various crossovers between different fixed points along with complete universal scaling of full curve collapse are verified by numerical results.

What carries the argument

Renormalization-group scaling analysis of the driven dynamic mean-field quartic model, which locates the fixed points that dictate the distinct scaling regimes for hysteresis under different driving rates and noise conditions.

If this is right

  • Hysteresis curves for any driving rate and noise level collapse onto universal forms once the appropriate fixed-point scaling is applied.
  • At large driving rates without noise the exponents are those of critical phenomena rather than the usual first-order fixed point.
  • Smooth crossovers connect the different scaling regimes as driving rate or noise strength is varied continuously.
  • Full numerical verification of curve collapse confirms the fixed-point predictions across all identified classes.

Where Pith is reading between the lines

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

  • The same fixed-point structure may appear in experimental first-order transitions if the effective dynamics remain close to mean-field quartic form.
  • In lower dimensions spatial fluctuations could destabilize the mean-field fixed points and produce new exponents not captured here.
  • The scaling framework offers a template for analyzing hysteresis in non-equilibrium systems outside physics, such as abrupt shifts in ecological or economic models.
  • Measuring how the area or width of observed hysteresis loops scales with sweep rate in a noisy physical system would test for the predicted crossovers.

Load-bearing premise

The dynamic mean-field quartic model together with its renormalization-group treatment captures the dominant physics of hysteresis without substantial corrections from spatial fluctuations or higher-order terms.

What would settle it

Numerical solution of the model stochastic differential equation for a sequence of driving rates, both with and without additive noise, must produce families of hysteresis loops that collapse onto master curves when rescaled with the predicted exponents and crossover functions; absence of collapse or inconsistent exponents would refute the classification.

Figures

Figures reproduced from arXiv: 2605.00567 by Fan Zhong, Jiapeng Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Coercivity view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Magnetization curves for a series of view at source ↗
read the original abstract

Phase transitions are divided into first-order phase transitions and continuous ones in current classification. While the latter shows striking phenomena of scaling and universality, the former is generically characterized by discontinuous jumps in extensive variables and pronounced hysteresis. Recent studies have demonstrated universal scaling behavior controlled by a cubic fixed point in first-order phase transitions. However, more recent investigations into the hysteresis in a dynamic mean-field quartic model driven through its first-order phase transitions have revealed new scaling exponents for different driving rates. Here, we discover a new exponent for large driving rates arising surprisingly from critical phenomena and show that, depending on the magnitude of the driving rates and on the absence or presence of noise, the same mean-field model remarkably exhibits several universality classes with definite universal scaling exponents governed by their corresponding fixed points through a systematic scaling analysis based on renormalization group theory. The theories and their various crossovers between different fixed points along with complete universal scaling of full curve collapse are verified by numerical results. This further confirms universal scaling in first-order phase transitions.

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 / 2 minor

Summary. The manuscript analyzes hysteresis scaling in a dynamic mean-field quartic model driven through first-order phase transitions. It uses renormalization-group methods to identify multiple fixed points that govern distinct universality classes depending on driving-rate magnitude and the presence or absence of noise; a new exponent is reported for the large-driving-rate regime, with crossovers between classes and full curve-collapse scaling verified numerically.

Significance. If the central RG analysis and numerical verifications hold, the work would establish a systematic framework for multiple scaling regimes in first-order hysteresis, extending beyond the cubic fixed point and providing concrete exponents and crossover predictions that could be tested in other mean-field and low-dimensional systems.

major comments (3)
  1. [RG scaling analysis] The dynamic mean-field model employs a quartic Landau potential with negative quartic coefficient to produce the first-order jump. This potential is unbounded below, yet the scaling analysis does not compute the scaling dimension of a stabilizing sixth-order term at any of the claimed fixed points (including the large-driving-rate critical fixed point) or demonstrate its irrelevance. This leaves the truncation assumption unsecured for the reported universality classes.
  2. [Fixed-point analysis] The abstract and introduction state that the new large-driving-rate exponent 'arises surprisingly from critical phenomena,' but the manuscript does not provide the explicit beta-function fixed-point solution or eigenvalue spectrum that isolates this exponent independently of the numerical data. Without this derivation, the claim that the exponent is governed by a distinct fixed point rather than fitted remains unverified.
  3. [Numerical results] Numerical verification of curve collapse and crossovers is asserted, but the text supplies neither the precise data-exclusion criteria nor the error-analysis protocol used to confirm that the observed collapses are not post-hoc adjustments. This weakens the support for the multiple-universality-class claim.
minor comments (2)
  1. Notation for the driving-rate parameter and noise strength is introduced without a consolidated table of symbols; readers must hunt through the text to recall definitions when comparing regimes.
  2. Figure captions for the scaling collapses should explicitly state the range of driving rates and noise amplitudes used in each panel to facilitate direct comparison with the RG crossover predictions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments, which have helped us identify areas where the manuscript can be strengthened. We address each major comment point by point below, indicating the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [RG scaling analysis] The dynamic mean-field model employs a quartic Landau potential with negative quartic coefficient to produce the first-order jump. This potential is unbounded below, yet the scaling analysis does not compute the scaling dimension of a stabilizing sixth-order term at any of the claimed fixed points (including the large-driving-rate critical fixed point) or demonstrate its irrelevance. This leaves the truncation assumption unsecured for the reported universality classes.

    Authors: We agree that the quartic potential with negative coefficient is formally unbounded below and that a sixth-order stabilizing term is required for global stability. Our RG analysis focused on the relevant scaling operators near the fixed points of interest, but we did not explicitly evaluate the scaling dimension of the sixth-order term. In the revised manuscript we will add this calculation for each fixed point (including the large-driving-rate critical fixed point), demonstrating that the sixth-order operator is irrelevant at the reported fixed points and thereby securing the truncation. revision: yes

  2. Referee: [Fixed-point analysis] The abstract and introduction state that the new large-driving-rate exponent 'arises surprisingly from critical phenomena,' but the manuscript does not provide the explicit beta-function fixed-point solution or eigenvalue spectrum that isolates this exponent independently of the numerical data. Without this derivation, the claim that the exponent is governed by a distinct fixed point rather than fitted remains unverified.

    Authors: The beta functions for the dynamic mean-field model are derived in the manuscript, and the fixed points are obtained by solving the RG flow equations. To make the identification of the large-driving-rate fixed point fully explicit and independent of the numerical data, we will include in the revision the complete set of beta-function equations, the analytic fixed-point solutions, and the associated eigenvalue spectrum that isolates the new exponent. revision: yes

  3. Referee: [Numerical results] Numerical verification of curve collapse and crossovers is asserted, but the text supplies neither the precise data-exclusion criteria nor the error-analysis protocol used to confirm that the observed collapses are not post-hoc adjustments. This weakens the support for the multiple-universality-class claim.

    Authors: We acknowledge that the numerical section would benefit from greater transparency. In the revised manuscript we will specify the exact data-exclusion criteria (ranges of driving rates, system sizes, and noise strengths retained for the scaling analysis), the fitting procedures employed to extract exponents, and the error-analysis protocol, including bootstrap or jackknife estimates of uncertainties in the collapse quality, to demonstrate that the reported universal curves are robust. revision: yes

Circularity Check

0 steps flagged

RG scaling analysis derives fixed points independently of numerics

full rationale

The paper applies standard renormalization-group methods to the dynamic mean-field quartic model, locating fixed points that govern hysteresis scaling for varying drive rates and noise levels. Exponents and crossovers are obtained from the RG flow equations and stability analysis; numerical curve collapses serve solely as post-derivation verification. No quoted step reduces a claimed prediction to a fitted input, self-citation, or definitional tautology. The derivation chain therefore remains self-contained against external RG benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of renormalization group theory to the driven dynamic mean-field model and the assumption that fixed points control the observed scaling behaviors.

axioms (1)
  • domain assumption Renormalization group theory can be applied to identify fixed points and scaling in the dynamic mean-field quartic model for hysteresis
    The paper performs a systematic scaling analysis based on RG theory to find the governing fixed points.

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Reference graph

Works this paper leans on

90 extracted references · 90 canonical work pages

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    This gives rise to the exact mean-field 2 / 3 hysteresis exponent [ 16]

    can be analytically solved by Airy’s functions whose argument is propor- tional to R− 2/ 3h for h = Rt. This gives rise to the exact mean-field 2 / 3 hysteresis exponent [ 16]. To see how this and other exponents emerge generally, we make a scale transformation to the dynamic equation, Eq. (

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    This is to change every quantity O to O′ through O = O′b− [O] for a scaling factor b, where we have utilized the square brackets to denote the scale dimension

    [ 5, 43, 58]. This is to change every quantity O to O′ through O = O′b− [O] for a scaling factor b, where we have utilized the square brackets to denote the scale dimension. To keep the transformed dynamic equation identical with the original one, one must have [ t] = − [a2], [H] = [ φ] + [ a2], [ ζ] = [ φ] − [t]/ 2 = [ σ ]/ 2, and [ an] = [a2] − (n − 2)[...

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    ( 8)) is an externally imposed finite driving timescale, in direct analogy with the finite system size in finite-size scaling

    is referred to as an FTS form is that R[t]/r = R− z/r (derived from tR− [t]/ [r] by substi- tuting b = R− 1/r into Eq. ( 8)) is an externally imposed finite driving timescale, in direct analogy with the finite system size in finite-size scaling. This controllable finite timescale allows one to circumvent critical slowing down, just as the finite size allows on...

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    The dynamic equation reduces to dφ/dt = − a2φ + H + ζ

    in which the highest order term needed is a2 corresponding to a quadratic φ 2 model. The dynamic equation reduces to dφ/dt = − a2φ + H + ζ. (16) Indeed, one sees from Eq. ( 11) that a4 is irrelevant and becomes negligible for R → 0. A normal situation this model describes is a paramagnet (the disordered phase) for a2 > 0 or T > T c, the Curie temperature....

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    for H = Rt. Another situation in which this model is applicable occurs when the noise amplitude σ is large enough so that the system makes a transition to a regime in which it can freely jump from one potential well to the other for a2 < 0 [ 28]. As a result, the averaged magnetization is so small that the quadratic φ 2 term is sufficient in this high-noise...

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    ( 4), and twice as large as the coefficient of the original quadratic φ 2 model, Eq

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    As constructed in Eq

    for the cubic φ 3 theory where a3 is marginal. As constructed in Eq. ( 4), this theory characterizes the behavior of the transition in the vicinity of the spinodal point ( Hs, M s) within the noiseless quartic φ 4 model. For the corre- sponding noisy model, the transition can occur prior to reaching Hs due to noise activation. Crucially, any finite noise c...

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    This indicates that the observed noise-strength dependent hysteresis expo- nents [ 57] are effective, because contributions from scaled variables other than hR− 2/ 3 in Eq

    [ 58]. This indicates that the observed noise-strength dependent hysteresis expo- nents [ 57] are effective, because contributions from scaled variables other than hR− 2/ 3 in Eq. ( 12) or ( 15) were not considered. Additionally, the 2 / 3 hysteresis expo- nent has also been observed in the intermediate R range by expanding the dynamic equation, Eq. ( 2), ...

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    For very large R, M is large and hence the highest order term a∞ is dominant, which in practice may be approximated by using a very large n

    in which a∞ is marginal. For very large R, M is large and hence the highest order term a∞ is dominant, which in practice may be approximated by using a very large n. The leading hysteresis exponent ought to be 1 / 2. How- ever, for the φ 4 theory, the highest-order term is only a4. As a result, the FTS of the hysteresis must be described by Eq. (

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    We see that different R ranges may be controlled by different fixed points

    and the leading hysteresis exponent must be 3/ 5 instead. We see that different R ranges may be controlled by different fixed points. This indicates that crossovers be- tween different regimes—controlled by their respective fixed points—are inevitable over a wide range of R values. For a noiseless quartic φ 4 model, two distinct regimes ex- ist: one as R → 0, ...

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    ( 14), and in Eqs.( 20) and ( 21) two different representations for n = 3

    the description employed n = ∞ , Eq. ( 14), and in Eqs.( 20) and ( 21) two different representations for n = 3. In fact, they can be consolidated into HR − 2 3 vs a2 a4 R− 2 3 →      ( a2 a4 R− 2 3 ) 1 , H → Hs, R → 0, ( a2 a4 R− 2 3 ) 1 10 , H → R 3 5 , R → ∞ . (24) In all the above representations, the powers of the paren- theses are termed crossove...

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