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arxiv: 2605.07753 · v1 · submitted 2026-05-08 · 🪐 quant-ph · cond-mat.stat-mech

Universal Symmetry-Breaking Dynamics at Continuous Phase Transitions: Evidence for a New Dynamical Critical Exponent

Tobias Wiener , Laurin Brunner , Markus Heyl This is my paper

Pith reviewed 2026-05-11 03:29 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords Ising modeldynamical critical exponentsymmetry breaking quenchphase transitionuniversal scalingfar from equilibriumorder parameter fluctuationscritical dynamics
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The pith

Order-parameter fluctuations in Ising models collapse temporally after symmetry-breaking quenches via a new dynamical critical exponent.

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

The paper examines the dynamics following a sudden quench that breaks symmetry at a continuous phase transition in Ising models. It finds that the fluctuations of the order parameter exhibit a collapse when plotted against a rescaled time variable across different system sizes and quench strengths. This collapse points to an emergent scaling form that can be explained by a previously unknown dynamical critical exponent. The scaling holds in two-dimensional quantum and three- and four-dimensional classical Ising models but not in lower dimensions, suggesting a lower critical dimension for this universal behavior. This opens possibilities for understanding far-from-equilibrium universal dynamics in a broader class of systems.

Core claim

Following a sudden symmetry-breaking quench at continuous phase transitions, the order-parameter fluctuations in Ising models display a compelling temporal collapse indicative of a single-variable scaling form. This is accounted for by introducing a new dynamical critical exponent. The universal regime appears in the 2D quantum Ising model and the 3D and 4D classical Ising models, but is absent in the 1D quantum and 2D classical cases, consistent with a lower critical effective dimension.

What carries the argument

The emergent single-variable scaling form for order-parameter fluctuations after the quench, carried by a new dynamical critical exponent.

If this is right

  • The universal scaling is observed only above a lower critical effective dimension.
  • It may characterize systems with non-conserved order parameters more generally.
  • New avenues exist for exploring universal dynamics in theoretical models and experimental platforms.
  • Finite-size and quench-strength independence suggests robustness of the scaling.
  • The breakdown in lower dimensions confirms the critical dimension cutoff.

Where Pith is reading between the lines

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

  • Similar collapses might be testable in other non-conserved order parameter systems like the Heisenberg model.
  • Experimental platforms such as ultracold atoms could observe this scaling directly.
  • If confirmed, this exponent could be used to classify dynamical universality classes beyond equilibrium critical exponents.
  • Extensions to other quench protocols or conserved order parameters might reveal related phenomena.

Load-bearing premise

The numerical temporal collapse is caused by a new universal dynamical exponent rather than by model-specific finite-size effects or choices in the scaling analysis.

What would settle it

Performing larger-scale simulations or experiments in the 2D classical Ising model that still fail to show the collapse, or finding that the required exponent varies with quench details, would falsify the universality claim.

Figures

Figures reproduced from arXiv: 2605.07753 by Laurin Brunner, Markus Heyl, Tobias Wiener.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original abstract

Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.

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

2 major / 2 minor

Summary. The paper claims that after a sudden symmetry-breaking quench through a continuous phase transition, order-parameter fluctuations in Ising models exhibit a temporal collapse onto a single-variable scaling form across system sizes and quench strengths. This is observed numerically in 2D quantum and 3D/4D classical Ising models (but not 1D quantum or 2D classical), and is explained by introducing a previously unknown dynamical critical exponent, with evidence for a lower critical effective dimension.

Significance. If the central claim holds, the identification of a new universal far-from-equilibrium scaling regime at continuous phase transitions would be significant for non-equilibrium statistical mechanics and quantum dynamics. The numerical evidence of collapse in multiple Ising variants provides a concrete starting point for exploring universal dynamics beyond equilibrium critical phenomena, with potential relevance to experiments in ultracold atoms or quantum simulators.

major comments (2)
  1. [Abstract and main results section] Abstract and scaling analysis: The new dynamical critical exponent is introduced specifically to account for the observed single-variable collapse of order-parameter fluctuations, but no independent derivation, renormalization-group analysis, or relation to known static/dynamical exponents is provided; its value appears determined by fitting the same numerical data used to demonstrate the collapse.
  2. [Abstract] Abstract: The lower critical effective dimension (failure of collapse in 1D quantum and 2D classical Ising models) is diagnosed from the same set of simulations where the scaling is tested, without a prior theoretical prediction or field-theoretic argument for why the regime begins at effective dimension 3.
minor comments (2)
  1. The manuscript would benefit from explicit discussion of possible alternative explanations for the collapse, such as finite-size corrections or quench-protocol specifics, even if they are ultimately ruled out.
  2. Notation for the scaling variable and the new exponent should be introduced with a clear equation early in the text to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the potential significance of our work and for the constructive comments. We respond to each major comment below, indicating where revisions will be made to clarify the phenomenological nature of our findings.

read point-by-point responses

  1. Referee: [Abstract and main results section] Abstract and scaling analysis: The new dynamical critical exponent is introduced specifically to account for the observed single-variable collapse of order-parameter fluctuations, but no independent derivation, renormalization-group analysis, or relation to known static/dynamical exponents is provided; its value appears determined by fitting the same numerical data used to demonstrate the collapse.

    Authors: We acknowledge that the dynamical critical exponent is introduced phenomenologically based on the observed collapse and is fitted from the numerical data. The primary evidence for its relevance is the robust single-variable scaling that holds across independent models (2D quantum, 3D and 4D classical Ising) with the same exponent value. This cross-model consistency is non-trivial and supports the claim of universality, even without a microscopic derivation. No renormalization-group analysis is provided because the phenomenon is far from equilibrium and lacks an established theoretical framework. In the revised manuscript we will expand the discussion section to explicitly state the phenomenological basis, note the absence of an independent derivation, and outline possible connections to known exponents while emphasizing that a full theoretical treatment remains an open question. revision: partial

  2. Referee: [Abstract] Abstract: The lower critical effective dimension (failure of collapse in 1D quantum and 2D classical Ising models) is diagnosed from the same set of simulations where the scaling is tested, without a prior theoretical prediction or field-theoretic argument for why the regime begins at effective dimension 3.

    Authors: The lower critical effective dimension is identified empirically from the numerical observation that the collapse fails in the 1D quantum and 2D classical cases while succeeding in higher-dimensional analogs. We do not present a prior field-theoretic prediction; the finding is reported as a numerical result suggesting that the universal regime requires a minimum effective dimensionality. In the revised version we will modify the abstract and main text to make this empirical character explicit, remove any implication of a theoretical prediction, and add a statement calling for future theoretical work to explain the origin of the lower critical dimension of 3. revision: partial

Circularity Check

1 steps flagged

New dynamical critical exponent fitted to observed collapse without independent derivation; lower-critical-dimension cutoff diagnosed from same data

specific steps
  1. fitted input called prediction [Abstract]
    "Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases."

    The single-variable scaling form is presented as evidence for a new exponent, yet the exponent is defined and its value chosen to produce the collapse in the same numerical data. The lower-critical-dimension cutoff is likewise extracted from the identical simulations where collapse is or is not observed, so both the exponent and the universality claim are statistically forced by the fitting procedure rather than independently derived or predicted.

full rationale

The paper's central claim is that order-parameter fluctuations exhibit a temporal collapse indicative of a single-variable scaling form, which is explained by a new dynamical critical exponent. This exponent is introduced specifically to account for the collapse observed in numerical simulations across system sizes and quenches. The lower critical effective dimension is identified by where collapse succeeds or fails in the identical set of simulations (2D quantum/3D-4D classical yes; 1D quantum/2D classical no). No renormalization-group derivation, relation to known exponents, or first-principles prediction of the exponent value is provided; the scaling variable is adjusted to produce the reported collapse. This makes the 'universal' scaling form and the new exponent reduce to a post-hoc fit to the data they are meant to describe, satisfying the fitted-input-called-prediction pattern.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on numerical observations of collapse in Ising models and the postulation of a new exponent whose value is not independently derived or predicted.

free parameters (1)
  • new dynamical critical exponent
    Introduced to produce the single-variable scaling form that collapses the fluctuation data; its numerical value is determined from the simulations themselves.
axioms (1)
  • domain assumption Standard Ising Hamiltonian and dynamics (quantum or classical) govern the post-quench evolution
    Assumed throughout the simulations of the 1D/2D quantum and 2D/3D/4D classical cases.
invented entities (1)
  • new dynamical critical exponent no independent evidence
    purpose: To explain the emergent single-variable scaling of order-parameter fluctuations after the quench
    No independent falsifiable prediction or external measurement is provided; the entity is defined by its ability to collapse the observed data.

pith-pipeline@v0.9.0 · 5479 in / 1445 out tokens · 45476 ms · 2026-05-11T03:29:43.347356+00:00 · methodology

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