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arxiv: 2503.16796 · v1 · submitted 2025-03-21 · ❄️ cond-mat.str-el

Finite-time scaling with two characteristic time scales: Driven critical dynamics with emergent symmetry

Yu-Rong Shu , Li-Ying Yang , Shuai Yin This is my paper

Pith reviewed 2026-05-22 23:26 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords finite-time scalingdriven critical dynamicsemergent symmetryclock modelKibble-Zurek mechanismdangerously irrelevant variableangular order parameter
0
0 comments X

The pith

In clock models with emergent symmetry, the angular order parameter switches between two distinct finite-time scaling regimes as drive speed changes.

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

The paper studies driven critical dynamics in a three-dimensional q-state clock model where the ordered phase breaks discrete Z_q symmetry but an emergent U(1) symmetry appears at criticality. It finds two finite-time scaling regions with time scales ζ_d proportional to v to the power -z/r and ζ_d' proportional to v to the power -z/r', the second arising from the dangerously irrelevant scaling variable. The squared order parameter M squared follows the usual finite-time scaling form, but the angular order parameter φ_q is dominated by the first time scale at small drive velocities v and by the second at large v. A sympathetic reader would care because the result extends the Kibble-Zurek mechanism to cases with emergent symmetries and shows how an extra time scale becomes visible in the dynamics. The same two-regime behavior holds for both isotropic and anisotropic couplings.

Core claim

Traversing the critical point by increasing temperature at finite velocity v produces two finite-time scaling regions with characteristic time scales ζ_d ∝ v^{-z/r} and ζ_d' ∝ v^{-z/r'}, where r' is the exponent tied to the dangerously irrelevant scaling variable. While M² obeys the conventional finite-time scaling form, the angular order parameter φ_q is controlled by ζ_d for small v and by ζ_d' for large v. The scaling properties remain the same in models with isotropic and anisotropic couplings.

What carries the argument

The angular order parameter φ_q together with the two driving-induced time scales ζ_d and ζ_d' generated by the usual scaling exponent r and the additional exponent r' of the dangerously irrelevant variable.

If this is right

  • M² always follows the standard finite-time scaling form across all drive speeds.
  • φ_q is governed by the time scale ζ_d in the slow-drive regime.
  • φ_q is governed by the time scale ζ_d' in the fast-drive regime.
  • The two-regime scaling holds identically for isotropic and anisotropic couplings.
  • The results supply a basis for experiments in hexagonal RMnO₃ materials.

Where Pith is reading between the lines

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

  • Dangerously irrelevant variables can generate measurable dynamic time scales under finite-time driving.
  • The same two-scale structure may appear in other models where discrete symmetry breaking gives way to emergent continuous symmetry.
  • Dynamic experiments could be used to extract the value of the new exponent r' directly.

Load-bearing premise

The dangerously irrelevant scaling variable generates a new independent exponent r' that produces a separate observable time scale ζ_d' cleanly separable from the usual one in the driven dynamics.

What would settle it

A measurement showing that the effective power-law dependence of φ_q on drive velocity v changes from the exponent set by -z/r to the one set by -z/r' when v crosses from small to large values.

Figures

Figures reproduced from arXiv: 2503.16796 by Li-Ying Yang, Shuai Yin, Yu-Rong Shu.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of heating critical dynamics with emer [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Heating critical dynamics of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dynamic scaling of the angular order parameter [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Crossover scaling property of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Dynamic scaling of the angular order parameter [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Dynamic scaling of the angular order parameter [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Crossover scaling property of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in a three-dimensional $q$-state clock model, in which the ordered phase breaks the $Z_q$ discrete symmetry, while an emergent $U(1)$ symmetry appears at the critical point. By increasing the temperature at a finite velocity $v$ to traverse the critical point from the ordered phase, we uncover rich dynamic scaling properties beyond the celebrated Kibble-Zurek mechanism. Our findings reveal the existence of two finite-time scaling (FTS) regions, characterized by two driving-induced time scales $\zeta_d\propto v^{-z/r}$ and $\zeta_d'\propto v^{-z/r'}$, respectively. Here $z$ is the dynamic exponent, $r$ is the usual critical exponent of $v$, and $r'$ represents an additional critical exponent of $v$ associated with the dangerously irrelevant scaling variable. While the square of the order parameter $M^2$ obeys the usual FTS form, the angular order parameter $\phi_q$ shows remarkably distinct scaling behaviors controlled by both FTS regions. For small $v$, $\phi_q$ is dominated by the time scale $\zeta_d$, whereas for large $v$, $\phi_q$ is governed by the second time scale $\zeta_d'$. We verify the universality of these scaling properties in models with both isotropic and anisotropic couplings. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal RMnO$_3$ (R=rare earth) materials.

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 examines driven critical dynamics in the three-dimensional q-state clock model, where the ordered phase breaks Z_q symmetry but an emergent U(1) symmetry appears at criticality. By ramping temperature across the critical point at finite velocity v, the authors identify two distinct finite-time scaling (FTS) regimes controlled by time scales ζ_d ∝ v^{-z/r} and ζ_d' ∝ v^{-z/r'}, with the second exponent r' arising from a dangerously irrelevant scaling variable. While M² obeys conventional FTS, the angular order parameter φ_q is reported to cross over from ζ_d-dominated scaling at small v to ζ_d'-dominated scaling at large v. The universality of these features is claimed to hold for both isotropic and anisotropic couplings, with suggested relevance to hexagonal RMnO₃ materials.

Significance. If the separation into two cleanly distinguishable FTS regimes for φ_q is robustly established, the work would extend Kibble-Zurek scaling to systems with emergent continuous symmetries and dangerously irrelevant operators, providing a concrete mechanism by which an additional relevant exponent r' controls observable dynamics. This could open new avenues for interpreting nonequilibrium experiments in materials with similar symmetry structures.

major comments (3)
  1. [Numerical results / scaling collapse figures (around the discussion of φ_q vs. v)] The central claim that φ_q exhibits two separable FTS regimes (small-v controlled by r, large-v controlled by r') is load-bearing for the paper's novelty. The manuscript must demonstrate explicitly that the large-v data for φ_q cannot be collapsed by the standard scaling function of r together with analytic corrections in v; without such a test (e.g., via attempted single-r collapse or residual analysis), the identification of an independent r' remains an assumption rather than a derived result.
  2. [Theoretical framework / scaling ansatz section] The derivation of how the dangerously irrelevant operator generates a distinct relevant eigenvalue r' that couples differently to the angular operator φ_q than to M² is not shown in detail. The abstract and main text present r' as producing an independent time scale ζ_d', but the RG flow or scaling-function construction that isolates this effect from ordinary corrections should be provided, especially since dangerously irrelevant variables typically enter through sub-leading corrections rather than new leading powers.
  3. [Methods / data analysis subsection] Error analysis and statistical robustness of the regime separation are not addressed in the provided abstract or summary. The identification of distinct small-v and large-v regimes for φ_q requires quantitative criteria (e.g., goodness-of-fit metrics, crossover location with uncertainty) to rule out artifacts from partitioning the v-range.
minor comments (2)
  1. [Introduction / abstract] Notation for the two time scales (ζ_d and ζ_d') and the exponents (r and r') should be introduced with explicit definitions and distinguished from standard Kibble-Zurek notation to avoid reader confusion.
  2. [Results section] The statement that the findings are verified in both isotropic and anisotropic models would benefit from a brief table or figure caption summarizing the extracted exponents r and r' in each case for direct comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments, which help clarify the presentation of the two distinct FTS regimes. We address each point below and will revise the manuscript accordingly to strengthen the evidence and derivations.

read point-by-point responses

  1. Referee: [Numerical results / scaling collapse figures (around the discussion of φ_q vs. v)] The central claim that φ_q exhibits two separable FTS regimes (small-v controlled by r, large-v controlled by r') is load-bearing for the paper's novelty. The manuscript must demonstrate explicitly that the large-v data for φ_q cannot be collapsed by the standard scaling function of r together with analytic corrections in v; without such a test (e.g., via attempted single-r collapse or residual analysis), the identification of an independent r' remains an assumption rather than a derived result.

    Authors: We agree that an explicit test against single-r scaling (with analytic corrections) is necessary to establish the second regime as independent rather than an artifact. In the revised manuscript we will add a dedicated panel (or supplementary figure) showing attempted collapses of the large-v φ_q data using only the conventional r exponent, together with residual plots and goodness-of-fit metrics. This will quantify the systematic deviations that motivate the introduction of r'. revision: yes

  2. Referee: [Theoretical framework / scaling ansatz section] The derivation of how the dangerously irrelevant operator generates a distinct relevant eigenvalue r' that couples differently to the angular operator φ_q than to M² is not shown in detail. The abstract and main text present r' as producing an independent time scale ζ_d', but the RG flow or scaling-function construction that isolates this effect from ordinary corrections should be provided, especially since dangerously irrelevant variables typically enter through sub-leading corrections rather than new leading powers.

    Authors: The referee is correct that the RG argument linking the dangerously irrelevant variable to a distinct leading exponent r' for φ_q requires more explicit construction. We will expand the scaling-ansatz section with a step-by-step RG-flow sketch showing how the irrelevant operator, when coupled to the angular sector, generates an additional relevant eigenvalue r' that dominates the leading scaling of φ_q at large v, while remaining sub-leading for M². This will distinguish the new time scale from ordinary correction-to-scaling terms. revision: yes

  3. Referee: [Methods / data analysis subsection] Error analysis and statistical robustness of the regime separation are not addressed in the provided abstract or summary. The identification of distinct small-v and large-v regimes for φ_q requires quantitative criteria (e.g., goodness-of-fit metrics, crossover location with uncertainty) to rule out artifacts from partitioning the v-range.

    Authors: We acknowledge that quantitative criteria for regime identification were not provided. In the revised methods section we will include (i) χ²/dof values for the scaling collapses in each velocity window, (ii) a data-driven estimate of the crossover velocity v* together with bootstrap-derived uncertainty, and (iii) a sensitivity analysis showing that the extracted r' remains stable under modest changes in the partitioning threshold. These additions will demonstrate that the separation is statistically robust. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling forms are independent of fitted inputs

full rationale

The paper extends standard Kibble-Zurek/FTS scaling by positing an additional exponent r' tied to the dangerously irrelevant variable, then reports that M² follows the usual form while φ_q exhibits two regimes controlled by ζ_d and ζ_d'. No equation or claim reduces a reported scaling function or regime separation to a parameter fitted from the same φ_q or M² data; the two time scales are introduced as theoretical consequences rather than outputs of a self-consistent fit. Verification across isotropic and anisotropic models supplies independent content. No self-citation chain is shown to be load-bearing for the central distinction, and no ansatz is smuggled via prior work in a way that collapses the result to its own definition. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard dynamic scaling assumptions plus the existence of a dangerously irrelevant variable that generates an independent exponent r'; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Standard dynamic scaling relations hold for the driven system, including the usual relation between dynamic exponent z and the velocity exponent r.
    Invoked to define the first time scale ζ_d ∝ v^{-z/r}.
  • domain assumption A dangerously irrelevant scaling variable produces a second independent exponent r' that controls a distinct time scale ζ_d'.
    This is the key additional premise needed for the second FTS region.

pith-pipeline@v0.9.0 · 5826 in / 1451 out tokens · 30711 ms · 2026-05-22T23:26:43.159085+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    two driving-induced time scales ζ_d ∝ v^{-z/r} and ζ_d' ∝ v^{-z/r'} ... r' represents an additional critical exponent of v associated with the dangerously irrelevant scaling variable

  • Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    M^{2} obeys the usual FTS form ... ϕ_q shows remarkably distinct scaling behaviors controlled by both FTS regions

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Optimized adiabatic-impulse protocol preserving Kibble-Zurek scaling with attenuated anti-Kibble-Zurek behavior

    quant-ph 2026-01 unverdicted novelty 5.0

    An optimized adiabatic-impulse protocol reduces total time for crossing quantum critical points while preserving Kibble-Zurek scaling and attenuating noise-induced anti-Kibble-Zurek defects in the transverse Ising chain.

Reference graph

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