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arxiv: 2509.04556 · v2 · submitted 2025-09-04 · ❄️ cond-mat.stat-mech

Domain coarsening in fractonic systems: a cascade of critical exponents

Jacopo Gliozzi , Federico Balducci , Giuseppe De Tomasi This is my paper

Pith reviewed 2026-05-18 18:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords domain coarseningfractonic systemsmultipole conservationdynamical critical exponentsIsing modelnon-equilibrium universality
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0 comments X

The pith

Conservation of the m-th multipole moment causes domains to grow as R(t) ∼ t^{1/(2m+3)}

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

The paper studies domain growth in the Ising model after a quench when higher moments of the order parameter are conserved in addition to the usual cases. In the absence of conserved quantities, the typical domain size grows as the square root of time, slowing to the cube root when the order parameter is conserved. Conservation of the m-th multipole moment further modifies the coarsening to follow R(t) ∼ t^{1/(2m+3)}. This result, obtained both analytically and numerically, points to a cascade of critical exponents that define new universality classes in fractonic systems.

Core claim

We analytically and numerically show that conservation of the m-th multipole moment causes domains to grow as R(t) ∼ t^{1/(2m+3)}. This cascade of dynamical critical exponents characterizes a new family of non-equilibrium universality classes for fractonic systems.

What carries the argument

The m-th multipole moment conservation applied to curvature-driven domain coarsening.

If this is right

  • For m=0 this recovers the known Lifshitz-Slyozov growth law of t^{1/3}.
  • Higher m leads to progressively slower growth with exponents 1/5, 1/7, and so on.
  • The scaling applies to fractonic systems where multipole moments are conserved due to restricted mobility.

Where Pith is reading between the lines

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

  • If fractonic mobility constraints add further restrictions beyond multipole conservation, the growth law could be modified further.
  • Analogous cascades might appear in other systems with higher-order conservation laws, such as in active matter or glassy dynamics.
  • Direct measurement of the growth exponent in a dipole-conserving fracton model would test the prediction for m=1.

Load-bearing premise

The derivation assumes that the standard model of curvature-driven coarsening remains valid once multipole conservation is imposed, without additional dynamical constraints from fractonic mobility fundamentally altering the scaling form.

What would settle it

A simulation or experiment showing domain size growing with a time exponent inconsistent with 1/(2m+3) for a given conserved multipole moment m would disprove the central claim.

Figures

Figures reproduced from arXiv: 2509.04556 by Federico Balducci, Giuseppe De Tomasi, Jacopo Gliozzi.

Figure 1
Figure 1. Figure 1: FIG. 1. Spin dynamics in the kinetic Ising model follow [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Two examples of spin updates that conserve dipole [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Domain growth with Kawasaki dynamics. (a) Long [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Domain growth for dipole-conserving dynamics, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of spins in the dipole-conserving case, with [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Domain growth for quadrupole-conserving dy [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Growth of domains for dipole-conserving dynamics at [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

We study the dynamics of domain growth when multipole moments of the order parameter are conserved. Following a quench into the ordered phase of the Ising model, the typical size of domains grows with time as $R(t) \sim t^{1/2}$ in the absence of conserved quantities. When the order parameter is conserved, the domain growth slows to $R(t) \sim t^{1/3}$. Conservation of higher moments of the order parameter fundamentally modifies this behavior: coarsening proceeds via anomalously slow growth. We analytically and numerically show that conservation of the $m$-th multipole moment causes domains to grow as $R(t) \sim t^{1/(2m+3)}$. This cascade of dynamical critical exponents characterizes a new family of non-equilibrium universality classes for fractonic systems.

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 / 1 minor

Summary. The manuscript examines domain coarsening after a quench in Ising-like models subject to conservation of the m-th multipole moment of the order parameter, a setting relevant to fractonic systems. It claims that this conservation law produces anomalously slow growth with the scaling R(t) ∼ t^{1/(2m+3)}, in contrast to the t^{1/2} (non-conserved) and t^{1/3} (conserved order parameter) cases. The result is presented as following from the imposed conservation laws and is supported by both analytical arguments and numerical simulations, thereby defining a cascade of dynamical critical exponents and a new family of non-equilibrium universality classes.

Significance. If the central scaling relation holds, the work would be significant for non-equilibrium statistical mechanics and fracton physics. It supplies a concrete, m-dependent family of growth exponents that could organize coarsening behavior across systems with restricted mobility. The combination of an analytical derivation tied directly to the conservation laws and accompanying numerical evidence is a strength that, if fully detailed and robust, would help establish these universality classes.

major comments (2)
  1. [Abstract / analytical section] Abstract and analytical derivation: the scaling R(t) ∼ t^{1/(2m+3)} is obtained by assuming that m-th multipole conservation modifies only the continuity equation (or effective chemical potential) while the kinetic coefficient and interface velocity law remain those of the standard curvature-driven Allen-Cahn/Cahn-Hilliard model. In fractonic systems, however, single charges are immobile and motion requires coordinated higher-multipole rearrangements; this can couple mobility to additional gradients or higher-order operators. The manuscript must explicitly show, via the effective dynamical equation or a renormalization argument, that these mobility constraints remain irrelevant in the long-wavelength limit and do not generate new length scales or corrections to the exponent.
  2. [Numerical evidence] Numerical section: the simulations are stated to confirm the analytical exponent, but the microscopic update rules used to enforce multipole conservation must be shown to be representative of realistic fractonic dynamics rather than an artifact of the chosen dynamics. Provide details on how the exponent is extracted (including error bars, fitting procedure, and finite-size scaling) and demonstrate that the growth law is insensitive to the precise form of the local mobility rule.
minor comments (1)
  1. [Model definition] Clarify the precise definition of the m-th multipole moment and its conservation law in the model Hamiltonian or dynamics, including any notation for the associated continuity equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments help clarify the presentation of our results on the cascade of growth exponents arising from multipole conservation. We address each point below and will revise the manuscript to incorporate additional details and arguments.

read point-by-point responses

  1. Referee: [Abstract / analytical section] Abstract and analytical derivation: the scaling R(t) ∼ t^{1/(2m+3)} is obtained by assuming that m-th multipole conservation modifies only the continuity equation (or effective chemical potential) while the kinetic coefficient and interface velocity law remain those of the standard curvature-driven Allen-Cahn/Cahn-Hilliard model. In fractonic systems, however, single charges are immobile and motion requires coordinated higher-multipole rearrangements; this can couple mobility to additional gradients or higher-order operators. The manuscript must explicitly show, via the effective dynamical equation or a renormalization argument, that these mobility constraints remain irrelevant in the long-wavelength limit and do not generate new length scales or corrections to the exponent.

    Authors: We agree that an explicit demonstration of the irrelevance of additional mobility constraints is needed for clarity. Our analytical derivation starts from the imposed m-th multipole conservation, which directly yields a generalized continuity equation with a higher-order Laplacian acting on the chemical potential. This modifies the bulk transport but leaves the local interface velocity law curvature-driven, as the conservation law does not introduce new relevant operators or length scales at long wavelengths. To address the referee's concern, we will add to the revised manuscript an explicit derivation of the effective dynamical equation together with a brief renormalization-group argument showing that higher-gradient mobility terms are irrelevant under coarse-graining, thereby confirming that the scaling R(t) ∼ t^{1/(2m+3)} remains robust. revision: yes

  2. Referee: [Numerical evidence] Numerical section: the simulations are stated to confirm the analytical exponent, but the microscopic update rules used to enforce multipole conservation must be shown to be representative of realistic fractonic dynamics rather than an artifact of the chosen dynamics. Provide details on how the exponent is extracted (including error bars, fitting procedure, and finite-size scaling) and demonstrate that the growth law is insensitive to the precise form of the local mobility rule.

    Authors: We agree that further details on the numerical implementation and robustness checks are warranted. The microscopic update rules are constructed to enforce exact conservation of the m-th multipole moment at every step while permitting only local rearrangements consistent with the restricted mobility of fractonic charges. In the revised manuscript we will expand the numerical section to include: (i) a precise description of the update algorithm, (ii) the fitting procedure used to extract the growth exponent together with statistical error bars, (iii) finite-size scaling analysis confirming the scaling regime, and (iv) additional simulations employing alternative local mobility rules that preserve the same conservation law. These checks demonstrate that the measured exponent is insensitive to the precise form of the local rule and remains consistent with the analytical prediction. revision: yes

Circularity Check

0 steps flagged

Derivation of multipole-conservation exponent is self-contained with independent scaling content

full rationale

The paper derives R(t) ~ t^{1/(2m+3)} by imposing m-th multipole conservation into the continuity equation for the order parameter and then applying standard curvature-driven coarsening scaling (modified only by the higher-order conservation constraint) to obtain the dynamical exponent. This step is presented as following directly from the conservation law plus the usual Allen-Cahn/Cahn-Hilliard interface velocity assumption; numerical quenches are used to confirm the scaling independently. No quoted reduction equates the final exponent to a fitted input, a self-citation chain, or a renamed known result. The central analytic claim therefore retains independent content from the imposed conservation and does not collapse by construction to its premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the central claim rests on standard assumptions of phase-ordering kinetics plus the imposition of multipole conservation, with no free parameters or new entities explicitly introduced in the provided text.

axioms (1)
  • domain assumption Standard curvature-driven coarsening dynamics remain applicable when higher multipole moments are conserved.
    Invoked implicitly when extending the known t^{1/2} and t^{1/3} laws to the m-th moment case.

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