Response to the comment on The inconvenient truth about flocks by Chat\'e and Solon

Ananyo Maitra; Chiu Fan Lee; John Toner; Leiming Chen; Patrick Jentsch; Sriram Ramaswamy

Response to the comment on The inconvenient truth about flocks by Chat\'e and Solon

T0 review · 0 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read There is no argument that rotation invariance forces the deterministic dynamics of flock Goldstone modes into a conservation-law form, so the published critique of the authors' hydrodynamic theory fails.

desk verdict The reply's core point is right—rotation invariance does not force a conservation law—but it is a dependent, mostly negative contribution that should be judged alongside its parent preprint. read the letter →

arxiv 2505.21602 v1 pith:QCYHXDBR submitted 2025-05-27 cond-mat.soft cond-mat.stat-mech

Leiming Chen , Patrick Jentsch , Chiu Fan Lee , Ananyo Maitra , Sriram Ramaswamy , John Toner This is my paper

classification cond-mat.softcond-mat.stat-mech

keywords flocksGoldstonemodesrotationsymmetryhydrodynamicsconservationlawnematicliquidcrystalsactivematterrenormalizationgroup

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 response defends the authors' earlier hydrodynamic theory of flocks against a published comment. Its central claim is that spontaneously breaking rotation symmetry does not require the deterministic dynamics of the Goldstone mode—the slow angular fluctuations of the ordering direction—to take a conservation-law form, that is, to be a total divergence. The response says the comment offers no proof of such a requirement, and that nematic liquid crystals are a direct counterexample: they are rotation invariant, yet their angle dynamics are not conserved. It also argues that the mass seen in the commenters' simulations of the immortal flock (a flock without births or deaths) arises from solving truncated equations that break rotation symmetry, not from the true dynamics. If this is right, the analytic objection fails even though the numerical data themselves are not disputed.

What carries the argument

The central object is the Goldstone mode of the flock: the slow angle field $\theta$ that records local fluctuations of the broken-symmetry direction. The argument turns on whether a term like $A^\alpha_{ij}\partial_j v_i$ can be rewritten as $\partial_j(A^\alpha_{ij} v_i)$. That rewriting is valid only while the coefficient $A^\alpha_{ij}$ is independent of the fields; once the equations go beyond linear order, the coefficients depend on the fields, so rotation invariance does not force the deterministic part to be a total divergence. The response carries the argument with the full explicitly rotation-symmetric equations (IV.5)–(IV.6) of its earlier article, and with the counterexample of nematic liquid crystals, where angle dynamics are non-conserved despite rotation symmetry.

What would settle it

Numerically integrate the full rotation-symmetric equations (IV.5)–(IV.6) from the authors' earlier article and measure the long-wavelength angle correlator; if a finite mass appears despite the full symmetry, the claim that the simulated mass comes only from truncated symmetry-violating equations is false.

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Extended reading notes

Core claim

The paper's claim, stated on its own terms, is that "there is no argument whatsoever for the dynamics of Goldstone modes to have a conservation law form." Rotation invariance alone does not force the deterministic part of the angle-field equation to be a total divergence, because a term like $A^\alpha_{ij}\partial_j v_i$ can be written as $\partial_j(A^\alpha_{ij} v_i)$ only when the coefficient $A^\alpha_{ij}$ does not depend on the fields; beyond linear order it does. The response therefore rejects the comment's requirement that the deterministic Goldstone dynamics obey a continuity equation, and it attributes the mass found in simulations of the immortal flock to rotation-symmetry breaking introduced by truncating the equations: only the full equations (IV.5)–(IV.6) of the earlier article are symmetric under joint rotation of space and orientation, and only for parameter values with $h_{x,2}=\lambda_1$ does the correlator remain massless. The response also notes an internal inconsistency in the comment: its noise correlations do not vanish as $q\to 0$, even though it insists the deterministic part be a total divergence.

Load-bearing premise

The rebuttal assumes that the full, explicitly rotation-symmetric set of long-wavelength equations from the authors' earlier paper is the correct description of the immortal flock, so the mass seen in simulations is an artifact of truncated equations rather than real physics.

Editorial extensions

If this is right

The analytic prediction of the criticized article, which rests on the conservation-law requirement, is unsupported.
The mass reported in simulations of the immortal flock does not count as evidence against the hydrodynamic theory, because it appears only when the simulated equations are truncated and thereby break rotation symmetry.
Simulations intended to respect the symmetry should integrate the full, explicitly symmetric equations rather than truncated versions.
Without the conservation-law constraint, the noise strength and nonlinearities in the angle equation may acquire graphical corrections, so the exact exponent values remain open.
The nematic liquid crystal counterexample shifts the burden: anyone who asserts a general conservation-law rule for Goldstone modes must prove it.

Reading between the lines

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

One testable extension is to simulate the full, explicitly rotation-symmetric equations (IV.5)–(IV.6) and compare the angle correlator with the truncated version; the response predicts a massless correlator only in the full equations.
If the conservation-law constraint is truly absent, the renormalization-group flow for two-dimensional flocks may show asymptotic exponents only at system sizes well beyond current simulations, which would reconcile the dispute with the numerical data.
The same "beyond linear order the coefficients depend on fields" argument should apply to other spontaneously broken continuous symmetries in active matter, such as active nematics, where field-dependent couplings are generic.
The debate could be settled analytically by finding any genuinely rotation-invariant model whose Goldstone-mode equation is exactly a conservation law at all orders; the response implies no such model exists.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This reply lands one solid punch: rotation invariance alone does not force the deterministic part of the angle field's dynamics into a conservation-law form. The nematic liquid crystal counterexample is standard and decisive, and the point about Chaté and Solon's noise violating their own conservation requirement is well taken. If you are following the flocking universality class debate, this reply clears the air on that specific issue.

What is actually new: not much. The authors state their position is unchanged from [3], and they lean on that preprint for the key equations, the RG statements, and the hx,2 = λ1 relation. The density-gradient argument—that a fixed gradient breaks isotropy and should generate a restoring torque—is asserted, not derived. It is plausible, but it is a sketch. There is also an overstatement: the claim that A_ij ∂_j v_i can be written as a total divergence iff A_ij is field-independent is too strong, since there are field-dependent forms where that rearrangement holds in irrotational cases. That does not wound the central conclusion, because the nematic counterexample stands independently.

The self-referential chain is real but not fatal. The key conceptual point is anchored in Martin–Parodi–Pershan and standard nematic hydrodynamics, not in [3] alone. What is genuinely missing is a derivation of the density-gradient mass term and a demonstration that the truncated equations used in Chaté and Solon's simulations break rotation invariance in precisely the way that generates the mass. The reply says this is so, but does not show the algebra. And the authors concede they cannot compute the exponents for the immortal flock, so the reply is negative in character: it knocks down the conservation-law argument without offering alternative predictions.

Bottom line: worth a serious referee if the venue publishes back-and-forth exchanges, but it must be evaluated together with [3] and the original comment. The full argument lives in all three documents. I would not cite it as a standalone result, only as part of the exchange. Yes, send it to peer review; the central claim is correct and the reply makes a genuine point that the comment gets wrong.

Referee Report

0 major / 6 minor

Summary. This manuscript is a reply to Chaté and Solon's comment on the authors' earlier preprint "The inconvenient truth about flocks". The reply defends the claim that rotation invariance does not force the deterministic part of the orientational Goldstone-mode dynamics to have a conservation-law form. It supports this claim with a nematic liquid-crystal counterexample, a discussion of the Martin-Parodi-Pershan hydrodynamic formulation, a density-gradient argument, and a critique of the noise statistics in the original comment. It also addresses the numerical mass observed in Chaté and Solon's simulations, attributing it to simulating truncated, rotation-symmetry-violating equations. The reply concludes that the original critique is incorrect, while explicitly admitting that the authors cannot compute the exponents for the immortal flock.

Significance. If the central claim holds, it invalidates the main analytical objection raised by Chaté and Solon. The nematic liquid-crystal counterexample is standard and well established, and the density-gradient argument provides a clear physical reason why a fixed density gradient should yield a mass term that is not a total derivative. The reply also makes a good point about the inconsistency of requiring only the deterministic part to be a conservation law while the noise does not satisfy a corresponding requirement. The admission of inability to compute exponents is honest and does not undermine the negative claim. The paper is a useful contribution to the debate, though its positive program remains incomplete. The reply relies heavily on the authors' own prior preprint [3] for several key equations, and a few statements are overstated, which are presentation issues rather than fatal flaws.

minor comments (6)

[Paragraph beginning "Further, their claims regarding [4]"] The statement that "A_ij ∂_j v_i can be written as ∂_j(A_ij v_i) if and only if A_ij is not a function of the fields" is too strong; there exist field-dependent A_ij (for example, an A_ij that is a divergence of an antisymmetric tensor built from fields) for which ∂_j A_ij vanishes identically, making the expression a total divergence despite the field dependence. The underlying point that the Martin-Parodi-Pershan term is not a total divergence beyond linear order is correct, but the "if and only if" formulation should be softened.
[Paragraph on density-gradient mass] The implication that a fixed density gradient requires a term ∝ θ ∂_x ρ which cannot come from a total derivative is asserted rather than demonstrated; a brief symmetry argument or a more explicit citation to the standard polar liquid-crystal literature would make this step clearer.
[Throughout] The reply repeatedly refers to Eqs. (III.5), (IV.5), and (IV.6) as being "of this article", but these equations appear in the authors' earlier preprint [3], not in the reply itself; this confusion should be resolved by explicitly referring to "Ref. [3]" in those passages.
[Density-gradient paragraph] The symbol Θ is used for the global direction in the density-gradient argument while θ is used elsewhere; the notation should be unified to avoid ambiguity.
[Paragraph about advective term] The sentence "the advective term v · ∇θ is not a total divergence" is true, but it would be helpful to state explicitly that this is another counterexample to the conservation-law claim, since this term appears in the angle dynamics.
[Minor grammatical points] There are a few grammatical slips, such as "their h_x,2 has to be equal to λ1" (the possessive is unnecessary) and "it can't since" (should be "it cannot, since"); these should be corrected.

Circularity Check

1 steps flagged · score 3.0 of 10

Central claim is independently supported; one load-bearing self-citation for the numerical-mass sub-claim.

self citation load bearing [Paragraph beginning 'Furthermore, in [2], Chaté and Solon again claim...']
"For Chaté and Solon’s [1] Eq. (18) to be rotation invariant, their hx,2 has to be equal to λ1 (or equivalently, hx,2 = −σ in (22), (23) in their SI); see Eq. (III.5), and Eqs. (IV.5) and (IV.6) of this article."

The response explains away the numerical mass seen by Chaté and Solon by asserting that their equations break rotation invariance unless hx,2 = λ1, and the sole justification given for this relation is a reference to the authors' own Eqs. (III.5), (IV.5) and (IV.6) from their prior preprint [3]. This is not an independent derivation within the response; it adopts the very hydrodynamic framework that Chaté and Solon dispute. Consequently, the conclusion that the mass is a symmetry-breaking artifact is forced only if one already accepts the authors' own equations. This is a self-referential chain for that sub-claim, although it does not underpin the central assertion about the conservation-law form, which is independently supported.

full rationale

The paper's principal thesis—that rotation invariance does not force the deterministic part of the Goldstone-mode dynamics to have a total-divergence/conservation-law form—is supported by three independent lines of reasoning: the nematic liquid crystal counterexample (standard in the literature and backed by Refs. [5,6,7,8]), the analysis of the Martin-Parodi-Pershan term beyond linear order, and the density-gradient mass argument presented in the text itself. None of these reduces to the authors' prior equations. The only load-bearing self-citation appears in the numerical-mass discussion, where the relation hx,2 = λ1 is asserted by reference to the authors' own Eqs. (III.5), (IV.5) and (IV.6) of [3], making that sub-claim dependent on their contested framework. This warrants a moderate circularity score, but not a high one because the central conceptual point stands on independent evidence.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or ad hoc constants are introduced in the response; the arguments are conceptual and refer to equations from [3] and standard texts. No new entities are invented. The loaded assumptions are the applicability of nematic hydrodynamics to flocks, the density-gradient mass argument, and the correctness of the authors' own equations from [3].

assumptions (4)

domain assumption Nonlinear hydrodynamic couplings in ordered phases can be field-dependent, so a term like A^α_ij∂_j v_i is not a total divergence beyond linear order.
The reply uses this to reject the conservation-law inference from Martin, Parodi and Pershan [4], citing standard texts [5,6,7], but no explicit nonlinear derivation is given in the response.
domain assumption Nematic liquid crystals are a legitimate counterexample showing that rotation invariance does not require conserved Goldstone-mode dynamics.
The entire refutation rests on this counterexample; it is asserted with references to [3,4,5,6,7] and not re-derived in the reply.
domain assumption A constant density gradient explicitly breaks rotational isotropy and therefore generates a mass for orientational fluctuations, forcing a term proportional to θ∂xρ in the angle equation.
Used in the paragraph beginning 'Finally, the explicit argument...' to show a non-total-divergence term must exist; no derivation is supplied.
ad hoc to paper The full equations (IV.5)-(IV.6) of [3] are the correct rotation-invariant hydrodynamics for the immortal flock, and rotation invariance forces h_{x,2}=λ1.
This identification is used to explain why Chaté and Solon's simulations find a mass (truncated equations), but it is only asserted here and referred to [3].

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Cite this review

Pith. "Pith review of Response to the comment on The inconvenient truth about flocks by Chat\'e and Solon." pith.science (2026). https://pith.science/paper/QCYHXDBR

@misc{pith2026250521602,
  author       = {Pith},
  title        = {Pith review of: Response to the comment on The inconvenient truth about flocks by Chat\'e and Solon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCYHXDBR}},
  note         = {Machine review of arXiv:2505.21602}
}

read the original abstract

This is our response to the comment arXiv:2504.13683 posted by Chat\'e and Solon in reference to our preprint arXiv:2503.17064

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

22 extracted references · 16 canonical work pages

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arXiv 2025
[2]

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work page
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• We reiterate that we cannot calculate the exponents for either the immortal flock or the Malthusian flock analytically

agree that their equation of motion for Malthusian flocks does not actually have pseudo-Galilean invari- ance. • We reiterate that we cannot calculate the exponents for either the immortal flock or the Malthusian flock analytically. Therefore, the exponents of the Malthu- sian flock may be arbitrarily close to the ones numeri- cally obtained in [1], thoug...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

(3) with respect to time, yields time-non-local con- tributions to linear order inθ0 in the transformations of ∂xθ and ∂xθ (see Eq

Integrating their Eq. (3) with respect to time, yields time-non-local con- tributions to linear order inθ0 in the transformations of ∂xθ and ∂xθ (see Eq. (III.62) and (III.63) of [3]) that are not cancelled out. Further, plugging Eqs. (III.61), (III.62) and (III.63) into (III.59), yields terms linear in θ0 that are absent in (III.58). We have also not bee...

work page

[2] [1]

global direction cannot relax deterministically to an- other direction

is incorrect, and that their comment [2] on our ar- ticle [3] in no way refutes this. Ref. [2] identified two main points of contention. Their issue (ii) is easily dis- posed of: they do not identify a true invariance of the flock, as they now agree. Their first point (i) amounts to the claim that Goldstone modes must obey conserved dynamics. As we noted ...

arXiv 2025

[3] [2]

relevant in the RG sense

and the comment [2] present no evidence for this claim; indeed, itcan ’tsince the claim itself is incorrect as shown by an enormous number of counterexamples. • We did not refer to [14] in our article because it is not directly relevant. This work examines a flocking model in the presence of a global constraint. The dynamics for the Goldstone mode of this...

work page

[4] [4]

• We reiterate that we cannot calculate the exponents for either the immortal flock or the Malthusian flock analytically

agree that their equation of motion for Malthusian flocks does not actually have pseudo-Galilean invari- ance. • We reiterate that we cannot calculate the exponents for either the immortal flock or the Malthusian flock analytically. Therefore, the exponents of the Malthu- sian flock may be arbitrarily close to the ones numeri- cally obtained in [1], thoug...

work page

[5] [5]

Chaté and A

H. Chaté and A. Solon, Phys. Rev. Lett. 132, 268302 (2024)

work page 2024

[6] [6]

the incon- venient truth about flocks

H. Chaté and A. Solon, Comment on "the incon- venient truth about flocks" by chen et al (2025), arXiv:2504.13683 [cond-mat.soft]

arXiv 2025

[7] [7]

L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ra- maswamy, and J. Toner, The inconvenient truth about flocks (2025), arXiv:2503.17064 [cond-mat.soft]

arXiv 2025

[8] [8]

P. C. Martin, O. Parodi, and P. S. Pershan, Physical Review A 6, 2401 (1972)

work page 1972