REVIEW 3 major objections 3 minor 13 references
Continuing the discussion on "The inconvenient truth about flocks" by Chen et al
T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read In two-dimensional flocks, the deterministic dynamics of the Goldstone mode must be a continuity equation, and the paper argues that all known counterexamples rely on non-hydrodynamic terms.
desk verdict A clear restatement of the symmetry argument for the continuity form, but the paper leaves the real sticking point—whether non-divergence nonlinearities are irrelevant at the fixed point—asserted rather than demonstrated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the Goldstone mode θ(r,t), the local angle of the broken-symmetry direction in a polar flock. The load-bearing identity is ∂tθ = −∇·J + η, where J is a conserved current and η is non-conserved white noise; it is derived from the requirement that the global direction have pure-noise dynamics. The second piece of machinery is the statement that hydrodynamic equations are coarse-grained fixed points of the renormalization group, which justifies discarding irrelevant nonlinearities and field-dependent transport coefficients.
What would settle it
A decisive calculation would take a concrete microscopic or continuum model in the 2D polar-flock symmetry class, include a field-dependent coefficient multiplying a non-conserved term such as (∂xθ)^2 or a θ-dependent prefactor, and run a controlled renormalization-group analysis to the fixed point. If any such term is relevant and changes the fixed-point scaling relations, the paper's central claim fails; if all such terms flow to zero, the continuity equation is confirmed as the unique hydrodynamic form.
Extended reading notes
Core claim
The central claim is that Eq. (1) is the only valid large-scale deterministic equation for the Goldstone mode associated with spontaneously broken rotational symmetry in two dimensions. The proof idea is to integrate the local angle over the whole system: the global direction Θ(t) must have purely stochastic dynamics, because any deterministic relaxation toward another direction would violate the fact that the chosen direction is the only one selected by symmetry. Repeating the argument for a finite subvolume, the interaction with the exterior is a surface integral, which in local form is exactly the divergence of a current. The paper further holds that hydrodynamic equations are fixed points of the renormalization-group flow, so candidate equations containing an infinite number of nonlinearities are not directly comparable to Eq. (1); only the relevant terms with constant coefficients survive at large scales. It then presents two examples in which RG flows restore the continuity-equation symmetry, and argues that mistakenly writing an equation for the velocity phase instead of a density-corrected Goldstone mode produces a mass and destroys the Goldstone behavior.
Load-bearing premise
The argument relies on the assumption that at the renormalization-group fixed point only a small number of relevant terms with constant transport coefficients survive; if arbitrary field-dependent nonlinearities remain relevant, the continuity-equation form could fail, and the paper gives no general proof that they do not.
Editorial extensions
If this is right
- Any valid hydrodynamic description of a two-dimensional polar flock must be written with a conserved current for the Goldstone mode; field-dependent transport coefficients are not part of the large-scale theory.
- Continuum equations that violate the continuity equation are not automatically wrong; they are pre-hydrodynamic descriptions whose offending nonlinearities must be irrelevant under renormalization.
- A Goldstone mode constructed from the velocity phase alone acquires a mass, so the correct mode must include density corrections to satisfy the continuity equation.
- Known counterexamples from solids, smectics, chiral rotors, and externally forced gradients fall outside the symmetry class for which Eq. (1) is claimed.
Reading between the lines
- If the paper is right, then any alleged counterexample to the continuity equation should be testable by a controlled renormalization-group calculation: a field-dependent nonlinearity that is relevant at the fixed point would be a genuine counterexample, and the paper's argument predicts none exists in this symmetry class.
- The same global-direction reasoning could be transferred to other spontaneously broken continuous symmetries with non-conserved order parameters, such as certain active nematic or magnetic models, though the paper does not make this extension.
- The underlying disagreement is about whether hydrodynamics is a closed small-parameter theory or an open-ended continuum description; this paper takes the former position, and the debate's resolution hinges on which notion survives coarse-graining.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript is a reply to Chen et al.'s response to a previous comment by the same authors. It restates the argument that, in two-dimensional systems spontaneously breaking rotational symmetry, the deterministic part of the Goldstone mode dynamics must be a continuity equation (Eq. (1)), with a non-conserved Gaussian white noise. The manuscript then asserts that the large-scale hydrodynamic equation at the RG fixed point contains only a small number of relevant terms with constant transport coefficients, so the non-divergence nonlinearities written by Chen et al. are irrelevant. It supports this with two illustrative RG calculations (Nelson-Pelcovits and Jentsch-Lee) and with an example from the authors' own work showing that accidentally breaking the continuity symmetry leads to a massive Goldstone mode. The conclusion is that the symmetry Eq. (1) is supported by analytical and numerical evidence.
Significance. If the argument were complete, it would settle the dispute in favor of the authors: it would provide a symmetry-based justification for the continuity form and would relegate Chen et al.'s nonlinearities to irrelevant operators. The global zero-mode argument is elegant and valid as far as it goes; the paper also correctly emphasizes that the noise in Eq. (1) is non-conserved, which is a point of confusion in the exchange. However, the load-bearing step from the symmetry constraint to 'constant coefficients' is not established, so the significance of the paper as a refutation is currently limited.
major comments (3)
- [Hydrodynamic equation] The central claim that 'the hydrodynamic description should contain a small number of relevant terms with associated constant coefficients and not arbitrary dependencies in the fields' is asserted without proof. The zero-mode argument leading to Eq. (1) permits a field-dependent current J, and it does not rule out deterministic non-divergence terms such as the (K1 - K3) term in Eq. (3). To refute Chen et al., the manuscript must show, by a controlled RG calculation that retains those terms, that they are irrelevant at the flocking fixed point; otherwise the claim is an assumption and the paper's conclusion is not established.
- [Hydrodynamic equation, first example] The Nelson-Pelcovits example concerns the relaxational dynamics of a two-dimensional nematic, not an active polar flock. The manuscript does not explain why the flow to K1 = K3 in that equilibrium model carries over to the active, non-Hamiltonian case where the noise in Eq. (1) is non-conserved and where density and momentum fluctuations are coupled. A statement of transferability is needed; as written, the example is suggestive but not a proof.
- [Hydrodynamic equation, second example] The Jentsch-Lee calculation is presented as evidence for the scaling relation, but the manuscript itself states that the calculation was made 'at the price of neglecting several non-linearities.' Since the neglected terms include the very nonlinearities in dispute, citing this calculation to prove their irrelevance is circular. The paper's challenge that Chen et al. 'do not support' their claim does not supply the missing calculation; the burden is on the authors to demonstrate irrelevance with the terms retained.
minor comments (3)
- [Title and text] There are typographical errors: 'Chen at al' on the first page should read 'Chen et al.', and the title contains 'Chenet al.' without a space.
- [Hydrodynamic equation] The equivalence 'Eq.(22-23) in the SM of [3] or equivalently Eq.(2.18,2.28) of [10]' is not self-contained; a reader without access to the supplementary material cannot check the equivalence without the equations being reproduced.
- [Eq. (3)] The notation ∂_∥ and ∂_⊥ is introduced only after Eq. (3), but it would be clearer to define these operators before presenting the equation.
Circularity Check
No circularity: the continuity-equation argument is self-contained, and the contested RG-irrelevance step is an asserted assumption, not a constructional identity.
full rationale
The paper's central claim—that the deterministic part of the Goldstone-mode dynamics must be a divergence—is derived from the spontaneous breaking of rotational symmetry. The argument tracks the global mode Θ(t), whose deterministic relaxation is forbidden by symmetry, and then applies the same reasoning to a finite subvolume, where interactions with the rest of the system enter only through a surface integral; passing to local form gives Eq. (1). This is a self-contained kinematic derivation with no fitted parameters and no reliance on the authors' prior results. The subsequent hydrodynamic claim that non-divergence nonlinearities such as Eqs. (IV.5-6) and (IV.17-18) of [2] are irrelevant at the RG fixed point is indeed an assumption, and the examples cited are not fully transferable: Nelson-Pelcovits treats relaxational nematics, and Jentsch-Lee itself proceeds 'at the price of neglecting several non-linearities to keep the problem tractable.' But an unproven or contested assumption is a scientific weakness, not circularity: the paper does not define Eq. (1) in terms of that assumption, nor does it fit a parameter and then predict the same parameter. Self-citations [3,12] appear only as numerical evidence in the final sentence and as an illustrative example of what happens when Eq. (1)'s symmetry is broken during coarse-graining; the analytical argument for Eq. (1) is independent of them. The paper also explicitly acknowledges the argument's limitations (rotating steady states, explicitly broken rotational symmetry), which do not affect the claimed derivation in the symmetry class considered. No step reduces to its input by construction.
Assumptions & free parameters
assumptions (4)
- domain assumption For a spontaneously broken continuous symmetry, the global Goldstone mode has no deterministic relaxation.
- domain assumption Interactions are local, so the influence of the rest of the system on a subvolume enters only through a surface integral.
- domain assumption At the hydrodynamic fixed point, transport coefficients are constant and field dependencies are irrelevant.
- domain assumption The coarse-grained noise η is Gaussian white and delta-correlated in space.
Cite this review
Pith. "Pith review of Continuing the discussion on "The inconvenient truth about flocks" by Chen et al." pith.science (2026). https://pith.science/paper/S525UBSR
@misc{pith2026250613437,
author = {Pith},
title = {Pith review of: Continuing the discussion on "The inconvenient truth about flocks" by Chen et al},
year = {2026},
howpublished = {\url{https://pith.science/paper/S525UBSR}},
note = {Machine review of arXiv:2506.13437}
}
read the original abstract
We hereby reply concisely and hopefully clearly to the ongoing claims of incorrectness made by Chen et al. about our work on two-dimensional flocks.
Reference graph
Works this paper leans on
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[1]
H. Chaté and A. Solon, Comment on the inconve- nient truth about flocks by chen et al, arXiv preprint arXiv:2504.13683 (2025)
arXiv 2025
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[2]
L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ra- maswamy, and J. Toner, The inconvenient truth about flocks, arXiv e-prints 10.48550/arXiv.2503.17064 (2025)
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[3]
H. Chaté and A. Solon, Dynamic scaling of two- dimensional polar flocks, Physical Review Letters132, 268302 (2024)
work page 2024
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[4]
L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ra- maswamy, and J. Toner, Response to the comment on the inconvenient truth about flocks by Chaté and Solon, arXiv preprint arXiv:2505.21602 (2025)
arXiv 2025
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[6]
As is well known [13], there is then no additional Gold- stone mode associated with rotational symmetry break- ing
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[7]
P. C. Martin, O. Parodi, and P. S. Pershan, Unified hy- drodynamic theory for crystals, liquid crystals, and nor- mal fluids, Physical Review A6, 2401 (1972)
work page 1972
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[8]
D. R. Nelson and R. A. Pelcovits, Momentum-shell re- cursion relations, anisotropic spins, and liquid crystals in 2+epsilon dimensions, Physical Review B16, 2191 (1977), publisher: APS
work page 1977
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[9]
Jentsch and C
P. Jentsch and C. F. Lee, New Universality Class De- scribes Vicsek’s Flocking Phase in Physical Dimensions, Physical Review Letters133, 128301 (2024)
2024
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[10]
Toner, Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks, Physical Review E86, 031918 (2012)
J. Toner, Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks, Physical Review E86, 031918 (2012)
2012
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[11]
Dupuis, L
N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonper- turbative functional renormalization group and its appli- cations, Physics Reports910, 1 (2021)
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[12]
Mahault, F
B. Mahault, F. Ginelli, and H. Chaté, Quantitative as- sessmentoftheTonerandTutheoryofpolarflocks,Phys- ical Review Letters123, 218001 (2019)
2019
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[13]
P. M. Chaikin, T. C. Lubensky, and T. A. Witten, Principles of condensed matter physics, Vol. 10 (Cam- bridge university press Cambridge, 1995)
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Reviewed August 15, 2026 · model on record in the stance chip above.
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