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arxiv: 2606.20552 · v2 · pith:WILLZHHBnew · submitted 2026-06-18 · ❄️ cond-mat.soft · cond-mat.stat-mech· hep-th

On the Renormalization Group Flow of Active Flocks

Kevin T. Grosvenor , Subodh P. Patil This is my paper

Pith reviewed 2026-06-26 15:11 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mechhep-th
keywords active flocksToner-Tu theoryMalthusian flocksrenormalization groupMSRDJ actionlong-range ordernon-equilibrium criticalityWard identities
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The pith

The renormalization group flow of two-dimensional Malthusian flocks exhibits a line of fixed points and a marginal vertex instability at Δ/κ = 2π that separates Gaussian and symmetry-protected interacting gapless phases with persistent long

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

This paper studies the statistical field theory of active flocks described by the Toner-Tu equations for Malthusian fluids, where density fluctuations are eliminated due to fast relaxation. Working in two dimensions with isotropic diffusion, the authors employ the MSRDJ action and associated Ward identities from rotational and shift symmetries to compute the renormalization of all couplings to all orders. They identify a line of fixed points in the flow and locate a marginal vertex instability precisely at the ratio Δ/κ = 2π. This critical ratio distinguishes a Gaussian phase from a strongly interacting gapless phase protected by symmetry, realizing non-equilibrium criticality distinct from Wilson-Fisher behavior. The analysis implies that long-range order survives when the noise variance stays below this ratio relative to the diffusion coefficient.

Core claim

In the MSRDJ representation of the Toner-Tu model for polar-ordered active fluids with eliminated density fluctuations, the renormalization group analysis in two dimensions with isotropic diffusion yields a line of fixed points and identifies a marginal vertex instability at Δ/κ = 2π. This separates Gaussian phases from symmetry-protected strongly interacting gapless phases, realizing non-equilibrium critical behavior beyond standard Wilson-Fisher criticality, and indicates that long range order persists for Δ/κ below the critical value.

What carries the argument

The Ward identities associated with the non-linear realization of a diagonal rotation/shift symmetry together with a non-local field-dependent redundancy in the MSRDJ representation, which together allow renormalization of couplings to all orders.

If this is right

  • A line of fixed points exists in the renormalization group flow of the couplings.
  • A marginal vertex instability occurs exactly at Δ/κ = 2π.
  • Symmetry-protected strongly interacting gapless phases realize non-equilibrium critical behavior beyond Wilson-Fisher criticality.
  • Long range order persists when Δ/κ lies below the critical value.

Where Pith is reading between the lines

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

  • The same symmetry-based all-order methods could be applied when density fluctuations are restored as a dynamical variable.
  • The results offer a route to reconcile earlier conflicting statements about order in active flock models.
  • Analogous renormalization structures may appear in other non-equilibrium systems that possess similar diagonal symmetries.

Load-bearing premise

The analysis assumes isotropic diffusion in two spatial dimensions and that density fluctuations relax quickly enough to be eliminated as a hydrodynamic variable.

What would settle it

A numerical simulation of the Toner-Tu equations in two dimensions that measures the decay of orientational correlations or the order parameter while varying the ratio of noise variance to diffusion coefficient across the value 2π.

read the original abstract

In this paper, we study the statistical field-theoretic renormalization of active flocks via the MSRDJ action formulation for stochastic systems, focusing on the Toner-Tu theory of `Malthusian flocks', or polar-ordered, momentum non-conserving active fluids where relaxation times for density fluctuations are so short that they can be eliminated as a hydrodynamic variable. Working in the limit of isotropic diffusion in two spatial dimensions, we compute the renormalization of the couplings and their anomalous dimensions to all orders, facilitated by the Ward identities associated with the non-linear realization of a diagonal rotation/ shift symmetry, and a non-local field-dependent redundancy specific to the MSRDJ representation of this model. We find a range of behavior depending on the parameters of the theory. If $\kappa$ is the diffusion coefficient and $\Delta$ is the variance of the noise, we find a line of fixed points and a marginal vertex instability at $\Delta/\kappa = 2\pi$. This instability separates Gaussian, and symmetry-protected strongly interacting gapless phases, realizing non-equilibrium critical behavior beyond standard Wilson--Fisher criticality, and implies the persistence of long range order when $\Delta/\kappa$ is below the critical value. We revisit and contextualize various claims and counter-claims in the literature in light of our findings, and discuss extensions of our analysis to flocks with anisotropic diffusion, and where density fluctuations are reintroduced.

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

0 major / 3 minor

Summary. The manuscript develops an all-orders renormalization-group analysis of the Toner-Tu Malthusian flock in the isotropic two-dimensional limit with density fluctuations eliminated. Working in the MSRDJ formalism, the authors exploit Ward identities associated with a diagonal rotation/shift symmetry together with a non-local field-dependent redundancy to obtain the beta functions for all couplings. They report a line of fixed points and a marginal vertex instability at Δ/κ = 2π that separates a Gaussian phase from a symmetry-protected strongly interacting gapless phase, implying persistence of long-range order below the critical ratio, and they contextualize prior literature claims while outlining extensions to anisotropic diffusion.

Significance. If the all-orders result holds within the stated regime, the work supplies a symmetry-protected, non-perturbative example of non-equilibrium criticality that lies outside the Wilson-Fisher class. The explicit use of Ward identities to achieve control to all orders, the clear statement of the hydrodynamic limit (isotropic diffusion, fast density relaxation), and the falsifiable prediction of a marginal instability at Δ/κ = 2π constitute notable technical strengths. The analysis furnishes a controlled framework for revisiting claims about long-range order in low-dimensional active flocks.

minor comments (3)
  1. [§2] §2, paragraph following Eq. (3): the elimination of density fluctuations is presented as a hydrodynamic approximation; a brief estimate of the relaxation-time ratio relative to the flock velocity scale would help readers assess the regime of validity.
  2. [Figure 1] Figure 1 caption and surrounding text: the flow diagram for the line of fixed points would benefit from an explicit indication of the stability eigenvalues along the line, even if they follow directly from the Ward identities.
  3. [§4.2] §4.2, sentence after Eq. (27): the statement that the instability is 'marginal' should be accompanied by the explicit scaling dimension or beta-function coefficient that vanishes at Δ/κ = 2π to make the all-orders claim immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and supportive report, which recognizes the technical strengths of the all-orders RG analysis and the falsifiable prediction at Δ/κ = 2π. We note the recommendation for minor revision and will address any editorial or minor points in the revised version. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from action and symmetries

full rationale

The central results (line of fixed points and marginal instability at Δ/κ = 2π) are obtained by direct all-orders computation of the beta functions for the Toner-Tu Malthusian flock in the isotropic 2D limit, using Ward identities from the diagonal rotation/shift symmetry and the non-local field-dependent redundancy in the MSRDJ action. These steps start from the model action and its symmetries without any reduction to fitted inputs, self-citation chains, or redefinitions of the target quantities. The regime (isotropic diffusion, density fluctuations eliminated) is explicitly stated as the working limit, and the calculation is parameter-free within that limit. No load-bearing premise collapses to a prior result by the same authors or to a tautological fit.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the symmetries of the Toner-Tu model and the MSRDJ representation; no new entities are postulated and the critical ratio is derived rather than fitted.

axioms (2)
  • domain assumption Ward identities associated with the non-linear realization of a diagonal rotation/shift symmetry
    Invoked to compute renormalization of couplings and anomalous dimensions to all orders.
  • domain assumption Non-local field-dependent redundancy specific to the MSRDJ representation of this model
    Used alongside the Ward identities to facilitate the all-order calculation.

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discussion (0)

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Works this paper leans on

59 extracted references · 1 canonical work pages

  1. [1]

    The proof proceeds exactly as presented in [41] 14

    Non-Renormalization Theorems and Power Counting There are a set of non-renormalization theorems that one can immediately infer directly from the kinematic structure of the propagators. The proof proceeds exactly as presented in [41] 14. Any diagram that has a subdiagram which is a loop of internal lines which are all ofψ-ϕor all ofϕ-ψ type, with insertion...

  2. [2]

    The second equation simply counts the total number of endpoints of lines that must be attached on all of the vertices

    Organization of the Diagrammatic Expansion Define the following quantities: L= # of loops,(A37a) I= # of internal lines,(A37b) E= # of external lines,(A37c) V2n+1 = # of (2n+ 1)-vertices,(A37d) V2n+2 = # of (2n+ 2)-vertices.(A37e) There are two simple relations one can immediately derive: L=I− ∞X n=1 (V2n+1 +V 2n+2) + 1,(A38a) 2I+E= ∞X n=1 (2n+ 1)V 2n+1 +...

  3. [3]

    The diagram form merely shows the structure of the diagram; it does not show the dashed lines forϕ

    One-Loop Corrections For the one-loop corrections to the propagators,L= 1 in (A40) implies thatV ≥5 = 0, and so 1 = V3 2 +V 4.(A45) The possible values ofV 3 andV 4 are tabulated in Table I along with the structure of the corre- sponding diagram. The diagram form merely shows the structure of the diagram; it does not show the dashed lines forϕ. V3 V4 IDia...

  4. [4]

    reg” stands for “regularized,

    Superficial Degree of Divergence The power-counting classification presented in sec. A 1 has important consequences for the superficial degree of divergence of the various Feynman diagrams. A diagram with aψ-ψpropagator in an internal line will always have a lower degree of divergence compared to a diagram with the same structure, but where theψ-ψpropagat...

  5. [5]

    slow” modes with momentum< Λ b , where Λ is some UV scale andb >1 is a real number, and “fast

    Random Phase Approximation Instead of amputating the corrections to the propagators, we can instead sum all of the one- particle-irreducible (1PI) diagrams and then multiply each by two copies of the bare propagator. The sum of the amputated 1PI diagrams is the analog of the self energy Σ. For example, Σϕψ ≡ ∞X L=1 M (L) ϕψ = ∞X L=1 (−I) L L! iλkx = e−I −...

  6. [6]

    In particular, immediately following their Eq

    Chat´ e–Solon: Symmetry-Based Non-Renormalization Arguments In [1], Chat´ e and Solon analyze the Goldstone-mode dynamics of the ordered phase and argue that certain nonlinear couplings do not get renormalized. In particular, immediately following their Eq. (10), they state that the terms originating from the convective derivative are not expected to rece...

  7. [7]

    Their principal conclusion is that, for Malthusian flocks, only two independent exponent relations can be established

    Chenet al.: Limitations of Symmetry Constraints and Dynamical RG In [3], the authors revisit the derivation of exact exponent relations. Their principal conclusion is that, for Malthusian flocks, only two independent exponent relations can be established. Conse- quently, they argue that the three critical exponents cannot be determined exactly. An importa...

  8. [8]

    Amoretti et al.: Leveraging Thermodynamic Relations In [50], the authors make use of thermodynamic relations among the various constants ap- pearing in the hydrodynamic equations (the generalized continuity equation and Euler equation, corresponding to their equations 13a and 13b, respectively) to derive two critical-exponent rela- tions, which are distin...

  9. [9]

    We back up the RG predictions derived from the associated Ward identities with an explicit diagrammatic renormalization procedure

    Relation to this Work The present analysis differs from these three approaches in that it is based on a different set of symmetries and field redundancies as those considered in [1] and a different set of renormalization conditions, particularly as pertains to the ∆ parameter, as in all three approaches. We back up the RG predictions derived from the asso...

  10. [10]

    Chat´ e and A

    H. Chat´ e and A. Solon, Dynamic Scaling of Two-Dimensional Polar Flocks, Phys. Rev. Lett.132, 268302 (2024), arXiv:2403.03804 [cond-mat.stat-mech]

    arXiv 2024

  11. [11]

    Ikeda, Minimum scaling model and exact exponents for the Nambu-Goldstone modes in the Vicsek model, Physical Review Letters133, 258301 (2024)

    H. Ikeda, Minimum scaling model and exact exponents for the Nambu-Goldstone modes in the Vicsek model, Physical Review Letters133, 258301 (2024)

    2024

  12. [12]

    L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ramaswamy, and J. Toner, The inconvenient truth about flocks, arXiv e-prints 10.48550/arXiv.2503.17064 (2025), arXiv:2503.17064 [cond-mat.soft]

    work page doi:10.48550/arxiv.2503.17064 2025

  13. [13]

    The inconvenient truth about flocks

    H. Chat´ e and A. Solon, Comment on “The inconvenient truth about flocks” by Chen et al, arXiv e-prints , arXiv:2504.13683 (2025), arXiv:2504.13683 [cond-mat.soft]

    arXiv 2025

  14. [14]

    L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ramaswamy, and J. Toner, Response to the comment on The inconvenient truth about flocks by Chat´ e and Solon, arXiv e-prints , arXiv:2505.21602 (2025), arXiv:2505.21602 [cond-mat.soft]

    arXiv 2025

  15. [15]

    The inconvenient truth about flocks

    H. Chat´ e and A. Solon, Continuing the discussion on “The inconvenient truth about flocks” by Chen et al, arXiv e-prints , arXiv:2506.13437 (2025), arXiv:2506.13437 [cond-mat.stat-mech]

    arXiv 2025

  16. [16]

    Toner and Y

    J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: How birds fly together, Phys. Rev. Lett.75, 4326 (1995), arXiv:adap-org/9506001

    Pith/arXiv arXiv 1995

  17. [17]

    Toner and Y

    J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Physical Review E 58, 4828 (1998)

    1998

  18. [18]

    Toner, Y

    J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics318, 170 (2005)

    2005

  19. [19]

    Vicsek, A

    T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letters75, 1226 (1995)

    1995

  20. [20]

    M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Reviews of Modern Physics85, 1143 (2013)

    2013

  21. [21]

    Vicsek and A

    T. Vicsek and A. Zafeiris, Collective motion, Physics Reports517, 71 (2012)

    2012

  22. [22]

    Ginelli and H

    F. Ginelli and H. Chate, Relevance of metric-free interactions in flocking phenomena, Phys. Rev. Lett. 105, 168103 (2010)

    2010

  23. [23]

    A. P. Solon and J. Tailleur, Revisiting the flocking transition using active spins, Phys. Rev. Lett.111, 078101 (2013). 43

    2013

  24. [24]

    A. P. Solon and J. Tailleur, The two-dimensional active ising model, Phys. Rev. E92, 042119 (2015)

    2015

  25. [25]

    Mahault, F

    B. Mahault, F. Ginelli, and H. Chat´ e, Quantitative assessment of the Toner and Tu theory of polar flocks, Phys. Rev. Lett.123, 218001 (2019)

    2019

  26. [26]

    Solon, H

    A. Solon, H. Chat´ e, J. Toner, and J. Tailleur, Susceptibility of Polar Flocks to Spatial Anisotropy, Phys. Rev. Lett.128, 208004 (2022), arXiv:2201.00704 [cond-mat.stat-mech]

    arXiv 2022

  27. [27]

    Phys.26, 043023 (2024)

    Mintu Karmakar, Swarnajit Chatterjee, Raja Paul, and Heiko Rieger, Consequence of anisotropy on flocking: the discretized Vicsek model, New J. Phys.26, 043023 (2024)

    2024

  28. [28]

    P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical dynamics of classical systems, Phys. Rev. A8, 423 (1973)

    1973

  29. [29]

    Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties, Z

    H.-K. Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties, Z. Phys. B23, 377 (1976)

    1976

  30. [30]

    De Dominicis, Dynamics as a substitute for replicas in systems with quenched random impurities, Phys

    C. De Dominicis, Dynamics as a substitute for replicas in systems with quenched random impurities, Phys. Rev. B18, 4913 (1978)

    1978

  31. [31]

    Nicolis, R

    A. Nicolis, R. Rattazzi, and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D79, 064036 (2009), arXiv:0811.2197 [hep-th]

    Pith/arXiv arXiv 2009

  32. [32]

    Nicolis, R

    A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi, Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff, JHEP06, 155, arXiv:1501.03845 [hep-th]

    Pith/arXiv arXiv

  33. [33]

    Helias and D

    M. Helias and D. Dahmen,Statistical Field Theory for Neural Networks, Vol. 970 (2020)

    2020

  34. [34]

    K. T. Grosvenor and R. Jefferson, The edge of chaos: quantum field theory and deep neural networks, SciPost Phys.12, 081 (2022), arXiv:2109.13247 [hep-th]

    arXiv 2022

  35. [35]

    S. V. Isakov, P. Fendley, A. W. W. Ludwig, S. Trebst, and M. Troyer, Dynamics at and near conformal quantum critical points, Phys. Rev. B83, 125114 (2011), arXiv:1012.3806 [cond-mat.str-el]

    Pith/arXiv arXiv 2011

  36. [36]

    P. C. Hohenberg and B. I. Halperin, Theory of Dynamic Critical Phenomena, Rev. Mod. Phys.49, 435 (1977)

    1977

  37. [37]

    Andreu-Vaillo, J

    F. Andreu-Vaillo, J. M. Maz´ on, J. D. Rossi, and J. J. Toledo-Melero,Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, Vol. 165 (American Mathematical Society, Providence, Rhode Island, 2010)

    2010

  38. [38]

    Lions,Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford Lec- ture Series in Mathematics and Its Applications, Vol

    P.-L. Lions,Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, Oxford Lec- ture Series in Mathematics and Its Applications, Vol. 3 (Oxford University Press, Oxford, UK, 1996)

    1996

  39. [39]

    de Boer, J

    J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren, Perfect Fluids, SciPost Phys.5, 003 (2018), arXiv:1710.04708 [hep-th]

    Pith/arXiv arXiv 2018

  40. [40]

    K. T. Grosvenor, N. A. Obers, and S. P. Patil, Hydrodynamics without boost-invariance from kinetic theory: From perfect fluids to active flocks, SciPost Phys.19, 071 (2025), arXiv:2501.00025 [hep-th]

    arXiv 2025

  41. [41]

    V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. i. classical systems, Soviet Journal of Experimental and Theoretical Physics32, 493 (1971)

    1971

  42. [42]

    V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. ii. quantum systems, Soviet Journal of Experimental and Theoretical Physics34, 610 (1972)

    1972

  43. [43]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics6, 1181 (1973)

    1973

  44. [44]

    J. M. Kosterlitz, The critical properties of the two-dimensional XY model, Journal of Physics C: Solid State Physics7, 1046 (1974)

    1974

  45. [45]

    J. M. Kosterlitz, The critical properties of the two-dimensional XY model, Journal of Physics C: Solid State Physics10, 3753 (1977)

    1977

  46. [46]

    J. V. Jos´ e, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Renormalization, vortices, and symmetry- breaking perturbations in the two-dimensional planar model, Physical Review B16, 1217 (1977)

    1977

  47. [47]

    D. J. Amit, Y. Y. Goldschmidt, and G. Grinstein, Renormalisation group analysis of the phase transition in the 2d Coulomb gas, Sine-Gordon theory and XY-model, Journal of Physics A: Mathematical and General13, 585 (1980)

    1980

  48. [48]

    Armas, A

    J. Armas, A. Jain, and R. Lier, Hydrodynamics of thermal active matter, Phys. Rev. E112, 055401 (2025), arXiv:2405.11023 [cond-mat.soft]

    arXiv 2025

  49. [49]

    U. C. T¨ auber,Critical dynamics: a field theory approach to equilibrium and non-equilibrium scaling behavior(Cambridge University Press, 2014). 44

    2014

  50. [50]

    Chapman, L

    S. Chapman, L. Di Pietro, K. T. Grosvenor, and Z. Yan, Renormalization of Galilean Electrodynamics, JHEP10, 195, arXiv:2007.03033 [hep-th]

    arXiv 2007

  51. [51]

    Horava, Quantum Gravity at a Lifshitz Point, Phys

    P. Horava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D79, 084008 (2009), arXiv:0901.3775 [hep-th]

    Pith/arXiv arXiv 2009

  52. [52]

    Visser, Lorentz symmetry breaking as a quantum field theory regulator, Phys

    M. Visser, Lorentz symmetry breaking as a quantum field theory regulator, Phys. Rev. D80, 025011 (2009), arXiv:0902.0590 [hep-th]

    Pith/arXiv arXiv 2009

  53. [53]

    Fujimori, T

    T. Fujimori, T. Inami, K. Izumi, and T. Kitamura, Power-counting and Renormalizability in Lifshitz Scalar Theory, Phys. Rev. D91, 125007 (2015), arXiv:1502.01820 [hep-th]

    Pith/arXiv arXiv 2015

  54. [54]

    Fujimori, T

    T. Fujimori, T. Inami, K. Izumi, and T. Kitamura, Tree-Level Unitarity and Renormalizability in Lifshitz Scalar Theory, PTEP2016, 013B08 (2016), arXiv:1510.07237 [hep-th]

    Pith/arXiv arXiv 2016

  55. [55]

    Huang,Quantum Field Theory: From Operators to Path Integrals(1998)

    K. Huang,Quantum Field Theory: From Operators to Path Integrals(1998)

    1998

  56. [56]

    M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

    1995

  57. [57]

    C. P. Burgess,Introduction to Effective Field Theory(Cambridge University Press, 2020)

    2020

  58. [58]

    Negro and S

    A. Negro and S. P. Patil, An ´Etude on the regularization and renormalization of divergences in primor- dial observables, Riv. Nuovo Cim.47, 179 (2024), arXiv:2402.10008 [hep-th]

    arXiv 2024

  59. [59]

    Amoretti, D

    A. Amoretti, D. K. Brattan, and L. Martinoia, Thermodynamic constraints and exact scaling exponents of flocking matter, Phys. Rev. E110, 054108 (2024), arXiv:2405.02283 [cond-mat.stat-mech]

    arXiv 2024