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REVIEW 3 major objections 4 minor 39 references

New universality classes govern the critical and multicritical behavior of an active Ising model

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The generic critical behavior of the active Ising model is set by a new universality class, not the Wilson-Fisher class, in a one-loop dynamic renormalization-group calculation.

desk verdict A careful one-loop DRG calculation that proposes new universality classes for the active Ising model, but the generic-attractor claim rests on an unshown full-space flow analysis and a real discrepancy between the main-text and Supplemental flow equations. read the letter →

arxiv 2507.06068 v2 pith:7654NJ7F submitted 2025-07-08 cond-mat.stat-mech cond-mat.soft

Matthew Wong , Chiu Fan Lee This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.soft
keywords activeIsingmodeluniversalityclassdynamicrenormalizationgroupepsilonexpansioncriticalphenomenasoftmodenonequilibriumphasetransitionmulticriticalpoints
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

The paper asks whether the spin number density, which is necessarily a soft hydrodynamic mode when spins move according to their orientation, changes the critical behavior of the active Ising model. It argues that it does: at the order-disorder critical point the generic universality class is no longer the Wilson-Fisher class of the equilibrium Ising model. Using a one-loop dynamic renormalization group with an epsilon expansion about four spatial dimensions, the authors find six fixed points, three of them new. One new fixed point, FP I, is stable within the critical manifold and therefore governs generic critical behavior, while the Wilson-Fisher fixed point survives only on a finely tuned multicritical subspace. If correct, this replaces the standard expectation for the critical exponents of active Ising systems.

What carries the argument

The load-bearing object is the pair of coarse-grained equations coupling the mass density $\rho$ to the longitudinal momentum density $\phi$: the conservation law $\partial_t\rho=-\gamma\partial_x\phi+K\nabla_\perp^2\rho$ and the momentum equation with nonlinearities $\lambda\partial_x\phi^2$, $\kappa_2\partial_x\rho^2$, $\alpha_2\rho^2\phi$, and $\beta\phi^3$. The one-loop dynamic renormalization group converts these into flow equations for the four dimensionless couplings $g_\beta$, $g_\lambda$, $g_{\kappa_2}$, $g_{\alpha_2}$, whose fixed points and stability eigenvalues on the critical manifold $\alpha_0=\alpha_1=0$ determine the universality classes.

What would settle it

Integrate the full one-loop flow equations (25)-(28) from many generic initial conditions and look for a stable fixed point with nonzero $g_\lambda$ or $g_{\kappa_2}$; if such a fixed point attracts trajectories, the generic class is not FP I. Alternatively, direct simulation of the active Ising model near criticality in $d=3$ can measure the anisotropic correlation exponents and test whether $\zeta=1$ and $\chi=-1+\epsilon/2$ hold.

Watch

Extended reading notes

Core claim

For an active Ising model whose hydrodynamic description is a polar active fluid with an easy axis, the conserved density field does not decouple from the order-parameter criticality. At one-loop order in the epsilon expansion about the upper critical dimension $d_c=4$, the renormalization-group flow of the four dimensionless nonlinear couplings $g_\beta$, $g_\lambda$, $g_{\kappa_2}$, and $g_{\alpha_2}$ admits six fixed points. Three are known—Gaussian, Wilson-Fisher, and the Hwa-Kardar universality class of self-organized criticality—and three are new. The generic fixed point FP I, located at $(g_\beta^*,g_\lambda^*,g_{\kappa_2}^*,g_{\alpha_2}^*)=(\epsilon/9)(2,0,0,6)$, has one unstable direction within the critical manifold and therefore attracts the generic critical flow. At this fixed point the exponents are $z=2$, $\zeta=1$, $\chi_\phi=\chi_\rho=-1+\epsilon/2$, and $y_{\alpha_0}=2-\epsilon/3$, with the Wilson-Fisher fixed point recovered only after the additional fine tuning $\alpha_2=0$.

Load-bearing premise

The assignment of the generic universality class rests on the claim that the full four-coupling renormalization-group flow drives two of the nonlinear couplings, $\lambda$ and $\kappa_2$, to zero almost immediately; this is asserted from solving the four flow equations simultaneously, but a complete four-dimensional phase portrait is not shown.

Editorial extensions

If this is right

  • Numerical simulations of the active Ising model at its order-disorder critical point should see anisotropic scaling with $\zeta=1$ and roughness exponent $\chi=-1+\epsilon/2$, rather than Wilson-Fisher exponents.
  • The Wilson-Fisher universality class applies only after the density-induced coupling $\alpha_2$ is tuned to zero; generically the soft density mode changes the critical exponents.
  • Fine tuning combinations of $\beta$, $\kappa_2$, and $\alpha_2$ produces a hierarchy of multicritical classes, including the Hwa-Kardar class, a second new universality class, and the Gaussian class.
  • The upper critical dimension remains four, so the one-loop $\epsilon$-expansion results are controlled predictions near $d=4$ and can be refined by higher-loop or nonperturbative methods.

Reading between the lines

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

  • The same mechanism—a conserved density mode coupled to the ordering field—could generically modify Wilson-Fisher criticality in other conserved-species active or driven systems with an easy axis, beyond the specific model studied here.
  • The stability of FP I could be tested directly by integrating the full four-dimensional flow equations from many generic initial conditions; if any trajectory is attracted to a fixed point with $g_\lambda^*$ or $g_{\kappa_2}^*$ nonzero, the generic class would differ from FP I.
  • At physical dimension $d=3$, where $\epsilon=1$, the one-loop roughness exponent becomes negative, suggesting that connecting these predictions to real-space measurements may require a nonperturbative renormalization-group treatment.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies a hydrodynamic active Ising model (TT model with an easy axis) and performs a one-loop dynamic renormalization-group analysis around the upper critical dimension d_c=4. It reports six fixed points, three of which (FP I, FP III, FP V) are claimed to be new universality classes. The central claim is that FP I, not the Wilson-Fisher fixed point, is the generic attractor governing the critical behavior of the active Ising model when the spin-number density is a soft mode. The Supplemental Material contains a detailed one-loop derivation of the flow equations, the fixed-point candidates, and the resulting critical exponents z, ζ, χ_φ, χ_ρ, and y_{α0}.

Significance. If the central claim holds, this is a substantive result: it would show that coupling the Ising order parameter to a conserved density changes the generic critical universality class, which is directly relevant to active Ising models and more broadly to equilibrium-like transitions in dry polar active fluids. The paper also provides a complete one-loop derivation with explicit Feynman-diagram integrals, and the fixed-point table is internally consistent in the reduced subspaces. The identification of FP II with Wilson-Fisher and FP IV with Hwa-Kardar gives useful anchors, and the three new fixed points are plausible and potentially testable by simulation. However, the paper's flagship claim—that FP I is the generic attractor—is not yet supported by a full analysis of the four-coupling flow, and one fixed-point equation appears in two inconsistent forms in the main text and the Supplemental Material.

major comments (3)
  1. [Main text, 'RG fixed points' and SM, 'RG FIXED POINTS'] The claim that FP I is the generic universality class is load-bearing and rests on an unshown assertion. The SM states, 'Solving all four ODE simultaneously indicates the RG flow drives λ and κ2 to zero almost immediately,' but no four-dimensional phase portrait, basin-of-attraction analysis, or eigenvalue spectrum for the 4×4 stability matrix at FP I is provided. This matters because the flow equations (25)–(28) contain the ratios gκ2 gβ/gλ and gλ gα2/gκ2, so the behavior near the gλ=0, gκ2=0 boundary is singular and cannot be inferred by simply setting λ=κ2=0 in the fixed-point equations. The polynomial forms (118)–(121) contain cross terms that control boundary stability, and a stable fixed point with nonzero gλ or gκ2 elsewhere in the full space would change the generic universality class. The Nunst column in Table I is likewise presented without a shown stability calculation. The authors should display the full fixed-point analysis or explicitly reduce the problem by a justified argument.
  2. [Main text Eq. (16) vs. SM Eq. (103)] The one-loop flow of κ2 is written inconsistently. Main-text Eq. (16) gives the correction term −(1/8)gα2 gβ/gκ2, while SM Eq. (103) gives −(1/8)gλ gα2/gκ2 (before the K-absorption step). These two forms lead to different fixed-point conditions for gκ2, and the resolution is not explained. Since the classification of FP I, FP III, and FP IV depends on the subspace gκ2=0 and on the stability along the gκ2 direction, this inconsistency must be resolved before the fixed-point table can be accepted.
  3. [SM, 'EVALUATION OF SCALING EXPONENTS'] The equality χρ=χφ is imposed in all fixed-point evaluations, but the flow equations (14) and (15) give d lnγ/dℓ = z−ζ+χφ−χρ and d lnκ1/dℓ = z−ζ−χφ+χρ. At the quoted fixed-point values (e.g., z=2, ζ=1 for FP I), these equations are not separately satisfied; they both vanish only if χφ−χρ=z−ζ=1, not if χφ=χρ. The paper does not explain whether γ and κ1 are redundant parameters whose scaling can be absorbed by a field redefinition, or whether they are allowed to run. This is a second load-bearing point that affects the exponent table and should be clarified.
minor comments (4)
  1. [SM, 'ACHIEVING CRITICALITY AT THE LINEAR LEVEL'] There is a typo in the first sentence: 'liner stability analysis' should read 'linear stability analysis.'
  2. [SM, 'VARIATION I' and elsewhere] There are several typographical slips in the integral calculations, e.g., 'udner RG flow' after Eq. (228) and 'structure' truncations in Eqs. (82) and (91). These do not affect the final results but should be corrected.
  3. [Main text, Fig. 2] Figure 2 is labeled 'schematic RG flow diagrams,' but the paper makes quantitative claims about which fixed point is the generic attractor. Since the full four-dimensional flow is not shown, a quantitative plot of the reduced flows or at least a table of the full stability eigenvalues would make the figure more informative.
  4. [SM, 'RG FLOW EQUATIONS'] The absorption step that replaces gβ+gβK by 2gβ and sets gα2−gα2K=0 is not fully explained, given that gβK and gα2K differ from gβ and gα2 by a factor K/μ⊥. A brief justification of this approximation (e.g., the μ⊥+K→μ⊥ replacement used in the integrals) would improve transparency.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: RG fixed-point derivation is self-contained from the stated EOM; the generic-FP claim's main weakness is an unshown full-flow stability assertion, which is a correctness gap rather than a circular reduction.

full rationale

The paper's central derivation is a one-loop DRG calculation starting from an explicit active-Ising EOM (Eqs. 29-30), with nonlinear terms selected by power counting, diagrammatic integrals tabulated in the SM, and fixed points solved from the flow equations (114)-(121). No parameter is fitted to data and no prediction is an input renamed. The claim that FP I is the generic UC does rest on the SM assertion, 'Solving all four ODE simultaneously indicates the RG flow drives λ and κ2 to zero almost immediately,' and the displayed equations do not include the 4x4 stability analysis or full phase portrait; there is also an internal inconsistency between MT Eq. (16) and SM Eq. (103) in the κ2 flow term. These are genuine gaps in the demonstration, but they are omitted proofs/correctness risks, not circularity: the reduced two-coupling flow is obtained from a dynamical claim about the paper's own ODEs, not from a definition or fit that already contains FP I. The self-citations for the existence of the critical point ([12,13]) and for the DRG method ([26]) are load-bearing premises but are supported by lattice-Boltzmann simulations and prior independent RG analyses, so they are external evidence rather than circular self-support. The recovery of the Wilson-Fisher, Hwa-Kardar, and Gaussian fixed points provides independent contact with known results.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard hydrodynamic-modeling assumptions and one unproved assertion about the RG flow. No new particles or fitted constants are introduced. The fine-tuning alpha0 = alpha1 = 0 is a condition for criticality, not a fitted parameter.

assumptions (5)
  • domain assumption The active Ising model's hydrodynamics is described by the Toner-Tu model with an easy axis (Eqs. 1-6).
    The paper assumes the coarse-grained dynamics of the AIM are captured by the TT equations with a damping term for perpendicular momentum.
  • domain assumption The perpendicular momentum g_perp is a fast mode and can be adiabatically eliminated (Eq. 7).
    The paper integrates out g_perp using g_perp approximately equal to Gamma^{-1}(-nabla_perp P_perp + f_perp), which requires Gamma to be large compared to relevant frequencies; noise from g_perp is dropped.
  • domain assumption At criticality both alpha0 and alpha1 must vanish.
    Linear stability analysis (SM Eqs. 33-35) shows the hydrodynamic mode becomes soft only when alpha0 = alpha1 = 0; this is a two-parameter fine-tuning.
  • ad hoc to paper The full four-coupling RG flow drives lambda and kappa2 to zero, so the generic fixed point lies in the subspace with lambda = kappa2 = 0.
    Stated in the SM as 'Solving all four ODE simultaneously indicates...' without showing the full phase portrait or fixed-point search; this underpins the identification of FP I as the generic attractor.
  • domain assumption The density and momentum roughness exponents remain equal (chi_rho = chi_phi).
    The paper lists chi_phi/chi_rho as a single column and assumes the two exponents coincide, motivated by the linear relation <rho rho> = (gamma/kappa1) <phi phi>; this is not derived beyond linear order.

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Pith. "Pith review of New universality classes govern the critical and multicritical behavior of an active Ising model." pith.science (2026). https://pith.science/paper/7654NJ7F

@misc{pith2026250706068,
  author       = {Pith},
  title        = {Pith review of: New universality classes govern the critical and multicritical behavior of an active Ising model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7654NJ7F}},
  note         = {Machine review of arXiv:2507.06068}
}
read the original abstract

The Ising model is one of the most well known models in statistical physics, with its critical behavior governed by the Wilson-Fisher universality class (UC). When active motility is incorporated into the Ising model by, e.g., dictating that the spins' directional movements follow their orientations, the spin number density necessarily constitutes a soft mode in the hydrodynamic description, and can therefore modify the scaling behavior of the system. Here, we show that this is indeed the case in a critical active Ising model in which density can impede the system's collective motion. Specifically, we use a perturbative dynamic renormalization group method to the one-loop level to uncover three new UCs, one of which supersedes the Wilson-Fisher UC to become the generic UC that governs the critical behavior of the active Ising model.

Figures

Figures reproduced from arXiv: 2507.06068 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Graphical notation for the Feynman diagrams. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Symmeterised internal momenta loops. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

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Reference graph

Works this paper leans on

39 extracted references · 35 canonical work pages

  1. [1]

    coarse-graining

    the upper critical dimension dc is 4, and 2) the rel- evant nonlinear terms are precisely those appearing in Eq. (30). In particular, there are now four nonlinearities in the EOM, two of which (κ2∂xρ2 and α2ρ2ϕ) result from the incorporation of the density field which, as we shall see, are crucial to the correct description of the critical behavior of the...

    work page

  2. [2]

    K. G. Wilson and M. E. Fisher, Critical Exponents in 3.99 Dimensions, Physical Review Letters 28, 240 (1972)

    work page 1972

  3. [3]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liv- erpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynam- ics of soft active matter, Reviews of Modern Physics 85, 1143 (2013)

    work page 2013

  4. [4]

    J¨ ulicher, S

    F. J¨ ulicher, S. W. Grill, and G. Salbreux, Hydrodynamic theory of active matter, Reports on Progress in Physics 81, 076601 (2018)

    2018

  5. [5]

    A. P. Solon and J. Tailleur, Revisiting the flocking tran- sition using active spins., Physical review letters 111, 078101 (2013)

    work page 2013

  6. [6]

    A. P. Solon and J. Tailleur, Flocking with discrete sym- metry: The two-dimensional active Ising model., Physical review. E, Statistical, nonlinear, and soft matter physics 92, 042119 (2015)

    work page 2015

  7. [7]

    Scandolo, J

    M. Scandolo, J. Pausch, and M. E. Cates, Active Ising Models of flocking: a field-theoretic approach, The Euro- pean Physical Journal E 46, 103 (2023)

    work page 2023

  8. [8]

    Bandyopadhyay, S

    S. Bandyopadhyay, S. Chatterjee, A. K. Dutta, M. Kar- makar, H. Rieger, and R. Paul, Ordering kinetics in the active Ising model, Physical Review E 109, 064143 (2024)

    work page 2024

Show all 39 references
  1. [9]

    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 Letters 75, 1226 (1995)

    1995

  2. [10]

    Gr´ egoire and H

    G. Gr´ egoire and H. Chat´ e, Onset of Collective and Cohe- sive Motion, Physical Review Letters 92, 025702 (2004)

    2004

  3. [11]

    Chat´ e, F

    H. Chat´ e, F. Ginelli, G. Gr´ egoire, and F. Raynaud, Col- lective motion of self-propelled particles interacting with- out cohesion, Physical Review E 77, 46113 (2008)

    2008

  4. [12]

    Bertin, M

    E. Bertin, M. Droz, and G. Gregoire, Boltzmann and hy- drodynamic description for self-propelled particles, Phys- ical Review E 74, 022101 (2006)

    2006

  5. [13]

    Nesbitt, G

    D. Nesbitt, G. Pruessner, and C. F. Lee, Uncovering novel phase transitions in dense dry polar active fluids us- ing a lattice Boltzmann method, New Journal of Physics 23, 043047 (2021)

    2021

  6. [15]

    Jentsch and C

    P. Jentsch and C. F. Lee, Critical phenomena in com- pressible polar active fluids: Dynamical and functional renormalization group studies, Physical Review Research 5, 023061 (2023)

    2023

  7. [16]

    S. K. Schnyder, J. J. Molina, Y. Tanaka, and R. Ya- mamoto, Collective motion of cells crawling on a substrate: roles of cell shape and contact inhibition, Scientific Re- ports 7, 5163 (2017)

    2017

  8. [17]

    Toner and Y

    J. Toner and Y. Tu, Long-Range Order in a Two- Dimensional Dynamical XY Model: How Birds Fly To- gether, Physical Review Letters 75, 4326 (1995)

    1995

  9. [18]

    Toner and Y

    J. Toner and Y. Tu, Flocks, herds, and schools: A quan- 5 titative theory of flocking, Physical Review E 58, 4828 (1998)

    1998

  10. [19]

    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

  11. [21]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics 49, 435 (1977)

    1977

  12. [22]

    Agranov, R

    T. Agranov, R. L. Jack, M. E. Cates, and ´E. Fodor, Ther- modynamically consistent flocking: from discontinuous to continuous transitions, New Journal of Physics 26, 063006 (2024)

    2024

  13. [23]

    K. E. Bassler and B. Schmittmann, Critical Dynamics of Nonconserved Ising-Like Systems, Physical Review Letters 73, 3343 (1994)

    1994

  14. [24]

    Kardar, Statistical Physics of Fields (Cambridge Uni- versity Press, 2007)

    M. Kardar, Statistical Physics of Fields (Cambridge Uni- versity Press, 2007)

    2007

  15. [25]

    Forster, D

    D. Forster, D. R. Nelson, and M. J. Stephen, Large- distance and long-time properties of a randomly stirred fluid, Physical Review A 16, 732 (1977)

    1977

  16. [26]

    Toner, The Physics of Flocking: Birth, Death, and Flight in Active Matter , 1st ed

    J. Toner, The Physics of Flocking: Birth, Death, and Flight in Active Matter , 1st ed. (Cambridge University Press, 2024)

    2024

  17. [27]

    Jentsch and C

    P. Jentsch and C. F. Lee, New Universality Class De- scribes Vicsek’s Flocking Phase in Physical Dimensions, Physical Review Letters 133, 128301 (2024)

    2024

  18. [28]

    Hwa and M

    T. Hwa and M. Kardar, Dissipative transport in open sys- tems: An investigation of self-organized criticality, Physi- cal Review Letters 62, 1813 (1989)

    1989

  19. [29]

    Hwa and M

    T. Hwa and M. Kardar, Avalanches, hydrodynamics, and discharge events in models of sandpiles, Physical Review A 45, 7002 (1992)

    1992

  20. [30]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J.M. Pawlowski, M. Tissier, and N. Wschebor, The nonpertur- bative functional renormalization group and its applica- tions, Physics Reports 910, 1 (2021)

    2021

  21. [31]

    3 32 D µ1/2 x µ3/2 ⊥ Sd−1 (2π)d−1 Λd−4dℓ + 3 32 KD µ1/2 x µ5/2 ⊥ Sd−1 (2π)d−1 Λd−4dℓ # , (83) I1.2 =

    See Supplemental Material at ... for further analytical details. 6 Supplemental Material ACTIVE ISING MODEL As discussed in the main text, our starting point is the TT model [1–4] with an easy axis, arbitrarily chosen to be the x axis, resulting in the reduced equations of mot...

  22. [32]

    (138) The exponential runaway from a fixed point embedded in the critical manifold is characterised by eyα0 dℓ, where yα0 is termed as the instability exponent

    = 0 , =⇒ α∗ 0 ≈ [− 1 3 ϵ + O(ϵ2)]µ⊥Λ2 . (138) The exponential runaway from a fixed point embedded in the critical manifold is characterised by eyα0 dℓ, where yα0 is termed as the instability exponent. Taking α0’s flow equation dα0 dℓ = zα0 + 1 4 (2µ⊥Λ2 − α0)(3gβ + gα2 ) , (139...

  23. [33]

    (160) With the growth equation dδα0 dℓ = 2 − 1 4 (3gβ + gα2 ) δα0 , (161) dδα0 dℓ = 2 − 1 6 ϵ δα0

    = 0 , α ∗ 0 = − 1 6 ϵ + O(ϵ2) µ⊥Λ2 . (160) With the growth equation dδα0 dℓ = 2 − 1 4 (3gβ + gα2 ) δα0 , (161) dδα0 dℓ = 2 − 1 6 ϵ δα0 . (162) The instability exponent is yα0 = 2 − 1 6 ϵ . (163) FP IV FP IV occurs when we tune β, κ2, α2 to zero. The surviving flow equations ar...

  24. [34]

    1 µ1/2 ⊥ Λ 1 α0 − 1 2µ⊥Λ2 # = α0

    = 0 , α 0 = − 1 5 ϵ + O(ϵ2) µ⊥Λ2 (176) With the growth equation, dδα0 dℓ = 2 − 1 4 (3gβ + gα2 ) δα0 , (177) dδα0 dℓ = 2 − 1 5 ϵ δα0 . (178) The instability exponent is, yα0 = 2 − 1 5 ϵ . (179) 19 CALCULA TION OF THE CRITICAL EXPONENTyα0 This section involves solving the β and ...

  25. [35]

    Toner and Y

    J. Toner and Y. Tu, Long-Range Order in a Two-Dimensional Dynamical XY Model: How Birds Fly Together, Physical Review Letters 75, 4326 (1995)

    1995

  26. [36]

    Toner and Y

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

    1998

  27. [37]

    Toner, Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks, Physical Review E 86, 031918 (2012)

    J. Toner, Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks, Physical Review E 86, 031918 (2012)

    2012

  28. [38]

    L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ramaswamy, and J. Toner, The inconvenient truth about flocks (2025), Eprint: arXiv:2503.17064

    2025 arXiv

  29. [39]

    Nesbitt, G

    D. Nesbitt, G. Pruessner, and C. F. Lee, Uncovering novel phase transitions in dense dry polar active fluids using a lattice Boltzmann method, New Journal of Physics 23, 043047 (2021)

    2021

  30. [40]

    Bertrand and C

    T. Bertrand and C. F. Lee, Diversity of phase transitions and phase separations in active fluids, Physical Review Research 4, L022046 (2022)

    2022

  31. [41]

    Kardar, Statistical Physics of Fields (Cambridge University Press, 2007)

    M. Kardar, Statistical Physics of Fields (Cambridge University Press, 2007)

    2007

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Reviewed August 6, 2026 · model on record in the stance chip above.