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REVIEW 3 major objections 5 minor 145 references

Scaling behavior in non-reciprocal and odd conserved dynamics near criticality

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Near the critical point of a non-reciprocally coupled conserved mixture, static and dynamic correlations are controlled by two different diverging length scales: a Wilson-Fisher length set by temperature and a new dynamic length set by…

desk verdict Serious RG paper with a plausible two-length-scale picture; the new dynamic exponent is real at leading order but the O(ε²) claim rests on an unverified scaling-hypothesis step. read the letter →

arxiv 2608.05027 v1 pith:RNSRY7PZ submitted 2026-08-05 cond-mat.soft cond-mat.stat-mechphysics.bio-ph

Martin Kj{o}llesdal Johnsrud , Giulia Pisegna , Ramin Golestanian This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mechphysics.bio-ph
keywords non-reciprocalCahn-HilliardmodelcriticaldynamicsdynamiccorrelationlengthrenormalizationgroupoddmobilityWilson-Fisherexponenttime-reversalsymmetrybreakingactivematter
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 paper studies the non-reciprocal Cahn-Hilliard (NRCH) model, a minimal description of two conserved species that interact without action-reaction symmetry, close to its phase-separation critical point. It argues that static and dynamic correlations are controlled by two distinct diverging length scales: the static correlation length $\xi_s$ follows the equilibrium Wilson-Fisher exponent $\nu = 1/2+\epsilon/10$, while the dynamic correlation length $\xi_d$ diverges with a new exponent $\nu_n = 1/2 + O(\epsilon^2)$ in the manifestly non-reciprocal regime. The non-reciprocal coupling $\alpha_0$, set by catalytic reaction rates, acts as a second tuning parameter alongside temperature, so a system can change its dynamic response without changing its structure. The paper's significance is that it identifies a concrete mechanism—broken time-reversal symmetry combined with a conservation law—by which living or active materials could control the speed of phase separation and pattern formation using metabolic activity.

What carries the argument

The load-bearing object is the pair of correlation lengths defined from the equal-time correlation function and from the total linear response, $\xi_s$ and $\xi_d$. In the linear theory they satisfy $\xi_s = \sqrt{K/r}$ and $\xi_d = \sqrt{(K+i\beta_0)/(r+i\alpha_0)}$, so any nonzero non-reciprocal coupling makes them differ. The argument is carried by a response-field path-integral representation of the NRCH Langevin dynamics, renormalized with a momentum-shell scheme at one loop and a Callan-Symanzik scheme at two loops, using dimensional regularization. The key identity is the potential condition $f_{\rm eq} \propto f_{\rm nr}$, i.e. $\alpha_1/u = \beta_0/K = \alpha_0/r = \theta$, which defines the odd Cahn-Hilliard (OCH) model with odd mobility $\Upsilon_{ab} = \delta_{ab} + \theta \varepsilon_{ab}$; on this line the system has a Boltzmann-like steady state and a fluctuation-dissipation-like identity. The central result follows from the eigenvalues of the RG flow at the OCH fixed point: $r$ and $\sigma = \alpha_0 - \theta r$ are both relevant directions, with $\lambda_r = 2 - 2\epsilon/5$ and $\lambda_\sigma = 2$, producing the scaling hypothesis $\xi_d(\alpha_0,r) = r^{-\nu} G_{\sigma r}(\sigma/r^\Delta)$ with gap exponent $\Delta = \nu/\nu_n = 1+\epsilon/5$.

What would settle it

Simulate the NRCH Langevin equations numerically near the critical point in $d=3$ and measure the decay of the equal-time correlation $C^{ET}(x)$ and of the total linear response $\chi^T(x)$; if $\xi_d$ does not scale as $r^{-\nu_n}$ or $|\alpha_0-\theta r|^{-\nu_n}$ with $\nu_n \approx 1/2$ while $\xi_s \sim r^{-\nu}$ with $\nu \approx 0.6$, the central claim fails. A cheaper check is to recompute the two-loop integrals $H_1$ and $H_2$ with an independent regularization scheme and verify that the OCH fixed point remains unstable with $\omega_\mu = 2\epsilon/5$.

Watch

Extended reading notes

Core claim

Using one-loop momentum-shell and two-loop Callan-Symanzik renormalization group calculations, the paper shows that the NRCH critical point is characterized by two correlation lengths. The equal-time correlation function gives $\xi_s \sim r^{-\nu}$ with the Wilson-Fisher exponent $\nu = 1/2 + \epsilon/10$ and anomalous dimension $\eta = \epsilon^2/50$, matching equilibrium model B. The linear response function instead gives $\xi_d$, which in the effective-equilibrium regime also scales as $r^{-\nu}$, but in the manifestly non-reciprocal regime scales as $r^{-\nu_n}$ or $|\alpha_0-\theta r|^{-\nu_n}$ with the new exponent $\nu_n = 1/2 + O(\epsilon^2)$. This difference is a direct signature of broken time-reversal symmetry: the fluctuation-dissipation theorem no longer ties correlation and response together. The RG analysis identifies an odd Cahn-Hilliard (OCH) fixed point, where the dynamics obeys the potential conditions and has a Boltzmann-like steady state with odd mobility, but this fixed point is unstable; generic systems flow to stable non-reciprocal fixed points NR$\pm$, at which $\alpha_0$ scales as its own relevant direction. Because $\nu_n < \nu$, the dynamic correlation length grows more slowly than the static one, and the crossover between regimes occurs at a length $\xi_\times \sim |K\alpha_0/(\beta_0 r)-1|^{-1/\omega_\mu}$ with $\omega_\mu = 2\epsilon/5$.

Load-bearing premise

The two-loop renormalization-group calculation—in particular the flow of the non-reciprocal stiffness $\beta_0$ and the anomalous dimension $\eta$—is assumed correct without independent numerical or experimental verification; an error in those integrals would change the stability of the OCH fixed point and the value of $\nu_n$.

Editorial extensions

If this is right

  • In the manifestly non-reciprocal regime, the dynamic correlation length diverges with exponent $\nu_n = 1/2+O(\epsilon^2)$, so the response length and the static structure length no longer coincide; this is an observable time-reversal-breaking signature.
  • The non-reciprocal coupling $\alpha_0$ is a relevant tuning parameter, meaning catalytic reaction rates can drive the system across its critical region just as temperature does.
  • Since the dynamic exponent is $z = 4 - \eta$ with $\eta = \epsilon^2/50$, the response time $T_d \sim \xi_d^z$ is extremely sensitive to the tuning parameters, so a modest change in $\alpha_0$ can sharply speed up or slow down the response.
  • In the effective-equilibrium regime the model recovers model B scaling ($\xi_d \sim \xi_s \sim r^{-\nu}$, FDT satisfied), so the new behavior is confined to scales beyond the crossover length $\xi_\times$.
  • The OCH fixed point describes a conserved, equilibrium-like odd-mobility dynamics with a Boltzmann steady state, but it is unstable; generic conserved non-reciprocal systems flow to the NR$\pm$ fixed points at large scales.

Reading between the lines

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

  • Editorial extension: if $\nu_n = 1/2 + O(\epsilon^2)$ holds beyond two loops, then in $d=3$ the dynamic correlation length diverges with a smaller exponent than the static one, so near criticality the response function would appear short-ranged even where equal-time correlations look critical; a measurement separating these two decays would be a sharp test.
  • Editorial extension: the same two-length mechanism should appear in any conserved field theory with broken time-reversal symmetry, for example active model B+ or chiral active matter; the paper's potential-condition analysis suggests the OCH line is the only place FDT-like behavior survives.
  • Editorial extension: the crossover length $\xi_\times$ could be probed experimentally in chemically active colloid or enzyme mixtures by measuring the linear response and the static structure factor simultaneously and locating the scale where their decay lengths split.
  • Editorial extension: a higher-order or non-perturbative RG check of the two-loop integrals (the $\beta_\kappa$ function and $\eta$) is the natural next step; if those integrals are wrong, the stability of the OCH fixed point and the value of $\nu_n$ would change, but the qualitative two-length picture would likely survive.
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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 / 5 minor

Summary. The paper studies the non-reciprocal Cahn-Hilliard (NRCH) model near its critical point using perturbative dynamical renormalization group (RG) techniques, with both one-loop momentum-shell and two-loop Callan-Symanzik calculations. The authors identify a line of fixed points (the odd Cahn-Hilliard, OCH, fixed points) that satisfy a potential condition, and a pair of stable manifestly non-reciprocal fixed points (NR±). They argue that the structural correlation length ξ_s always scales with the Wilson-Fisher exponent ν = 1/2 + ε/10, while the dynamic correlation length ξ_d exhibits multiple scaling regimes: an effective-equilibrium regime where ξ_d ∼ ξ_s, and manifestly non-reciprocal regimes where ξ_d ∼ |σ|^{-ν_n} with ν_n = 1/2 + O(ε²), where σ = α0 − θr. The crossover between regimes is controlled by the gap exponent Δ = 1 + ε/5. The paper also derives two-loop results for the β-function of the nonreciprocal stiffness κ and the anomalous dimension η.

Significance. If the results hold, the prediction of two distinct correlation lengths with different scaling exponents in a conserved, non-reciprocal active system is conceptually important and potentially relevant for biological phase separation controlled by metabolic activity. The paper makes a serious technical effort: it presents two complementary RG schemes, provides explicit two-loop integral evaluations in the appendices, and produces falsifiable predictions (e.g., ξ_d ≠ ξ_s in the NR regimes, with a specific crossover exponent). The RG calculation is internally consistent at leading order, and the identification of the OCH fixed line and its instability is a useful contribution. However, the central claim that the NR-regime dynamic exponent is ν_n = 1/2 + O(ε²) rests on a scaling hypothesis around the OCH fixed point, not on a calculation at the NR± fixed points that actually control the NR regime; the paper itself calls for numerical or higher-order verification. This gap materially limits the current strength of the main claim.

major comments (3)
  1. [§IV B, Eqs. (125)–(126) and Appendix B 3] The exponent ν_n governing ξ_d in the manifestly non-reciprocal regime is taken from the OCH fixed point (Eq. (94), ν_n = 1/λ_σ with λ_σ = 2), but the NR regime is controlled by the NR± fixed points, not by OCH. At NR±, the paper gives only leading-order exponents (Eq. (126): γ'_r = 2ε/5, γ'_α0 = 0) and does not compute the scaling dimension of the σ direction at that fixed point. The crossover scaling form Eq. (90) and its asymptote G(s) ∼ |s|^{-ν_n} are therefore an assumption, not a derivation from the fixed point that governs the NR regime. Since the separation of length scales and the novel exponent are the central results, the authors must either compute the two-loop exponents at NR± (or otherwise derive the asymptote from the RG flow) or clearly state in the abstract and Table I that ν_n in the NR regime is a conjecture. As written, the claim exceeds what the calculation demonstrates.
  2. [Appendix B 3, sentence beginning 'We therefore surmise that X_s ∼ O(1)'] The structural correlation length is asserted to scale as ξ_s ∼ ξ_r = r^{-ν} with no dependence on σ. This is justified only by a surmise that the scaling function X_s(y) is O(1) over the relevant range of y. Because the paper's key qualitative conclusion is that ξ_s and ξ_d behave differently, an unverified O(1) assumption for ξ_s is load-bearing. The authors should at least compute the linear-level correction to X_s(y) and argue why it remains O(1) under renormalization, or explicitly frame the structural-length scaling as part of the same conjecture.
  3. [Table I and Section V] Table I lists ν_n = 1/2 + O(ε²) as a definite result for the αNR and r(S)NR regimes, while Section V admits that this is a conjecture requiring numerical or higher-order verification. This inconsistency between the presentation in the table and the stated status of the result should be resolved. The authors should either provide a derivation of the NR-regime exponent from the NR± fixed points, or mark the NR-regime entries as conjectural and adjust the abstract and conclusions accordingly.
minor comments (5)
  1. [Section III B, Eq. (68)] The notation D_d defined in Eq. (68) is easily confused with the noise amplitude D and the dimension d; consider using a different symbol or explicit wording.
  2. [Figure 2 caption] The caption refers to a 'yellow-and-black line' and a 'black dashed line' but the colors in the figure are described as black-and-yellow and dashed black; please make the descriptions consistent and unambiguous.
  3. [Appendix B 3, Eq. (B24) and Eq. (92)] The definition Δ ≡ 1 + ω_μ ν in Eq. (B24) is followed by Eq. (92) where Δ = ν/ν_n; the equivalence follows from ν_n = 1/λ_σ but is not stated explicitly, which can confuse the reader.
  4. [Throughout] Several references are to arXiv preprints (e.g., [20], [53], [58], [60], [128]); if the journal requires published references, please update where possible.
  5. [Section IV B, Eq. (121)] In Eq. (121), the OCH exponents are written as γ*_r = γ*_α0 = 2ε/5, but the text later notes that α0 and r have different eigenvalues; please clarify that these are the flow functions, not the eigenvalues, and that the eigenvalues differ as given in Eqs. (123)–(124).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scaling exponents are computed from the paper's own RG flow equations and eigenvalue analysis; the scaling hypothesis is confirmed by Callan-Symanzik matching rather than fitted or defined into the result.

full rationale

The central exponents are derived, not fitted: ν=1/λr and νn=1/λσ from the Jacobian eigenvalues (Eqs. 93-94, 81, 123-124), and η=ε²/50 from the two-loop anomalous dimension (Eq. 119). The scaling hypothesis Eq. (90) is presented as a hypothesis, but it is independently re-derived in App. B3 from the Callan-Symanzik equation and the matching condition δμ(ℓr)=σ̂/r̂^Δ, which fixes Δ=1+ωμν and yields ξd=Λ^{-1} r̂^{-ν} X_d(σ̂/r̂^Δ) (Eq. B37). The NR-regime asymptote ξd∼|σ|^{-νn} follows from this scaling form with νn=ν/Δ=1/(λr+ωμ)=1/λσ; it is a consistency relation, not a fit. The paper cites prior work by the same group for the OCH model and restricted FDT (refs. [90,91]), but those results only characterize the OCH fixed point and are not used to fix the exponent values; the β-functions (Eqs. 114-117) and the two-loop integrals are computed in this paper. The caveat in Sec. V that numerical or higher-order verification is needed concerns the robustness of the two-loop/NR-regime conjecture and is an honest limitation, not a circular step.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The central claim rests on standard perturbative RG methods plus a few model-specific assumptions. The most fragile input is the explicit 'surmise' that the structural correlation length scaling function is O(1), and the free parameter θ labels the OCH fixed line and enters the scaling variable σ.

free parameters (1)
  • θ = β_0/K (OCH fixed-line parameter)
    Labels the line of OCH fixed points satisfying α1/u = β0/K = α0/r = θ; the paper states that θ is a free parameter set by the initial conditions (Section III.B). It enters the scaling variable σ = α0 - θr and the crossover line, though the leading universal exponents are independent of θ.
assumptions (4)
  • standard math Validity of perturbative ε-expansion and two-loop RG
    The paper uses Wilson momentum-shell and Callan-Symanzik RG with dimensional regularization and ε-expansion, assuming that the leading and two-loop results capture the fixed-point structure.
  • domain assumption NRCH with SO(2) symmetry and conserved dynamics describes the physical systems of interest
    The model is introduced as a minimal description for enzyme/colloid mixtures; the biological implications in the introduction and conclusion depend on this mapping.
  • ad hoc to paper The structural correlation length scaling function is assumed to be O(1), i.e., X_s ~ 1
    In Appendix B 3, after deriving the dynamic scaling form, the authors 'surmise' that X_s ~ O(1) based on the linear level, which supports the claim ξ_s ~ r^{-ν} in all regimes. This is not derived from the RG flow.
  • domain assumption Identification of the OCH fixed point with effective equilibrium via potential conditions and FDT from cited works [90,91]
    The restricted FDT relation Eq. (76) is taken as a non-perturbative result from prior papers by the same group; it is used to interpret the OCH critical point.
invented entities (1)
  • Odd Cahn-Hilliard (OCH) fixed point / odd mobility independent evidence
    purpose: Identifies the critical point of NRCH with a conserved gradient dynamics with odd mobility, satisfying potential conditions; serves as the unstable fixed point controlling the scaling crossover.
    The OCH model is a special case of NRCH (Eq. 19) with a Boltzmann steady state and restricted FDT (Eq. 76); it makes falsifiable predictions for the scaling exponents and response functions, testable in simulations. Independent experimental evidence is not available, but the predictions are checkable.

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Cite this review

Pith. "Pith review of Scaling behavior in non-reciprocal and odd conserved dynamics near criticality." pith.science (2026). https://pith.science/paper/RNSRY7PZ

@misc{pith2026260805027,
  author       = {Pith},
  title        = {Pith review of: Scaling behavior in non-reciprocal and odd conserved dynamics near criticality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RNSRY7PZ}},
  note         = {Machine review of arXiv:2608.05027}
}
read the original abstract

In recent years, non-reciprocity has been explored as a ubiquitous manifestation of non-equilibrium activity at the microscopic scales for various active matter systems, from mixtures of chemically active enzymes, colloids, and droplets, to engineered light-controlled active colloids and robotic meta-materials. A commonly used minimal model to describe the dynamics of a binary mixture of conserved species with non-reciprocal interactions is the non-reciprocal Cahn-Hilliard (NRCH) model. The model is characterized by a temperature-like tuning parameter, which can trigger phase separation, and a non-reciprocal coupling, which can lead to the formation of spatio-temporal patterns, as it represents an intrinsic source of non-equilibrium activity and breaks parity and time-reversal symmetries. Here, we study the scaling behavior of the NRCH model near the critical point using perturbative dynamical renormalization group techniques. We find that structural and dynamical correlations are controlled by different correlation lengths, both of which diverge at the critical point, but governed by different scaling laws. In particular, while the structural correlations are always controlled by temperature and the classical Wilson-Fisher critical exponent, the dynamical correlation length exhibits multiple scaling regimes in which either temperature or non-reciprocal coupling can dominate as the key tuning parameter, with a new critical exponent characterizing the divergence. The critical point corresponds to a conserved equilibrium-like dynamics with odd mobility, which we denote as the odd Cahn-Hilliard (OCH) model. Our findings may have important implications on how living systems can control phase separation and spatio-temporal pattern formation using the rates of catalytic reactions, and in general metabolism, as control parameters.

Figures

Figures reproduced from arXiv: 2608.05027 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the hierarchical structure of the sys [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The scaling behavior of the NRCH for various values of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The different scaling regimes are connected to different RG fixed-points, OCH and NR [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The flow of the effective tuning-parameter [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The scaling behavior of the Ising universality class, as a function of the two tuning-parameters [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The flow of a small sample of points in parameter space. The points first flow towards the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The function [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The functions [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

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Reference graph

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