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Coupling an Ising order parameter to a passive conserved density drives a new non-equilibrium critical class in which correlation and response split at large scales.

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load-bearing objection Solid dynamical RG paper that cleanly identifies a new non-equilibrium fixed point with an observable FDT-violating signature and all-orders exact relations; the only real caveat is the usual ε-expansion uniqueness assumption the authors already flag.

arxiv 2607.02667 v1 pith:VH5NKLZZ submitted 2026-07-02 cond-mat.stat-mech cond-mat.soft

Non-equilibrium phase transition in the Brownian Ising Model: field theory, renormalization group, and exact results

Mattia Scandolo , Luca Di Carlo This is my paper
classification cond-mat.stat-mech cond-mat.soft
keywords Brownian Ising modelnon-equilibrium critical phenomenarenormalization groupfluctuation-dissipation violationconserved densityuniversality classMartin-Siggia-Roseepsilon expansion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper analyzes the Brownian Ising model: a Z2 order parameter whose local mass is shifted by a freely diffusing conserved density, with no feedback from the order parameter onto the density. That one-sided coupling breaks detailed balance and, under renormalization-group flow below four dimensions, is relevant. It drives the system to a new infrared fixed point, distinct from both ordinary Ising and quenched diluted-Ising criticality. At the new fixed point the density acts as effective noise that is white in time but long-range in space, producing a negative anomalous dimension for correlations and, crucially, unequal anomalous dimensions for correlation and response—an observable signature of fluctuation-dissipation violation that equilibrium fixed points cannot produce. Exact identities from the linear density dynamics and an emergent shift symmetry yield the relation ν = 2/(d+z−2) and prove that both Ising and diluted-Ising fixed points are unstable in three dimensions for any nonzero diffusion constant, so the new class is the unique large-scale attractor.

Core claim

A passive conserved density coupled only through the mass term of a Z2 order parameter is renormalization-group relevant below four dimensions and drives the system to a new non-equilibrium fixed point (the BIM fixed point). There the correlation and response functions acquire different anomalous dimensions η ≠ η′, the density generates long-range-in-space white-in-time noise that makes η negative, and the exact scaling relation ν = 2/(d+z−2) holds. Both the pure Ising and diluted-Ising fixed points are unstable in d = 3, establishing the BIM fixed point as the unique infrared attractor for any nonzero diffusion constant.

What carries the argument

Martin–Siggia–Rose field theory with an ε = 4−d expansion, closed by all-orders identities among renormalization factors that follow from the linearity of the density equation and an emergent shift symmetry; these identities fix the β-functions, give the exact relation ν = 2/(d+z−2), and produce a modified Harris criterion that rules out Ising and diluted-Ising stability in three dimensions.

Load-bearing premise

The claim that the BIM fixed point remains the unique attractor all the way down to three dimensions assumes that no extra fixed point appears at some intermediate dimension between three and four.

What would settle it

In a large three-dimensional lattice realization of the Brownian Ising model, measure whether the equal-time correlation exponent η and the integrated-response exponent η′ stay unequal as system size grows, whether pure Ising and diluted-Ising scaling are ruled out, and whether the measured ν and z obey ν = 2/(d+z−2).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any three-dimensional microscopic model with BIM structure and nonzero diffusion should flow to BIM exponents rather than Ising or diluted-Ising exponents.
  • A measured splitting η ≠ η′ between equal-time correlations and integrated response is a direct large-scale signature of broken fluctuation-dissipation that equilibrium classes cannot produce.
  • The correction-to-scaling exponent is small (ω ≈ 0.02 in d = 3), so finite-size corrections decay slowly and asymptotic exponents require very large systems or careful correction accounting.
  • The exact relation ν = 2/(d+z−2) reduces the number of independent exponents and lets improved measurements of z fix ν at two-loop order.
  • The quenched (diluted-Ising) fixed point is unstable to any finite defect motility, so frozen disorder is only a transient regime.

Where Pith is reading between the lines

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

  • Bistable motile-particle systems and catalyst-modulated reaction networks are natural experimental platforms in which to hunt for negative η and a correlation–response split.
  • Functional renormalization-group or large-scale numerical studies are the direct way to test whether an intermediate fixed point appears between three and four dimensions.
  • The same one-sided density coupling could generate analogous non-equilibrium classes for other order-parameter symmetries beyond Z2.
  • Slow crossover from mean-field or Ising-like preasymptotics may explain why some active-matter simulations still look Ising-like at currently accessible sizes.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The manuscript presents a complete Martin–Siggia–Rose field theory and ε=4−d renormalization-group analysis of the Brownian Ising Model (BIM): a Z2 order parameter ψ coupled nonreciprocally to a passive conserved density ρ that breaks detailed balance. Power counting establishes dc=4; one-loop β- and γ-functions identify a new IR-stable BIM fixed point (XR*=0, uR*=ε/6, f̃R*=ε/8) distinct from Model A and quenched (diluted) Ising. Critical exponents are obtained to lowest nontrivial order, with a dedicated two-loop calculation of z, η′ and ω at XR=0. Exact all-orders Z-factor identities from density linearity and an emergent shift symmetry yield the scaling relation ν=2/(d+z−2) and a modified Harris criterion that rules out both Ising and diluted-Ising fixed points in d=3 for any nonzero diffusion constant. A defining signature is the FDT-violating splitting η≠η′ already at one loop, together with a small correction-to-scaling exponent ω≈0.020 in d=3.

Significance. If the results hold, the paper identifies a clean, structurally robust non-equilibrium route out of Ising universality that has no equilibrium analog and is controlled by a conserved density without feedback. The exact Z-factor identities, the modified Harris criterion, and the relation ν=2/(d+z−2) are all-orders statements that go beyond the ε-expansion and immediately constrain d=3 physics using known Model-A and diluted-Ising exponents. The predicted splitting η≠η′ and the negative η are sharp, falsifiable signatures of large-scale FDT violation. Explicit Feynman rules, Z-factor expressions, and the two-loop diagrams for Γψψ̃ at XR=0 (Appendices A–B) make the calculation reproducible and provide concrete benchmarks for numerics and NPRG studies. The small ω is a useful practical warning for finite-size tests.

minor comments (5)
  1. In Sec. IV G and Table II the two-loop expressions for z, η′ and ω are given; a short numerical evaluation at ε=1 (already stated in the text) could be repeated in the table caption for reader convenience.
  2. Fig. 4 is a qualitative sketch of the full (XR,f̃R) flow. A sentence clarifying that the crossover between quenched and BIM fixed points is not captured by the standard Callan–Symanzik scheme (as noted in Sec. IV H) would help readers who might otherwise expect a continuous trajectory on that plane.
  3. The companion short paper [26] is cited for the summary of results; a single sentence in the introduction stating which results are new to the present long manuscript versus the companion would improve self-containedness.
  4. Notation: the same symbol τ is used both for the reduced temperature and for timescales (τρ, τψ). A brief local reminder when the timescale ratio is introduced (Sec. II D) would avoid momentary confusion.
  5. Appendix A 2 lists the one-loop Z-factor relations; a parenthetical cross-reference to the exact all-orders identities of Sec. V A would make the logical continuity clearer.

Circularity Check

0 steps flagged

No significant circularity: exponents, exact relations, and stability criteria are derived from the MSR action, one-/two-loop diagrams, and Z-factor identities, not forced by definition or load-bearing self-citation.

full rationale

The paper constructs the BIM from hydrodynamic equations (Sec. II), casts them into the MSR action (Sec. III), performs standard power-counting and Callan–Symanzik renormalization (Secs. IV–V), evaluates one- and two-loop diagrams for the anomalous dimensions and β-functions, and extracts fixed points and exponents. The exact relation ν = 2/(d + z − 2) follows directly from β_f̃* = 0 at finite f̃* together with the all-orders identities Z_f = Z_ρ and Z_f = Z_τ² that arise from density-sector non-renormalization and the emergent shift symmetry; it is not assumed as an input. Stability of the BIM fixed point and instability of Ising/diluted-Ising fixed points in d = 3 likewise follow from the closed-form β-functions plus literature values of Model-A/diluted-Ising exponents, not from a self-referential uniqueness theorem. Self-citations ([26] companion summary, [35] prior hydrodynamics) supply motivation or a short-form preview and are not required for the diagrammatic or exact results. No parameters are fitted to data and then re-presented as predictions; no ansatz is smuggled in via citation; no known empirical pattern is merely renamed. The sole disclosed limitation (possible non-perturbative fixed points for 3 < d* < 4) is external to the derivation chain and does not create circularity. Score 1 reflects only the minor, non-load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 1 invented entities

The central claim rests on standard dynamical field theory (MSR, ε-expansion, minimal subtraction) plus two structural domain assumptions about the BIM: (i) the density equation is linear and receives no ψ-dependent current, and (ii) near the BIM fixed point an emergent shift symmetry (ρ, τ) → (ρ+ϕ, τ−ϕ) holds. No free parameters are fitted to data. The BIM itself is a modeling idealization of existing microscopic systems rather than a new particle or force.

axioms (6)
  • standard math Martin–Siggia–Rose/Janssen–De Dominicis path-integral representation of the Langevin equations is valid for the BIM hydrodynamics.
    Used throughout Sec. III to define the action and 1PI vertices; standard for dynamical critical phenomena.
  • standard math ε = 4 − d expansion with minimal subtraction controls the IR fixed-point structure near the upper critical dimension.
    Power counting in Sec. III E sets dc = 4; all β-functions and exponents are organized as series in ε.
  • domain assumption The density current carries no ψ-dependent terms (no feedback), so Γρ̃ψψ = 0 to all orders and the density sector does not renormalize.
    Defining structural feature of BIM (Sec. II, Sec. V A 1); yields exact Zρ̃ = Zρ−1, Zw = Zλ, Zf = Zρ.
  • domain assumption Near the BIM fixed point the renormalized theory is invariant under the simultaneous shift ρ → ρ + ϕ, τ → τ − ϕ.
    Emergent Ward identity used in Sec. V A 2 to obtain Zf = Zτ2 and the exact ν–z relation; stated not to hold at the quenched fixed point.
  • domain assumption Multiplicative noise corrections √(2D0 δρ) and density dependence of λ, ũ, λ̃ beyond the mass coupling g0 are RG-irrelevant near d = 4.
    Sec. II B; used to truncate the hydrodynamic equations to the BIM form before the RG analysis.
  • ad hoc to paper No additional non-perturbative fixed point appears for intermediate dimensions 3 < d* < 4 that would capture the flow away from the perturbative BIM fixed point.
    Explicitly acknowledged in Introduction and Sec. VI as beyond the scope of the ε-expansion; required for uniqueness of BIM as IR attractor in d = 3.
invented entities (1)
  • Brownian Ising Model (BIM) fixed point / universality class no independent evidence
    purpose: Names the new IR-stable non-equilibrium fixed point of the one-sided density–order-parameter coupling and organizes its exponents and FDT-violation signature.
    The fixed point is an output of the RG calculation, not an extra postulated mediator. Independent evidence would come from measuring η ≠ η′ and the predicted exponents in microscopic realizations (active Ising weak-advection, catalyst-modulated reactions).

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read the original abstract

We present a complete field-theoretical renormalization-group (RG) analysis of the Brownian Ising Model (BIM), in which a $\mathbb{Z}_2$ order parameter is coupled to a passive conserved density, breaking detailed balance. Using the Martin-Siggia-Rose formalism and an $\epsilon=4-d$ expansion, we show that this density-order parameter coupling is RG-relevant below four dimensions and drives the system to a new non-equilibrium fixed point, distinct from the Ising universality class. Critical exponents are computed at lowest nontrivial order, some of which require a dedicated two-loop analysis. At large scales, the density acts as an effective noise that is white in time but long-range in space, enhancing order-parameter fluctuations and producing a negative anomalous dimension $\eta$. A defining feature of the new class is that the correlation and response functions acquire different anomalous dimensions, $\eta \neq 2 - \gamma / \nu$ - a direct, observable signature of fluctuation-dissipation-theorem violation at large scales that cannot occur in equilibrium. We also find a small correction-to-scaling exponent, implying large preasymptotic corrections that must be accounted for in numerical and experimental tests. We further derive a set of relations among renormalization factors that hold to all orders in perturbation theory, following from the linearity of the density dynamics and an emergent shift symmetry. These yield an exact scaling relation $\nu = 2/(d+z-2)$ at the BIM fixed point and establish that the Ising universality class, as well as that of quenched diluted-Ising, is unstable in $d=3$. This establishes the BIM fixed point as the unique infrared attractor for any nonzero diffusion constant.

Figures

Figures reproduced from arXiv: 2607.02667 by Luca Di Carlo, Mattia Scandolo.

Figure 1
Figure 1. Figure 1: FIG. 1. One-loop RG flow in the ( [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (A) shows an example of such diagrams; only two diagrams survive, shown in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. One-loop diagrams contributing to the four-point [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

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Reference graph

Works this paper leans on

68 extracted references · 1 linked inside Pith

  1. [1]

    In the absence of external potentials or alignment forces, the rel- evant dynamics is purely diffusive

    The conserved density The conserved densityρ(x, t) evolves autonomously, without any feedback from the ordering fieldψ. In the absence of external potentials or alignment forces, the rel- evant dynamics is purely diffusive. The continuity equa- tion reads ∂tρ=−∇·J,J=−D 0 ∇ρ+ p 2D0ρζ,(1) whereD 0 is the bare diffusion constant andζ(x, t) is a Gaussian whit...

  2. [2]

    The order parameter and the RG-relevance of the mass coupling In the absence of density fluctuations (uniform and staticρ=ρ 0), the order parameterψof a bistable system is expected to relax according to the standard Model A dynamics [20]. Performing a Landau expansion inψand ∇ψnear the ordering transition yields λ−1 0 ∂tψ=∇ 2ψ−r 0ψ−˜u0ψ3 + q 2˜λ0λ−2 0 ξ ,...

  3. [3]

    The timescale ratio Near the critical point, bothψandδρare hydro- dynamic slow variables whose characteristic frequencies vanish asq→0. The order parameterψundergoes crit- ical slowing down, with modes at wave-vectorqrelaxing on a timescale τψ ∼λ −1 0 Λz−2 q−z ,(11) where Λ is the UV momentum scale (e.g., inverse lattice spacing) andzis the dynamic critic...

  4. [4]

    This would be the case, for example, ifz <2 andqis sufficiently small com- pared to Λ, or ifw 0 → ∞

    The quenched-disorder limit Wheneverw 0(q/Λ)z−2 ≫1, the density modes relax on timescales much longer thanτ ψ. This would be the case, for example, ifz <2 andqis sufficiently small com- pared to Λ, or ifw 0 → ∞. From the perspective of the order parameter, in this regime the density is effectively frozen. In other words, when observed on timescales t∼τ ψ,...

  5. [5]

    This would be the case, for example, ifz >2 andq is sufficiently small compared to Λ, or ifw 0 →0

    The fast-diffusion limit and the effective noise In the opposite regime,w 0(q/Λ)z−2 ≪1, density fluc- tuations relax much faster than the order parameter. This would be the case, for example, ifz >2 andq is sufficiently small compared to Λ, or ifw 0 →0. Set- tingw 0 = 0 in Eq. (4) drivesC ρ →0, suggesting thatδρ decouples fromψwheneverw 0(q/Λ)z−2 ≪1. Alth...

  6. [6]

    TheGaussianfixed point (u ∗ R = ˜f ∗ R = 0) controls the critical behavior ford >4; belowd c = 4 it is unstable in both theu R and ˜fR directions

    The solutions are collected in Table I. TheGaussianfixed point (u ∗ R = ˜f ∗ R = 0) controls the critical behavior ford >4; belowd c = 4 it is unstable in both theu R and ˜fR directions. The value ofX ∗ R is undefined at this point, sinceβ X vanishes trivially when ˜fR = 0. TheModel Afixed point ( ˜f ∗ R = 0,u ∗ R =ϵ/9) repro- duces the Ising universality...

  7. [7]

    III C): it couples only bi- linearly toρthrough the Gaussian density propagator

    Non-renormalization of the density sector The response field ˜ρdoes not appear in any interac- tion vertex of the action (Sec. III C): it couples only bi- linearly toρthrough the Gaussian density propagator. Consequently, there are no 1PI diagrams with ˜ρas an external leg, and the 1PI functions Γ ρ˜ρand Γ ˜ρ˜ρreceive no perturbative corrections beyond tr...

  8. [8]

    The effective action, 14 containing only the relevant operators, is invariant under the simultaneous uniform shift ρ→ρ+ϕ , τ→τ−ϕ ,(104) for any constantϕ

    Shift symmetry A further exact relation follows from a shift symmetry of the renormalized field theory. The effective action, 14 containing only the relevant operators, is invariant under the simultaneous uniform shift ρ→ρ+ϕ , τ→τ−ϕ ,(104) for any constantϕ. This holds because the coupling be- tweenψandρenters only through the mass termτ ˜ψψ, and the dens...

  9. [9]

    Instability of the quenched fixed point The exact relation (103) implies that theβ-function forX R can be written in closed form, see Eq. (89). At any fixed point,z= 2 +γ ∗ λ, andβ ∗ X = 0 requires β∗ X =X ∗ R(X ∗ R −1)(z−2) = 0.(106) The three solutions determine the allowed fixed-point val- ues ofz:    z >2 ifX ∗ R = 0, z= 2 if 0< X ∗ R <1, z <2 if...

  10. [10]

    UsingZ f =Z 2 τ one finds β ˜f = ˜fR(ϵ−2γ τ + (XR −1)γ λ),(108) exact to all orders

    Instability of the Ising fixed point and modified Harris criterion The exact relation (105) allows us to write theβ- function for ˜fR in closed form. UsingZ f =Z 2 τ one finds β ˜f = ˜fR(ϵ−2γ τ + (XR −1)γ λ),(108) exact to all orders. At a fixed point,γ ∗ τ = 2−ν −1 and γ∗ λ =z−2, so β∗ ˜f =ν −1 ˜f ∗ R[2−dν+ν(X ∗ R −1)(z−2)].(109) The stability of a ˜f ∗ ...

  11. [11]

    This fixed point is however unstable with respect to finite defect motilityX R ̸= 1, since z >2 at the quenched fixed point

    controls the critical dynamics. This fixed point is however unstable with respect to finite defect motilityX R ̸= 1, since z >2 at the quenched fixed point. The BIM fixed point (red circle) is the only stable fixed point in both theX R and ˜fR directions, and controls the critical behavior of the Brownian Ising model ind <4. VI. DISCUSSION AND CONCLUSIONS...

  12. [12]

    Feynman rules Diagrammatically, the bare propagators and correla- tors in Eqs. (32)–(35) will be represented by lines, joining the two respective fields: G0 ψ : ˜ψ(˜k) ψ(˜q) =⟨ ˜ψ(˜k)ψ( ˜q)⟩0 (A1) C0 ψ : ψ(˜k) ψ(˜q) =⟨ψ( ˜k)ψ( ˜q)⟩0 (A2) G0 ρ : ˜ρ(˜k) ρ(˜q) =⟨˜ρ(˜k)ρ( ˜q)⟩0 (A3) C0 ρ : ρ(˜k) ρ(˜q) =⟨ρ( ˜k)ρ( ˜q)⟩0 (A4) The arrow convention follows from th...

  13. [13]

    We then also pro- vide their divergent parts, expressed as poles inϵ= 4−d, and how they are reabsorbed in the renormalization fac- tors

    One-loop divergent diagrams Here we list the diagrammatic contributions up to one loop to the diverging vertex-functions. We then also pro- vide their divergent parts, expressed as poles inϵ= 4−d, and how they are reabsorbed in the renormalization fac- tors. 17 a. Diagrammatic expansion of vertex functions The diagrams contributing to Γ ψ ˜ψ are: Γψ ˜ψ =−...

  14. [14]

    We shall therefore perform the calculation directly atw 0 = 0, but with ˜f0 = f0w0/(1 +w 0) (see Sec

    Two-loop The two-loop calculation is performed to assess the stability of thew ∗ R = 0 fixed point. We shall therefore perform the calculation directly atw 0 = 0, but with ˜f0 = f0w0/(1 +w 0) (see Sec. IV D). In this limit,ρbecomes a random noise with delta-correlations in time. Hence, anyρcorrelation function Cρ : (A20) forces the two connected points to...

  15. [15]

    M. E. Fisher, The theory of equilibrium critical phenom- ena, Reports on Progress in Physics30, 615 (1967)

    1967

  16. [16]

    M. V. Kompaniets and E. Panzer, Minimally subtracted six-loop renormalization ofo(n)-symmetricϕ 4 theory and critical exponents, Physical Review D96, 036016 (2017)

    2017

  17. [17]

    Pelissetto and E

    A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Physics Reports368, 549 (2002)

    2002

  18. [18]

    U. C. T¨ auber,Critical dynamics: a field theory ap- proach to equilibrium and non-equilibrium scaling behav- ior(Cambridge University Press, 2014)

    2014

  19. [19]

    U. C. Tauber and Z. Racz, Critical behavior of o (n)- symmetric systems with reversible mode-coupling terms: Stability against detailed-balance violation, Physical Re- view E55, 4120 (1997)

    1997

  20. [20]

    K. E. Bassler and B. Schmittmann, Critical dynamics of nonconserved ising-like systems, Physical Review Letters 73, 3343 (1994)

    1994

  21. [21]

    Grinstein, C

    G. Grinstein, C. Jayaprakash, and Y. He, Statistical me- chanics of probabilistic cellular automata, Physical Re- view Letters55, 2527 (1985)

    1985

  22. [22]

    ´Odor, Universality classes in nonequilibrium lattice systems, Reviews of Modern Physics76, 663 (2004)

    G. ´Odor, Universality classes in nonequilibrium lattice systems, Reviews of Modern Physics76, 663 (2004)

    2004

  23. [23]

    Di Carlo, Off-equilibrium kinetic ising model: The metric case, Physical Review Research7, 013250 (2025)

    L. Di Carlo, Off-equilibrium kinetic ising model: The metric case, Physical Review Research7, 013250 (2025)

    2025

  24. [24]

    Garc´ es and D

    A. Garc´ es and D. Levis, Phase transitions in single species ising models with non-reciprocal couplings, Jour- nal of Statistical Mechanics: Theory and Experiment 2025, 043205 (2025)

    2025

  25. [25]

    Akritidis, A

    M. Akritidis, A. Garc´ es, A. Vasilopoulos, M. Carosi, D. Levis, and N. G. Fytas, Fate of the ising uni- versality class under nonreciprocal interactions (2026), arXiv:2606.06981

    Pith/arXiv arXiv 2026

  26. [26]

    M. E. Cates and J. Tailleur, Motility-induced phase sep- aration, Annual Review of Condensed Matter Physics6, 219 (2015)

    2015

  27. [27]

    Caballero, C

    F. Caballero, C. Nardini, and M. E. Cates, From bulk to microphase separation in scalar active matter: a pertur- bative renormalization group analysis, Journal of Statis- tical Mechanics: Theory and Experiment2018, 123208 (2018)

    2018

  28. [28]

    Maggi, N

    C. Maggi, N. Gnan, M. Paoluzzi, E. Zaccarelli, and A. Crisanti, Critical active dynamics is captured by a colored-noise driven field theory, Communications Physics5, 55 (2022)

    2022

  29. [29]

    V. K. Akkineni and U. C. T¨ auber, Nonequilibrium crit- ical dynamics of the relaxational models c and d, Phys. Rev. E69, 036113 (2004)

    2004

  30. [30]

    A. B. Harris, Effect of random defects on the critical behaviour of ising models, Journal of Physics C: Solid State Physics7, 1671 (1974)

    1974

  31. [31]

    Khmelnitskii, Second-order phase transition in inho- mogeneous bodies, Zh

    D. Khmelnitskii, Second-order phase transition in inho- mogeneous bodies, Zh. Eksp. Teor. Fiz41, 1960 (1975)

    1960

  32. [32]

    T. C. Lubensky, Critical properties of random-spin mod- els from theϵexpansion, Physical Review B11, 3573 (1975)

    1975

  33. [33]

    Grinstein and A

    G. Grinstein and A. Luther, Application of the renormal- ization group to phase transitions in disordered systems, Phys. Rev. B13, 1329 (1976)

    1976

  34. [34]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys.49, 435 (1977)

    1977

  35. [35]

    A. P. Solon and J. Tailleur, Revisiting the flocking tran- sition using active spins, Physical Review Letters111, 078101 (2013)

    2013

  36. [36]

    A. P. Solon and J. Tailleur, Flocking with discrete sym- metry: The two-dimensional active ising model, Phys. Rev. E92, 042119 (2015)

    2015

  37. [37]

    Chen and G

    L. Chen and G. Pruessner, Numerical investigation of the brownianq= 2 potts model, Physical Review Research 7, 033295 (2025)

    2025

  38. [38]

    J. R. S. Lima, F. W. S. Lima, T. F. A. Alves, G. A. Alves, and A. Macedo-Filho, Diffusive majority-vote model, Physical Review E105, 034105 (2022)

    2022

  39. [39]

    Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics29, 255 (1966)

    R. Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics29, 255 (1966)

    1966

  40. [40]

    Scandolo, J

    M. Scandolo, J. Pausch, M. E. Cates, and L. Di Carlo, 20 Non-equilibrium coupling to a diffusing density breaks ising universality (2026), short companion paper

    2026

  41. [41]

    P. C. Martin, E. Siggia, and H. Rose, Statistical dynamics of classical systems, Physical Review A8, 423 (1973)

    1973

  42. [42]

    H.-K. Janssen, On a lagrangean for classical field dynam- ics and renormalization group calculations of dynamical critical properties, Zeitschrift f¨ ur Physik B Condensed Matter23, 377 (1976)

    1976

  43. [43]

    De Dominicis, C., Techniques de renormalisation de la th´ eorie des champs et dynamique des ph´ enom` enes cri- tiques, J. Phys. Colloques37, C1 (1976)

    1976

  44. [44]

    Vojta and R

    T. Vojta and R. Dickman, Spatiotemporal generalization of the harris criterion and its application to diffusive dis- order, Physical Review E93, 032143 (2016)

    2016

  45. [45]

    Berges, N

    J. Berges, N. Tetradis, and C. Wetterich, Non- perturbative renormalization flow in quantum field the- ory and statistical physics, Physics Reports363, 223 (2002)

    2002

  46. [46]

    Delamotte, An introduction to nonperturbative renor- malization, inRenormalization Group and Effective Field Theory Approaches to Many-Body Systems, edited by A

    B. Delamotte, An introduction to nonperturbative renor- malization, inRenormalization Group and Effective Field Theory Approaches to Many-Body Systems, edited by A. Schwenk and J. Polonyi (Springer, Berlin Heidelberg,

  47. [47]

    49–132, lect

    pp. 49–132, lect. Notes Phys.852

  48. [48]

    Peliti, Path integral approach to birth-death processes on a lattice, Journal de Physique46, 1469 (1985)

    L. Peliti, Path integral approach to birth-death processes on a lattice, Journal de Physique46, 1469 (1985)

    1985

  49. [49]

    Kourbane-Houssene, C

    M. Kourbane-Houssene, C. Erignoux, T. Bodineau, and J. Tailleur, Exact hydrodynamic description of active lat- tice gases, Phys. Rev. Lett.120, 268003 (2018)

    2018

  50. [50]

    Scandolo, J

    M. Scandolo, J. Pausch, and M. E. Cates, Active ising models of flocking: a field-theoretic approach, The Euro- pean Physical Journal E46, 103 (2023)

    2023

  51. [51]

    Grinstein, S.-k

    G. Grinstein, S.-k. Ma, and G. F. Mazenko, Dynamics of spins interacting with quenched random impurities, Physical Review B15, 258 (1977)

    1977

  52. [52]

    M. J. P. Nijmeijer, A. Parola, and L. Reatto, Ferromag- netic order-disorder transition in an ising fluid, Physical Review E57, 465 (1998)

    1998

  53. [53]

    M. J. P. Nijmeijer and J. J. Weis, Monte carlo simulation of the ferromagnetic order-disorder transition in a heisen- berg fluid, Physical Review Letters75, 2887 (1995)

    1995

  54. [54]

    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 Physics26, 063006 (2024)

    2024

  55. [55]

    J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. New- man,The theory of critical phenomena: an introduction to the renormalization group(Oxford University Press, 1992)

    1992

  56. [56]

    M. E. Peskin and F. Halzen,An Introduction to Quan- tum Field Theory, by Michael E. Peskin and Daniel V. Schroeder(Addison-Wesley, 1995)

    1995

  57. [57]

    D. J. Amit and V. Martin-Mayor,Field theory, the renor- malization group, and critical phenomena: graphs to computers(World Scientific Publishing Company, 2005)

    2005

  58. [58]

    Cardy,Finite-size Scaling, Current physics (North- Holland, 1988)

    J. Cardy,Finite-size Scaling, Current physics (North- Holland, 1988)

    1988

  59. [59]

    E. Frey, Crossover scaling functions and an extended min- imal subtraction scheme, Physica A: Statistical Mechan- ics and its Applications221, 52 (1995), proceedings of the Second IUPAP Topical Conference and the Third Taipei International Symposium on Statistical Physics

    1995

  60. [60]

    Cavagna, L

    A. Cavagna, L. Di Carlo, I. Giardina, L. Grandinetti, T. S. Grigera, and G. Pisegna, Renormalization group crossover in the critical dynamics of field theories with mode coupling terms, Physical Review E100, 062130 (2019)

    2019

  61. [61]

    R. Kree, B. Schaub, and B. Schmittmann, Effects of pol- lution on critical population dynamics, Physical Review A39, 2214 (1989)

    1989

  62. [62]

    Hasenbusch, A

    M. Hasenbusch, A. Pelissetto, and E. Vicari, Relaxational dynamics in 3d randomly diluted ising models, Journal of Statistical Mechanics: Theory and Experiment2007, P11009 (2007)

    2007

  63. [63]

    M. Hasenbusch, Dynamic critical exponentzof the three- dimensional ising universality class: Monte carlo simula- tions of the improved blume-capel model, Physical Re- view E101, 022126 (2020)

    2020

  64. [64]

    L. T. Adzhemyan, D. A. Evdokimov, M. Hnatiˇ c, E. V. Ivanova, M. V. Kompaniets, A. Kudlis, and D. V. Za- kharov, The dynamic critical exponent z for 2d and 3d ising models from five-loopϵexpansion, Physics Letters A425, 127870 (2022)

    2022

  65. [65]

    Z. Liu, E. Vatansever, G. T. Barkema, and N. G. Fy- tas, Critical dynamical behavior of the ising model, Phys. Rev. E108, 034118 (2023)

    2023

  66. [66]

    Parisi and N

    G. Parisi and N. Sourlas, Random magnetic fields, su- persymmetry, and negative dimensions, Physical Review Letters43, 744 (1979)

    1979

  67. [67]

    A. J. Bray and M. A. Moore, Scaling theory of the random-field ising model, Journal of Physics C: Solid State Physics18, L927 (1985)

    1985

  68. [68]

    Cardy and U

    J. Cardy and U. C. T¨ auber, Theory of branching and annihilating random walks, Physical Review Letters77, 4780 (1996)

    1996