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REVIEW 3 major objections 4 minor 20 references

Stabilization of long-range order in low-dimensional nonequilibrium $O(N)$ models

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

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Pith's one-line read This paper establishes that long-range order in low-dimensional nonequilibrium O(N) models is stabilized by the interplay of power-law (non-Markovian) bath memory and conserved angular momenta, yielding a Goldstone correlation exponent…

desk verdict A genuinely new mechanism -- non-Markovian dissipation plus conserved angular momentum -- with a solid Gaussian-level exponent, but the nonlinear-stability argument is thin enough that the central LRO claim is not fully closed. read the letter →

arxiv 2507.01959 v1 pith:O6IAPXXC submitted 2025-07-02 cond-mat.stat-mech cond-mat.mes-hallcond-mat.quant-gas

Oriana K. Diessel , Jaewon Kim , Ehud Altman This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mes-hallcond-mat.quant-gas MSC 82B2782C2682C31 PACS 05.40.-a05.70.Ln
keywords non-Markoviandissipationlong-rangeorderMermin-WagnertheoremO(N)modelGoldstonemodesconservedangularmomentumnonequilibriumsteadystateLangevindynamics
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 claims that the equilibrium rule known as the Mermin-Wagner theorem — continuous symmetries cannot break spontaneously in one or two dimensions — can be bypassed in a generic class of driven systems by combining two ingredients: a bath with a long power-law memory and a conserved angular momentum. Memory alone is shown to be insufficient; the conserved charges provide slow diffusive modes that translate temporal memory into effective long-range spatial couplings. The paper derives the momentum-space singularity of Goldstone-mode fluctuations, $\eta_\varphi = 2\alpha_n/(\alpha_n+\alpha_L)$, and shows that it drops below the divergence threshold ($\eta_\varphi<2$ in $d=2$, $\eta_\varphi<1$ in $d=1$) when the bath kernels are non-Markovian and the angular momentum is conserved. Nonlinearities are argued to be irrelevant in the ordered windows, and a fluctuation-dissipation test shows the steady state is genuinely nonthermal, so the mechanism is not a hidden equilibrium ordering.

What carries the argument

The machinery is the coupled set of linearized Langevin equations for the Goldstone modes and the conserved angular-momentum densities, together with the scaling analysis of their Martin-Siggia-Rose (MSR) action, a path-integral reformulation of the stochastic equations. Under rescaling $x\to x/b$, the requirement that all non-diffusive terms stay marginal fixes the field dimensions in Table I and gives the dynamical exponent $z=2/(\alpha_n+\alpha_L)$ and the correlation exponent $\eta_\varphi=2\alpha_n/(\alpha_n+\alpha_L)$. The same scaling powers evaluated on the two leading nonlinear vertices determine the stability of the Gaussian fixed point, which is where the linear prediction becomes a claim about the actual (nonlinear) model.

What would settle it

A direct test is an exact numerical solution of the full nonlinear Langevin equations in $d=1$ with conserved angular momentum and kernels satisfying $\alpha_n<\alpha_L$: measuring $C_{\varphi\varphi}(q)$ and finding it grows faster than $1/q$, or seeing the order parameter vanish with system size, would refute the predicted one-dimensional order. Measuring an exponent different from $2\alpha_n/(\alpha_n+\alpha_L)$ anywhere in the $(\alpha_n,\alpha_L)$ plane would likewise refute the scaling analysis.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the linearized dynamics of a single Goldstone mode $\varphi_\alpha$ and its conserved angular-momentum partner $\ell_\alpha$, with memory kernels $\gamma_n(t)\propto \theta(t)/t^{\alpha_n}$ and $\gamma_L(t)\propto \theta(t)/t^{\alpha_L}$, is governed at the Gaussian fixed point by the exponent $\eta_\varphi = 2\alpha_n/(\alpha_n+\alpha_L)$ for the equal-time correlation $C_{\varphi\varphi}(q)\sim 1/q^{\eta_\varphi}$. Because $\eta_\varphi<d$ is the condition for finite order-parameter fluctuations, order survives in $d=2$ whenever either kernel is sub-Ohmic and in $d=1$ whenever $\alpha_n<\alpha_L$; without angular-momentum conservation the exponent returns to $\eta_\varphi=2$ and the standard divergences apply. The same framework shows the leading nonlinear vertices are irrelevant in the regions where linear analysis allows order, and that the effective temperature of the conserved system vanishes as $|\omega|^{\alpha_L}$ at low frequency, an unambiguous departure from thermal equilibrium.

Load-bearing premise

The prediction assumes that the only long-wavelength modes that matter are the Goldstone mode and its conserved partner, and that the amplitude mode and all other angular-momentum components stay gapped or decoupled; if any of those omitted modes becomes soft or couples resonantly, the exponent and the ordered windows could change.

Editorial extensions

If this is right

  • In $d=2$, any sub-Ohmic memory ($\alpha_n<1$ or $\alpha_L<1$) pushes $\eta_\varphi$ below 2, so Goldstone fluctuations are finite and the broken-symmetry state survives.
  • In $d=1$, order is stable when $\alpha_n<\alpha_L$ since $\eta_\varphi<1$; at $\alpha_n\ge\alpha_L$ the correlations diverge faster than $1/q$ and order is lost.
  • If angular momentum is not conserved ($a=0$), the mechanism fails: $\eta_\varphi=2$ and the Mermin-Wagner divergences reappear in both one and two dimensions.
  • The steady state in the ordered regime is not a Gibbs state; the low-frequency effective temperature vanishes as $|\omega|^{\alpha_L}$, so the long-wavelength modes equilibrate at zero effective temperature.
  • The leading nonlinearities are irrelevant exactly where the linear analysis predicts order, so the Gaussian fixed point is stable and the long-range order should survive beyond the linearized approximation.

Reading between the lines

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

  • A design principle the paper leaves implicit is that one can engineer such order by coupling a low-dimensional system to a bath with a broad, power-law distribution of relaxation times (for example, diffusive wires) while keeping the relevant angular momentum conserved; the exponents $\alpha_n$ and $\alpha_L$ are then set by the bath's spectral density.
  • Since $\eta_\varphi$ varies continuously with $\alpha_n/\alpha_L$, the analysis implies a tunable family of correlation exponents that could be measured in cold-atom or trapped-ion simulators with engineered non-Markovian reservoirs, a connection the paper does not make.
  • The mechanism may also support quasi-long-range order or algebraic phases at the boundary $\alpha_n=\alpha_L$ in $d=1$, and it may persist under weak additional Markovian dissipation, though those regimes are not analyzed in the paper.
  • A natural numerical test beyond the paper is to integrate the full nonlinear Langevin equations and extract $C_{\varphi\varphi}(q)$ across the $(\alpha_n,\alpha_L)$ plane; matching $2\alpha_n/(\alpha_n+\alpha_L)$ would confirm the mechanism, while any deviation would reveal missing soft modes.
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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 proposes a new mechanism for circumventing the Mermin-Wagner theorem in low-dimensional driven O(N) models: the combination of non-Markovian (power-law) dissipation with conserved angular momentum. The authors analyze the linearized Langevin equations for the Goldstone modes and the coupled conserved density, derive by Gaussian scaling the exponent eta_phi = 2 alpha_n/(alpha_n+alpha_L) for the equal-time correlation C_phi_phi(q) ~ q^{-eta_phi}, and conclude that eta_phi<2 gives long-range order in d=2, while eta_phi<1 gives order in d=1 for alpha_n<alpha_L. The phase diagram in Fig. 2 summarizes these regimes. The paper also argues by tree-level power counting that the leading nonlinearities do not destabilize the Gaussian fixed point. The numerical solution of the linearized equations in App. E is reported to agree with the analytic exponent.

Significance. If the result holds, the paper identifies a new and fairly general route to nonequilibrium long-range order, with a parameter-free prediction for the anomalous exponent eta_phi as a function of the two memory exponents, and a concrete physical setup (Fig. 1) of a 2D system coupled to diffusive wires. The analytic exponent is not obtained by fitting, and the agreement with the numerical solution of the linearized equations in App. E is a genuine cross-check. The proposed mechanism is distinct from previously known ones (flocking, shear, anisotropic relaxation). However, the proof that nonlinearities do not alter the Gaussian result is the weakest part: the stability analysis is at tree level only, and one of the key scaling assignments used in that analysis appears to be incorrect. The paper is therefore potentially significant, but the central stabilization claim is not yet fully established.

major comments (3)
  1. [Nonlinearities, Eq. (8) and Table II] The assertion that the fields L_alpha_beta and \tilde L_alpha_beta have the same scaling dimensions as ell and \tilde ell is not correct. Equation (5d) is decoupled from the Goldstone sector; promoting it to a Gaussian MSR action with the same sub-Ohmic kernel gamma_L(t) ~ t^{-alpha_L} yields C_LL(q) ~ const as q -> 0, which implies [L_alpha_beta] = d/2, not alpha_n/(alpha_n+alpha_L) + d/2. Recomputing the scaling dimension of the vertex g \tilde phi_alpha L_alpha_beta phi_beta in Eq. (8) with [L] = d/2 gives [g] = 2 alpha_n/(alpha_n+alpha_L) - d/2. This is positive for alpha_n > alpha_L in d=2 and for alpha_n > alpha_L/3 in d=1, i.e. in regions where the linearized analysis (Fig. 2) predicts long-range order. The paper must either provide a correct derivation of the L_alpha_beta scaling or test the stability of these regions by an explicit one-loop or numerical calculation.
  2. [Nonlinearities and Appendix E] The stability of the Gaussian fixed point is justified only by tree-level power counting of the two operators in Eq. (8). No one-loop renormalization of the quadratic action (6) is presented, and Appendix E verifies only the linearized equations (5), not the full nonlinear Langevin equations (1). Since the central claim of the paper is the stabilization of long-range order, i.e. that nonlinearities do not change the Gaussian exponent (7), an explicit check of the beta-functions for g and g' (or an equivalent numerical simulation) is needed before the conclusion can be accepted.
  3. [Appendix A and Eq. (5)] The truncation that omits the amplitude mode sigma is not fully justified for the nonlinear stability analysis. Appendix A states that eliminating sigma adiabatically 'would lead to new interactions and renormalization of existing interactions,' but the paper does not analyze these terms; it only uses the truncated equations to show that a Markovian perturbation is generated. Because sigma is massive, this omission may be innocent, but in the absence of a one-loop calculation the reader cannot exclude the possibility that sigma generates a relevant operator that shifts eta_phi away from Eq. (7).
minor comments (4)
  1. [Abstract] The phrase 'consider the effect of non-Markovian dissipation, in stabilizing long range order' contains a misplaced comma; it should read 'consider the effect of non-Markovian dissipation in stabilizing long range order.'
  2. [Fig. 2] The caption does not explain the color coding of the LRO regions nor the status of the alpha_n = 0 line; the text discusses both, so a short caption note would help.
  3. [Fig. 5 caption] The phrase 'The gray line indicates the value above which order in d=1 can be sustained' is misleading; since order requires eta < 1, the line at eta = 1 is the threshold above which order is lost, not sustained. Please reword.
  4. [Main text, paragraph before Eq. (7)] The statement 'if alpha_i > 1 we should set it to alpha_i = 1 to obtain the correct scaling' is abrupt; it should reference Appendix B, where the super-Ohmic limit is treated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (7) is a parameter-free scaling result, and the numerical and fitting content is used only as validation or for subleading corrections.

full rationale

The central exponent η_φ = 2α_n/(α_n+α_L) is derived from the Gaussian scaling dimensions in Table I, which are assigned by requiring the terms of the quadratic MSR action (6) to be marginal under x → x/b. This is a power-counting calculation whose inputs are the model parameters α_n and α_L; no fitted parameter is renamed as a prediction. The numerical solution of the linearized equations in App. E is used after the fact to validate the analytic exponent, not to set it, and the fitted prefactors A_φ and B_φ in Fig. 3 describe only the subleading q² term that dominates at the Markovian point and do not enter the long-range-order criterion. The only self-citation of note is Ref. [14] (O. K. Diessel et al.), used for an analogy with equilibrium long-range interactions; it is not load-bearing for the present derivation. The concerning assumption that the omitted amplitude mode σ and the L_{αβ} sector do not change the fixed point is an unproved stability assertion that could affect correctness, but it is not a circular reduction: the paper does not define its output in terms of its input, and a possible mismatch between the scaling dimensions of L_{αβ} and ℓ would be a scaling mistake rather than a self-consistent fit. Accordingly, no circular step can be exhibited under the required quote-and-reduction standard.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central result rests on the choice of power-law memory exponents alpha_n and alpha_L, and on the truncation of the fluctuating dynamics to Goldstone mode plus conserved density. No genuinely new physical entity is introduced; the bath degrees of freedom in App. D are auxiliary.

free parameters (2)
  • alpha_n
    Power-law exponent of the non-Markovian memory kernel for the order parameter; an input parameter of the model, not fitted to data. The whole phase diagram is a function of alpha_n and alpha_L.
  • alpha_L
    Power-law exponent of the non-Markovian memory kernel for the angular momentum; an input parameter, not fitted to data.
assumptions (4)
  • domain assumption The dynamics of the O(N) order parameter and its conjugate angular momentum obey the non-Markovian Langevin equations (1) with power-law memory kernels gamma(t)=theta(t)/t^alpha.
    This is the starting model; it is phenomenological and not derived from a microscopic Hamiltonian in the paper.
  • standard math The MSR path integral and Gaussian fixed point scaling analysis give the correct infrared exponents for the linearized equations.
    These are standard tools in critical dynamics; the paper applies them without proof of convergence.
  • domain assumption The system is genuinely out of equilibrium in the low-frequency limit, as shown by the violation of the fluctuation-dissipation relation (App. C) and the absence of thermal symmetry (App. D).
    The proofs in the appendices rely on the linearized or enlarged Markovian representation of the model.
  • ad hoc to paper The amplitude mode and the non-conserved angular momentum components decouple and can be neglected in the long-wavelength analysis.
    This truncation is used in Eqs. (5) and App. A; its validity is not fully established.

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Pith. "Pith review of Stabilization of long-range order in low-dimensional nonequilibrium $O(N)$ models." pith.science (2026). https://pith.science/paper/O6IAPXXC

@misc{pith2026250701959,
  author       = {Pith},
  title        = {Pith review of: Stabilization of long-range order in low-dimensional nonequilibrium $O(N)$ models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6IAPXXC}},
  note         = {Machine review of arXiv:2507.01959}
}
abstract

It is now well established that the Mermin-Wagner theorem can be circumvented in nonequilibrium systems, allowing for the spontaneous breaking of a continuous symmetry and the emergence of long-range order in low dimensions. However, only a few models demonstrating this violation are known, and they often rely on specific mechanisms that may not be generally applicable. In this work, we identify a new mechanism for nonequilibrium-induced long-range order in a class of $O(N)$-symmetric models. Inspired by the role of long-range spatial interactions in equilibrium, consider the effect of non-Markovian dissipation, in stabilizing long range order in low-dimensional nonequilibrium systems. We find that this alone is insufficient, but the interplay of non-Markovian dissipation and slow modes due to conservation laws can effectively suppress fluctuations and stabilize long-range order.

Figures

Figures reproduced from arXiv: 2507.01959 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics of a system that is coupled to a non [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Phase diagram of the model ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Dependence of the prefactor [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Definition of Feynman rules for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Extracted correlation exponent [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

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