Critical dimensions of the $O(n)$ model with spatiotemporally correlated noise in the strongly anticorrelated regime $\rho<-1/2$

The $O(n)$ model driven by noise with power-law spatiotemporal correlations $D(q,\omega)\propto|q|^{-2\rho}|\omega|^{-2\theta}$ violates detailed balance, and its lower critical dimension $d_l$ shifts away from the equilibrium value 2: anticorrelated noise ($\rho<0$, $\theta<0$) lowers it — permitting continuous symmetry breaking in $d\le 2$ — while positively correlated noise raises it. However, the two available determinations of the critical dimensions disagree in the strongly anticorrelated regime $\rho<-1/2$ (with $-1/2<\theta<1/2$): the exactly solved spherical ($n\to\infty$) model gives $d_l=2+2\rho+4\theta$ for model-A dynamics ($2+2\rho+8\theta$ for model-B dynamics) with no saturation in $\rho$, whereas an Imry–Ma-type dimensional analysis of the interacting model gives $d_l=2+2\max(\rho,-1/2)+4\theta$, which saturates once $\rho<-1/2$; at $(\rho,\theta)=(-1,0)$ the two predict $d_l=0$ and $d_l=1$, disagreeing even on whether one-dimensional continuous symmetry breaking is possible. The problem is to identify the origin of this discrepancy, determine the correct scaling of $d_l$ and $d_u$ with $\rho$ in that regime for the interacting model at finite $n$, and settle the related question of when hyperuniformity or giant number fluctuations imply a shifted lower critical dimension. The problem is open, as left unresolved by the source.

Background

In equilibrium, the Mermin–Wagner theorem forbids spontaneous breaking of a continuous symmetry in spatial dimension $d\le 2$ for systems with short-range interactions: long-wavelength fluctuations of the order parameter diverge and destroy long-range order. For the $O(n)$ model — an $n$-component order parameter $\phi=(\phi_1,\dots,\phi_n)$ with the rotation-invariant free energy $F[\phi]=\int dx\,[|\nabla\phi|^2/2+\epsilon|\phi|^2/2+g|\phi|^4/4]$ — this makes $d_l=2$ the lower critical dimension (the dimension below which no ordering transition occurs at any nonzero temperature), while $d_u=4$ is the upper critical dimension above which mean-field theory becomes exact. The spherical model, the $n\to\infty$ limit in which the constraint $\int dx\,|\phi|^2=N$ is imposed only on average through a Lagrange multiplier $\mu$, is exactly solvable in any dimension and reproduces $d_l=2$, $d_u=4$; it is the standard exactly solvable laboratory for statements about critical dimensions.

Out of equilibrium these bounds can fail. The source paper studies the $O(n)$ model under model-A dynamics (purely relaxational dynamics of a nonconserved order parameter, $\partial_t\phi_a=-\Gamma\,\delta F/\delta\phi_a+\xi_a$) and model-B dynamics (the conserved-order-parameter variant, $\partial_t\phi_a=\Gamma\nabla^2\,\delta F/\delta\phi_a+\nabla\cdot\xi_a$), but driven by Gaussian noise whose correlations are power-law in space and time: in Fourier variables $\langle\xi_a(q,\omega)\xi_b(q',\omega')\rangle\propto\delta_{ab}\,\delta(q+q')\delta(\omega+\omega')\,D(q,\omega)$ with $D(q,\omega)=|q|^{-2\rho}|\omega|^{-2\theta}$. Here $\rho=\theta=0$ is equilibrium white noise (the noise then obeys the fluctuation–dissipation theorem and the steady state is the equilibrium Boltzmann distribution); $\rho>0$ or $\theta>0$ gives positively correlated noise whose power grows at long wavelengths and long times; $\rho<0$ or $\theta<0$ gives anticorrelated noise whose integrated power vanishes, $D(0,0)=0$; and $\theta\to 1/2$ approaches a quenched (time-frozen) random field. For any $(\rho,\theta)\neq(0,0)$ detailed balance is violated and the steady state is genuinely nonequilibrium. Anticorrelated spectra of this type arise physically: noise entering through a conserved current has $D\propto q^2$, i.e. $\rho=-1$, which is the effective situation for particles driven by pairwise stochastic forces (as in elastic active solids and tissues) and is closely related to chiral active particles and periodically driven matter.

Two determinations of the critical dimensions exist. First, a dimensional analysis of the interacting $O(n)$ model — in the spirit of the Imry–Ma argument, which weighs the fluctuation induced by the noise in a space-time box, including subleading surface contributions $\sim l^{d-1}$, against the ordering tendency — yields, for model-A dynamics, $d_l=2+2\rho'+4\theta'$ and $d_u=4+2\rho'+4\theta'$, where $\rho'=\max(\rho,-1/2)$ and $\theta'=\max(\theta,-1/2)$: once $\rho<-1/2$ the surface terms dominate the box fluctuation and the dependence on $\rho$ saturates. Second, an exact solution of the spherical model (valid for $-1/2<\theta<1/2$ and any $\rho$), obtained from the self-consistency integral for the Lagrange multiplier, gives $d_l=2+2\rho+4\theta$ and $d_u=4+2\rho+4\theta$ for model-A dynamics — with no saturation in $\rho$ at all. (For model-B dynamics the same two results hold with $4\theta$ replaced by $8\theta$.) The two determinations agree for $\rho>-1/2$ and disagree for $\rho<-1/2$: at $\theta=0$, $\rho=-1$, for instance, the exact result gives $d_l=0$ — continuous symmetry breaking possible in every dimension, including $d=1$ — while the dimensional analysis gives $d_l=1$, forbidding it in one dimension.

The same exact solution shows that the steady-state fluctuations above the transition are anomalous: the static structure factor $S(q)$ (the equal-time two-point correlation function in Fourier space) behaves as $S(q)\sim q^{-2\rho}$ for model-A dynamics ($q^{-2\rho-4\theta}$ for model-B) at small $q$. When $S(q)$ vanishes as $q\to 0$ the system is hyperuniform (HU) — large-scale fluctuations are suppressed below the central-limit expectation; when $S(q)$ diverges the system shows giant number fluctuations (GNF) — fluctuations of the total order parameter in a region of volume $l^d$ grow faster than $l^d$. The natural expectation that HU lowers $d_l$ below 2 and GNF raises it holds in the purely correlated ($\rho>0,\theta>0$) and purely anticorrelated ($\rho<0,\theta<0$) quadrants, but already fails within the exact spherical model in the mixed-sign quadrants (for example $\rho<0$, $\theta>0$ with $2\rho+4\theta>0$ is hyperuniform yet has $d_l>2$), and the general status of this link is unsettled. Since the physically realized anticorrelated cases sit squarely in the disputed regime $\rho<-1/2$, and since a growing body of work on long-range order in driven and active matter uses these predictions as its reference theory, the correct critical dimensions in that regime — and the origin of the disagreement between the two calculations — constitute a well-defined open question.

Problem Statement

Consider the $n$-component $O(n)$ order parameter with free energy $F[\phi]=\int dx\,[|\nabla\phi|^2/2+\epsilon|\phi|^2/2+g|\phi|^4/4]$ with $g>0$, evolving by model-A dynamics

$\partial_t\phi_a(x,t)=-\Gamma\,\frac{\delta F}{\delta\phi_a}+\xi_a(x,t),$

and, in parallel, by model-B dynamics $\partial_t\phi_a=\Gamma\nabla^2\,\delta F/\delta\phi_a+\nabla\cdot\xi_a$, where the Gaussian noise has the power-law spatiotemporal spectrum

$\langle\xi_a(q,\omega)\xi_b(q',\omega')\rangle=2T\Gamma\,\delta_{ab}\,\delta(q+q')\,\delta(\omega+\omega')\,|q|^{-2\rho}|\omega|^{-2\theta},$

with $-1/2<\theta<1/2$ and $\rho$ allowed below $-1/2$ (with $\rho<d/2$ and microscopic cutoffs so that the noise exists; the noise can be generated by convolving white noise with a causal kernel $K$, i.e. one satisfying $K(t<0)=0$, of spectrum $|K(q,\omega)|\sim|q|^{-\rho}|\omega|^{-\theta}$). For model-B dynamics the noise is a $d$-component vector $\xi_a=\{\xi_{a,\mu}\}_{\mu=1,\dots,d}$ whose correlator carries an additional factor $\delta_{\mu\nu}$; the spectrum $D(q,\omega)=|q|^{-2\rho}|\omega|^{-2\theta}$ is that of $\xi$ itself, before the divergence. For $\rho>-1/2$ the exact $n\to\infty$ (spherical-model) critical dimensions and the Imry–Ma-type dimensional analysis of the interacting model agree; for $\rho<-1/2$ they conflict — for model-A dynamics, $d_l=2+2\rho+4\theta$ (exact) versus $d_l=2+2\max(\rho,-1/2)+4\theta$ (dimensional analysis), and likewise $d_u=4+2\rho+4\theta$ versus $d_u=4+2\max(\rho,-1/2)+4\theta$; for model-B dynamics the same formulas hold with $4\theta$ replaced by $8\theta$.

The problem, as left open by Ikeda (Physical Review E 108, 064119 (2023)), is to resolve this conflict. The following linked questions, all left open by the source, concern one common objective — the correct critical-dimension phase diagram $d_l(\rho,\theta)$, $d_u(\rho,\theta)$ of the $O(n)$ model with correlated noise:

(1) The origin of the disagreement for $\rho<-1/2$: which of the two calculations is flawed and why (for instance, whether the surface-term saturation entering the dimensional analysis is an artifact of the noise-scaling argument that does not transfer to the order-parameter fluctuations), or in which well-defined sense both are correct (for instance, non-commutativity of the $n\to\infty$ limit with the strongly anticorrelated regime).

(2) The correct scaling of $d_l$ and $d_u$ with $\rho$ throughout the regime $\rho<-1/2$, $-1/2<\theta<1/2$, for the interacting model at finite $n$ — whether the unsaturated (spherical-model) form, the saturated (dimensional-analysis) form, or a third form holds — stated as the full function of $(\rho,\theta)$ rather than at isolated points. The pair $(\rho,\theta)=(-1,1/8)$ in model-A dynamics, where the two published predictions give $d_l=1/2$ and $d_l=3/2$ so that $d=1$ cleanly separates them, is a natural discriminating test case, but is not a substitute for the general answer. Both model-A and model-B dynamics should be covered, or a reason given why they differ.

(3) As a corollary of the same finite-$n$ theory, the precise relation between hyperuniformity or giant number fluctuations (the sign of the small-$q$ exponent of the static structure factor above the transition) and the shift of $d_l$ away from its equilibrium value 2 — in particular, whether hyperuniformity implies $d_l<2$ and giant number fluctuations imply $d_l>2$ in the mixed-sign quadrants $\rho<0<\theta$ and $\theta<0<\rho$.

Parts (2) and (3) are the observable content of the mechanism identified in part (1).

References

1. Harukuni Ikeda, 'Correlated noise and critical dimensions', Physical Review E 108, 064119 (2023), doi:10.1103/PhysRevE.108.064119, https://doi.org/10.1103/PhysRevE.108.064119 (preprint arXiv:2302.13666, https://arxiv.org/abs/2302.13666).
2. Harukuni Ikeda and Yuta Kuroda, 'Continuous symmetry breaking of low-dimensional systems driven by inhomogeneous oscillatory driving forces', Physical Review E 110, 024140 (2024), doi:10.1103/PhysRevE.110.024140, https://doi.org/10.1103/PhysRevE.110.024140 (preprint arXiv:2304.14235, https://arxiv.org/abs/2304.14235).
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