Kardar-Parisi-Zhang Equation with temporally correlated noise: a non-perturbativerenormalization group approach

Davide Squizzato and Leonie CanetUniv. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporallycorrelated noise. The presence of time correlations in the microscopic noise breaks the statisticaltilt symmetry, or Galilean invariance, of the original KPZ equation with delta-correlated noise(denoted SR-KPZ). Thus it is not clear whether the KPZ universality class is preserved in this case.Conflicting results exist in the literature, some advocating that it is destroyed even in the limit ofinfinitesimal temporal correlations, while others find that it persists up to a critical range of suchcorrelations. Using non-perturbative and functional renormalization group techniques, we study theinfluence of two types of temporal correlators of the noise: a short-range one with a typical time-scale τ , and a power-law one with a varying exponent θ. We show that for the short-range noise withany finite τ , the symmetries (the Galilean symmetry, and the time-reversal one in 1 + 1 dimension)are dynamically restored at large scales, such that the long-distance and long-time properties aregoverned by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZfixed point is still stable for θ below a critical value θth, in accordance with previous renormalizationgroup results, while a long-range fixed point controls the critical scaling for θ > θth, and we evaluatethe θ-dependent critical exponents at this long-range fixed point, in both 1+1 and 2+1 dimensions.While the results in 1 + 1 dimension can be compared with previous studies, no other predictionwas available in 2 + 1 dimension. We finally report in 1 + 1 dimension the emergence of anomalousscaling in the long-range phase.

I. INTRODUCTION

The Kardar-Parisi-Zhang equation [1], originally de-rived to describe stochastic interface growth, standsas a fundamental model in non-equilibrium statisticalphysics to understand scaling and phase transitions out-of-equilibrium, akin the Ising model at equilibrium. Be-yond growing interfaces, the KPZ universality class ex-tends to many very different systems, such as directedpolymers in random media, randomly stirred fluids, par-ticle transport, driven-dissipative Bose-Einstein conden-sates, to cite a few [2–6].

An impressive breakthrough has been achieved in thelast decade regarding the characterization of the KPZuniversality class for a one-dimensional interface, sus-tained by a wealth of exact results [7]. A particularlystriking feature is the discovery of universality sub-classesfor the distribution of the height fluctuations, determinedby the nature of the initial conditions (flat, sharp-wedge,or stochastic), which has revealed a deep connection withrandom matrix theory [8–12]. Moreover, experiments inliquid crystals provided the first set-up to allow for quan-titative measurements of KPZ universal properties, andthey confirmed with a high precision the theoretical re-sults [13, 15].

However, for a higher-dimensional interface, or in thepresence of additional ingredients such as the presenceof correlations of the microscopic noise, the integrabilityof the KPZ equation is broken, and controlled analyti-cal methods to describe the rough phase are scarse. TheNon-Perturbative (also named functional) Renormaliza-tion Group (NPRG) is one of them [14], and is the one weemploy in this work. Our aim is to investigate the effectof temporal correlations in the microscopic noise on the

universal properties of the system. The interest is two-fold. First, strictly uncorrelated processes are a math-ematical idealization, any real physical system is likelyto exhibit some time correlations, at least over a smallfinite timescale. Hence, it is important to understandtheir role and assess the relevance of the delta-correlatedmodel. Second, some physical systems are characterizedby intrinsic long-range temporal correlations. An inter-esting example arises in cosmology, where the KPZ equa-tion with power-law time correlations in the noise wasshown to emerge as an effective model for matter dis-tribution in the Universe, starting from the dynamics ofself-gravitating Newtonian fluids [16, 17]. More generally,long-range time correlations can originate from impuri-ties which do not diffuse and impede the growth of thesurface, or from the coupling of the dynamics to somereservoir which is likely to introduce some memory ef-fects.

Let us now define the model. The original KPZ equa-tion describes the stochastic time evolution of a heightfield h(t, ~x), encompassing a smoothening diffusion anda non-linearity as a key ingredient:

∂th = ν∇2h+λ

2(~∇h)2 + η . (1)

The non-linear term takes into account a lateral growth ofthe height profile which tends to enhance the rougheningof the interface. The noise η is defined as a Gaussiannoise with zero mean and variance

〈η(t, ~x)η(t′, ~x ′)〉 = 2D δ(t− t′)δd(~x− ~x ′) , (2)

where d is the dimension of the interface, moving in a(d + 1)-dimensional space, and D the noise amplitude.

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As already mentioned, these delta correlations are a sim-plification, and this raises the question of the robustnessof the KPZ universality class with respect to the presenceof some microscopic correlations in the stochastic processdriving the growth. This question was first investigatedby Medina et al. [18], who considered the more generalform of noise correlator

〈η(t, ~x)η(t′, ~x ′)〉 = 2D(t− t′, |~x− ~x ′|) (3)

with long-range (LR) power-law correlations, defined inthe Fourier space as

D(ω,~k) = D0 +Dθk−2ρω−2θ . (4)

This modification of the noise structure breaks the in-tegrability of the original KPZ equation with noise (2),that we denote short-range (SR) KPZ. The effect of spa-tially correlated noise has been thoroughly investigated,both analytically and numerically [19–33]. It was shownthat for a SR enough noise, i.e. ρ < ρc, the standardSR-KPZ properties are preserved, while beyond ρc, a LRphase with ρ-dependent critical exponents emerges. Fora noise characterized by a finite correlation length ξ, itwas shown for a one-dimensional interface that the time-reversal symmetry, which is broken by the presence ofthe spatial correlations in the microscopic noise, is re-stored at large distance, and thus one also finds SR-KPZuniversality in this case [34].

In contrast, temporally correlated noise has receivedmuch less attention. The few existing analytical [18, 35–38] and numerical [39–41] studies yield conflicting results.One of the reasons is that the presence of temporal cor-relations is much more severe than spatial ones, in thatit breaks the constitutive KPZ symmetry, which is theGalilean invariance, also known as statistical tilt symme-try. Thus it is not clear a priori whether even an infinites-imal amount of time-correlation destroys or not KPZ uni-versal physics, and both answers have been given. Let ussummarize these results.

The problem of temporal correlations of the mi-croscopic noise was first investigated using DynamicalRenormalization Group (DRG) by Medina et al., focus-ing on d = 1 [18]. They found that the SR-KPZ fixedpoint is stable up to a threshold value θth = 1/6, andthus for θ ≤ θth, the critical exponents are the standardSR-KPZ ones zSR = 3/2 and χSR = 1/2. Above thethreshold θth, they determined from the one-loop flowequations an approximate expression of the critical ex-ponents:

χLR =1 + 4θ

3 + 2θ, zLR = 2− χLR , (5)

obtained by neglecting the corrections on the non-linearity induced by the violation of Galilean invariancedue to the temporal correlations. This expression is thusonly valid for small θ close to the threshold. Indeed, theexact relation χSR + zSR = 2 stemming from Galilean in-variance, in any d, only holds at the SR fixed point, and

is replaced at the LR fixed point by the exact relation

zLR(1 + 2θ)− 2χLR − d = 0 , (6)

which is violated by the estimate (5). The authors thensolved numerically a set of truncated flow equations ind = 1 which led to exponents, that could be approxi-mately fitted by

χLR = 1.69θ + 0.22 , zLR =2χLR + 1

1 + 2θ. (7)

At variance with this scenario, Ma and Ma [35] advocatedon the basis of a Flory-type scaling argument a smoothvariation of the critical exponents as functions of θ, withno threshold, following

χLR =2 + 4θ

2θ + d+ 3, zLR =

2d+ 4

d+ 3 + 2θ, (8)

such that the SR-KPZ exponents are only recovered atθ = 0. This alternative scenario was supported by aSelf-Consistent Expansion (SCE) developed by Katzavand Schwartz [36]. The authors found within the SCEtwo strong-coupling solutions, one which coincides withthe one-loop DRG result, and the other, considered asdominant, which leads to a smooth dependence on θ withno threshold, and with a decreasing zLR(θ), whereas thesolution (7) is increasing.

The problem was re-visited using perturbative func-tional RG within the framework of elastic manifolds incorrelated disorder [37]. In this context, a crossover froma SR behavior to a LR one beyond a certain thresholdwas confirmed. The two-loop LR exponents were calcu-lated in a perturbative expansion in ε = 4 − d where dis the dimension of the elastic manifold. However, theKPZ interface is equivalent to a d = 1 directed polymer,which implies ε = 3, and the extrapolation to such a largevalue is not reliable. Notwithstanding this limitation, thetwo-loop results indicate a decreasing zLR for small θ, atvariance with (7). Based on a stability criterion, the au-thor also derives bounds for the value of zLR in d = 1as

5

3 + 2θ≤ zLR ≤

3

2(9)

where the lower bound coincides with the one-loop result(5). This bound rules out both the second SCE solu-tion and the scaling solution. On the analytical side, thesituation is thus unclear.

On the numerical side, very few attempts exist in theliterature. Among them, Refs. [39, 40] cannot convinc-ingly discriminate between the two scenarii (presence orabsence of a threshold) nor on the sense of variation ofzLR. They essentially find a very weak dependence atsmall θ and are too scattered to settle whether zLR isdecreasing or increasing at larger values of θ. A progressin this direction was recently achieved in Ref. [41], wherethe authors simulate both the KPZ equation and ballis-tic deposition with temporal correlations with improved

3

accuracy. They find no threshold, that is the appearanceof a long-range phase for any non-zero temporal correla-tion. Moreover, they unveil the existence of anomalousscaling at large θ, which they relate to the emergence of“faceting” structures [41].

Note that the effect of temporal correlations is also cru-cial in the context of turbulence. In particular, field the-oretical approaches to turbulence are constructed fromNavier-Stokes equation with a stochastic large-scale forc-ing, which is delta-correlated in time to preserve Galileaninvariance, whereas a physical forcing cannot be com-pletely uncorrelated. The presence of temporal correla-tions in the forcing correlator was investigated in [42],and the results support the robustness of the SR proper-ties below a threshold value.

In this work, we analyze the effect of temporal correla-tions in the microscopic noise of the KPZ equation in theframework of the NPRG. Indeed, this method has turnedout to be successful to describe KPZ interfaces since theNPRG flow equations embed the strong-coupling fixedpoint in any dimensions [43], whereas the latter cannot bereached at any order from perturbative expansions [44].Moreover, a controlled approximation scheme, based onsymmetries, can be devised in this framework [45, 46].It was shown that it reproduces with very high accuracythe exact results in d = 1 for the scaling function [45]. Ityielded predictions for dimensionless ratios in d = 2 and3 [46] which were later accurately confirmed by large-scale numerical simulations [47]. This framework wasextended to study the effect of anisotropy [48], and alsoof spatial correlations in the noise, following a power-law[33] or with a finite length-scale [34].

We here study the influence of temporal correlationsboth in d = 1 and d = 2, and both for a finite correlationtime or for a LR power-law correlator

Dτ (ω,~k) = D0e− 1

2ω2τ2

, D∞(ω,~k) = D0 +Dθω−2θ .(10)

The Dτ correlator is studied to probe whether the SR-KPZ physics is destroyed as soon as Galilean invarianceis broken at the microscopic scale, even on a short finiterange. We find that this is not the case, and we showthat when τ is finite, this symmetry is always restoredat long distance and long time. We then investigate theeffect of the power-law temporal noise D∞, and find thatthe SR-KPZ fixed point is stable below a threshold θth =1/6 in d = 1 and θth ' 0.35 in d = 2. Beyond thisthreshold, a LR fixed-point takes over and we computethe θ-dependent critical exponents in this LR dominatedphase. We find that zLR(θ) is decreasing and satisfy thebound (9) in d = 1. We finally investigate in more detailsthe scaling properties of the LR phase in d = 1, whichshows the presence of anomalous scaling, in agreementwith the results from the numerical simulations of [41].

The remainder of the paper is organized as follows. Webriefly present the KPZ field theory and its symmetriesin Sec. II. We then introduce the NPRG framework, andthe approximation scheme used in Sec. III, and derive the

corresponding flow equations. The results are presentedand discussed in Sec. IV.

II. KPZ FIELD THEORY AND ITSSYMMETRIES

The KPZ equation (1) can be cast into a field the-ory following the standard response functional formalismintroduced by Martin-Siggia-Rose and Janssen-De Do-minicis [49–51]. The KPZ field theory reads

Z[j, j] =

∫D[h]D[h]e−S[h,h]+

∫t,~x

jh+jh

S[h, h] =

∫t,~x

h

(∂th− ν∇2h− λ

2(~∇h)2

)−∫ω,~q

h(−ω,−~q)D(ω, ~q)h(ω, ~q) (11)

where∫t,~x≡∫dtdd~x and

∫ω,~q≡∫dω2π

dd~q(2π)d

. Upon rescal-

ing the time and the fields, one finds that the SR part ofthe KPZ action is characterized by a single dimensionlesscoupling g = λ2D0/ν

3, while the LR correlation intro-duces another dimensionless coupling wθ = Dθ/(D0ν

2θ).These couplings have canonical dimensions

[g] = 2− d, [wθ] = 4θ . (12)

In the absence of temporal correlations, i.e. with a

noise correlator D0(ω,~k) = D0, the KPZ action possessesseveral symmetries. Besides the usual invariance underspace-time translations and space rotations, it is invari-ant under a shift in the height field and a Galilean trans-formation (or tilt of the interface). The latter enforcesthe exact relation z + χ = 2 in any dimension.

In fact, theses last symmetries admit extended forms,which correspond to the following infinitesimal fieldtransformations with time-dependent parameters:

h(t, ~x) −→ h(t, ~x) + c(t) (13)

for the height shift, and for the Galilean transformation

h(t, ~x) −→ h(t, ~x+ λ~ε(t)) + ~x · ∂t~εh(t, ~x) −→ h(t, ~x+ λ~ε(t)) . (14)

The choice ~ε(t) = ~v t yields the standard Galilean trans-

formation (for the velocity field ~∇h, which correspondsto a tilt for the height field). An arbitrary infinitesimal~ε(t) gives a local-in-time, or time-gauged Galilean trans-formation. The time-gauged symmetries (13) and (14)are extended symmetries, in the sense that the KPZ ac-tion is not strictly invariant under these transformations,but the induced variations are linear in the fields. Onecan also derive in the case of extended symmetries Wardidentities which, because of the locality in time of thecorresponding transformations, have a stronger content

4

than their non-gauged versions [45]. These exact identi-ties are very useful to constrain approximations. For ad = 1 interface, there exists an additional discrete sym-metry associated to the time-reversal transformation [52]

h(t, ~x)→ −h(−t, ~x)

h(t, ~x)→ h(−t, ~x) +ν

D0∇2h(−t, ~x) . (15)

Indeed the corresponding variation of the action is δS ∝∫~x(~∇h)2∇2h, which vanishes in one dimension only. The

existence of this additional symmetry in d = 1 in turncompletely fixes the SR-KPZ critical exponents in thisdimension to the values χSR = 1/2 and zSR = 3/2.

The presence of temporal correlations, either of theform Dτ or D∞, breaks the Galilean symmetry in alldimensions, and also the time-reversal symmetry in d =1. The consequences are studied within the NPRG, whichis presented in the next section.

III. NON-PERTURBATIVERENORMALIZATION GROUP FOR KPZ

A. Non-Perturbative Renormalization Groupformalism

Integrating out microscopic fluctuations plays a centralrole in understanding the long-distance and long-timeuniversal properties of a physical system. The NPRGis a modern implementation of Wilson’s original idea ofthe renormalization group [53], conceived to efficientlyaverage over fluctuations, even when they develop at allscales, as in standard critical phenomena [14, 54, 55]. It isa powerful method to compute the properties of stronglycorrelated systems, which can reach high precision levels[56–58], and can yield fully non-perturbative results, atequilibrium [59–61] and also for non-equilibrium systems[43, 45, 52, 62–64], restricting to a few classical statisticalphysics applications.

The progressive integration of fluctuation modes isachieved by introducing in the KPZ action (11) a scale-dependent quadratic term

∆Sκ =1

2

∫ω,~q

φi(ω, ~q)[Rκ(ω, ~q)]i,jφj(−ω,−~q) (16)

where κ is a momentum scale, and φ1 ≡ h, φ2 ≡ h.The matrix elements of Rκ are proportional to a cutofffunction r(q2/κ2), with q = |~q|, which ensures the selec-tion of fluctuation modes: r(x) is required to be largefor x . 1 such that the fluctuation modes φi(q . κ) areessentially frozen and do not contribute in the path inte-gral, and to be negligible for x & 1 such that the othermodes (φi(q & κ)) are not affected. ∆Sκ must preservethe symmetries of the original action and causality prop-erties. For the KPZ field theory, a suitable form is [43]

Rκ(ω, ~q)≡Rκ(~q)=r

(q2

κ2

)(0 νκq

2

νκq2 −2Dκ

), (17)

where the running coefficients νκ and Dκ are definedlater. Here we work with the cutoff function

r(x) = α/(exp(x)− 1) , (18)

where α is a free parameter. In the exact theory, theresults are independent of the precise form of the cut-off function. However, any approximation introduces a(typically small) spurious dependence on this choice. Theparameter α can thus be conveniently used to estimatethe error and optimize the results, as discussed in Ap-pendix C 3.

We emphasize that the regulator (17) does not de-pend on frequency. Whereas it would be desirable toalso regularize in frequency, it is much simpler not to,and it is the actual choice made in most applications tonon-equilibrium systems [62, 65]. It turns out that formost applications, regularizing in momentum is enoughto achieve the separation of fluctuation modes and toensure the analyticity of the flow. The implementationof a frequency regularization was studied in [66] on theexample of Model A, where it was shown that it does im-prove the results. However, the difficulty lies in formulat-ing a regulator which respects both causality and all thesymmetries of the model. For KPZ, the Galilean invari-ance precludes from having a (manageable) frequency-dependent regulator. This has implications for the studyof the power-law correlator D∞, since the latter bringsnon-analyticities in ω which would be cured (as theyshould) by a frequency regularization, whereas with onlya momentum regulator they can survive and have to bedealt with (as explained in the following).

The inclusion of ∆Sκ in (11) leads to a scale-dependentgenerating functional Zκ. Field expectation values in thepresence of the external sources j and j are obtained fromthe functional Wκ = logZκ as

ϕ(x) = 〈h(x)〉 =δWκ

δj(x), ϕ(x) = 〈h(x)〉 =

δWκ

δj(x), (19)

denoting x = (t, ~x). The effective average action is de-fined as the modified Legendre transform of Wκ as

Γκ[ϕ, ϕ] +Wκ[j, j] =

∫jiϕi −

1

2

∫ϕi [Rκ]ij ϕj . (20)

where ji are the sources associated with the fields ϕi,with ϕ1 = ϕ, ϕ2 = ϕ and similarly for ji. The lastterm in (20) ensures that at the microscopic scale Λ, theeffective average action coincides with the microscopicaction Γκ=Λ = S, provided that Rκ is very large whenκ→ Λ [14]. In the opposite limit κ→ 0, the cut-off Rκ isrequired to vanish such that one recovers the standard ef-fective action Γ (which would be Gibbs free energy for anequilibrium system). The scale-dependent effective aver-age action Γκ thus smoothly interpolates between themicroscopic action and the full effective action which en-compasses all the fluctuations. It obeys an exact flowequation, usually referred to as Wetterich equation [67]:

∂sΓκ =1

2Tr

∂sRκ

[Γ(2)κ +Rκ

]−1

(21)

5

where s = ln(κ/Λ) is the renormalization “time” and Γ(2)κ

is the 2× 2 Hessian matrix

[Γ(2)κ ]i,j =

δ2Γκ[ϕ]δϕi δϕj

. (22)

Tr· is the trace over all the internal degrees of freedom.

Eventhough the equation (21) is exact, it cannotbe solved exactly because of its non-linear functionalintegro-differential structure. One has to employ someapproximation scheme [14]. The key advantage of thisapproach is that these approximations do not have tobe perturbative in couplings or in dimensions, but theyare rather based on some controlled truncation of thefunctional space. There exist two main approximationschemes within the NPRG context: the derivative ex-pansion [14] and the Blaizot-Mendez-Wschebor (BMW)scheme [68, 69]. The derivative expansion, which is themost widely used, consists in expanding the effective av-erage action Γκ in powers of gradients and time deriva-tives, retaining a finite number of terms. It usually pro-vides a reliable description of large-distance and long-time properties (that is the small momentum and fre-quency sector), including critical exponents and phasediagrams. Furthermore, it can reach a high precisionlevel, competing with current boostrap methods for theIsing model [58]. On the other hand, the BMW schemeis designed to reliably obtain the full momentum andfrequency dependence of the correlation functions, notlimited to the small momentum and frequency sector. Itrather relies on an expansion in the vertices of the flowequations, which is controlled by the presence of the reg-ulator term. It was also shown to reach a high precision[57].

For the KPZ equation, the simplest approximation isthe first order of the derivative expansion, which is usu-ally called the Local Potential Approximation (LPA).This approximation was shown to be sufficient to accessthe strong-coupling KPZ fixed point in any dimensions d[70]. It thus already goes beyond perturbative RG, sincethe latter fails to capture this fixed point in d 6= 1 even toall orders in perturbation theory [44]. However, the crit-ical exponents are quite poorly determined within LPA,except in d = 1 where they are fixed exactly by the sym-metries. This lack of accuracy of the derivative expansionfor the KPZ problem is related to the derivative nature ofthe interaction in the KPZ equation. The BMW schemehas turned out to be more appropriate in this context,as evidenced in subsequent studies [45, 46]. In fact, thestandard BMW scheme has to be adapted in order notto spoil the KPZ symmetries. Its rationale is expoundedin more details in [45, 46]. In practice, it can be imple-mented using an ansatz for the effective average action,which is presented in the next section.

B. Effective average action for the pure SR-KPZ

To study the original KPZ equation, one can use anansatz for Γκ, such that i) it preserves the full momen-tum and frequency dependence of the two-point func-tions, and ii) it preserves the KPZ symmetries. Using anansatz, rather than performing a direct BMW expansionof the vertices, is a solution to concile i) and ii). Indeed,the (extended) Galilean symmetry yields constraints on

the vertices Γ(n)κ , under the form of exact Ward identities,

which relate a Γ(n+1)κ vertex with one vanishing momen-

tum on a ϕ leg to a lower order vertex Γ(n)κ . Introducing

the notation Γ(m,n)κ where the m first derivatives are with

respect to ϕ and the n last with respect to ϕ, they read[45]:

∂

∂~qΓ(m+1,n)κ (ω, ~q,$1, ~p1, . . . , $n+m−1, ~pn+m−1)

∣∣∣~q=0

= −iλn+m−1∑k=1

~pkω

[Γ(m,n)κ (· · · , $k + ω, ~pk, · · · )

− Γ(m,n)κ (· · · , $k, ~pk, · · · )

]. (23)

Expanding the vertices while satisfying these identitiesturns out to be complicated. A simpler way is to con-struct a general ansatz for the effective average action us-ing as building blocks invariants under the Galilean sym-metry. For this symmetry, one can define a field f(t, ~x) asa scalar density if its infinitesimal transform under (14)

is δf(t, ~x) = λ~ε(t) · ~∇f , since this implies that∫dd~xf

is invariant under a Galilean transformation. One cancheck that with this definition, the elementary Galileanscalar densities are h, ∂i∂jh, and

Dth ≡ ∂th−λ

2(~∇h)2 , (24)

but not ∂th alone. The scalar property is preserved by

the operator ~∇ and by the covariant time derivative

Dt = ∂t − λ~∇h · ~∇ , (25)

but not by ∂t. Combining these Galilean scalars andoperators, one can construct an ansatz which explicitlypreserves Galilean symmetry. At quadratic order in theresponse field, the most general ansatz obtained in thisway, called SO (for Second Order), was first proposed in[45] and reads:

Γκ[ϕ, ϕ] =

∫x

ϕfλκ (−D2

t ,−∇2)Dtϕ− ϕfDκ (−D2t ,−∇2)ϕ

−1

2

[∇2ϕfνκ (−D2

t ,−∇2)ϕ+ ϕfνκ (−D2t ,−∇2)∇2ϕ

](26)

with fXκ analytic functions of their arguments defined as

fXκ (−D2t ,−∇2) =

∞∑m,n=0

aXκ,mn(−D2t )m(−∇2)n . (27)

6

One notices that the term proportional to Dtϕ renor-malizes as a whole, with a unique function fλκ in (26),which is equivalent to stating that λ is not renormalized.The Ward identities (23) are automatically satisfied at all

scales κ by the vertices Γ(n)κ computed from the ansatz

(26) [45]. Furthermore, additional constraints stem fromthe other symmetries. The time-gauged shift symmetry(13) imposes that fλκ (ω, ~p = 0) = 1 at any scale κ. Ind = 1, the time-reversal symmetry further imposes thatfDκ = fνκ , and fλκ = 1, such that there is a single inde-pendent running function in one dimension.

The ansatz (26) truncates the functional dependencein ϕ at quadratic order, but it remains functional in ϕthrough the operators Dt. This ansatz provides a non-trivial frequency and momentum dependence for all ver-

tices Γ(n)κ . This dependence is the most general one for

the two-point functions, but it is not for higher order ver-tices. It was shown in [45] that this ansatz yields veryaccurate results. It reproduces in particular to a veryhigh precision level the exact results available in d = 1for the scaling functions associated with the two-pointcorrelation function, up to very fine details of the tails ofthese functions.

However, solving the flow equations at SO representsquite a heavy numerical task in d > 1. Thus, a simplifi-cation was proposed in [46], which consists in neglectingthe frequency dependence of the functions fXκ within theintegrands of the flow equations. This approximation,named Next-to-Leading Order (NLO), allows one to ex-plore higher spatial dimensions in a reasonable compu-tational time. Indeed, at NLO, all the n−point vertices

Γ(n)κ , with n > 2, vanish except the bare one Γ

(2,1)κ . The

NLO approximation leads to reliable estimates for thecritical exponents in d = 2 and d = 3, and it enables oneto determine non-trivial properties of the rough phase,such as scaling functions, and associated universal am-plitude ratios [46]. Some of the predictions obtained atNLO were accurately confirmed by subsequent numeri-cal simulations [47]. Note that this approximation turnsout to deteriorate when the dimension grows, and it be-comes unreliable above d & 3.5 [46]. Therefore it cannotbe used for instance to probe the existence or not of anupper critical dimension for KPZ, for which the full SOapproximation should be implemented.

In this work, we use approximations close to the NLOone, minimally extended to take into account violationsof Galilean invariance. For the LR noise, we simply in-clude the scale-dependent long-range coupling constantwθκ associated with the non-analytical frequency depen-dence of the effective noise, together with the inducedrenormalization of the non-linear coupling λκ. These twoquantities are enough to discriminate between a LR anda SR phase and to estimate the corresponding criticalexponents, as shown in the following. In this case, theanalytical frequency dependence, carried by the renor-malization functions fXκ (ω, p), is sub-dominant comparedto the non-analytical one, so it is sufficient to computeit within the NLO approximation. For the SR noise, all

the non-trivial frequency dependence generated by thisnoise is carried by the analytical function fDκ . In par-ticular, at the microscopic scale Λ, this function takesthe form Dτ in (10). To implement this initial condition,one cannot neglect the frequency dependence of fDκ in theright-hand side of the flow equations, as is done in NLO.Hence we devised a generalized version, denoted NLOω,which keeps the frequency dependence of the functionsfνκ and fDκ in the integrands of hte flow equations.

Since the NLO can be obtained as a simplification ofthe NLOω, we present first in the next section the latterapproximation, and then the subsequent simplifications.These different approximations, LPA, NLO, NLOω andSO, can be seen as four successive orders of our approxi-mation scheme, with increasing accuracy. We emphasizethat the NLO order is already a satisfactory (and quiteinvolved) one since a good accuracy can be obtained atthis order in the physical dimensions 1, 2 and 3 [46].

C. Effective average action with broken Galileaninvariance

Introducing a non-trivial frequency dependence in thenoise correlator of the KPZ equation breaks Galilean in-variance at the microscopic level. This means that theconstraints associated with this symmetry no longer ap-ply. In particular, the non-linear coupling can acquirea non-trivial RG flow λ ≡ λk, since it is no longer thestructure constant of a symmetry of the system. Thisrenormalization has to be taken into account. It impliesin particular that the covariant time derivative is splittedin two independent parts. This induces some modifica-tions of the ansatz. First, the term proportional to Dtϕseparates in two parts

fλκ Dtϕ→ fλκ ∂tϕ−λκ2fλκ (~∇ϕ)2 , (28)

and we keep, as a minimal extension of the NLO approx-imation, the same function fλκ for the two parts. Al-though they can in principles be different, it is clear thatthe dominant effect of the breaking of Galilean symme-try is described by the renormalization of λκ. Similarly,Dt decomposes in two independent parts. For simplic-ity, again as a minimal extension of NLO, we only retainat NLOω in the arguments of the functions fXκ the timederivative part when necessary, that is for fDκ and fνκ

fD,νκ (−D2t ,−∇2)→ fD,νκ (−∂2

t ,−∇2) . (29)

For fλκ , better resolving its frequency dependence is notneeded, so we compute it only in the NLO approximation.Thus, within the NLOω approximation, the functions fXκno longer depend on the field ϕ, which implies that only

the 3-point vertex Γ(2,1)κ is non-zero, as for NLO. The

corresponding ansatz NLOω, reads

Γκ[ϕ, ϕ] =

∫x

ϕfλκ ∂tϕ−

λκ2ϕfλκ (~∇ϕ)2 − ϕfDκ ϕ

7

−1

2

[∇2ϕfνκ ϕ+ ϕfνκ∇2ϕ

], (30)

where all functions depend on (−∂2t ,−∇2). With this

ansatz, the two-point functions are given by

Γ(1,1)κ ($, ~p) = i$fλκ ($, p) + ~p 2fνκ ($, p)

Γ(0,2)κ ($, ~p) = −2fDκ ($, p)

Γ(2,0)κ ($, ~p) = 0 , (31)

noting simply that the actual dependence of the functionsfXκ is on $2 and ~p 2, and that the frequency dependenceof fλκ is treated in the NLO approximation, i.e. it neverappears in the non-linear part of the flow equations (33).

Let us place again this approximation with respect tothe other ones, NLO and SO. In the NLO approxima-tion, the frequency dependence of all the functions fXκis neglected in the right-hand side of the flow equations,which amounts to the replacement fXκ ($, p)→ fXκ (p) inthe integrands of (33) [46]. The functions fXκ nonethe-less acquire a frequency dependence, which is generatedby the explicit dependence on the external frequency inthe flow equations (through Pκ(Ω, Q) in (33)). Withinthe NLOω approximation, this replacement is performedonly for the function fλκ , while the full frequency depen-dence of fDκ and fνκ is kept in the flow equations. Thisis the minimal approximation that allows one to studySR temporal correlations in the noise while limiting theexplicit breaking of the KPZ symmetries by the ansatz.The NLOω scheme induces an additional computationalcost compared to NLO (in particular, the integration overthe internal frequency ω can no longer be performed an-alytically, and additional interpolations in the frequencysector are needed, see Appendix C). However, since theNLOω approximation is actually quadratic in both ϕ andϕ, there remains only one non-zero 3-point vertex func-

tion, as in the NLO scheme, which is Γ(2,1)κ

Γ(2,1)κ ($1, ~p1, $2, ~p2) = λκ~p1 · ~p2f

λκ (($1 +$2)2, |~p1 + ~p2|2).

(32)

This implies that the expression of the flow equations isstill greatly simplified compared to the SO scheme [65],and thus remains numerically reasonable, in particularin d = 2. The price to pay is that the NLOω approxi-mation induces a small spurious breaking of the Galileaninvariance (even when this symmetry is present at the

microscopic level). Indeed, contrarily to the Dt opera-tor, the simple time derivative ∂t does not preserve theGalilean scalar property. In particular, the frequency de-pendence in the two-point functions is not accompaniedby a higher-order field dependence as it should to satisfythe Galilean Ward identities (23) and thus preserve thissymmetry. Treating the full frequency dependence with-out inducing any spurious breaking of Galilean invari-ance would require to work with the SO ansatz. How-ever, within the NLOω scheme, this spurious breakingremains very small, and does not prevent from identi-fying a “true” physical breaking, as shown in the nextsections. In fact, it provides an estimate of the error as-sociated with this order of approximation, which is small.

D. Flow equations and running anomalousdimensions

1. Flow equations of the renormalization functions fXκ

According to (31), the flow equation for the runningfunctions fνκ , fλκ , and fDκ respectively, can be deducedfrom the flow equations of the real part, imaginary part

of Γ(1,1)κ , and Γ

(0,2)κ , respectively. One has to take two

functional derivatives of the exact flow equation (21), andthen replace the vertex functions and the propagator inthis expression by the ones computed from the ansatz(30), evaluated at zero fields. The calculations are thesame as those reported in [46], where more details can befound. One obtains within the NLOω scheme

∂κfDκ ($, p) = 2gκf

λκ (p)2

∫ω,~q

(~q 2 + (~p · ~q))2 kκ(Ω, Q)

Pκ(ω, q)2Pκ(Ω, Q)

Pκ(ω, q) ∂κS

Dκ (q)− 2 ~q 2 `κ(ω, q) kκ(ω, q) ∂κS

νκ(q)

, (33a)

∂κfνκ ($, p) = −2

gκp2fλκ (p)

∫ω,~q

~q 2 + (~p · ~q)Pκ(ω, q)2Pκ(Ω, Q)

− ~p · ~q fλκ (Q) `κ(Ω, Q)Pκ(ω, q) ∂κS

Dκ (q)

+[2 ~p · ~q fλκ (Q) `κ(Ω, Q) `κ(ω, q) kκ(ω, q) + (~p 2 + ~p · ~q) fλκ (q) kκ(Ω, Q)(ω2 fλκ (q)2 − `κ(ω, q)2)

]~q 2 ∂κS

νκ(q)

,

(33b)

∂κfλκ ($, p) = 2

gκ$fλκ (p)

∫ω,~q

~q 2 + (~p · ~q)Pκ(ω, q)2Pκ(Ω, Q)

− Ω ~p · ~q fλκ (Q)2 Pκ(ω, q) ∂κS

Dκ (q)

+ 2[Ω ~p · ~q fλκ (Q)2 kκ(ω, q) + ω (~p 2 + ~p · ~q) fλκ (q)2 kκ(Ω, Q)

]~q 2 `κ(ω, q) ∂κS

νκ(q)

, (33c)

8

100

10−1 100 101 102 103

100

10−1 100 101 102 103

(a) (b)

fD κ(

,p=

0)

∼ −1/3

τ = 0τ = 0.01τ = 0.05τ = 0.1

∼ −1/3

fDκ (, p = 0)

fνκ (, p = 0)

s = 0 s = −3 s = −40

FIG. 1. (a) Evolution of the function fDκ ($, 0) with the RG scale for different values of τ = 0, 0.01, 0.05, 0.1. For each τ , thefunction is represented at successive RG times s = − log(κ/Λ): s = 0 where they are Gaussians of different width, s = 3 wherethe functions for the different τ are already almost superimposed (erasure of the initial conditions), and s = 40 where the

fixed-point shape is reached, characterized by a power-law decay very close to the KPZ one ∼ $1/3 (indicated as a guideline).(b) Evolution of the functions fDκ ($, 0) and fνκ ($, 0) with the RG scale for τ = 0.01, represented for the RG times s = 0, 3, 40.Although they start at s = 0 from different shapes fDκ=Λ($, 0) 6= fνκ=Λ($, 0), the time-reversal symmetry is restored at s . 3where they already coincide, up to the fixed point fD∗ ($, 0) = fν∗ ($, 0).

with Q ≡ |~p+ ~q|, Ω ≡ ω +$, and

`κ(ω, q) = q2(fνκ (ω, q) + νκ r(q2/κ2)), (34a)

kκ(ω, q) = fDκ (ω, q) +Dκr(q2/κ2) (34b)

Pκ(ω, q) = ω2 fλκ (q)2

+ `κ (ω, q)2

(34c)

SXκ (q) = Xκr(y) , y = q2/κ2 , X ∈ D, ν, (34d)

κ∂κSXκ (y) = −Xκ (ηXκ r(y) + 2y ∂yr(y)), (34e)

and where the anomalous dimensions ηXκ are defined be-low.

The breaking of the Galilean symmetry is encompassedby the flow of the non-linear coupling, which can be de-

fined from the 3-point vertex function Γ(2,1)κ given in (32)

as

λκ = limp→0

4

p2Γ(2,1)κ

(0,~p

2, 0,

~p

2

). (35)

The computation of the flow of λκ is reported in Ap-pendix B. We obtain within the NLOω approximation

∂sλκ = −Sd2gκd

∫ ∞0

dq

(2π)d

∫ ∞−∞

dω

(2π)

qd+3fλκ (q)2

Pκ(ω, q)4

∂sS

Dκ (q)Pκ(ω, q)

[Pκ(ω, q)− 4ω2fλκ (q)2

]− 4q2∂sS

νκ(q)kκ(ω, q)`κ(ω, q)

[Pκ(ω, q)− 6ω2fλκ (q)2

](36)

where we used∫

d~q (~p · ~q)2F (q) = Sd

d p2∫∞

0dq q2F (q),

with Sd = 2πd/2/Γ(d/2) the d-dimensional solid angle.

The flow equations within the NLO approximation canbe deduced from the ones at NLOω, by further neglectingthe frequency dependence of fDκ and fνκ in the integrands(33) and (36), i.e. fν,Dκ (ω, q) → fν,Dκ (q). With this re-placement, the integration over the internal frequency ωcan be performed analytically (see [46] for the explicit ex-pressions). Moreover, let us emphasize that one obtainsin this case, after integration on ω, that ∂sλκ is exactlyzero. This explains why the NLOω extension is neces-sary to account for a ”smooth”, i.e analytical, breakingof Galilean symmetry, as the one occurring in the SR casewith the correlator Dτ .

2. Anomalous dimensions and dimensionless flows

The global scaling of the renormalization functionscan be determined at a specific normalization point($NP, pNP). This is equivalent to the choice of a prescrip-tion point in standard perturbative RG. Within the con-text of NPRG, this normalization point can be in generalsimply chosen as (0, 0), because the flow is regularizedand no singularity occurs at vanishing momentum andfrequency. Here, this is more subtle in the case of power-law correlations which may introduce a non-analyticityat zero frequency, since the flow is not regularized in thefrequency sector. It is useful in this case to consider anon-zero normalization frequency. Hence, we define twoscale-dependent coefficients νκ and Dκ as the normaliza-tions of fνκ and fDκ at the point ($0, 0) according to

Dκ ≡ fDκ ($0, 0) , νκ ≡ fνκ ($0, 0) , (37)

9

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

1 10

−0.004

−0.002

0

0.002

0.004

0.006

1 10

(a) (b)

|ηD κ−ην κ|

−s

ηλ κ

−s

τ = 0.01 τ = 0.05 τ = 0.1

FIG. 2. Evolution with the RG time s = − log(κ/Λ) of (a) ηλκ and (b) |ηDκ −ηνκ|, for different values of τ = 0, 0.01, 0.05, 0.1. Oneobserves that: (a) the time-reversal symmetry is dynamically restored along the flow since ηD∗ = ην∗ at the fixed point, whichimplies that χ = 1/2, (b) the Galilean invariance is almost restored: ηλ∗ takes a very small value for all τ . As explained in thetext, this residual non-zero violation of Galilean invariance is induced by the NLOω ansatz, which implies that z = 2− χ− ηλ∗slightly deviates (by less than 0.5%) from the SR-KPZ value z = 3/2.

where $0 = 0 unless stated otherwise. We emphasizethat the choice of the normalization point is in principlearbitrary and the results should not depend on it. Thisis true in the exact theory. However, as for the choice ofthe cutoff function, once approximations are performed,one can expect a small residual dependence on the precisevalue of the normalization point. We checked that it isnegligible, see Appendix A.

The two coefficients Dκ and νκ encompass the renor-malization of the fields and the scaling between space andtime. Their flow can be simply obtained from the limit($, p)→ ($0, 0) in Eq. (33a) and Eq. (33b) respectively.One can define two running scaling dimensions associatedwith these coefficients as

ηDκ = −κ∂κ lnDκ , ηνκ = −κ∂κ ln νκ . (38)

One can show that the critical exponents can be ex-pressed in terms of the fixed point values of these scalingexponents as [45]

z = 2− ην∗ , χ = (2− d+ ηD∗ − ην∗ )/2 . (39)

For fλκ , the shift-gauged symmetry imposes thatfλκ ($, 0) = 1 for all $, and in particular for $0, so thisfunction does not introduce another scaling coefficient.

In the following, we are interested in the fixed pointsof the RG flow equations. Indeed, a fixed point meansthat all quantities do not depend on the scale κ anylonger, which physically implies that the system is scaleinvariant, critical. As common in RG approaches, theappropriate way to search for a fixed point is to switchto dimensionless quantities. We hence define dimen-sionless momenta, e.g. p = p/κ, and frequencies, e.g.$ = $/(νκκ

2), and consider the dimensionless functions

obtained as fXκ ($, p) = fXκ ($, p)/Xκ. Their flow equa-tion is thus given by

∂sfXκ ($, p) =

[ηXκ + (2− ηνκ)$∂$ + p∂p

]fXκ ($, p)

+ IXκ ($, p) (40)

where IXκ is the non-linear part of the flow equations∂sf

Xκ /Xκ, given in (33), expressed in terms of dimen-

sionless variables, and with Xκ = Dκ, νκ, 1 and ηXκ =ηνκ, η

Dκ , 0 for fDκ , fνκ and fλκ respectively.

Finally, we denote the flow of λκ as

κ∂κ lnλκ = −ηλκ . (41)

The flow of the dimensionless coupling gκ ≡κd−2λ2

κDκ/ν3κ can be expressed as

∂sgk = gk(d− 2− 2ηλκ + 3ηνκ − ηDκ

). (42)

At a non-gaussian fixed point gk 6= 0, this implies theexact relation

z + χ− 2 = ηλ∗ . (43)

Hence, if Galilean symmetry is present, then ηλ∗ = 0 andone recovers the standard relation z+χ = 2. A non-zeroηλ∗ quantifies the violation of Galilean invariance.

Flow of wθκ in the NLO approximation

The presence of a power-law noise correlator D∞(ω, ~q)in (10) introduces another coupling wθκ related to thenon-analytic part. The function fDκ is now composed oftwo parts

fDκ (ω, q) = fDκ (ω, q) + wθκω−2θ. (44)

In principle, the NPRG flow is analytic, such that nonon-analytic contribution can arise to renormalize thecoupling wθκ. The situation is more subtle here since the

10

frequency sector is not regularized, see discussion in Ap-pendix A, but the non-renormalization of wθκ is preserved.Defining the dimensionless running coupling wθκ as

wθκ = κ−4θwθκ1

Dκν2θκ

(45)

one obtains its flow as

∂swθκ = wθκ

(−4θ + ηDκ + 2θηνκ

). (46)

For any fixed-point solution for which wθ∗ 6= 0, one de-duces that

ηD∗ = 4θ − 2θην∗ (47)

which yields if g∗ 6= 0

ηλ∗ =1

2(2− d+ 4θ − (3 + 2θ)ην∗ ), (48)

which is non-zero in general. Hence, if a LR fixed-pointwith wθ∗ 6= 0 exists and is stable, it is associated with aviolation of Galilean symmetry. Assuming that the twofixed-points, the LR and the SR ones, exist and compete,then the transition from one to the other occurs whenthe corresponding dynamical exponents are equal, thatis for zLR = zSR and thus η∗λ = 0. One deduces that thecorresponding critical value θth is given by

θth(d) =1

2(η∗ν − 2)(2− d− 3η∗ν) . (49)

One then expects a transition from a SR to a LR domi-nated phase with critical exponents satisfying:

SR : z + χ = 2, θ < θth

LR : z + χ = 2− η∗λ(θ), θ > θth .

IV. RESULTS

In this section, we only consider dimensionless quanti-ties, so we omit the hat symbols to alleviate notations.

A. Temporal correlations with a finite correlationtime

We consider the KPZ action with the microscopic noisecorrelator Dτ defined in (10). As explained previously,Γκ coincides with the microscopic action at the micro-scopic scale κ = Λ. Comparing (11) and (30), one con-cludes that this initial condition corresponds to

fDκ=Λ($, p) = Dτ ($, p) = e−12$

2τ2

(50)

and

fνΛ($, p) = 1, fλΛ($, p) ≡ 1 . (51)

Hence at the microscopic level, both the Galilean in-variance (since fDΛ depends on frequency) and the time-reversal symmetry in d = 1 (since fDΛ 6= fνΛ) are broken.

Let us focus on d = 1. In this dimension, the functionfλκ is kept to one as imposed by the time-reversal symme-try. We integrated numerically the flow equations for thetwo functions fDκ ($, p) and fνκ ($, p) (NLOω approxima-tion), together with the flow equations for the couplinggκ, and for the coefficients νκ and Dκ, for different initialvalues of τ between 0 and 1. Details on the numericalprocedure are provided in Appendix C.

For all values of τ , we observed that the flow reaches afixed point, with stationarity in κ for all quantities. Thecoupling gκ tends to a fixed-point value g∗. At the sametime, the renormalization functions fDκ and fνκ smoothlyevolve to endow a fixed-point form, which does not de-pend on the value of τ , as illustrated for fDκ in Fig. 1(a). This means that the large distance physics is uni-versal, i.e. independent of the microscopic details, and itcorresponds to the SR-KPZ universality class (the samefixed-point is attained as for τ = 0).

Furthermore, although they start with very differentshapes, the two functions fDκ and fνκ become equal atthe fixed point fD∗ ($, p) ≡ fν∗ ($, p), as illustrated inFig. 1 (b). This means that the time-reversal symmetryis dynamically restored at large distances. This is furtherillustrated in Fig. 2 (b), which shows that the difference|ηνκ − ηDκ | vanishes at the fixed point for all τ . Accordingto Eq. (39), this implies that the χ exponent is exactlythe SR-KPZ one χSR = 1/2.

Moreover, the Galilean symmetry is also restored atthe fixed point, although only approximately at NLOω.This can be assessed by the value of η∗λ, which is rep-resented in Fig. 2 (a). One observes that it reaches aconstant value, which is not strictly zero but a smallnumber of order 0.0065. As explained before, this reflectsthe spurious violation of Galilean invariance induced bythe NLOω ansatz (dependence in ∂t rather than Dt).This value is the same as for the pure SR-KPZ case (forτ = 0) and yields an error of less than 0.5% on the expo-nent z. The NLOω approximation is hence still accuratedespite its simplification compared to SO. Furthermore,we observe that for any finite τ , the function fD∗ decays

at large frequency as a power law fD∗,τ ($, 0) ∼ $−ηD∗ /z,with z = 2 − χ − ηλ∗ , very close to the pure SR-KPZcase fD∗,τ=0($, 0) ∼ $−1/3. Hence one can conclude thatfor all τ , the universal properties of the interface are thestandard SR-KPZ ones.

In two dimensions, the NLOω approximation does notseem to suffice to properly describe the pure SR-KPZcase. We did not succeed in accessing the fixed pointwithin this scheme, probably because the violation ofGalilean symmetry induced by the ansatz (through ne-glecting all higher-order vertex functions) is too severein d = 2. On the other hand, the NLO scheme alonedoes not allow to implement an initial condition whichinvolves a functional frequency dependence of fDΛ as in(50). Hence, to study the effect of a temporal SR-

11

10−2

10−1

100

10−1 100 101 10210−4

10−3

10−2

10−1

100 101 102

(a) (b)

θ = 0

θ = 0.24

wθ κ

−s

ηλ κ

−s

FIG. 3. Evolution with the RG time s in d = 1 of (a) the LR coupling wθκ and (b) the violation of Galilean invariance ηλκ ,for different values of θ = 0, 0.1, 0.16, 0.166, 0.17, 0.22, 0.24 (from bottom to top) in d = 1. For θ < θth, the flow reaches theSR-KPZ fixed point with wθ∗ = 0, ηλ∗ = 0, while for θ > θth, a LR fixed point with wθ∗ 6= 0, ηλ∗ 6= 0 is reached. The critical valueis θth = 0.166 confirming the theoretical prediction θth = 1/6. Interestingly, it is clearly identified as the value leading to analgebraic decay of wθκ and ηλκ in the RG time s (bold orange line).

correlated noise in d = 2 would require to use the fullSO ansatz (which does not induce any artificial violationof Galilean symmetry). This is beyond the scope of thiswork. We will thus restrict in d = 2 to the study of theLR case, which can be studied at NLO.

To summarize on the temporally SR-correlated noise,we found in d = 1 that its presence does not changethe large-distance and long-time properties of the inter-face, which is still characterized by the SR-KPZ uni-versality class. Hence, although both the Galilean andtime-reversal symmetries are broken at the microscopiclevel, these symmetries are restored dynamically alongthe flow. This is the first analysis of the effect of SR time-correlations in the KPZ equation, which is here renderedpossible by the both functional and non-perturbative for-malism we use.

B. Power-law temporal correlations

We now investigate the presence of correlations in themicroscopic noise with no typical length-scale, i.e. thepower-law LR correlations D∞ in (10). This correspondsto the initial condition

fDκ=Λ($, p) = D∞($, p) = 1 + wθΛ$−2θ (52)

together with (51). As explained in Sec. III D, this LRnoise introduces the new dimensionless coupling constantwθκ and two different scenarii may now emerge: eitherthe SR part of the noise dominates, corresponding toa stable SR fixed point with wθ∗ = 0, or the LR partdominates, corresponding to a stable LR fixed point withwθ∗ 6= 0. Since for such a fixed point Galilean symmetryis broken, z + χ 6= 2, there is no simple way to computethe associated LR critical exponents even in d = 1.

To study the LR noise, it is enough to work withinthe NLO approximation, since the analytical dependence

in frequency of the functions fXκ is not essential (sub-dominant) in this case. The advantage is that there isno spurious (i.e. introduced by the ansatz) breaking ofGalilean symmetry at NLO. Indeed, the analytical part ofthe flow of λκ vanishes at NLO as explained in Sec. III D.The only contribution to the flow of λκ thus stems fromthe non-analytical part of fDκ and reads

∂sλκ = Sd8gκw

θκ

d

∫ ∞0

dq

(2π)dqd+5fλκ (q)2∂sS

νκ(q)`κ(q)

×∫ ∞−∞

dω

(2π)

ω−2θ

Pκ(ω, q)4

[Pκ(ω, q)− 6ω2fλκ (q)2

].

(53)

In this part, we also use the local potential approxi-mation, in order to compare the results stemming fromsuccessive orders of approximations. The LPA flow equa-tions can be simply deduced from the NLO ones by com-pletely neglecting the momentum and frequency depen-dence of the running functions, which is thus equivalentto simply considering the flow of the two dimensionlesscouplings gκ and wθκ, and the two anomalous dimensionsηDκ and ηνκ.

1. One dimensional case

As for the SR correlated noise, we fix fλκ = 1 in d = 1in order to satisfy the time-reversal symmetry. We inte-grated numerically the NLO flow equations for fDκ andfνκ together with the flow equations for the two dimen-sionless couplings gκ and wθκ and for the two anomalousdimensions. We find two distinct regimes depending onthe value of θ, as illustrated on Fig. 3. For θ < θth = 1/6,gκ flows to a finite fixed point value g∗ while wθκ flows tozero. At the same time, ηλκ also flows to zero, henceGalilean invariance is dynamically restored, and the crit-ical exponents take the SR-KPZ values χSR = 1/2 and

12

0.5

0.52

0.54

0.56

0.58

0.6

0.62

1.45

1.5

0 0.05 0.1 0.15 0.2

(a)

(b)

χ(θ)

RG 1-loop

RG num.

LPA

NLOωNLO

z(θ)

θ

FIG. 4. (a) Roughness exponent χ and (b) dynamical expo-nent z as a function of θ in d = 1 and for different approx-imation schemes. The exponents take the SR-KPZ valuesχ = 1/2 and z = 3/2 up to a critical value of θ very closeto the theoretical prediction θth = 1/6. Beyond this value,the exponents vary continuously with θ. The LPA resultsimproves the analytical RG result at one-loop given in (5),where we recall the authors set ηλκ = 0, but still differ fromthe NLO results. The NLO and NLOω results are very close,despite the slight shift in z at NLOω due the the residualbreaking a Galilean invariance within this sheme. At NLO,we find an almost linear behavior of χ(θ) ' 1.43θ + 0.26. Itturns out to be reasonably close to numerical estimation fromthe approximate RG equations (7) for χ, but not for z. Allthe estimates from NPRG lie within the bounds given in (9)(shaded region).

zSR = 3/2. The long-distance physics is hence the samefor all θ < θth and controlled by the SR-KPZ fixed-point.For θ > θth, both gκ and wθκ flow to a non-zero fixed-point value, and the violation of Galilean invariance ηλ∗increases with θ, as illustrated on Fig. 3. Hence in thisregime, the long-distance properties are controlled by aline of LR fixed points, with critical exponents depend-ing on θ. The critical value θth = 1/6 delimiting the tworegimes is clearly identified on Fig. 3 by the algrebraicdecay of wθκ and ηλκ with the RG time s.

These findings are in agreement with the results pre-sented in [18, 37], and show that a pure SR-KPZ regimeis not destroyed for an infinitesimal θ, contrary to thescenario advocated by SCE or Flory approaches. Thecritical exponents χ and z obtained at NLO are repre-sented on Fig. 4, and compared to the DRG approach of[18] for which explicit results are given.

We also performed the same analysis within different

approximations of NPRG to test the robustness of theresults. Within the NLOω scheme, the existence of thetwo regimes is confirmed. However, since in this approxi-mation a residual breaking of Galilean invariance even atθ = 0 (ηλ∗ ' 0.0065) induced by the ansatz subsists at theSR fixed point, the critical value of θth is slightly shifted,but the critical exponents are close to the NLO ones, seeFig. 4 (especially if the small shift 0.0065 is compensatedfor).

Within the LPA, we also find the two regimes withthe same critical value θth, but the critical exponentsslightly differ, they lie closer to the one-loop results (5)as could be expected. The NLO results fall in betweenthe numerical approximation of [18] and the LPA results.Let us emphasize that all our estimates for zLR(θ) aredecreasing with θ and lie within the bounds (9) derivedin [37].

Let us finally note that these theoretical results are notin agreement with the most recent numerical simulations[41]. In the simulations, the SR-KPZ phase is destroyedfor any non-zero θ (no threshold value) and the value ofthe dynamical critical exponent is found to be constantz ' 3/2 independently of θ. We have no clear under-standing of these discrepancies. The main differences liein the fact that the simulations deal with finite-size sys-tems in discretized space and time whereas we work inthe infinite-size limit in continuous space-time. In gen-eral, finite-size effects tend to smear out transitions. Aputative explanation could be that very large system sizesare required to clearly resolve the threshold. As for thevalue of z, we emphasize that the deviation from the SRvalue zSR = 3/2 in the LR phase is small, as manifest inFig. 4(b) [more quantitatively, zLR = 1.42(5) for = 0.47and zLR = 1.49(4) for = 0.24]. Indeed, z only dependson ην∗ , which is less sensitive than ηD∗ to the presenceof the LR noise. Since z is determined in [41] from thecollapse of the structure function, it is possible that thenumerical results are in fact still compatible with suchsmall variations. These points deserve further studies.

2. Two dimensional case

In two dimensions, the estimation of χ for the pureSR-KPZ within the NLO approximation is χSR ' 0.375,in good agreement with the value χ ' 0.3869(4) fromrecent numerical simulations [71]. Substituting the NLOvalue of ην∗ ≡ 2− z = χ in (49), one obtains a theoreticalestimate of the critical value θth = 0.346. In the work of[18], a non-physical divergence in ω appears for θ ≥ 1/4.Within the present work, a similar singularity at θ = 1/4arises in the limit of zero frequency. It occurs in the flowequation of fDκ , in the term proportional to ∂κS

Dκ , and

is proportional to (ω + $)−4θ. The singularity is thuspresent only at zero external frequency $, that is for thecalculation of the anomalous dimension ηDκ . This resid-ual divergence is due to the absence of a regulator in thefrequency sector. If we could use a frequency-dependent

13

10−2

10−1

100

10−1 100 10110−2

10−1

10−1 100 101

θ = 0

θ = 0.45

(a) (b)

wθ κ

−s

ηλ κ

−s

FIG. 5. (a) The LR coupling wθκ and (b) ηλκ , for different values of θ = 0, 0.1, 0.2, 0.3, 0.34, 0.35, 0.4, 0.45 (from bottom to top)and θth = 0.346 (bold orange line) as a function of the RG time s, in d = 2. Two distinct behaviors, corresponding to the SRand the LR fixed points are observed, separated by the critical value θth for which wθκ and ηλκ vanish algebraically with s.

regulator which do not break Galilean invariance, thisproblem would not exist. Without such a regulator, thisproblem can be nevertheless avoided by shifting the nor-malization point $0 of the anomalous dimensions (37), toa non-zero external frequency (see Appendix C). Hence,it can be quite simply dealt with, at variance with per-turbative RG.

We performed the same analysis as for the one-dimensional case. We observed that for θ < θth, theGalilean invariance is restored by the flow, and the largedistance physics is described by the pure SR-KPZ fixedpoint. This is illustrated on Fig. 5 which shows that theLR coupling wθκ vanishes at the fixed-point, and ηλκ alsovanishes (exactly at NLO). For θ > θth, the LR cou-pling wθκ and ηλκ both reach a non-zero fixed-point value,which depends on θ. This corresponds to a LR fixed-point, where Galilean invariance remains broken. Thecritical value θth can be identified on Fig. 5 by the al-gebraic decay of the flow, separating the two differentbehaviors.

The results for the critical exponents are shown on Fig.6, within two versions of NLO: either with a non-trivialflow for fλκ (p), or setting fλκ (p) = 1. This latter scheme issimilar to the one used in d = 1, although it is imposed inthis dimension by the time-reversal symmetry, whereas itis arbitrary in d = 2. Both results are in agreement.

C. Anomalous scaling at the LR fixed point

A recent numerical study of the KPZ equation withpower-law time correlation in d = 1 by Ales and Lopez[41] unveiled some anomalous scaling at large values of θ.In the NPRG framework, anomalous scaling can emergeas a consequence of a non-decoupling of the low- andhigh-momentum modes in the flow equations. This wasindeed evidenced in the context of turbulence from astudy of the stochastic Navier-Stokes equation for incom-pressible flows [64, 72] and was related in this case tointermittency.

Let us investigate this point for KPZ with LR corre-lations, focusing in d = 1. For this, we study in moredepth the scaling properties at the fixed point. The fixedpoint equation is given by (40) with ∂sf

X∗ = 0, that is

− IX∗ ($, p) =[ηX∗ + (2− ην∗ )$∂$ + p∂p

]fX∗ ($, p). (54)

For standard scale invariance, the non-linear partof the flow IXκ becomes negligible compared with

fX∗ ($, p)/p−ηX∗ at large frequency and/or momentum,

which means that the large momentum and frequencysector decouples from the low one. As a consequence,the fixed-point universal (IR) properties are insensitive tothe microscopic (UV) details. One can straightforwardlyshow that the solution of the fixed-point equation (54)when this decoupling property is satisfied is the scalingform [45]

fX∗ ($, p) = p−ηX∗ ζX

($

pz

), $, p 1 , (55)

where ζX has the asymptotic behavior

ζX(y) =

ζX0 , y → 0

ζX∞y−ηX∗ /z , y →∞ . (56)

One can then show that the physical (dimensionful)two-point correlation function, which can be expressedin terms of the fX∗ ($, p), also endows a scaling form[45, 46]. Thus, for standard scale invariance, the scalingof the dimensionless functions fXκ in the large-frequencyand large-momentum sector is fixed by the scaling di-mensions ηX∗ , calculated from the behavior in κ of thecoefficient Dκ and νκ (defined in the opposite sector($, p) = ($0 ' 0, 0)).

However, these conclusions do not hold if the decou-pling property is not verified. To assess the effectivenessof decoupling, we have computed the non-linear part ofthe flow of fDκ (which is related to the noise) at the fixedpoint ID∗ . The result is displayed in Fig. 7 (a), whichshows that in the LR phase ID∗ becomes less and less

14

0.35

0.4

0.45

0.5

0.55

0.6

1.6

1.62

1.64

1.66

1.68

1.7

1.72

0 0.1 0.2 0.3 0.4

0.3 0.35

1.662

1.663

(a)

(b)

χ(θ)

z(θ)

θ

SR-KPZ Num.

RG 1-loop

SR-KPZ, NLO

NLO, fλκ = 1NLO

FIG. 6. (a) The roughness χ(θ) and (b) dynamical z(θ) criti-cal exponents as a function of the LR exponent θ in d = 2, andfor different approximation schemes. The reference values forthe pure KPZ case, obtained within NLO and from recent nu-merical simulations [71], are represented as the dashed linesSR-KPZ NLO and SR-KPZ Num., respectively. We find ind = 2 at NLO two regimes, as in d = 1. The critical expo-nents coincide with the SR-KPZ ones below a critical valueθth ' 0.346, while beyond this value, we obtain θ-dependentcritical exponents. Within the simplified NLO approximationwith fλκ = 1, the qualitative picture is the same, althoughthe curves are shifted because the values for the exponents atthe SR-KPZ fixed point differ a bit (less than 10 %) withinthis scheme. The one-loop analytical result (5) from DRGis also represented for comparison, although in this case, thevalue for θth cannot be obtained from the perturbative analy-sis. It can be estimated here as the intersection between thisprediction and the SR-KPZ values for the exponents.

negligible as the temporal correlation is increased. De-pending on the form of ID∗ this phenomenon can modifythe scaling form (55) in different ways. In Fig. 7 (b) weshow that in the present case the non-decoupling mani-fests itself in the appearance of a non-zero slope in thescaling function ζD(y) (associated with the noise vertex)as the limit y → 0 is approached, that is

ζD(y) =

ζD0 y

−∆ηD , y → 0

ζD∞y−ηD∗ /z , y →∞ , (57)

where ∆ηD = ηD∗ − ηDp1, with ηD∗ the scaling exponent

computed from the running coefficient Dκ, and ηDp1 theexponent computed from a power-law fit at large p of the

function fD∗ : fD∗ ($, p 1) ∼ p−ηDp1 . The behavior (57)

corresponds to a generalized version of the Family-Vicsekscaling form, and is analogous to the one introduced in

[73] for the structure factor S(t,~k) = 〈h(t,~k)h(t,−~k)〉.The value of ηDp1 cannot be predicted by any scalingargument. This anomalous scaling corresponds to whatis termed intrinsic anomalous scaling in [73], for whichχ < 1 but a new independent exponent exists, by oppo-sition to superroughnening, for which χ > 1. This is alsoreminiscent of the anomalous scaling in turbulence gen-erated by intermittency effects [74]. We observed that inthe scaling function ζν(y) the anomalous scaling is lesspronounced. These anomalous effects increase with θ, asis shown in Fig. 8 (a) where the usual roughness expo-nent (denoted χ(κ) in the following) appearing in (39) iscompared with the one in which the scaling dimensionsηX∗ are replaced by the actual scaling exponents in mo-mentum space,

χ(p) = (2− d+ ηDp1 − ηνp1)/2 . (58)

Hence the quantity ∆χ = χ(κ)−χ(p) is a direct measure-ment of the presence of anomalous scaling in the system.

The appearance of a new critical exponent for strongenough time correlations is similar to the one exhibitedin [41]. In that work, this new critical exponent is linkedto the presence of emergent spatial structures in the sys-tem, referred to as faceting, which appears for θ & 0.23.In Fig. 8 (b) we compare our estimate for ∆χ with thenumerical results of [41]. The qualitative behavior seemsto be in agreement. Although in our case, the anomalousscaling appears in the whole LR phase θ > θth, it be-comes significant only at larger values of θ & 0.3, whichis consistent with the result of [41] which reports largeanomalous scaling for θ & 0.3 − 0.4. However, for suchlarge values, the numerical values for χ in [41] are greaterthan one, which would correspond to superroughening,whereas our results remains in the regime χ < 1, whichcorresponds to intrinsic anomalous scaling.

V. CONCLUSION

In this work, we studied the effect of temporal correla-tions in the microscopic noise of the KPZ equation, bothin d = 1 and d = 2. Their presence breaks the con-stitutive symmetry of the KPZ universality class, whichis the Galilean invariance. It is thus not clear a prioriwhether an infinitesimal amount of temporal correlationssuffice to destroy the KPZ universal properties, and thiswas debated in the literature. We investigated this is-sue within a non-perturbative renormalization group ap-proach, which is functional in both momentum and fre-quency, and thus allows one to precisely analyze non-delta correlations in the microscopic noise.

We first studied the case of SR temporal correlations,characterized by a finite time scale τ , in d = 1. Thistype of correlation breaks both the Galilean and the time-reversal symmetries in d = 1. However, we found that for

15

10−6

10−5

10−4

10−3

10−2

10−1

100

10−1 100 101 102

100

101

10−3 10−2 10−1 100 101

(a) (b)−ID ∗(

0,p)

p

θ = 0θ = 0.15θ = 0.2θ = 0.3θ = 0.4

ζD(y)

y = /pz

θ = 0θ = 0.15θ = 0.2θ = 0.3θ = 0.4

FIG. 7. (a) Non-linear part ID∗ ($0, p) of the flow of fDκ at the fixed point as a function of the momentum p. For θ = 0.4, thecontribution of ID∗ in the p 1 regime is almost three orders of magnitude larger than in the SR-KPZ case θ < θth, whichindicates that the decoupling is much less effective. (b) Scaling function ζD associated with fDκ as a function of the scalingvariable y = $/pz. The enhanced contribution of the non-linear part of the flow at large θ translates into a non-zero slope inthe y → 0 limit. The curves for θ = 0 and θ = 0.15, which both belong to the SR-KPZ phase, are superimposed.

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.40

0.1

0.2

0 0.1 0.2 0.3 0.4

(a) (b)

θ

χ(p)

χ(κ)

θ

χ(κ) − χ(p)

α− αs, [41]

FIG. 8. (a) Critical roughness exponent: χ(κ) determined from the scaling dimension of the renormalization functions fXκ and

χ(p) determined from their large momentum behavior. We observe that while χ(κ) displays a quasi-linear variation with θ inthe LR regime, χ(p) seems to saturate at large θ. (b) Comparison of the anomalous scaling computed in this work and foundin the numerical simulations of [41]. [The corresponding data points are courtesy of the authors].

any τ , these microscopic correlations are washed out bythe flow: both symmetries are dynamically restored aftera certain RG scale, and the pure SR-KPZ fixed point isreached. This means that the large distance propertiesof the system are still described by the KPZ class. Thisresult is reasonable since temporal correlations are hardlystriclty delta-correlated in any real system and still KPZuniversal properties can be observed with high accuracyin experiments [13, 15].

We then focused on the case of LR temporal corre-lations, embodied in a power-law with exponent θ. Inboth d = 1 and d = 2, we found that there exists a crit-ical value θth(d) separating two regimes, in agreement

with previous RG studies in d = 1. For θ < θth, theLR part flows to zero, the symmetries are restored andthe large distance and long time universal properties aredescribed by the pure SR-KPZ fixed point, while abovethis value, the LR part dominates and drives the sys-tem to a new LR fixed point, with θ-dependent criticalexponents χ(θ) and z(θ) and a breaking of Galilean sym-metry ηλ∗ (θ) = z(θ) + χ(θ) − 2 6= 0 increasing with θ.We computed these exponents with increased precisioncompared to previous approaches in d = 1, and providedfor the first time an estimate in d = 2. In one dimension,we also evidenced in the LR phase some anomalous scal-ing. As a result, we found that the function associated

16

to the noise vertex, and hence the two point correlationfunction, displays a scaling form which is a generalizedversion of the usual Family-Vicsek one, as proposed in[41]. This behavior may be related to the emergenceof intermittency effects in the system, which warrants adedicated study.

As a future development, it would be interesting tostudy the effect of temporal correlations within the nextlevel of approximation, termed the SO approximation,which allows one to fully describe the momentum andfrequency dependence of two-point functions without in-ducing any spurious breaking of symmetries. This willnot change qualitatively the results presented here butwould allow one to achieve more precision. However, theflow equations at SO are much more complicated sincethey include contributions from all higher-order vertex

functions Γ(n)κ and the numerical cost to integrate them

is increased. Implementing this scheme in d > 1 wouldbe desirable even for the pure case, since it would allowone to probe the existence of a upper critical dimensionfor KPZ. This is work in progress.

ACKNOWLEDGMENTS

The authors thank N. Wschebor and B. Delamotte foruseful discussions on this work, and the authors of [41]for kindly providing us with the data from their numer-ical simulations concerning the anomalous scaling. Thiswork received support from the French ANR through theproject NeqFluids (grant ANR-18-CE92-0019).

Appendix A: Non-analycities in the presence of apower-law correlator

In principles, the presence of the regulator in theNPRG flow ensures the analyticity of all vertex func-

tions Γ(n)κ at any finite scale κ. This is always true for

the momentum dependence because of the presence ofthe regulator (17). However, this is not guaranteed forthe frequency dependence since we have not included a

frequency regulator. In fact, in most systems, as long asthe initial condition is smooth, the integrands in the flowequations are generally well-behaved in frequency andlead to convergent integrals. A problem may arise whennon-analytic initial conditions are considered, as for thecase of LR correlations. The flow of the LR coupling wθκcan be extracted from the flow of fDκ as

∂swθκ = lim

$→0$2θ∂sf

Dκ ($, 0) . (A1)

Within the NLO scheme, one finds

∂swθκ = lim

$→02λκ

×∫ω,~p

(~q · ~Q)2

P (ω, q)2P (Ω, Q)

$2θ

($ + ω)2θ

×(P (ω, q)∂sS

Dk (q)− 2~q 2`κ(q)∂sS

νk (q)

($ω

)2θ).

(A2)

For values θ < 1/2, the contribution of the first term inthe right hand side always vanishes. The same holds truefor the second term for θ < 1/4. However, in the range1/4 ≤ θ < 1/2, an ambiguity arises in the second term,since this term vanishes only if the limit $ → 0 is takenbefore the integration on ω. This ambiguity is presentbecause the frequency sector is not properly regularized,and would disappear with a frequency-dependent regula-tor [66]. Since the result should not depend on the choiceof the regulator, we simply assume that this term is zerosince it would vanish with a frequency regulator. Underthis assumption, the coupling wθκ is indeed not renor-malized. The same conclusion can be reached by simplyshifting the normalization point to a non-zero externalfrequency $0, which also resolves the ambiguity.

Appendix B: Flow equation of λκ

The flow of the non-linear coupling λκ, defined by (35),can be extracted from the flow of the three-point function

Γ(2,1)κ which reads

∂κ[Γ(3)κ ]ijk(p1,p2) =

1

2Tr∂κRκ(q)Gκ(q)

[−Γ

(5)κ,ijk(p1,p2,−p1 − p2,q)

+Γ(4)κ,ij(p1,p2,q)Gκ(p1 + p2 + q)Γ

(3)κ,k(−p1 − p2,p1 + p2 + q)

−Γ(3)κ,i(p1,q)Gκ(p1 + q)Γ

(3)κ,j(p2,p1 + q)Gκ(p1 + p2 + q)Γ

(3)κ,k(−p1 − p2,p1 + p2 + q)

]Gκ(q)

(B1a)

17

>>

> >>

>>

>

>

>> >

>>

>>>

>

(B1b)

where the Γ(n)κ on the right-hand side are represented

as 2 × 2 matrices (i.e. only the external indices areindicated, summation over the internal indices is im-plied), and where the cross in the diagrams represents thederivative with respect to κ of the regulator, and Pn isthe permutation group associated to the set (i1, . . . , in).The trace operation in (B1a) also includes all non-trivialpermutations of the external vertices with associated mo-menta. Within the NLOω approximation, the 4- and 5-point vertex functions are zero, and only the 3-point ver-

tex Γ(2,1)κ ≡ [Γ

(3)κ ]ϕ,ϕ,ϕ gives a non-vanishing contribution

in the remaining diagrams. Evaluating this expression atexternal frequencies $1 = $2 = 0 and external momenta~p1 = ~p2 ≡ ~p/2 and taking the limit ~p→ 0 yields (36).

Appendix C: Numerical Integration

In this appendix, we only consider dimensionless quan-tities, so we omit the hat symbols to alleviate notations.

1. Integration scheme

We integrated numerically the flow equations for thedimensionless functions fDκ , fνκ and fλκ , for the dimen-sionless couplings gκ and wθκ, and for the running anoma-lous dimensions ηνκ and ηDκ , using standard procedures.The advancement in the RG time s is achieved with anadaptative time-step: a default time-step ∆s = 1 is usedas long as the ratio of the non-linear part IXκ ($, p) ofthe flow of a function fXκ (or a coupling) for all ex-ternal ($, p) does not exceed 1% of the function itselfIXκ ($, p)/fXκ ($, p) < 0.01, else the time-step is itera-

tively decreased by a factor√

10 until this constraint issatisfied.

In generic spatial dimension d there are three differentintegrals to perform: over the modulus of the internalmomentum q, over the internal frequency ω and over theangle ψ between the external and internal momenta. Aquadrature Gauss-Legendre method is used to computethe three of them,∫ π

0

dψ

∫ ∞0

dq

∫ ∞−∞

dω fXκ (ψ, q, ω)

→∑i,j,k

w(ψ)i w

(q)j w

(ω)k fXκ (ψ∗i , q

∗j , ω

∗k) (C1)

where (ψ∗i , q∗j , ω

∗k) are the quadrature grid-points and w

(·)i

their respective weights. For some specific points (typ-ically zero momentum or frequency) the integrand hasto be treated analytically to avoid spurious numericaldivergences. The domain of integration on the internalmomentum is q ∈ [0,∞). However, the presence of thederivative of the regulator ∂κRκ in IXκ ($, p) effectivelycuts exponentially the internal momentum to q . κ suchthat the integral can be performed without loss of preci-sion over a finite domain q ∈ [0, qmax]. We checked thatqmax = 10 suffices to obtain a converged value for theintegrals. On the other hand, as the frequency sector isnot regularized, the integral over the internal frequency isnot cut, such that the contribution of the high-frequencysector is not negligible. The integral over the internalfrequency is hence performed on a domain ω ∈ [0, ωmax]with ωmax = 103.

100

101

102

103

10−2 10−1 100 101 102

p

Integr. Grid

Full Flow

Dim.Flow

Spline

FIG. 9. Sketch of the numerical procedure: the blue trianglesand green circles represent the grid points where the flows ofthe functions fXκ ($, p) are computed from given initial con-ditions at s = 0. The orange squares represent the grid pointsused in the Gauss-Legendre algorithm to compute numericallythe integrals over the internal frequency and momentum (thegrid for the integration over the angle is not represented inthis sketch). The values of the functions in the whole orange-shaded domain are calculated using a spline interpolation.The flows of the functions on the boundaries (green circles)are approximated by only the dimensional (linear) flow.

18

0.4

0.5

0.6

0.7

10 20 30 40 50 60 70

χ

α

θ = 0θ = 0.35θ = 0.45

FIG. 10. Roughness exponent χ as a function of the cut-offparameter α in d = 2 for different values of θ.

2. Grids and Interpolation

The flow equations are computed for external fre-quency and momentum on grid points ($, p) with loga-rithmic spacing, represented by blue triangles and greencircles on Fig. 9. To compute the integrals over the inter-nal momentum q and frequency ω, the functions fXκ haveto be evaluated at values Q = |~p + ~q| and Ω = ω + $,which can fall outside grid points. In this work, theseintegrals are computed using another grid, representedin Fig. 9 by orange squares, which corresponds to theGauss-Legendre quadrature roots in their respective in-tegration domains, q ∈ [0, qmax] and ω ∈ [0, ωmax]. Theexternal momenta (respectively frequencies) are chosensuch that p ∈ [0, Qmax = pmax + qmax] (resp. $ ∈[0,Ωmax = $max + ωmax]), where pmax (resp. $max))is the last blue triangle and Qmax (resp. Ωmax) is thefollowing green circle. The functions can be evaluated inthe whole orange-shaded domain using a bi-cubic splineprocedure from the external points (represented by bluetriangles and green circles). The values of the deriva-tives ∂pf

Xκ ($, p) and ∂$f

Xκ ($, p) are also evaluated us-

ing the bi-cubic spline interpolation. With this choice,the flow equations of fXκ ($, p) for all grid points up to($max, pmax) can hence be evaluated since the values ofintegration all lie in the orange-shaded domain. To com-pute the flow on the boundaries, i.e. the points with$ = Ωmax or p = Qmax, represented by the green cir-cles on Fig. 9, we exploit the decoupling property of theflow: for sufficiently high momenta and frequencies, thenon-linear part of the flow IXκ ($, p) become negligiblecompared to the function fXκ itself when the fixed-pointis approached. For these points, we thus approximate theflow by the linear contribution only:

∂sfXκ ($, p) =

(ηXκ + (2− ηνκ)$∂$ + p ∂p

)fXκ ($, p).

(C2)

3. Choice of the cutoff parameter andnormalization point

The cutoff function (18) depends on a free parameterα, which can be varied to assess the precision within achosen approximation level. Indeed, if the flow equationswere solved exaclty, the results would not depend on theregulator. Any approximation induces a spurious resid-ual dependence on α, and a minimal sensitivity principlecan be exploited to select the optimal value of α [62]. Inboth d = 1 and d = 2, we studied the influence of α onthe critical exponents obtained at both NLO and NLOω.We observed that their values only weakly depend on α,as illustrated for χ in Fig. 10. We thus fixed α = 10 forall the results presented in this work. This value cor-responds to an optimal value for the pure KPZ case ind = 1 at NLO [46].

Let us finally discuss the choice of the normalizationpoint $0 in d = 2. The normalization point corre-sponds to a specific configuration of the external mo-mentum and frequency. While the RG flow equationsfor the running parameters Xκ depend on the particularconfiguration chosen to define it Xκ = fXκ ($NP, pNP),the running dimensions ηXκ at the fixed point do not[75, 76]. It is interesting to note that in the frameworkof NPRG, one can in general safely take the simplestchoice ($NP, pNP) = ($ = 0, p = 0), which is a prob-lematic configuration in the dimensional-regularizationscheme due to the mixing of infra-red and ultra-violetsingularities for massless theories [77].

In this work, as the power-law exponent θ for the timecorrelations approaches the value θ = 1/4, a divergenceemerges in the flow of fDκ ($ = 0, p). As mentioned inthe main text, this divergence is a pure artifact due tothe absence of a frequency regulator. As explained inSec. IV B 2, one can simply avoid such a singularity byshifting the normalization point to a small but non-zerovalue $0. As we are working in practice with some ap-proximations and not with the exact theory, varying thenormalization point introduces a small spurious varia-tion of the anomalous dimensions ηX∗ at the fixed point.However, we checked that for small enough $0, this resid-ual dependence is indeed negligible. This is illustrated

0.5

0.55

0.6

10−4 10−3 10−2 10−1

χ

NP

FIG. 11. Roughness exponent χ for θ = 0.45 as a function ofthe normalization point $0 in d = 2.

19

on Fig. 11 which shows the variation of χ with $0 forθ = 0.45, for which the singularity at the origin is the

steepest. We hence fixed $0 = 0.01 in d = 2 for allvalues of θ.

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