Anomalous scaling at non-thermal fixed points of Burgers' and Gross-Pitaevskii turbulence
AAnomalous scaling at non-thermal fixed pointsof Burgers’ and Gross-Pitaevskii turbulence
Steven Mathey, ∗ Thomas Gasenzer, † and Jan M. Pawlowski ‡ Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany,Heidelberg Center for Quantum Dynamics, Universit¨at Heidelberg, INF 226, 69120 Heidelberg, Germany,ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany (Dated: October 10, 2018)Scaling in the dynamical properties of complex many-body systems has been of strong interestsince turbulence phenomena became the subject of systematic mathematical studies. In this article,dynamical critical phenomena far from equilibrium are investigated with functional renormalisationgroup equations. The focus is set on scaling solutions of the stochastic driven-dissipative Burgersequation and their relation to solutions known in the literature for Burgers and Kardar-Parisi-Zhang dynamics. We furthermore relate superfluid as well as acoustic turbulence described bythe Gross-Pitaevskii model to known analytic and numerical results for scaling solutions. In thisway, the canonical Kolmogorov exponent 5 / PACS numbers: 03.65.Db 05.10.Cc 05.70.Jk, 47.27.ef 47.37.+q,
I. INTRODUCTION
The concept of scaling has been of tremendous interestsince the days of Galilei to whom the insight is attributedthat the effects of physical laws may change considerablyunder a mere rescaling of object sizes [1]. Scale transfor-mations became important in studies of turbulent flow,beginning with Reynolds’ work and culminating in Kol-mogorov’s seminal papers of 1941 [2, 3]. In the contextof phase transitions between equilibrium states of mat-ter, scaling behaviour is a hallmark of criticality and theappearance of macroscopic structure independent of mi-croscopic details. When a system near a phase transitionis driven away from equilibrium, critical properties alsoshow up in the ensuing dynamical evolution [4]. The cor-relation length grows large, and relaxation time scales di-verge. Critical dynamics away from thermal equilibriumhas been studied extensively in the physics of non-linearsystems, in particular in the context of pattern forma-tion [5], phase-ordering kinetics [6], and turbulence [7],but the full structure of non-thermal criticality is still farfrom being satisfactorily understood.Here, we study, by means of functional renorma-lisation-group methods, non-thermal critical states ofdriven and dissipative hydrodynamics in view of possiblescaling. We consider classical Burgers’ turbulence [8–10],and relate our results to scaling solutions of the Kardar-Parisi-Zhang (KPZ) equation [11] as well as to quantumturbulence described by the Gross-Pitaevskii (GP) model[12]. We study, specifically, possible scalings at fixed ∗ [email protected] † [email protected] ‡ [email protected] points of the driven-dissipative Burgers equation and dis-cuss their relevance in the context of turbulent cascades.By comparing with semi-classical simulations of the GPequation, we show that critical exponents known for theKPZ equation can be used to quantify anomalous scalingof acoustic turbulence in a superfluid.The dynamics of many-body systems driven out ofequilibrium can become stationary by means of dissipa-tion and be characterised by a local flux of energy, both inposition and momentum space. Such dynamics has manyrealisations in nature since it is virtually impossible tofully suppress contact to the environment in any realisticsetting. If the driving, the flux of energy in momentumspace, and the dissipation have suitable characteristics,the stationary dynamics can exhibit scale invariance anduniversality distinct from any thermal equilibrium state.Examples of such driven-dissipative stationary systemsare realised in a wide range of systems, from exciton-polariton condensates [13–19], through pattern formationin non-linear media [5, 20] all the way to classical hydro-dynamic turbulence [2, 3].In this article, we study scaling solutions of thestochastic Burgers equation [8–10], which is a model forfully compressible fluid dynamics. We compute station-ary correlation functions of Burgers turbulence driven bya random Gaussian forcing. We set up functional renor-malisation group (RG) flow equations to look for non-perturbative fixed points, applying ideas from Refs. [21–23] in the context of classical hydrodynamic turbulence[24–26]. For this, we take momentum dependence of low-order correlations into account.We identify a range of critical scalings corresponding tosolutions regular in the ultraviolet. These scalings com-prise that at known fixed points of the KPZ equation[11]. The respective exponents are shown to corroborate a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug numerical results for sound-wave turbulence in GP super-fluids, giving rise to first estimates of anomalous expo-nents at non-thermal fixed points [27–29]. Fixed pointsoutside this range, which require a UV regulator to beimplemented in the integrals, can, to a certain extent, berelated to direct cascades of energy. Kolmogorov scalingof incompressible fluids as well as superfluid turbulence[30–38] belong to this regime.For the GP model, non-thermal fixed points are knownto exist [29] which are related to strong phase excitations,including ensembles of quasi-topological defects such asvortices [39–42] or solitons [43, 44], as well as local den-sity depressions and sound-wave turbulence [40]. Here,we use an additional constraint set by Galilei invarianceto analytically obtain the canonical scaling laws at thesefixed points: While the quasi-particle cascade exhibitsthe known p − scaling of random vortex-antivortex en-sembles, the energy cascade exhibits Kolmogorov- p − / scaling known from simulations of the GPE [34, 35].Comparing simulation results for superfluid turbulence[34, 35] and non-thermal fixed points [39–41] with an-alytic predictions of the present work as well as of thestrong-wave-turbulence analysis [29] we conjecture thatanomalous exponents for the respective scaling solutionsof the GP model are close to zero. Finally, we find sig-natures of an additional yet unknown fixed point whichis associated with spatially non-local forcing. For d = 1our results are in agreement with perturbative [45] andnon-perturbative [46–49] RG calculations.Our results should be of relevance for future exper-iments with driven-dissipative systems such as excitonpolaritons [13–19] as well as ultracold atomic gases [50]which have the potential to measure particle number dis-tributions in momentum space and thus observe powerlaws directly.Our paper is organised as follows: In Sect. II, we dis-cuss driven-dissipative dynamics of the stochastic Burg-ers and GP equations. In Sect. III, we introduce thefunctional RG approach before setting up flow equationsin Sect. III B. The RG fixed point equations are derivedand discussed in Sect. IV. Analytical constraints on theirproperties are presented in Sect. IV D. In Sect. V, we dis-cuss the physical implications of our results for classicaland quantum turbulence. II. TURBULENCE IN DRIVEN DISSIPATIVESYSTEMSA. Burgers and KPZ turbulence
In this article, we study scaling solutions of thestochastic Burgers equation [8–10], ∂ t v + ( v · ∇ ) v − ν ∇ v = f . (1) v is the position and time dependent velocity field and ν is the kinematic viscosity. f is a force with zero average, (cid:104) f (cid:105) = 0, and Gaussian fluctuations (cid:104) f i ( t, x ) f j ( t (cid:48) , x (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) F ij ( | x − x (cid:48) | ). Latin indices denote spatial dimen-sions. To distinguish different types of forcing we choosethe power-law ansatz (cid:104) f i ( ω, p ) f j ( ω (cid:48) , p (cid:48) ) (cid:105) = δ ij δ ( ω + ω (cid:48) ) δ ( p + p (cid:48) ) F p β (2)for the force correlator in Fourier-space, where p = | p | .Hence, the exponent β determines the degree of non-locality of the forcing. For β > β < β = 0 corresponds to aforcing delta correlated in space.Burgers’ equation is equivalent to the Navier-Stokesequation if the equation of state is assumed to imposea constant pressure, P = const. [51]. For applica-tions of Burgers’ equation see Ref. [10] and referencestherein. The irrotationally forced Burgers equation canbe mapped onto the Kardar-Parisi-Zhang (KPZ) equa-tion [11], ∂ t θ + λ ∇ θ ) = ν ∇ θ, (3)with v = ∇ θ . The KPZ equation is typically used to de-scribe non-linear interface growth but can also be appliedto the dynamics of phase fluctuations in an ultracold Bosegas described by the stochastic GP model [13, 52], or todirected polymers in random media [53–55]. In the limit ν →
0, Burgers’ equation possesses shock-wave solutionswith discontinuities in the velocity field [56, 57]. Theseshocks can also appear in the GP model but, due to thedefinition of the phase on a compact circle, lead to thecreation of (quasi) topological defects, e.g., dissolve intosoliton trains [58, 59].An extensive range of studies of critical dynamics existsfor the models studied here. To characterize a turbulentstate, an important quantity is the second moment of thevelocity increment,∆ v ( τ, r ) = (cid:104) [ v ( t + τ, x + r ) − v ( t, x )] (cid:105) , (4)which takes the scaling form∆ v ( τ, r ) = r χ − g ( τ /r z ) , (5)with r = | r | and roughness and dynamical critical expo-nents χ and z , respectively. We will in particular considerthe kinetic energy spectrum (cid:15) kin ( p ) = 12 (cid:90) ω (cid:104) v ( ω, p ) · v ( − ω, − p ) (cid:105) , (6)and derive the scaling exponent ξ defined by (cid:15) kin ( s | p | ) = s − ξ (cid:15) kin ( | p | ) . (7)Here and in the following we use the short-hand notation (cid:82) t, x = (cid:82) d t d d x , (cid:82) ω, p = (2 π ) − d − (cid:82) d ω d d p .The values of χ and z are, so far, only known for spatialdimension d = 1 and specific choices of the forcing andinitial conditions, see Refs. [60, 61]. For d ≥
2, the an-alytical studies [62–65] assume that the forcing (and/orvelocity field) is the gradient of a potential and exploitthe mapping to the KPZ equation. Numerical studiesexist for the one-dimensional case [10, 56, 57, 66].Scaling of correlation functions as described by theKPZ equation has been studied in Ref. [45] by meansof a one-loop perturbative RG calculation. A scaling (cid:104)| f ( ω, p ) | (cid:105) ∼ p β of the force correlator is assumed, andthe scaling exponent χ of the correlation function (5)is uniquely related to the exponent β of the forcing,for 0 < β < d = 1. Outside this range, non-perturbative effects become important and a more so-phisticated approach is necessary.An exact expression for the time-dependent velocityfield probability distribution was obtained in the case β = 2 and d = 1. For reviews, see [60, 61] and referencestherein. Its asymptotic limit confirms the predictions ofRefs. [11, 67] concerning the scaling exponents and yields χ = 1 / z = 3 /
2. However, due to the limitations ofperturbative methods, possible scaling forms of the corre-lator are unknown in d > d >
1. Mostof the literature concentrate on the case β = 2, corre-sponding to white-noise forcing in space. In this context,predictions for the scaling exponents, scaling functionsand upper critical dimension have been made, using, e.g.,the mode coupling approximation [68], the self-consistentexpansion [69], or the weak-noise scheme [70]. The case β < d → ∞ , and bi-fractal scalingof the velocity increments was obtained. The tails ofthe probability distribution of velocity differences wereaddressed in [71] using an operator product expansionin d = 1, and in [56, 64] within an instanton approach.Decaying Burgers turbulence was studied in [65].The stochastic KPZ equation, for the case β = 2,has been studied within the functional RG frameworkin Refs. [46–48]. Non-perturbative RG fixed points werefound for d ≤
3, and scaling exponents and functionswere computed. Furthermore, the perturbative resultsof [11] were recovered, and the form of the scaling func-tion g ( x ) derived compared well with the exact results ofRefs. [60, 61]. B. Driven dissipative Gross-Pitaevskii systems
Driven-dissipative superfluid dilute Bose gases can bedescribed in terms of the stochastic Gross-Pitaevskiiequation (SGPE), i∂ t ψ = (cid:20) − (cid:18) m − iν (cid:19) ∇ − µ + g | ψ | (cid:21) ψ + ζ. (8) Here and in the following, (cid:126) = 1. Driven-dissipativenon-linear equations of the type (8) have been studiedin the literature, see, e.g., [20, 35, 72–83]. Superfluidturbulence can be formulated in a setting similar to theabove Burgers turbulence problem, with the additionalconstraint that the forcing must conserve the propertyof the velocity to be a potential field. In the SGPE,we allow for the necessary dissipation, loss, and gainof energy and particles by allowing µ = µ + iµ and g = g − ig to become complex, including an effectiveparticle gain or loss µ , as well as two-body interactionand loss parameters g , . The diffusion term ∝ ν is gener-ated through the coarse graining of high-frequency modes[78–82]. ζ is a Gaussian, delta-correlated white noise, i.e., (cid:104) ζ ∗ ( t, x ) ζ ( t (cid:48) , x (cid:48) ) (cid:105) = γδ ( t − t (cid:48) ) δ ( x − x (cid:48) ), induced by the lossand gain of particles.Superfluid turbulence [30–38] manifests itself in self-similar field configurations in the domain of long-wavelength hydrodynamic excitations. The hydrody-namic formulation of the SGPE results by introducingthe parametrisation ψ = √ n exp[i θ ] in terms of the fluiddensity n and velocity fields v = m − ∇ θ . The phaseangle θ then obeys a Langevin equation of the KPZtype which is equivalent to Burger’s equation (1) forthe curl-free velocity field v , under the condition that f = m − ∇ U , with a random potential field U . See Ap-pendix A for details.We note that the cubic non-linearity in the KPZ Hamil-tonian can lead to an instability. For typical parame-ter choices, however, the KPZ equation describes surfacegrowth and smoothing [11], and a steady state is reachedwhen the driving and dissipation compensate each other.Shocks in the velocity field, corresponding to cusps inthe surface, develop and grow. The dynamics describedby the SGPE is different insofar the phase θ lives onthe compact circle. Moreover, the GPE supports soli-tary wave solutions and (quasi) topological defects suchas vortices. Velocity shock waves created due to the non-linearity typically lead to the formation of such defects. III. RENORMALISATION-GROUP APPROACH
The functional RG [84] provides a non-perturbativeframework to implement the coarse graining inherent tothe RG. See Refs. [85–93] for reviews.
A. Wetterich’s flow equation
A functional renormalisation-group (RG) analysis al-lows to determine the effective turbulent dynamics of theinfrared (IR) modes by applying a Wilson-type averagingprocedure to the ultraviolet (UV) modes. This leads toan effective action Γ k [ v ] describing the IR modes with p ≡ | p | < k , by integrating out the higher momenta,e − Γ k [ v ] = (cid:90) (cid:89) p>kω d v ( ω, p ) e − S [ v ] . (9)In the case of a Langevin field equation with a Gaus-sian forcing, the weight of the field configurations can beexpressed in terms of the exponential of the action S [ v ] = 12 (cid:90) t, x , y E i ( t, x ) E j ( t, y ) F − ij ( | x − y | ) . (10)Here and in the following repeated indexes are to besummed over. E ( t, x ) = E ( v ( t, x )) is defined as E ( t, x ) = ∂ t v + ( v · ∇ ) v − ν ∇ v . (11)The expression in Eq. (10) is obtained within the Martin-Siggia-Rose/Janssen-de Dominicis formalism [94–98] byintegrating out the response field. This field is usuallyexplicitly kept in the action for having access to responsefunctions which we do not consider here. We implementthe cutoff k by adding a term∆ S k [ v ] = 12 (cid:90) ω, p v i ( ω, p ) R k,ij ( | p | ) v j ( − ω, − p ) (12)to the action S , choosing the regulator R k,ij ( p ) = δ ij R k ( p ) diagonal in frequencies and momenta which di-verges for | p | = p (cid:28) k , damping out the velocity fluctu-ations on scales larger than 1 /k . For p (cid:29) k , the regula-tor vanishes. In the limit k → v ] results, which takes into account all fluctuations andgenerates the physical correlation functions. The regu-lator term allows to extend the functional integral overall momenta p and define the coarse-grained Schwingerfunctional W k [ J ],e − W k [ J ] = (cid:90) d v e − S [ v ] − ∆ S k [ v ]+ (cid:82) t, x J · v . (13)Functional derivatives of W k [ J ] generate the coarsegrained connected correlations functions. From this, thescale dependent effective action Γ k [ v ] is defined throughthe Legendre transform of W k [ J ],Γ k [ v ] = − W k [ J ] + (cid:90) t, x J · v − ∆ S k [ v ] . (14)It interpolates between the bare action Eq. (10), for k → ∞ , and the full effective action Γ[ v ], for k → k [ v ] with the cutoff k is determined byWetterich’s flow equation [84] k∂ k Γ k [ v ] = 12 Tr (cid:32) (2) k [ v ] + R k k∂ k R k (cid:33) ≡ I k [ v ] . (15) This involves the second moment of the effectiveaction, with matrix elements Γ (2) k,ij [ v ]( ω (cid:48) , k (cid:48) ; ω, k ) = δ Γ k / ( δv i ( ω (cid:48) , k (cid:48) ) δv j ( ω, k )). Evaluated on the solution ofΓ (1) k [ v ] = 0, it is related to the inverse of the k -dependenttwo-point velocity cumulant, G k → ,ij ( ω, p ) = (2 π ) d +1 (cid:104) v i ( ω, p ) v j ( − ω, − p ) (cid:105) = [Γ (2) k [0] + R k ] − ij | k → . (16)Details are deferred to Appendix B. B. Approximation scheme
The RG flow equation (15) connects effective actionfunctionals for different cutoff scales k to each other.Hence, in order to find self-similar turbulent configura-tions we look for IR fixed points of the flow, i.e., forsolutions of Eq. (15) which are scaling in ω and p inthe limit k →
0. Eq. (15) relates the action with its sec-ond moment and therefore creates an infinite hierarchy ofintegro-differential equations for its n th order moments.It is a functional integro-differential equation. Its full so-lution is equivalent to solving the path integral. In thepresent case, we are interested in turbulent scaling solu-tions. Hence, the ansatz for the effective action shouldallow for these solutions and respect the Galilei symme-try of the underlying theory. We chooseΓ k [ v ] = 12 (cid:90) ω, p E k ( ω, p ) · E k ( − ω, − p ) F − k ( p ) , E k ( ω, p ) = (cid:0) iω + ν k ( p ) p (cid:1) v ( ω, p ) − i (cid:90) ω (cid:48) , q [ v ( ω − ω (cid:48) , p − q ) · q ] v ( ω (cid:48) , q ) (17)for the effective average action, in terms of the effectiveinverse force correlator F − k and the kinematic viscos-ity ν k . Γ k [ v ] has the same form as the bare action S [ v ]of the underlying Burgers equation (10), but with the in-verse force correlator and the kinematic viscosity allowedto be k -dependent. We anticipate that ν k ( p ) will become p -dependent as a result of the RG flow because the ad-vective derivative renders the cubic and quartic couplingsof the velocity field momentum dependent. Eq. (17) isthe minimal ansatz consistent with Galilei symmetry andincluding the necessary momentum and frequency depen-dence for turbulence solutions. Extensions of the presentapproximation and the discussion of the systematic errorwill be presented elsewhere.With the above ansatz, we keep a general dependenceof the inverse propagator on p while taking into accountthe ω -dependence in an expansion to first order in ω ,Γ (2) k,ij ( ω, p ) ≡ δ ij Γ (2) k ( ω, p )= δ ij (cid:2) ν k ( p ) p + ω (cid:3) F − k ( p ) . (18)The sole dependence on the norm p reflects the assumedrotational invariance. The truncation of the frequencyexpansion ensures that the integrand on the right handside of the flow equation (15) is a rational function of ω .See Appendix C for details.Note that our ansatz is chosen such that the inverseforce correlator F − k and thus the inverse propagator arediagonal in momentum p and in the field indices i, j . As aconsequence, the correlator describes field configurationswhere the forcing injects vorticity. In contrast to this,Gaussian random forcing that keeps the velocity field curlfree can be written as the gradient of a potential, f = ∇ U , and therefore obeys a noise correlator of the form (cid:104) f i ( ω, p ) f j ( ω (cid:48) , p (cid:48) ) (cid:105) = δ ( ω + ω (cid:48) ) δ ( p + p (cid:48) ) p i p j u ( ω, p ) , (19)where u is the scalar correlator of the potential. C. Flow equations for correlators
The truncated differential equations describing the RGflow of F − k ( p ) and ν k ( p ) are obtained by taking field andfrequency derivatives of Eq. (15) and evaluating them at v = 0 and ω = 0. The diagonal ansatz (18) implies thatthe right-hand side of the resulting flow equations is pro-jected onto the part diagonal in frequencies, momenta,and field indices, i.e., δ I k δv i ( ω, p ) δv j ( ω (cid:48) , p (cid:48) ) [ v = 0] ≡ I (2) k,ij [0]( ω, p ; ω (cid:48) , p (cid:48) ) → δ ij δ ( ω + ω (cid:48) ) δ ( p + p (cid:48) )(2 π ) d +1 I (2) k [0]( ω, p ) . (20)After the expansion (18) to order ω the flow equationsformally read k∂ k (cid:2) F − k ( p ) ν k ( p ) p (cid:3) = I (2) k [0]( ω = 0 , p ) ,k∂ k F − k ( p ) = ∂∂ω I (2) k [0](0 , p ) . (21)The second derivative of I k has the diagrammatic repre-sentation 3 3 4 − I (2) k [0]( ω, p ) = , (22)where the thick lines denote G k = (Γ (2) k + R k ) − , thethin lines external momenta and frequencies, and theblack dots insertions of the derivative k∂ k R k of the reg-ulator. The 3- and 4-vertices are given in the appendix,in Eqs. (C5) and (C7), respectively.Since we have inserted the truncated effective actionΓ k [ v ] on both sides of the flow equations, the right hand sides of Eqs. (21) are again functionals of ν k ( p ) and F − k ( p ) which are given explicitly in App. C, Eqs. (C10)and (C11). The resulting closed set of RG flow equa-tions describes the change of ν k ( p ) and F − k ( p ) under ashift of the cutoff scale k . For k → ∞ , all velocity fluc-tuations are suppressed, and we set (bare) initial condi-tions: ν ∞ ( p ) = ν , F − ∞ ( p ) = F − ( p ). In the infrared limit ν k → ( p ) and F − k → ( p ) describe the physically observableviscosity and forcing as functions of p . IV. FIXED POINTS OF THERENORMALISATION-GROUP FLOW
Universal scaling regimes are described by fixed pointsof the renormalisation-group (RG) flow: At an RG fixedpoint, the system becomes invariant under scaling trans-formations and exhibits scaling properties of correlationssuch as (5). In turn, fixed points encode the turbulentsolutions of our model. In the following we apply thefixed point approach developed in [21] for momentumand frequency dependent correlation functions.
A. Parametrisation in the infrared
As we approach the fixed points at vanishing cut-off, k = 0, we parametrise the inverse propagator in terms ofa scaling form. Moreover, we make use of the rescaledvariables, using units where the viscosity is dimension-less,ˆ p = pk , ˆ ω = 1 k (cid:114) z z ω, ˆ v (ˆ ω, ˆ p ) = k d +1 v ( ω, p ) . (23)Using these, the scaling forms of the propagator and itsfrequency derivative read [21]:Γ (2) k (0 , p ) = Γ (2)0 (0 , p ) [1 + δZ (ˆ p )] ,∂ ω Γ (2) k (cid:12)(cid:12)(cid:12) ( ω =0 ,p ) = ∂ ω Γ (2)0 (cid:12)(cid:12)(cid:12) ( ω =0 ,p ) [1 + δZ (ˆ p )] , (24)where δZ and δZ are the deviations of the two-pointcorrelators from those at vanishing cutoff. Here and inthe following, indices i = 1 , δZ i , etc., refer to the0th and 1st-order derivatives with respect to ω . In thelimit ˆ p → ∞ , the conditions δZ i (ˆ p → ∞ ) = 0 must befulfilled. Since this is the scaling limit, our propagatortakes the formΓ (2)0 (0 , p ) = k d z ˆ p η ,∂ ω Γ (2)0 (cid:12)(cid:12)(cid:12) ( ω =0 ,p ) = k d − z ˆ p η , (25)which, by definition, is independent of the scale k . Hence, z ∼ k η − d , z ∼ k η − d +4 . (26)Note that Eqs. (25) are used as definitions of z i and η i .Then Eqs. (24) define δZ i ( p ). The powers of k in front ofEqs. (24) indicate the dimensions of Γ (2) k and its deriva-tive. The rescaled propagator,ˆΓ (2) k (ˆ ω, ˆ p ) = Γ (2) k ( ω, p ) / ( k d z ) , (27)depends on k only implicitly through ˆ p . The fixed-pointparametrisation (24) exhibits the IR scaling defined bythe exponents η i . We find that the scale dependent func-tions F − k ( p ) and ν k ( p ) are related to the variables intro-duced above by ν k ( p ) = (cid:112) z /z ˆ p ( η − η − / (cid:115) δZ (ˆ p )1 + δZ (ˆ p ) ,F − k ( p ) = k d − z ˆ p η [1 + δZ (ˆ p )] . (28)The RG flow equations determine the possible values ofthe exponents η , at the fixed points of the RG flow. Atthe fixed point, i.e., in situations where the parametri-sation (24) is valid for all k ≥
0, the functions δZ i (ˆ p )characterise the difference between coarse-grained effec-tive actions Γ k , which show scaling for p (cid:29) k , and the(fully scale invariant) effective action obtained for k → k → [ v ] ≡ Γ[ v ] does not depend on k and generates phys-ical observables. Eq. (5) implies that χ = η + η − d , z = η − η . (29) B. Fixed-point equations
We obtain the fixed points by inserting Eqs. (28) intothe flow equations (21) and expressing the resulting equa-tions for the δZ i in terms of the rescaled variables (23),d δZ i dˆ p = − h ˆ I (2) i (ˆ p )ˆ p η i +1 , i = 1 , . (30)Here, the rescaled flow integrals ˆ I (2) i (ˆ p ) are related to thevelocity-squared derivative of I k [ v ] by I (2) k [0] ( ω, p ) = k d (cid:114) z z (cid:104) ˆ I (2)1 (ˆ p ) + ˆ ω ˆ I (2)2 (ˆ p ) + O (cid:0) ˆ ω (cid:1)(cid:105) (31)and are given explicitly in Eqs. (D1)–(D4), with (D5)–(D11). We note that the indices 1 , I (2) i which referto the order of the ω expansion replace the notation ofthe explicit cutoff dependence of I (2) k . Before we discuss,in the next section, the properties of these flow integralsin further detail, we take a look at the prefactor h ≡ (cid:114) z z z (32) which equally appears in both flow equations. h mustbe independent of k since all other terms in Eqs. (30)depend on k only implicitly through ˆ p . h is the effectivecoupling constant the theory assumes at the fixed-point.A vanishing coupling h = 0 implies that the fixed pointis Gaussian.The coupling’s independence of k , together with theasymptotic scaling relations (26), implies the relation η + 4 + 2 d = 3 η (33)between the η , . Using this, we find the exponents (29)at the fixed point to be related to η and d only, z = 2 + d − η , (34) χ = η − d. (35)Adding these equations one obtains χ + z = 2 . (36)This can be attributed to Galilean invariance whichprohibits an anomalous scaling of the velocity field[45, 48, 67, 99, 100]: Since the non-linearity of E is partof the advective derivative of the fluid velocity, it mustscale in the same way as the partial time derivative. Thisis only possible if the velocity scales as position dividedby time ( √ ∆ v ∼ r/r z ), which implies, with the defini-tion of χ and z by Eq. (5), the relation (36). Of the fourparameters η , and z , we have introduced only two areindependent. Besides the relation (33), the overall scal-ing of the two-point function G − k ( ω, p ) is not determinedby the fixed-point equations and leaves a free choice ofthe system of units for g . C. Velocity correlations and kinetic energy
Once the scaling exponents, η i , are known, the scal-ing function, g ( x ) can be derived. This gives the scalingbehaviour (5) of the moment (4) which, with (17), reads∆ v k ( τ, r ) = d (cid:90) p − e −| τ | ν k ( p ) p e − i p · r F − k ( p ) ν k ( p ) p . (37)Inserting Eq. (28) and taking the limit k → δZ i (ˆ p ) = 0 and obtains∆ v ( τ, r ) = dh k z × (cid:90) ˆp − exp (cid:2) − ( | τ | k /z )(ˆ p d +2 − η /h ) − i ˆ p · k r (cid:3) ˆ p η − − d . (38)One can check, by substituting back p as integration vari-able, that the apparent dependence on the cutoff scale k cancels out. We are free to choose k = 1 /r and write∆ v ( τ, r ) = r η − d − (cid:18) k η − d z (cid:19) ˆ g (cid:18) τr d +2 − η k η − d z (cid:19) , ˆ g ( x ) = dh (cid:90) ˆ p − exp (cid:2) − ˆ p d +2 − η x/h − i ˆ p z (cid:3) ˆ p η − − d , (39)where ˆ p z is the z -component of ˆ p . According to the def-inition (5) the above scaling form is consistent with therelations (29), (34)–(36) between χ , z , η , and d . More-over, the factor k η − d /z , which makes the argument ofˆ g dimensionless, does not depend on the cutoff scale k ,see Eq. (26), and is used to normalise the scaling func-tion. See the discussion in Sect. IV B. Finally, insertingEq. (18) into Eq. (6) and performing the frequency inte-gration, we obtain the momentum scaling of the kineticenergy density (cid:15) kin ( p ) = 12 (cid:90) ω (cid:104) v ( ω, p ) · v ( − ω, − p ) (cid:105) = d F − ( p ) ν ( p ) p ∼ p − η +2+ d . (40)Hence, according to Eq. (7), ξ = 2 η − − d . D. Non-Gaussian fixed points
In the remainder of this section we focus on non-Gaussian fixed points at which the coupling h is non-vanishing, see Eq. (32). We derive two further relationsdetermining, together with the constraints discussed inthe previous section, the four parameters η , and z , .This is possible without explicitly solving Eqs. (30).The expressions of the flow integrals ˆ I (2) i ( y ) inEqs. (30), in terms of the functions δZ i , are given inthe appendix, in Eqs. (D1)–(D4), with (D5)–(D11). Tounderstand the relevance of the different contributionsadding to these integrals let us consider in more detailboth, the limits ˆ p → p (cid:28) k ) and ˆ p → ∞ ( p (cid:29) k ).Any set of RG flow equations that is local in momen-tum scale must satisfylim p/k →∞ k∂ k Γ (2) k ( ω, p )Γ (2) k ( ω, p ) = 0 . (41)This means that, for p far away from the cutoff k , thechange of the effective action with k does not affect thephysics. We will see in the following that the flow equa-tions at small ˆ p provide constraints on their solutionstogether with an explicit range of values for η . The op-posite limit, ˆ p (cid:29)
1. Limit of momenta p (cid:28) k Before we discuss the physical limit where p is muchgreater than the cutoff scale k , we consider the “bare”limit of Eqs. (30) of small ˆ p where none of the fluctu-ations are integrated out. We do this in order to beconsistent with the order of the detailed derivations inApp. D which is chosen such as to give simpler argu-ments first. Even though we do not know the exact formof the effective action Γ k →∞ [ v ] ≡ ˜ S [ v ] in this regime,we can extract information on the asymptotic scaling ofΓ (2) k →∞ ( ω, p ). Indeed, all fluctuations being suppressed bythe cutoff, it is natural to assume that the physical scal-ing of the propagator as given by η , , cf. Eqs. (24) and(25), is absent. This is only possible if δZ i (ˆ p ) assumesthe asymptotic form δZ i (ˆ p →
0) = − a i ˆ p − η i + α i [1 + F i (ˆ p )] . (42)Here, a i ˆ p − η i + α i is the leading contribution to δZ i (ˆ p →
0) + 1, with constants a i and α i to be determined, and F i (ˆ p ) contains the sub-leading parts. One can check byinserting Eq. (42) into Eq. (24) thatΓ (2) k →∞ (0 , p ) ∼ = a p α ,∂ ω Γ (2) k →∞ (cid:12)(cid:12)(cid:12) (0 , p ) ∼ = a p α . (43)Inserting Eqs. (42) into Eqs. (30) one finds that α i − η i − δZ i / dˆ p (ˆ p →
0) on the left-hand sides. On the right-hand sides, the integrals fall into several terms, each witha different scaling behaviour in the limit ˆ p →
0. Theseterms are given explicitly in Eqs. (D3)–(D11) of App. D,their infrared scaling in Eqs. (D14) and (D15). The con-dition that the leading infrared scaling powers must beidentical on both sides on Eqs. (30) leads to a closed setof equations for the exponents α i .Due to the absence of angular integrations the case d =1 is special. In d = 1 dimension, the resulting constraintson the α i read α = min (cid:16) α + α + 2 , α α , (cid:17) ,α = min (cid:16) α , α + α , α α , α (cid:17) + 2 . (44)This restricts the values of α , to the combinations( α , α ) ∈ ([ − , , − , ( α , α ) = (1 , . (45)In d (cid:54) = 1 dimensions, the leading IR scaling of the flowintegrals is modified, but the same procedure leads to − | α | | | | − – – α – – FIG. 1. (Color online) Non-Gaussian fixed points: A graphi-cal representation of the values of the bare exponents α and α , defined in Eq. (42) in blue and red for d = 1 and blackand red for d (cid:54) = 1, which characterise the non-Gaussian fixedpoints. The scaling exponents of the bare stochastic Burg-ers equation, as described by S [ v ], Eq. (10), are related by α = α + 4 and are shown as a black line. The blue dot at( α , α ) = (1 , −
2) corresponds to the fixed point investigatedin [45, 48], while the top half of the blue line (for α > η = 2 − α / η = 2 − α / d along the black line. η is related to η through Eq. (33). constraints on the α i : α = min (cid:16) , α α , α + α (cid:17) ,α = min (cid:16) α , α α , α + α , α (cid:17) . (46)Solutions of these equations are the combinations( α , α ) ∈ ([0 , , , ( α , α ) = (1 , . (47)The above results suggest that the allowed combinations( α , α ), summarised in Fig. 1, correspond to the differ-ent possible non-Gaussian fixed points. For the differ-ent dimensions, we find, for the scaling exponents rel-evant in the limit p (cid:28) k , a connected interval for α and an additional point at ( α , α ) = (1 , p scalings of ˜ S [ v ] corre-sponds to that of the bare action S [ v ] for Burgers’ equa-tion. In the bare action, ν is p -independent, which im-plies that the ratio of Γ (2) k →∞ (0 , p ) and ∂ Γ (2) k →∞ /∂ω (0 , p )scales as p , see Eq. (18). Hence, taking the ratio ofEqs. (43) gives α = α + 4. Analogously, Eq. (2) im-plies that ∂ Γ (2) k →∞ /∂ω (0 , p ) scales as p − β and thus that α = − β . The resulting possible combinations ( α , α ) =(4 − β, − β ) are marked by the black (dashed/solid) linein Fig. 1. As expected, there is no choice of the forcingexponent β that makes the stochastic Burgers equationsit at a non-Gaussian RG fixed point for all values of k .
2. Limit of momenta p (cid:29) k In the opposite limit of vanishing cutoff all fluctuationsare integrated out and the full effective theory emerges.Inserting Eqs. (24), the local-flow requirement (41) readslim ˆ p →∞ ˆ p δZ (cid:48) i (ˆ p )1 + δZ i (ˆ p ) = 0 , (48)with the notation δZ (cid:48) i (ˆ p ) = d δZ i (ˆ p ) / dˆ p . Assuming that δZ i (ˆ p (cid:29)
1) behaves as a power law, one finds that theabove requirement is only fulfilled if lim ˆ p →∞ δZ i (ˆ p ) = 0.In this case, once the cutoff scale is sent to zero, seeEq. (24), one finds fixed points with correlations given byscaling functions across all momentum scales. If, on theother hand, lim ˆ p →∞ δZ i (ˆ p ) = ∞ , an RG fixed point canonly exist if the scaling range is restricted to momentasmaller than some upper cutoff Λ. Then scaling onlyarises within the range of physical momenta and δZ i (1 (cid:28) ˆ p < Λ /k ) ∼ = 0. In this situation of a UV-divergent fixedpoint, the theory is not well defined exactly at the fixedpoint but the latter can be approached arbitrarily bychoosing Λ accordingly large.Here, we consider UV-finite fixed points and take thelimit ˆ p → ∞ in the flow integrals. The boundary condi-tion δZ i (ˆ p → ∞ ) = 0 then allows us to write Eq. (30) inthe integral form δZ i (ˆ p ) = h (cid:90) ∞ ˆ p d y ˆ I (2) i ( y ) y η i +1 , i = 1 , . (49)For ˆ p (cid:29)
1, also the integration variable y exceeds 1 by farsuch that we can approximate δZ i ( y ) = 0 in the integralsˆ I (2) i ( y ), see Appendix D for a detailed discussion. Theflow integrals can be further approximated by keepingonly their leading term as ˆ p → ∞ ,ˆ I (2) i (ˆ p → ∞ ) ∼ ˆ p β i . (50)We determine the exponents β i in App. D, see Eqs. (D52)and (D53). In order to obtain finite integrals on the righthand side of Eqs. (49), it is necessary that β i < η i . Thisimplies that η has to be within the range d = 1 : dd (cid:54) = 1 : ( d + 4) / (cid:27) < η < d + 1 (51)which is shown as the white area in Fig. 2.
3. Implications for driving, and turbulent cascades
The bounds (51) on η have distinct physical interpre-tations: While the lower bound can be expressed as aregularity condition on the type of forcing that is sam-pled by the stochastic process, the upper bound marksthe onset of a direct energy cascade.As the forcing is a Gaussian random variable, it follows1 | d | | – η – – FIG. 2. (Color online) The range of values of η (defined inEqs. (25) and related to η in Eq. (33) and to χ and η inEqs. (29)), which correspond to UV-convergent non-Gaussianfixed-points for different spatial dimensions d (white area).The stripe at d = 1 only applies to this single dimension. Inthe regions shaded in light and dark grey any potential fixedpoint is UV-divergent such that the RG flow integrals mustbe regularised in the UV. The blue dots correspond to theaverage literature values for the exponent η , cf. Refs. [101–107]. Their values, as given in Table I. of Ref. [48], are η =1 . d = 1), η = 2 . d = 2), and η = 3 . d = 3). The vertical blue line marks the range of possibleexponents found in Ref. [45] for different forcings, with theexponent β , defined in Eq. (2), chosen between 0 and 2. Thedark-grey shaded area marks the range of η correspondingto the sets ( α , α ) shown as the blue and black solid lines at α = − η > /
2. In between the two dotted lines thereis a direct cascade of kinetic energy. In the black area atthe left top, the dynamical critical exponent z is negative.See Sect. IV D 3 for further details. the probability distribution P k [ f ] ∼ exp (cid:26) − (cid:90) ω, p | f ( ω, p ) | F − k ( p ) (cid:27) . (52)This distribution implies that the probability of a spa-tially local force field f ( t, x ) ∼ δ ( x − x f ) can be finiteif (cid:82) p F − k ( p ) is finite and is necessarily zero otherwise.Furthermore, for the latter integral to be finite, the crit-ical exponent at a UV-finite fixed point needs to fulfil η < ( d + 4) /
3, as one finds by inserting the parametri-sation Eq. (28) for F − k . We conclude that in the lowergrey shaded area of Fig. 2, η < ( d + 4) /
3, local Gaussianforcing f ( t, x ) is included at a UV-finite fixed point whileit is suppressed for η > ( d + 4) /
3. We have shown abovethat UV-finite non-Gaussian fixed points require η > ( d + 4) /
3. Hence, for an RG fixed point to be UV-finite,the forcing needs to be sufficiently regular in space-time,specifically lim p →∞ | f ( ω, p ) | = lim ω →∞ ∂ ω | f ( ω, p ) | = 0.Note that, in the case d = 1, the regularity conditionis modified. The fixed points are UV-finite above therelatively lower limit η >
1. This is a consequence of the fact that in d = 1 dimension, point-like shocks are stablesolutions. Applying a force such as f ( t, x ) ∼ δ ( x − x s ) iscomparable to inserting a shock at the position x s .The upper bound, η = d + 1, can be related to the ap-pearance of a direct cascade of energy. We briefly sketchthe argument leading to this result in the following butleave a detailed derivation to a forthcoming publication.In order for the system to be stationary, the equation ofmotion (1), multiplied by v and averaged over statisti-cally, gives an evolution equation for the mean kineticenergy, ∂ t (cid:104) v ( t, x ) (cid:105) = ν (cid:104) v ( t, x ) · ∆ v ( t, x ) (cid:105) + (cid:104) f ( t, x ) · v ( t, x ) (cid:105)− (cid:104) v ( t, x ) · [( v ( t, x ) · ∇ ) v ( t, x )] (cid:105) = 0 . (53)The third term on the right-hand side can be written as (cid:15) adv = 12 (cid:104) ( v ( t, x ) · ∇ ) v ( t, x ) (cid:105) (54)and describes the advective increase of compressible en-ergy at x . Furthermore, (cid:15) ν = − ν (cid:104) v ( t, x ) · ∆ v ( t, x ) (cid:105) = (cid:90) ω, p ν k ( p ) p (cid:104)| v ( ω, p ) | (cid:105) ,(cid:15) f = (cid:104) f ( t, x ) · v ( t, x ) (cid:105) (55)capture energy dissipation and injection rates, respec-tively. Inserting the truncation (17) into the energy dis-sipation rate, the frequency integration can be performedby means of the residue theorem, and one obtains, in thelimit of removed regulator, (cid:15) ν = d (cid:90) p F k =0 ( p ) ≡ (cid:90) ∞ d p (cid:15) ν ( p ) . (56)The cubic advective term involves two convolutions inmomentum space, (cid:15) adv = (cid:90) ∞ d p (cid:90) ∞ d q (cid:15) adv ( p, q ) . (57)The advective transport kernel (cid:15) adv ( p, q ) describes therate of kinetic energy that is transported from the mo-mentum q to p . By means of the ansatz (17), (cid:15) adv ( p, q )can be expressed in terms of ν k ( p ) and F − k ( p ). In thelimit k → h and η only. Evalu-ating (cid:15) adv ( p, q ) in this way, a direct energy cascade, i.e.,transport which is local in momentum space on a loga-rithmic scale, can be identified for d + 1 < η < d + 3 / (cid:15) ( p, q ) is non-vanishing only for p ∼ = q (lo-cality), positive for p < q and negative for p > q (positivedirectionality), and (cid:15) ( p, q ) (cid:39) − (cid:15) ( p, − q ) (balance of driv-ing and dissipation, i.e., inertial turbulent transport).Note that it is natural to have a direct cascade requir-ing a UV regulator: Physically, a cascade is realised onlyin a given inertial range. For example, in turbulence ofan incompressible fluid in three dimensions, energy is in-0jected on the largest scales and transported to smallerscales by the non-linear dynamics which leads to largereddies feeding into smaller ones. The kinetic energy isdissipated into heat once it reaches the end of the in-ertial range set by the viscosity. At an RG fixed point,the inertial range by definition extends over all momenta.Hence, the UV cutoff of the dissipation scale is absent.As a result, energy in a direct cascade is transported toinfinitely large momenta, leading to a UV divergence ofthe fixed-point theory.Finally, using the solutions of Eqs. (44) and (46), themonomials in the flow integrals dominating in the limitˆ p → a i are obtained by matching the correspond-ing pre-factors. Other than Eqs. (44) and (46) theseare not closed and can not be solved independently ofEq. (30). However, for the interval sets of α i marked bythe black and blue lines in Fig. 1, the ratio of Eqs. (D18)(Eqs. (D26) for d (cid:54) = 1) determining the pre-factors a i givenew equations (D20) and (D28), respectively, which areindependent of the a i and provide the following relationsbetween α and η , η = (cid:26) − α / , d = 12 − α / d, d (cid:54) = 1 . (58)Specifically, for d = 1, we get 3 / < η < d (cid:54) = 1, 3 / d < η < d , intervals which we mark bydark grey shading in Fig. 2. This regime, for d (cid:54) = 1, isdisjunct with the regime where a direct energy cascademay occur. Only for d = 1, it allows for such a cascade,but does not require it. In view of Refs. [49, 108], we notethat we find an upper bound to the regime of allowed η .Specifically, Eq. (34) implies that above this bound, thedynamical critical exponent would become negative. V. IMPLICATIONS FOR CLASSICAL ANDQUANTUM TURBULENCEA. Burgers and KPZ scaling in d = 1 Our approach gives us access to the full set of fixedpoints that are consistent with the chosen truncation. Inthe following we discuss results known in the literaturefor Burgers and KPZ scaling solutions [45, 48] in view ofthe conditions (45), (51), and (58). η has been related,within 1-loop perturbation theory [45], to the exponent β of the force correlation function (2), for d = 1 and0 < β ≤ η = max (cid:18) , − β (cid:19) . (59)Hence, η = 2 for β = 0, from where it decreases linearlyas β increases and saturates at η = 3 / β ≥ / (cid:15) kin ∼ p − for a − | | β | | − − ξ – ← Non-local forcingLocal forcing → FIG. 3. (Color online) The scaling exponent of (cid:15) kin ( p ) ∼ p − ξ for d = 1, for different values of β . The grey area correspondsto values excluded for UV-finite fixed points. The linearlyrising part corresponds to the values of η associated withthe blue line of Fig. 1, and the saturation value to the upperblue dot, cf. Eq. (59). moderately local forcing ( β = 0), and its exponent growswith β , i.e., (cid:15) kin ∼ p for β ≥ /
2, see Fig. 3.It was found numerically for d = 1 and different val-ues of β [57, 66] and for d ≥ α , α ), derived in the pre-vious section, cf. Eq. (45), allow us to predict the ex-istence of new non-Gaussian RG fixed points, they arealso fully consistent with the known perturbative results[45]. Relation (58), which applies on the solid blue line − < α < / ≤ η ≤ β of the bare action is considered as a fixed parameterand governs the existence and properties of the differentfixed points. The resulting bare actions correspond tothe set of ( α , α ) marked as a dashed line in Fig. 1. It isconsistent with the constraint α = α + 4 resulting fromthe condition that the bare viscosity is p -independent,see the discussion after Eq. (47). In contrast to theseactions, the set of ( α , α ) marked as a blue line in thefigure represents bare actions which remain at the fixedpoint throughout the RG flow.When the forcing is strongly non-local, i.e., β <
0, ad-ditional non-linearities become relevant and the loop ex-pansion of Ref. [45] fails. Our results show that negativevalues of β can nevertheless be considered and that thereis another, yet unobserved fixed point, ( α , α ) = (1 , β < h > α .For d = 2 ,
3, we find that the literature values for η ,see Refs. [101–107], and their averages quoted in Table I.of Ref. [48] and marked as blue dots in Fig. 2, are within1the white range defined in (51). They are hence outsidethe dark grey area in Fig. 2 which we ascribe to the driv-ing of velocity fields of non-vanishing curl. We recall that,however, the Burgers and KPZ equations are equivalentonly for curl-free fields, which in turn is not ensured bythe truncation (17) of the action as it involves a diagonalforcing correlator F ij ∼ δ ij F , see the discussion at theend of Sect. III B. B. Strong wave and quantum turbulence
In the following we discuss results of the present workin the context of quantum turbulence in dilute Bose gasesas described by the Gross-Pitaevskii equation. We focus,in particular, on non-thermal fixed points identified bymeans of a strong-wave-turbulence analysis on the basisof two-particle irreducible (2PI) dynamic field equations[27–29] as well as semi-classical field simulations [39–43].We begin with a brief summary of the results ofRef. [29]. Stationary scaling solutions for the statisticaland spectral two-point correlators, F ( s z ω, s p ) = s − − κ F ( ω, p ) ,ρ ( s z ω, s p ) = s − η ρ ( ω, p ) , (60)respectively, were predicted by means of a non-perturbative wave-turbulence analysis of the 2PI dy-namic equations for these correlation functions.These solutions constitute non-thermal fixed points ofthe far-from-equilibrium dynamics [27–29]. Here, F and ρ are defined in terms of the time-ordered Greens func-tion G ( x − y ) = (cid:104)T ψ † ( x ) ψ ( y ) (cid:105) of the on the averagetranslationally invariant Bose gas as G ( x ) = F ( x ) − i sgn( x ) ρ ( x ) /
2, where x = ( t, x ). The critical behaviouris characterised by the exponents κ and η , as well asthe dynamical exponent z . η is an anomalous criticalexponent which determines the deviation of the spectralscaling from the free behaviour. In Ref. [29] two possiblesolutions were found, corresponding to different strong-wave-turbulence cascades, with scaling exponents κ P = d + 2 z − η P ,κ Q = d + z − η Q , (61)between κ , η , z , and d . ( κ P , η P ) correspond to an en-ergy cascade while ( κ Q , η Q ) reflect a quasi-particle cas-cade in the wave turbulent system. Both represent non-thermal fixed points of the non-equilibrium Bose gas [27–29]. The scaling of the statistical correlation function F implies scaling of the single-particle momentum distribu-tion n ( p ) = (cid:82) dωF ( ω, p ), cf. [29]: n ( p ) ∼ p − ζ , with ζ = κ − z + 2 . (62)The scaling of the single-particle kinetic energy, seeEq. (40), implies ζ = ξ + 2 = 2 η − d . Comparing thiswith (62), and making use of the constraint z = 2+ d − η , which is a consequence of Galilei invariance, cf. Eq. (34),one obtains κ = η and thus, with Eq. (33), the relations κ P = η ,κ Q = η , η P = d − η ,η Q = 2( d − − η ) / κ P = d + 4 / − η P / ,κ Q = d + 1 − η Q / , (64)and, with Eqs. (62) and (34), ζ P = d + 8 / − η P / ,ζ Q = d + 2 − η Q . (65)In turbulence theory, one considers the scaling of the ra-dial kinetic energy distribution E ( p ) ∼ p d − (cid:15) kin ( p ) ∼ p d − − ξ . Combining the above results, one finds that theenergy and particle cascades have radial single-particlekinetic energy distributions E P ( p ) ∼ p − / η P / ,E Q ( p ) ∼ p − η Q , (66)respectively. We find that, for the energy cascade, thestrong-wave-turbulence scaling [29] of E P ( p ) is equiva-lent to the classical Kolmogorov law [2, 3], with an in-termittency correction 2 η P /
3. Kolmogorov-5 / η Q = 0, the distribution E Q ( p ) corresponds,for d = 2 , v ∼ r − with the distance r from a vortex core [109] and, equiv-alently, of a random distribution of vortices [39, 40], aswe will discuss in more detail in the following. This in-verse cascade plays an important role in the equilibrationand condensation process [110, 111] after a strong coolingquench in a Bose gas [112, 113].In Refs. [39–43], the above non-thermal fixed pointsof the dilute superfluid gas were discussed in the con-text of topological defect formation and superfluid turbu-lence. A key result is that nearly degenerate Bose gases in d = 2 , d = 2 BKT) transition,can evolve quickly to a quasi-stationary state exhibitingcritical scaling [40] and slowing-down behaviour [41]. Thecritical scaling exponents ζ of the single-particle momen-tum spectra n ( p ) ∼ p − (cid:15) kin ( p ) ∼ p − ζ corroborated thepredictions ζ Q = d +2 − η Q of the strong-wave-turbulenceanalysis of Ref. [29] for a quasi-particle cascade, with avery small value of η Q . Within the numerical precisionit was found that ζ Q = 4 in d = 2 and ζ Q = 5 in d = 3[40]. These exponents turned out to be related to ran-domly distributed vortices and (large) vortex rings oc-2curring during the approach of the critical state [39, 40].Given the relation (66) of the scaling laws with hydro-dynamics and topological and geometric properties of thesuperfluid gas, we call the exponents − / − η P / η Q , respectively. C. Acoustic turbulence in a superfluid
Let us return to the KPZ dynamics. In order to makecontact with scaling in acoustic turbulence in a super-fluid, we insert the average literature values for η , asgiven in Table I. of Ref. [48], cf. also Fig. 2, which corre-spond to a forcing potential field delta-correlated in space( β = 2), into Eq. (40) and obtain (cid:15) kin ( p ) ∼ p − ξ , with ξ = 0 , for d = 1 ,ξ = 0 . , for d = 2 ,ξ = 1 . , for d = 3 . (67)These results can be compared with scaling behaviourobserved in acoustic turbulence in ultracold Bose gases,as summarised in the following.Results related to the quantum turbulence discussedin the previous section were obtained for a d = 1-dimensional Bose gas in Ref. [43]. There, the relationbetween critical scaling of the single-particle momentumspectrum and the appearance of solitary wave excitationswas pointed out. It was found that this spectrum, as fora thermal quasi-condensate, has a Lorentzian shape if thesolitons are distributed randomly in the system, with thewidth of the Lorentzian being related to the mean den-sity of solitons. The latter is in general different fromand independent of the thermal coherence length of agas with the same density and energy. The kinetic en-ergy spectrum, in the regime of momenta larger than theLorentzian width, correspondingly shows a momentumscaling (cid:15) kin ( p ) ∼ p n ( p ) ∼ p . This, in turn, is in fullagreement with the above result quoted in Eq. (67), cor-responding to a white-noise forcing, i.e., β = 2. Thepower law is consistent with that occurring in the single-particle spectrum of a random distribution of grey andblack solitons in a one-dimensional Bose gas [43].The fixed points found in Ref. [48], at which the expo-nents (67) apply, describe critical dynamics according tothe KPZ equation describing, e.g., the unbounded propa-gation of an interface moving with coordinates ( θ, x ) in atwo-component statistical system. On the contrary, theKPZ equation derived for the phase angle θ of the com-plex field ψ evolving according to the GPE, see Sect. II B,is subject to the additional constraint that the range ofangles 0 < θ ≤ π is compact. This constraint plays animportant role if the phase excitations are large enough toallow for (quasi) topological defects. Hence, one can notexpect the predictions (67) to necessarily match the scal-ings occurring when defects such as vortices are present.To make contact with the scalings (67), we note that, while the strong-wave-turbulence prediction ζ Q = d + 2is consistent with vortices dominating the infrared be-haviour of the single-particle spectrum [40], it does notapply to the ( d = 1)-dimensional case where there areno vortex defects, since ζ Q = d + 2 = 3 is by 1 largerthan the exponent ζ = 2 appearing in the Lorentziandistribution at large momenta. However, also in d = 2and d = 3, a scaling ζ c (cid:39) d + 1 appears as a re-sult of kink-like structures and longitudinal, compressiblesound excitations. In Ref. [40], it was demonstrated thatthe single-particle spectrum of the compressible compo-nent can show power-law behaviour, with an exponent n c ( p ) ∼ p − ζ c . This power-law was ascribed to soundwave turbulence on the background of the vortex gas, inparticular to the density depressions remaining for sometime in the gas after a vortex and an anti-vortex havemutually annihilated [114], cf. Fig. 15 of Ref. [40].We compare the predictions (67) with the scalingsfound in [40]. Using ζ = 2 η − d = ξ + 2 one finds ζ = 2 , for d = 1 ,ζ = 2 . , for d = 2 ,ζ = 3 . , for d = 3 . (68)Defining an anomalous exponent η by means of the rela-tion ζ = ζ c − η = d + 1 − η gives η = 0 , for d = 1 ,η = 0 . , for d = 2 ,η = 0 . , for d = 3 . (69)For d = 1, the scaling (68) corresponds to that of theLorentzian of a random soliton gas as discussed above.Furthermore, within the numerical precision, the powerlaws seen in Fig. 15 of Ref. [40] are found to be consistentwith the values (68). We reproduce the data in Fig. 4,comparing it with the IR scaling exponents (68) for d =2 , n ( p ) ∼ p − (cid:15) kin ( p ) obtainedfrom a numerical simulation of the GPE. The particularscalings occur shortly after the decay of the last topolog-ical excitations, i.e., the last vortex-antivortex pair for d = 2 or vortex ring for d = 3. At the time the pictureis taken, the compressible excitations dominate and theirscaling exponent can be measured. The relatively largeanomalous predictions of Eq. (69) fit the data very well.We remark that the grey solitary-wave excitations aswell as the density depressions remaining after vortex-anti-vortex annihilation are consistent with the absenceof the compactness constraint on θ in the KPZ equation.The soliton gas can be dominated by grey solitons [43]which imply only weak density depressions at the posi-tion of the phase jump. The weaker the depression, thesmaller the phase kink and the less relevant the compact-ness of the range of possible θ . Similarly the density de-pressions leading to ζ c (cid:39) d + 1 do not require the phaseto vary over the full circle. Hence, we expect in thesecases that KPZ predictions for critical exponents apply3 O cc up a t i o nnu m b e r n ( k ) Radial momentum k k ξ n ( k ) n q ( k ) n c ( k ) n i ( k ) k − k − k − . O cc up a t i o nnu m b e r n ( k ) Radial momentum k k ξ n ( k ) n q ( k ) n c ( k ) n i ( k ) k − k − k − . FIG. 4. (Color online) Acoustic turbulence in a d = 2 (up-per) and d = 3 (lower panel) dilute superfluid Bose gas:Shown are occupation number spectra of the late-stage evo-lution of a closed system after an initial quench, briefly af-ter the last vortex-antivortex pair ( d = 2) or vortex ring( d = 3) has disappeared, cf. Fig. 15 of Ref. [40]. The dis-tributions show a snapshot of a time evolution in a closedsystem. The blue solid lines correspond to the predictions ob-tained in the present work, given in Eq. (68), the black solidlines mark the canonical scaling, n ( p ) ∼ p − d − . The latterscaling is expected on geometrical grounds, e.g., results fromrandomly distributed plane-wave density depression waves orsolitons [40]. It is emphasised that the comparatively largedeviations ( ∼ p − . for d = 2 and ∼ p − . for d = 3) arefound independently and contrast the small deviation fromthe canonical scalings (66) of quantum turbulent spectra inthe presence of vortices. The radial momentum is given inlattice units, k = [2 (cid:80) di =1 sin (2 πn i a/L )] / , with n i ∈ Z , − L/ (2 a ) ≤ n i ≤ L/ (2 a ), a being the grid constant and L itsside length. k ξ = 2 sin( π/ (2 ξ )) is the momentum correspond-ing to the inverse healing length. also to the GPE, as defects do not play a role. VI. SUMMARY
In this article, we have investigated the scaling prop-erties of the stochastic Burgers equation within the func-tional renormalisation group (RG) approach. For thispurpose, a set of fixed-point equations has been estab-lished for the two-point correlators, including, in partic-ular, the velocity cumulants. They are solved in both, theultraviolet and infrared asymptotic regimes. A regime ofUV-convergent fixed points was identified. It was found that direct cascades of energy require a UV regulator asit is set by a dissipation term. On the other hand, the IRasymptotic form of the integral equations was found toallow for a continuous set of solutions. In one spatial di-mension, the continuous set of fixed points was shown tocoincide with perturbative results for the KPZ equation[45].Moreover, we found interesting implications for strongwave and quantum turbulence, as well as acoustic turbu-lence in a dilute Bose gas. In particular, we have shownthat the canonical Kolmogorov-type 5 / ACKNOWLEDGMENTS
We thank E. Altman, J. Berges, S. Bock, L. Canet,I. Chantesana, S. Diehl, S. Erne, M. Karl, T. Kloss, A.Liluashvili, D. Mesterhazy, M. Mitter, M. M¨uller, M. K.Oberthaler, A. Samberg, C. Wetterich, and N. Wscheborfor discussions, and B. Nowak and J. Schole for simula-tions related to Fig. 4. This work was supported by theDeutsche Forschungsgemeinschaft (GA677/7,8), the Eu-ropean Commission (ERC-AdG-290623), the HelmholtzAssociation (HA216/EMMI), and the University of Hei-delberg (LGFG and Center for Quantum Dynamics).
Appendix A: Superfluid hydrodynamics
In the following we give details on the relation betweenthe driven dissipative Gross-Pitaevskii equation (8) andthe stochastic Burgers equation (1). These are relatedthrough the hydrodynamic decomposition of the complexfield, ψ = √ n exp { i θ } , ∂ t θ + 12 m ( ∇ θ ) − ν ∇ θ = U,∂ t n + 1 m ∇ · ( n ∇ θ ) = S. (A1)This is formally similar to the equations that arise fromthe conservative GPE, with the addition that the con-tinuity equation is in-homogeneous and that the KPZ4equation has a non-zero dissipative term, U = 14 m √ n ∇ · (cid:18) ∇ n √ n (cid:19) + νn ∇ n · ∇ θ + µ − g n − Re( ζ e − iθ ) √ n ,S = ν √ n ∇ · (cid:18) ∇ n √ n (cid:19) − νn ( ∇ θ ) − µ n − g n + 2 √ n Im( ζ e − iθ ) . (A2)These equations are coupled non-linear Langevin equa-tions. If the fluctuations of the field amplitude are sub-dominant the former can be decoupled by assuming that U plays the role of the potential of the stochastic forcing f = m − ∇ U , with noise correlator (cid:104) U ( ω, p ) U ( ω (cid:48) , p (cid:48) ) (cid:105) = δ ( ω + ω (cid:48) ) δ ( p + p (cid:48) ) u ( ω, p ) . (A3)This describes particles being injected and removed asamplitude fluctuations, such that the system reaches astate where they can be described by a (not necessarilythermal) distribution and feed energy to the phase fluctu-ations. Note that, contrarily to the correlations of ζ , wedo not require U to be delta correlated in space. Burgersequation is obtained by setting v = m − ∇ θ .The kinetic energy spectrum is defined in terms of thetwo-point correlation function of ψ . It can be decom-posed into three parts, E kin ≡ − m (cid:90) x (cid:104) ψ † ∇ ψ (cid:105) = ρ m (cid:90) x (cid:104) ( ∇ θ ) (cid:105) + 12 m (cid:90) x (cid:104) ( ∇ n ) n (cid:105) + 12 m (cid:90) x (cid:104) δn ( ∇ θ ) (cid:105) = E phase + E amplitude + E exchange . (A4)The amplitude of ψ is separated into n = (cid:104) n (cid:105) + δn := ρ + δn . At sufficiently low energies, the average value ofthe amplitude is much larger than its fluctuations andthe major contribution to the kinetic energy is E phase .Then, E kin ∼ = ρ m (cid:90) x (cid:104) ( ∇ θ ) (cid:105) := V (cid:90) p (cid:15) kin ( p ) , (A5)where V is the volume of the system. Hence, (cid:15) kin ( p ) = mρ (cid:90) ω (cid:104) v ( ω, p ) · v ( − ω, − p ) (cid:105) . (A6) Appendix B: Effective action and observables
In the following we give details of the implementationof the coarse graining by means of the cutoff function R k . The Schwinger functional at scale k is defined ase W k [ J ] = (cid:90) Π p>k d v ( ω, p ) e − S [ v ]+ (cid:82) t, x J ( t, x ) · v ( t, x ) = (cid:90) D v e − S [ v ] − ∆ S k [ v ]+ (cid:82) t, x J ( t, x ) · v ( t, x ) , (B1)∆ S k [ v ] = 12 (cid:90) ω, p v ( ω, p ) · v ( − ω, − p ) R k ( p ) , (B2)with R k ( p ) > R k ( p (cid:29) k ) = 0 and R k ( p (cid:28) k ) = ∞ . Γ k → [ v ] ≡ Γ[ v ] is the Legendre transform of theSchwinger functional, W [ J ] which generates all the cor-relation functions of the velocity field. This informationis equivalently contained in Γ[ v ] which allows to derivecorrelation functions. For this one determines the aver-age field ¯ v = (cid:104) v ( t, x )) (cid:105) by solving the equation of motion δ Γ /δ v [¯ v ] = 0. In our case, we get ¯ v = 0. See, e.g.,Ref. [98] for the more general cases. The two-point cor-relation function is the inverse of the second derivativeof Γ[ v ] evaluated at ¯ v , (cid:104) v i ( t, x ) v j ( t (cid:48) , x (cid:48) ) (cid:105) = (cid:18) δ Γ δv i ( t, x ) δv j ( t (cid:48) , x (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ v (cid:19) − . (B3)The inverse is defined through (cid:90) τ, z (cid:104) v i ( t, x ) v k ( τ, z ) (cid:105) δ Γ δv k ( τ, z ) δv j ( t (cid:48) , x (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ v = δ i,j δ ( t − t (cid:48) ) δ ( x − x (cid:48) ) . (B4)Higher order moments are computed from the higher or-der derivatives of Γ[ v ] multiplied by external legs. Appendix C: RG flow equations
In this appendix we give explicit expressions for theintegral terms in the RG flow equations (21). We use thesharp cutoff R k ; ij ( p ) = δ ij k d z ˜ R k ( p ), with˜ R k ( p ) (cid:26) = 0 if p ≥ k, → ∞ if p < k. (C1)As a result we can use the identity1Γ (2) k + R k k ∂R k ∂k (2) k + R k = 2 k δ ( p − k ) 1Γ (2) k (C2)to evaluate the flow integrals. To proceed, we read off,from the ansatz (17), the two-point function δ Γ k [ v = 0] δv i ( ω (cid:48) , p (cid:48) ) δv j ( ω, p ) = Γ (2) k,ij ( ω, p )(2 π ) d +1 δ ( ω + ω (cid:48) ) δ ( p + p (cid:48) ) , (C3)5with Γ (2) k,ij ( ω, p ) defined in Eq. (18), the 3-vertex δ Γ k [ v = 0] δv i ( ω (cid:48) , p (cid:48) ) δv j ( ω, p ) δv l ( ω (cid:48)(cid:48) , q ) = (2 π ) − d +1) × δ ( ω + ω (cid:48) + ω (cid:48)(cid:48) ) δ ( p + p (cid:48) + q ) Γ (3) k ; ijl ( ω (cid:48) , p (cid:48) ; ω, p ) , (C4)withΓ (3) k ; ijl ( ω (cid:48) , p (cid:48) ; ω, p ) = − F − k ( | p + p (cid:48) | ) × (cid:16) ω + ω (cid:48) + iν k ( | p + p (cid:48) | ) | p + p (cid:48) | (cid:17) (cid:2) δ il p (cid:48) j + δ jl p i (cid:3) − F − k ( p (cid:48) ) (cid:0) − ω (cid:48) + iν k ( p (cid:48) ) p (cid:48) (cid:1) (cid:2) δ ij p l − δ il ( p (cid:48) j + p j ) (cid:3) − F − k ( p ) (cid:0) − ω + iν k ( p ) p (cid:1) [ δ ij p (cid:48) l − δ jl ( p (cid:48) i + p i )] , (C5)and the 4-vertex δ Γ k [ v = 0] δv i ( ω, p ) δv j ( ω (cid:48) , p (cid:48) ) δv l ( ω (cid:48)(cid:48) , q ) δv m ( ω (cid:48)(cid:48)(cid:48) , q (cid:48) ) = (2 π ) − d +1) × δ ( ω + ω (cid:48) + ω (cid:48)(cid:48) + ω (cid:48)(cid:48)(cid:48) ) δ ( p + p (cid:48) + q + q (cid:48) ) × Γ (4) k,ijlm ( p , p (cid:48) , q ) , (C6)withΓ (4) k,ijlm ( p , p (cid:48) , q )= F − k ( | p + p (cid:48) | ) (cid:2) ( δ mi p j + δ mj p (cid:48) i ) ( p l + p (cid:48) l + q l ) − ( δ li p j + δ lj p (cid:48) i ) q m (cid:3) + F − k ( | p + q | ) (cid:2) ( δ im p l + δ lm q i ) (cid:0) p j + p (cid:48) j + q j (cid:1) − ( δ ij p l + δ jl q i ) p (cid:48) m (cid:3) + F − k ( | p (cid:48) + q | ) (cid:2) ( δ jm p (cid:48) l + δ lm q j ) ( p i + p (cid:48) i + q i ) − ( δ ij p (cid:48) l + δ il q j ) p m (cid:3) . (C7)Using these, the flow integral (22) can be written as I (2) k [0]( ω, p ) = k d (cid:90) ω (cid:48) , p (cid:48) δ ( p (cid:48) − k )Γ (2) k ; ni − ( ω (cid:48) , p (cid:48) ) × (cid:104) (2) k,jm − ( ω (cid:48) − ω, p (cid:48) − p ) θ (( p (cid:48) − p ) − k ) × Γ (3) k ; ijl ( − ω (cid:48) , − p (cid:48) ; ω (cid:48) − ω, p (cid:48) − p ) × Γ (3) k ; mnl ( ω − ω (cid:48) , p − p (cid:48) ; ω (cid:48) , p (cid:48) ) − Γ (4) k ; nijj ( p , − p , p (cid:48) ) (cid:105) . (C8)The theta function arises because of the sharp cutoff.Since the cutoff diverges for p < k , the propagator G k vanishes in this regime, cf. Eq. (16). This is irrelevant inthe diagram depending on the 4-point vertex, cf. Eq. (22),where the single appearing propagator carries a ∂ k R k insertion and is thus evaluated at p = k . Because ofrotational symmetry, δ I k /δv i [0]( ω, p ) does not dependon i . Hence, taking the trace of δ I k /δv i δv j [0]( ω, p ) anddividing it by d makes the integrals rotationally invariant. Because of (18), the integrand is a rational function of ω (cid:48) such that the ω integration can be done. Introducingthe short-hand notation ˜ ν p = ν k ( p ) p , etc., we obtain I (2) k [0]( ω, p ) = − k d (cid:90) q , r (2 π ) d δ ( r − k ) δ ( p − q − r ) × (cid:8) F − k ( q ) F − k ( r )˜ ν q ˜ ν r [ ω + (˜ ν q + ˜ ν r ) ] (cid:9) − × (cid:104) F − k ( q ) ˜ ν q (cid:2) ω + (˜ ν q + ˜ ν r ) (cid:3) (cid:2) d ( p + r ) − p · r (cid:3) − θ (cid:0) q − k (cid:1) × (cid:16) F − k ( q ) ˜ ν q (cid:2) ω + (˜ ν q + ˜ ν r ) (cid:3) (cid:2) d ( p + r ) − p · r (cid:3) + F − k ( r ) ˜ ν r (cid:2) ω + (˜ ν q + ˜ ν r ) (cid:3) (cid:2) d ( p + q ) − p · q (cid:3) + F − k ( p ) (cid:0) ˜ ν q + ˜ ν r (cid:1) (cid:2) ω + ˜ ν p (cid:3) (cid:2) d ( q + r ) + 2 q · r (cid:3) + F − k ( p ) F − k ( q )˜ ν q (cid:2) ω − ˜ ν p (cid:0) ˜ ν q + ˜ ν r (cid:1)(cid:3) d p · q + F − k ( p ) F − k ( r )˜ ν r (cid:2) ω − ˜ ν p (cid:0) ˜ ν q + ˜ ν r (cid:1)(cid:3) d p · r (cid:17)(cid:105) . (C9)Note that the terms ∝ d and independent of d orig-inate from contractions of the types δ ij δ ji p k q k and δ ij δ jl p i q l , respectively. We can integrate radially, (cid:82) r = (cid:82) ∞ r d − dr (cid:82) Ω , which, for d = 1, reduces to (cid:82) Ω f ( p ) =[ f ( p ) + f ( − p )] / (2 π ). The delta distributions allow to set r = k e r and q = p − k e r , with | ˆ r | = | e r | = 1. Expansionin powers of ω gives I (2) k [0](0 , p ) = − k d d (cid:90) Ω (cid:2) F − k ( k ) F − k ( q )˜ ν k ˜ ν q (˜ ν k + ˜ ν q ) (cid:3) − × (cid:104) F − k ( q ) ˜ ν q (˜ ν q + ˜ ν k ) (cid:2) d ( p + k ) − k e r · p (cid:3) − θ (cid:0) q − k (cid:1) × (cid:16) F − k ( q ) ˜ ν q (˜ ν k + ˜ ν q ) (cid:2) d ( p + k ) − k p · e r (cid:3) + F − k ( k ) ˜ ν k (˜ ν k + ˜ ν q ) (cid:2) d ( p + q ) − p · q (cid:3) + F − k ( p ) ˜ ν p (cid:2) d ( k + q ) + 2 k q · e r (cid:3) − F − k ( p ) F − k ( q )˜ ν p ˜ ν q d p · q − F − k ( p ) F − k ( k )˜ ν p ˜ ν k d k p · e r (cid:17)(cid:105) , (C10)and ∂ ω I (2) k [0] (cid:12)(cid:12)(cid:12) (0 , p ) = k d d (cid:90) Ω θ (cid:0) q − k (cid:1) F − k ( p ) × (cid:2) F − k ( k ) F − k ( q )˜ ν k ˜ ν q (˜ ν k + ˜ ν q ) (cid:3) − × (cid:16) F − k ( p ) (cid:2) (˜ ν k + ˜ ν q ) − ˜ ν p (cid:3) (cid:2) d ( q + k ) + 2 k q · e r (cid:3) + F − k ( q ) ˜ ν q (cid:2) ˜ ν k + ˜ ν q + ˜ ν p (cid:3) d p · q + F − k ( k ) ˜ ν k (cid:2) ˜ ν k + ˜ ν q + ˜ ν p (cid:3) d k p · e r (cid:17) . (C11)6 Appendix D: Scaling analysis of flow integrals1. Classical limit ( p (cid:28) k ) To analyze the scaling of the flow integrals in the limit p (cid:28) k we go over to rescaled variables ˆ p = p/k , ˆ q = q /k = ˆ p − e r , cf. Eq. (23), and insert the parametrisation(28) into Eqs. (C10) and (C11). This allows to definedimensionless flow integralsˆ I (2)1 (ˆ p ) ≡ k − d (cid:114) z z I (2) k [0](0 , k ˆ p ) , ˆ I (2)2 (ˆ p ) ≡ k − d +4 (cid:18) z z (cid:19) / ∂ ω I (2) k [0] (cid:12)(cid:12)(cid:12) (0 ,k ˆ p ) , (D1)where the indices 1 , ω . We split the integrals ˆ I (2)1 and ˆ I (2)2 into three and four parts, respectively, accordingto their leading scaling behaviour in the limit ˆ p (cid:28) S i (ˆ p ) = (cid:112) δZ i (ˆ p ) , (D2)the rescaled flow integrals take the formˆ I (2)1 (ˆ p ) = ˆ p F , (ˆ p )+ ˆ p ( η + η ) / S (ˆ p ) S (ˆ p ) F , (ˆ p )+ ˆ p η + η +2 δ d [ S (ˆ p ) S (ˆ p )] F , (ˆ p ) , (D3)ˆ I (2)2 (ˆ p ) = ˆ p ( η + η ) / S (ˆ p ) S (ˆ p ) F , (ˆ p )+ ˆ p η +2 S (ˆ p ) F , (ˆ p )+ ˆ p η + η +2 δ d [ S (ˆ p ) S (ˆ p )] F , (ˆ p )+ ˆ p η +2 δ d S (ˆ p ) F , (ˆ p ) . (D4)Using, furthermore, the abbreviation (˜ ν k + ˜ ν q ) − = k − ( z /z ) / T (ˆ q ), the functions F i,j (ˆ p ) are given by F , (ˆ p ) = 12 d ˆ p (cid:90) Ω × (cid:110) θ (cid:0) ˆ q − (cid:1) ˆ q − ( η + η ) / S (1) [ S (ˆ q ) S (ˆ q )] − × (cid:2) d (ˆ p + ˆ q ) − p · ˆ q (cid:3) − θ (cid:0) − ˆ q (cid:1) ˆ q η S (ˆ q ) [ S (1) S (1)] − × (cid:2) d (ˆ p + 1) − p · e r (cid:3) (cid:111) , (D5) F , (ˆ p ) = − ˆ p − (cid:90) Ω θ (cid:0) ˆ q − (cid:1) T (ˆ q ) (cid:8) [ S (1) S (1)] − ˆ p · ˆ q + ˆ q − ( η + η ) / [ S (ˆ q ) S (ˆ q )] − ˆ p · e r (cid:9) , (D6) F , (ˆ p ) = ˆ p − δ d d (cid:90) Ω θ (cid:0) ˆ q − (cid:1) T (ˆ q ) (cid:2) d (ˆ q + 1) + 2ˆ q · e r (cid:3) × [ S (ˆ q ) S (ˆ q ) S (1) S (1)] − ˆ q − ( η + η ) / , (D7) F , (ˆ p ) = ˆ p − (cid:90) Ω θ (cid:0) ˆ q − (cid:1) T (ˆ q ) (cid:8) [ S (1) S (1)] − ˆ p · ˆ q + ˆ q − ( η + η ) / [ S (ˆ q ) S (ˆ q )] − ˆ p · e r (cid:9) , (D8) F , (ˆ p ) = ˆ p − (cid:90) Ω θ (cid:0) ˆ q − (cid:1) T (ˆ q ) (cid:8) [ S (1) S (1)] − ˆ p · ˆ q + ˆ q − ( η + η ) / [ S (ˆ q ) S (ˆ q )] − ˆ p · e r (cid:9) , (D9) F , (ˆ p ) = − ˆ p − δ d d (cid:90) Ω θ (cid:0) ˆ q − (cid:1) T (ˆ q ) ˆ q − ( η + η ) / × ( S (ˆ q ) S (ˆ q ) S (1) S (1)) − (cid:2) d (ˆ q + 1) + 2ˆ q · e r (cid:3) , (D10) F , (ˆ p ) = F , (ˆ p ) . (D11)Recall that, in the above expressions, ˆ q = | ˆ p − e r | . Hence,in integer d >
1, ˆ q = ˆ p + 1 − p cos θ , and ˆ p · e r =ˆ p cos θ , ˆ q · e r = ˆ p cos θ −
1, ˆ p · ˆ q = ˆ p − ˆ p cos θ , where θ is the angle between ˆ p and ˆ r = e r . In d = 1, one hasˆ q = (ˆ p − , etc. The different terms in Eqs. (D3),(D4) are defined in such a way that the functions F i,j (ˆ p )are analytic and non-vanishing at ˆ p = 0, i.e., can beexpanded as F i,j (ˆ p →
0) = F i,j (0) + F (cid:48) i,j (0) ˆ p + O(ˆ p ).Note that, for d = 1, an additional factor ˆ p has to beextracted from F , (ˆ p ), F , (ˆ p ), and F , (ˆ p ) because inthis case d (ˆ q + 1) + 2 e r · ˆ q = ˆ p . We have taken this intoaccount by inserting Kronecker deltas δ d . Then, in thelimit ˆ p →
0, where δZ i (ˆ p ) behaves as in Eq. (42), i.e.,where S i (ˆ p →
0) = √ a i ˆ p ( − η i + α i ) / [1 + F i (ˆ p )] / , (D12)one can use that F i (ˆ p →
0) contains sub-dominantterms, involving possibly logarithms. Neglecting theseterms, S i (ˆ p → ∼ = √ a i ˆ p ( − η i + α i ) / F i,j (ˆ p → ∼ = F i,j (0) , (D13)in Eqs. (D3) and (D4) gives the asymptotic relationsˆ I (2)1 (ˆ p (cid:28) ∼ = ˆ p F , (0) + √ a a ˆ p ( α + α +4) / F , (0)+ a a ˆ p α + α +2 δ d F , (0) (D14)ˆ I (2)2 (ˆ p (cid:28) ∼ = √ a a ˆ p ( α + α +4) / F , (0)+ a ˆ p α +2 F , (0) + a a ˆ p α + α +2 δ d F , (0)+ a ˆ p α +2 δ d F , (0) . (D15)Eqs. (44) and (46) then follow by inserting Eqs. (D14),(D15), together with Eq. (42), into Eq. (30) and requiring7that the exponent of the term leading in the IR on theright matches that on the left-hand side.The coefficients a i can be determined by inserting thesolutions of Eqs. (44) and (46) back into Eqs. (D14) and(D15) and identifying the leading monomials. Equationsfor the a i are extracted by matching the correspondingpre-factors. As Eqs. (44) and (46) have multiple solu-tions, each case must be handled separately.We start with d = 1. In the following, we use thenotation δZ (cid:48) i (1) = d δZ i / dˆ p | ˆ p =1 . Such terms arise inEqs. (D5) and (D8) because the respective integrals van-ish at ˆ p = 0, and the integrand must be Taylor expandedup to leading order in p . The relevant term is propor-tional to the ˆ p -derivative of δZ i . The solutions of Eq. (44)are given in Eq. (45). If ( α , α ) = (1 ,
5) the terms pro-portional to F , (0) and F , (0) are dominating. Thecorresponding equations are a ( η −
1) = hF , (0) ,a ( η −
5) = h ( a a ) / F , (0) , (D16)with F , (0) = (8 πS S ) − × (cid:2) η − S S − S δZ (cid:48) (1) + S δZ (cid:48) (1) (cid:3) ,F , (0) = (32 πS ) − × (cid:2) η − S S + S δZ (cid:48) (1) + S δZ (cid:48) (1) (cid:3) . (D17)If − < α < α = −
2, the dominating termsare the ones that are proportional to F , (0) and F , (0),and thus a ( η − α ) = ha a F , (0) ,a ( η + 2) = ha F , (0) , (D18)with F , (0) = F , (0) = (8 πS S ) − . (D19)While the above equations for the a i are not closed, h canbe eliminated by dividing the equations by each other.This gives the closed equation a a = − a a η + 2 α − η , (D20)equivalent to Eq. (58).If α = − α = −
2, the term proportional to F , (0) still dominates ˆ I (2)1 (ˆ p ), but the term proportionalto F , (0) is of the same order as the one proportional to F , (0). Taking both into account gives a ( η + 2) = ha a F , (0) ,a ( η + 2) = h (cid:2) a a F , (0) + a F , (0) (cid:3) , (D21) with F , (0) = − S (32 πS ) − . (D22)Finally, for α = 1 and α = − F , (0)+ a a F , (0) and F , (0), i.e., a ( η −
1) = h ( F , (0) + a a F , (0)) ,a ( η + 2) = ha F , (0) . (D23)As a result, at the end points ( α , α ) ∈ ( {− , } , − a i are not the same as for − < α <
1. There isno constraint on the value of η arising there. However,under the assumption that η is a continuous function of α , one obtains η = 2 − α / η = 2 − α / < α <
1, and an additional fixed point with η = 3 / η to α i at the other solutions of Eq. (45) in a similar way. Atthese points, the full solution of Eq. (21) is necessary toverify the existence of the RG fixed points and extractthe values of the η i .We finally consider the case that d (cid:54) = 1. Eq. (46) hasthe solutions (47). If ( α , α ) = (1 , a ( η −
1) = hF , (0) ,a ( η −
5) = h ( a a ) / F , (0) , (D24)with F , (0) = 2 d Ω d Γ( d/ (16 π ( d − S S ) − × (cid:2) η − S S − S δZ (cid:48) (1) + S δZ (cid:48) (1) (cid:3) ,F , (0) = Ω d (32 dS ) − × (cid:2) η − S S + S δZ (cid:48) (1) + S δZ (cid:48) (1) (cid:3) . (D25)Here we have introduced the surface factor Ω d = (cid:82) Ω = dπ d/ [(2 π ) d Γ( d/ − . If 0 < α < α = 0, weget a ( η − α ) = ha a F , (0) ,a η = ha F , (0) , (D26)with F , (0) = F , (0) = Ω d ( d − dS S ) − . (D27)Again, while the above equations for the a i are not closed, h can be eliminated by dividing the equations by each8other. This gives the closed equation a a = a a η − α η , (D28)equivalent to Eq. (58).If α = α = 0, the equations are a η = ha a F , (0) ,a η = h (cid:2) a a F , (0) + a F , (0) (cid:3) , (D29)with F , (0) = − Ω d ( d − S (16 dS ) − . (D30)If α = 1 and α = 0, one obtains a ( η −
1) = h [ F , (0) + a a F , (0)] ,a η = ha F , (0) . (D31)While none of the above sets of equations is closed, wecan eliminate h for ( α , α ) ∈ (]0 , , η = 2 d + 4 − α .
2. Scaling limit ( p (cid:29) k ) In this subsection, the asymptotic behaviour of the in-tegrals ˆ I (2)1 , (ˆ p ) for ˆ p (cid:29) F i,j (ˆ p ). We discuss this foreach F i,j separately, taking into account spherical sym-metry. For ˆ p (cid:29)
1, then also ˆ q ∼ ˆ p , i.e., ˆ q − >
0, andwe can approximately set the theta functions and, since δZ i (ˆ q → ∞ ) = 0, also the S i (ˆ q ) to one. Separating outthe leading UV scaling, F i,j (ˆ p → ∞ ) ∼ ˆ p γ i,j , we writethe F i,j in the form F i,j (ˆ p ) = ˆ p γ i,j (cid:90) Ω f i,j (1 / ˆ p, ˆ p · e r / ˆ p ) . (D32)The f i,j are finite and non-vanishing at 1 / ˆ p = 0. Notethat, in Eqs. (D6), (D8) and (D9), different terms can beleading in the UV such that the above definition of the f i,j and γ i,j depends on the values of the η , . Moreover,the denominator of T (ˆ q ) in Eqs. (D6)–(D11) contains adivergence if η − η > p ( η − η ) / appears. This can be seen by recallingthe definition T (ˆ q ) = ( z /z ) / k (˜ ν k +˜ ν q ) − , which gives(recall ˆ q = | ˆ p − e r | ) the large-ˆ p asymptotic behaviour T ( | ˆ p − e r | ) ∼ = (cid:34)(cid:18) ˆ p + 1 − p p p · e r (cid:19) ( η − η ) / + S (1) S (1) (cid:35) − ∼ = (cid:26) ˆ p − ( η − η ) / if η − η > S (1) /S (1) if η − η < . (D33)Having identified the leading scaling behaviour, the in-tegrals can be computed in the limit ˆ p → ∞ by neglect- ing sub-leading contributions to the integrands. We canapproximate f i,j (1 / ˆ p, ˆ p · e r / ˆ p ) ∼ = f i,j (0 , ˆ p · e r / ˆ p ) in theintegrands and perform the angular integration whichgives, for those integrals where f i,j (0 , y ) does not de-pend on y = ˆ p · e r / ˆ p , a surface factor Ω d = (cid:82) Ω = dπ d/ [(2 π ) d Γ( d/ − . The asymptotic behaviour ofthe integrals F , (ˆ p ), F , (ˆ p ), and F , (ˆ p ) can be derivedin this way. The result is ( S i ≡ S i (1)) F , (ˆ p → ∞ ) ∼ = Ω d S [ δ d / d − /d ] ˆ p − δ d × ˆ p − ( η + η ) / , (D34) F , (ˆ p → ∞ ) ∼ = Ω d S S ˆ p − δ d × ˆ p − η if η > η ˆ p − η (1 + S /S ) − if η = η ˆ p − ( η + η ) / ( S /S ) if η < η , (D35) F , (ˆ p → ∞ ) ∼ = − Ω d S S ˆ p − δ d × ˆ p − η + η if η > η ˆ p − η (1 + S /S ) − if η = η ˆ p − ( η + η ) / ( S /S ) if η < η . (D36)The calculation of the asymptotic behaviour of the in-tegrals (D6), (D8) and (D9) can become more involved.Two possibilities arise. If η + η ≥ −
2, the asymptoticbehaviour is determined in the same way as for F , , F , and F , . However, for η + η < − f i,j ( (cid:15) → , ˆ p · e r / ˆ p ) is proportional to y = ˆ p · e r / ˆ p ,and thus vanishes under the angular integral. In thiscase, the asymptotically leading term is obtained by ex-panding yT (ˆ q ) ≡ yT (ˆ p, y ) to order y before the limitˆ p → ∞ is taken and the term that is linear in y is ne-glected. This ensures that we only consider terms thatcontribute to the angular integration. One can checkthat truncating at order y does not affect the asymp-totic behaviour. Indeed y enters through the combina-tion ˆ p − p · e r = (1 − y/ ˆ p )ˆ p . We see that the termof order y n is multiplied by 1 / ˆ p n and can only dominatein the asymptotic regime if all the lower order terms areirrelevant.We discuss the procedure for F , (ˆ p ) and state the re-sults for the two remaining integrals F , (ˆ p ) and F , (ˆ p ).To simplify the derivation we use that ( η + η ) / η − − d from Eq. (33). We start by approximating δZ i (ˆ q ) (cid:39) θ (cid:0) ˆ q − (cid:1) = 1 in Eq. (D6), which gives,defining (cid:15) = 1 / ˆ p such that ˆ p · e r = y/(cid:15) , F , (ˆ p ) ∼ = (cid:90) Ω (cid:2) ( S S ) − ( (cid:15)y − − ˆ q − η +2+ d (cid:15)y (cid:3) T (ˆ q ) , (D37)with ˆ q = (cid:112) (cid:15) − (cid:15)y/(cid:15) . We factor out (cid:15) − η +2+ d from9ˆ q − η +2+ d in the second term: F , (ˆ p ) ∼ = (cid:90) Ω T (ˆ q ) (cid:104) ( S S ) − ( (cid:15)y − − (cid:15) − d − η y (1 + (cid:15) − (cid:15)y ) ( − η +2+ d ) / (cid:105) . (D38)The asymptotic behaviour of T (ˆ q ) is determined by thesign of − ( η − η ) / η − − d , see Eq. (D33). Forboth signs, different η will render either of the terms inEq. (D38) dominating for large ˆ p ( (cid:15) → η < d + 2, T (ˆ q → ∞ ) ∼ ˆ p η − d − : We write T (ˆ q ) = (cid:15) − η +2+ d ˜ T ( (cid:15) ) such that ˜ T ( (cid:15) →
0) = 1 and F , (ˆ p ) ∼ = (cid:90) Ω (cid:2) (cid:15) − η + d +2 ( S S ) − ( (cid:15)y − − (cid:15) η +1 (1 + (cid:15) − (cid:15)y ) ( − η +2+ d ) / y (cid:105) ˜ T ( (cid:15) ) . (D39)There are three sub-cases to be distinguished: (a) For2 η < d + 1, the second term, providing an extra scalingfactor (cid:15) η +1 , is dominant. Then the leading exponentdefined in (D32) reads γ , = − η −
1, and the integrand f , ( (cid:15), y ) = (cid:2) (cid:15) − η + d +1 ( S S ) − ( (cid:15)y − − (1 + (cid:15) − (cid:15)y ) ( − η +2+ d ) / y (cid:105) ˜ T ( (cid:15) ) . (D40)The leading term f , (0 , y ) = − y does not contribute tothe angular integral. Taking the sub-leading factors intoaccount by expanding to second order in y , f , ( (cid:15), y ) ∼ = − ˜ T ( (cid:15) ) (cid:16) y [1 + (cid:15) ] ( − η +2+ d ) / + y (cid:15) [1 + (cid:15) ] ( − η + d ) / (cid:104) η − − d − ( η − − d )(1 + (cid:15) ) ( − η +2+ d ) / ˜ T ( (cid:15) ) (cid:105) + (cid:15) − η + d +1 / ( S S ) (cid:110) − (cid:15)y × (cid:104) η − − d )(1 + (cid:15) ) ( − η + d ) / ˜ T ( (cid:15) ) (cid:105) + (cid:15) y ( η − − d )(1 + (cid:15) ) ( − η − d ) / ˜ T ( (cid:15) ) × (cid:104) ( η − − d )(1 + (cid:15) ) ( − η +2+ d ) / ˜ T ( (cid:15) )+( d − η ) / (cid:15) (cid:105) (cid:111)(cid:17) , (D41)we find that two terms are competing, requiring a fur-ther case distinction: If η < d/
2, the contributions ∝ (cid:15) − η + d +1 are sub-leading and the quadratic term in y dominates. If η > d/
2, the term not depending on y dominates. Both must be account for if η = d/
2. Then f , ( (cid:15) → , y ) (cid:39) − y − S S (cid:15) × y η S S η < d/ (cid:0) y dS S / (cid:1) η = d/ (cid:15) − η + d d/ < η < ( d + 1) / , (D42) and, after angular integration, (cid:82) Ω y = Ω d /d , F , (ˆ p → ∞ ) (cid:39) − Ω d S S ˆ p − × ˆ p − η η S S /d η < d/ p − d/ (1 + S S / η = d/ p η − d d/ < η < ( d + 1) / . (D43)(b) For 2 η = d + 1, both terms under the integral (D39)are equally important. We obtain γ , = − ( d + 3) / f , ( (cid:15), y ) = (cid:20) (cid:15)y − S S − (1 + (cid:15) − (cid:15)y ) / y (cid:21) ˜ T ( (cid:15) ) . (D44)The relevant contribution is f , (0 , y ) = − y − ( S S ) − while the terms of order y are sub-dominant. As a result,the asymptotics (D43) is supplemented with F , (ˆ p → ∞ ) (cid:39) − Ω d S S ˆ p − ( d +3) / if η = ( d + 1) / . (D45)(c) For ( d + 1) / < η < d + 2, the leading terms areinterchanged. Eq. (D39) provides γ , = η − d − f , (ˆ p ) = (cid:2) ( (cid:15)y −
1) ( S S ) − − (cid:15) η − d − ( (cid:15) + 1 − (cid:15)y ) ( d +2 − η ) / y (cid:3) ˜ T ( (cid:15) ) . (D46)We find f , (0 , y ) = − ( S S ) − , and, together with rela-tion (D45), the last case of the asymptotics (D43) reads F , (ˆ p → ∞ ) (cid:39) − Ω d S S ˆ p η − d − , for d/ < η < d + 2 . (D47)2. η ≥ d +2, T (ˆ q → ∞ ) ∼ const.: Now, ˆ q − η +2+ d is fi-nite for (cid:15) →
0, such that no powers of 1 / ˆ p arise from T (ˆ q ).Two competing terms in (D38) imply three sub-cases.However, for η ≥ d + 2, the term ∝ (cid:15) − d − η is sub-dominant and can be neglected. The term ∝ ( S S ) − in (D38) is dominant, such that γ , = 0 and f , ( (cid:15), y ) = (cid:2) ( (cid:15)y − S S ) − − (cid:15) η − d − (1 + (cid:15) − (cid:15)y ) ( d +2 − η ) / y (cid:3) T (ˆ q ) . (D48)Taking the limit f , ( (cid:15) → , y ) and performing the angu-lar integral one obtains the final asymptotics F , (ˆ p → ∞ ) (cid:39) − Ω d S S ˆ p − × ˆ p − η η S S /d η < d/ p − d/ (1 + S S / η = d/ p η − d d/ < η < d + 2ˆ p (1 + S /S ) − η = d + 2ˆ p S /S d + 2 < η . (D49)0Using analogous arguments we find F , (ˆ p ) ∼ Ω d S S ˆ p − × ˆ p η − d S S ( η + 2 + d ) / (2 d ) η < d/ p − d/ [1 + S S (3 d + 4) / (4 d )] η = d/ p η − d d/ < η < d + 2ˆ p (1 + S /S ) − η = d + 2ˆ p ( S /S ) d + 2 < η , (D50) F , (ˆ p ) ∼ Ω d S S ˆ p − × ˆ p η − d S S (5 η − − d ) /d η < d/ p − d/ [1 − S S (16 + 3 d ) / (2 d )] η = d/ p η − d d/ < η < d + 2ˆ p (1 + S /S ) − η = d + 2ˆ p ( S /S ) d + 2 < η . (D51) The UV leading terms in of F i,j inserted back intoEqs. (D3) and (D4) allow to compute the asymptoticsof ˆ I (2)1 , (ˆ p ), separately for each case. The integrals on theright-hand side of Eq. (49) converge if β i < η i , corre-sponding to the range (51) for η . With Eq. (33), we findˆ I (2) i (ˆ p (cid:29) ∼ ˆ p β i with β = − δ d − η + d if (6 + 3 d − δ d ) / ≥ η η − d − d − δ d ) / < η ≤ d + 22 η − d if η > d + 2 , (D52) β = η − d −
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