Universal critical behavior of noisy coupled oscillators: A renormalization group study

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A renormalization group study

Thomas Risler, Jacques Prost, Frank Julicher

To cite this version:

Thomas Risler, Jacques Prost, Frank Julicher. Universal critical behavior of noisy coupled oscillators:

A renormalization group study. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics,

American Physical Society, 2005, 72, pp.016130. �10.1103/PhysRevE.72.016130�. �hal-00961028�

Universal critical behavior of noisy coupled oscillators: A renormalization group study

Thomas Risler,^1,2Jacques Prost,^3,2and Frank Jülicher¹

1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzerstrasse 38, 01187 Dresden, Germany

2Physicochimie Curie (CNRS-UMR 168), Institut Curie, 26 rue d’Ulm, 75248 Paris Cedex 05, France

3Ecole Supérieure de Physique et de Chimie Industielles de la Ville de Paris, 10 rue Vauquelin, 75231 Paris Cedex 05, France 共Received 17 April 2005; published 29 July 2005兲

We show that the synchronization transition of a large number of noisy coupled oscillators is an example for a dynamic critical point far from thermodynamic equilibrium. The universal behaviors of such critical oscil- lators, arranged on a lattice in ad-dimensional space and coupled by nearest-neighbors interactions, can be studied using field-theoretical methods. The field theory associated with the critical point of a homogeneous oscillatory instability共or Hopf bifurcation of coupled oscillators兲is the complex Ginzburg-Landau equation with additive noise. We perform a perturbative renormalization group 共RG兲 study in a 共4 −⑀兲-dimensional space. We develop an RG scheme that eliminates the phase and frequency of the oscillations using a scale- dependent oscillating reference frame. Within Callan-Symanzik’s RG scheme to two-loop order in perturbation theory, we find that the RG fixed point is formally related to the one of the model A dynamics of the real Ginzburg-Landau theory with an O共2兲 symmetry of the order parameter. Therefore, the dominant critical exponents for coupled oscillators are the same as for this equilibrium field theory. This formal connection with an equilibrium critical point imposes a relation between the correlation and response functions of coupled oscillators in the critical regime. Since the system operates far from thermodynamic equilibrium, a strong violation of the fluctuation-dissipation relation occurs and is characterized by a universal divergence of an effective temperature. The formal relation between critical oscillators and equilibrium critical points suggests that long-range phase order exists in critical oscillators above two dimensions.

DOI:10.1103/PhysRevE.72.016130 PACS number共s兲: 64.60.Ht, 05.45.Xt, 64.60.Ak, 05.10.Cc

I. INTRODUCTION

Equilibrium systems consisting of a large number of de- grees of freedom exhibit phase transitions as a consequence of the collective behavior of many components关1–3兴. The universal behaviors near critical points have been studied extensively using field-theoretical methods and renormaliza- tion group 共RG兲 techniques 关3–13兴. In systems driven far from thermodynamic equilibrium, collective behaviors can lead to dynamic instabilities and nonequilibrium phase tran- sitions 关14–17兴. While the study of nonequilibrium critical points has remained a big challenge, RG methods have in some cases been applied关18–22兴.

An important example of nonequilibrium critical behavior is the homogeneous oscillatory instability or Hopf bifurca- tion of coupled oscillators关23兴. Such instabilities are impor- tant in many physical, chemical, and biological systems 关24,25兴. From the point of view of statistical physics, phase- coherent oscillations result as the collective behavior of a large number of degrees of freedom in the thermodynamic limit. For a system of finite size, fluctuations destroy the phase coherence of the oscillations, and the singular behav- iors characteristic of a Hopf bifurcation are concealed.

This general idea can be illustrated by individual oscilla- tors arranged on a lattice in a d-dimensional space and coupled to their nearest neighbors. As a consequence of fluc- tuations in the system, each oscillator is subject to a noise source. For small coupling strength and as a result of fluc- tuations, the oscillators have individual phases and exhibit a limited coherence time of oscillations. In the thermodynamic limit, a global phase emerges beyond a critical coupling strength where oscillations become coherent over large dis-

tances. As this critical point is approached from the disor- dered phase, a correlation length, which corresponds to the characteristic size of the domains of synchronized oscillators, diverges. At the same time, oscillations become coherent over long periods of time and long-range phase order ap- pears. The divergence of the correlation length allows a de- scription of the critical behaviors of spatially extended sys- tems in terms of continuous field theories and is characteristic of scale invariance in the critical regime. It constitutes the foundation of the RG theory, which explains the emergence of universality at critical points关4–6,11–13兴.

In this paper, we perform an RG study of the critical behaviors of a collection of oscillators, distributed on a d-dimensional lattice and coupled via nearest-neighbor inter- actions. We approach the critical point from the disordered phase. Close to criticality, the large-scale properties of the array of oscillators are described by a dynamic field theory that is given by the complex Ginzburg-Landau equation with an additive noise term. We apply field-theoretical perturba- tion theory in a d=共4 −⑀兲-dimensional space and introduce an RG scheme that is appropriate for the study of coupled oscillators.

The outline of the paper is as follows. In Sec. II, we present the general field-theoretical framework for the com- plex Ginzburg-Landau equation. We introduce in Sec. III an oscillating reference frame that is essential to define the RG procedure for oscillating critical systems and in which the phase and frequency of the oscillations are eliminated. The correlation and response functions of critical oscillators in mean-field theory are discussed in Sec. IV. These mean-field results are relevant above the critical dimensiond_c= 4. For

d⬍4, mean-field theory breaks down. We discuss in Sec. V the renormalization group of the complex Ginzburg-Landau field theory using Wilson’s RG scheme for which the renor- malization procedure for critical oscillators can be intro- duced most clearly. One-loop order calculations in perturba- tion theory are presented, but further calculations are necessary to characterize the correct qualitative structure of the RG flow. In Sec. VI, we present Callan-Symanzik’s RG scheme adapted for systems of coupled oscillators and cal- culate its beta functions as well as its complete RG flow and fixed points to two-loop order in perturbation theory. The physical properties of the oscillating system are character- ized by correlation and response functions. In Sec. VII, we discuss the asymptotic behaviors of these functions in the critical regime using the RG flow and applying a matching procedure. The formal relation of the RG fixed point for critical oscillators to the fixed point of a real Ginzburg- Landau theory关which satisfies a fluctuation-dissipation共FD兲 relation兴leads to an emergent symmetry at the critical point.

However, the system operates far from thermodynamic equi- librium and breaks the FD relation. The degree of this viola- tion can be characterized by the introduction of a frequency- dependent effective temperature which diverges with a universal anomalous power law at the critical point. We con- clude our presentation with a discussion of the general prop- erties of critical oscillators, their relations to equilibrium critical points, and possible experimental systems for which the critical behaviors discussed here could be observed in the future.

II. FIELD THEORY OF COUPLED OSCILLATORS A. Complex Ginzburg-Landau field theory

The generic behavior of a nonlinear oscillator in the vi- cinity of a Hopf bifurcation can be described by a dynamic equation for a complex variableZ characterizing the phase and amplitude of the oscillations关24兴. This variable can be chosen such that its real part is, to linear order, related to a physical observable, e.g., the displacementX共t兲generated by a mechanical oscillator: X共t兲= Re关Z共t兲兴+ nonlinear terms. In the presence of a periodic stimulus forceF共t兲=F˜ e^−i␻twith a frequency␻close to the oscillation frequency ␻0 at the bi- furcation, the generic dynamics obeys

⳵tZ= −共r+i␻0兲Z−共u+iua兲兩Z兩²Z+⌳⁻¹e^i␪F共t兲. 共1兲 ForF= 0 andr⬎0, the static stateZ= 0 is stable. The system undergoes a Hopf bifurcation atr= 0 and exhibits spontane- ous oscillations forr⬍0. The nonlinear term, characterized by the coefficientsuandu_a, stabilizes the oscillation ampli- tude foru⬎0. The external stimulus appears linearly in this equation and couples in general with a phase shift␪关26,27兴.

In the case of a mechanical oscillator, the coefficient⌳ has units of a friction.

Coupling many oscillators in a field theoretic continuum limit leads to the complex Ginzburg-Landau equation关28,29兴 with additive noise and external forcing terms,

⳵tZ= −共r+i␻0兲Z+共c+ica兲⌬Z−共u+iua兲兩Z兩²Z

+⌳⁻¹e^i␪F+␩^. 共2兲

Here, the complex variableZ共x,t兲becomes a field defined at positions x in a d-dimensional space and ⌬ denotes the Laplace operator in this space. The coefficients c and c_a characterize the local coupling of oscillators and the effects of fluctuations are described via a complex random forcing term␩共x,t兲, which will be chosen Gaussian with zero mean value, i.e., 具␩共x,t兲典= 0. As far as long-time and long- wavelength properties are concerned, the correlations of the noise can be neglected and white noise can be used.

For a vanishing external fieldF共x,t兲and in the absence of fluctuations, Eq. 共2兲 is invariant with respect to phase changes of the oscillations,

Z→Ze^i␾. 共3兲

This symmetry reflects the fact that only phase-invariant terms contribute to the dominant critical behaviors studied here. Indeed, the Hopf bifurcation is associated with the emergence of a nonzero oscillatory mode, which dominates the critical behaviors and for which time-translational invari- ance and phase invariance are equivalent. The noise correla- tions can therefore be chosen such that they respect phase invariance in the problem,

具␩共x,t兲␩共x⬘^,t⬘兲典= 0,

具␩共x,t兲␩^*共x⬘^,t⬘兲典= 4D␦^d共x−x⬘兲␦共t−t⬘兲. 共4兲 Here D is a real and positive coefficient characterizing the amplitude of the noise, and␦ ^and␦^d represent Dirac distri- butions, respectively, in 1 andd dimensions.

B. Physical correlation and response functions Since the physical variables of interest are real, we de- compose the complex fields Z and F into their real and imaginary parts byZ=␺1+i␺2 andF=F₁+iF₂. We focus on two important functions which characterize the behavior of the system, namely the two-point autocorrelation function C_␣␤and the linear-response function␹␣␤to an applied exter- nal forcing term. They are defined as

C_␣␤共x−x⬘^,t−t⬘兲=具␺_␣共x,t兲␺_␤共x⬘^,t⬘兲典c,

具␺_␣共x,t兲典=

冕

^{ddx⬘dt⬘␹␣␤共x−x⬘,t−t⬘兲F␤共x⬘,t⬘兲+O共兩F兩2兲,}

共5兲 where 具¯典c denotes a connected correlation function 关51兴. Because of phase invariance, these functions obey the fol- lowing symmetry relations:

C₁₁=C₂₂ and C₂₁= −C₁₂, 共6兲 with similar relations for the linear-response function␹␣␤.

C. Field-theoretical representation

The correlation and response functionsC_␣␤ and␹_␣␤ ^can be conveniently expressed using a field-theoretical formal-

ism. We introduce the Martin-Siggia-Rose response field␺^˜␣

关30兴and apply the Janssen–De Dominicis formalism关31,32兴 to write the following generating functional:

ZP关I˜

␣,I_␣兴=

冕

^{D关␺␣兴D关−i␺˜␣兴}

⫻exp

再

^{SP关␺˜␣,␺␣兴+}

^冕

^{ddxdt关˜I␣␺˜␣+I␣␺␣兴}

冎

^,

共7兲 where the “physical” actionSPis given by关52兴

SP关␺^˜_␣,␺_␣兴=

冕

^{ddxdt兵D␺˜␣␺˜␣−␺˜␣关⳵t␺␣+共R␣␤+␻0␧␣␤兲␺␤}

+U_␣␤␺_␤␺_␥␺_␥兴其. 共8兲

HereR_␣␤=共r−c⌬兲␦␣␤−ca⌬␧_␣␤ andU_␣␤=u␦␣␤+ua␧_␣␤, with

␧21= −␧12= 1 and ␧11=␧22= 0. The field˜I_␣ is related to the external forceF_␣ in the equation of motion关Eq.共2兲兴by

˜I

␣=⌳⁻¹⍀_␣␤共␪兲F_␤, 共9兲 where⍀_␣␤共␪兲 denotes the rotation matrix by an angle ␪ ⁱⁿ two dimensions,

⍀_␣␤共␪兲=

冉

^cos共sin共␪^␪兲^{兲 − sin共}cos共␪^␪兲^兲

冊

^{. 共10兲}

Correlation and response functions are given by derivatives of the generating functional. We have

C_␣␤共x−x⬘^,t−t⬘^兲=

冏

␦^I␣共x,t兲^␦2␦^lnI␤^Z共x⬘^,t⬘^兲

冏

I_␣,I˜

␣=0

, 共11兲

␹␣␤共x−x⬘^,t−t⬘兲=⌳⁻¹⍀_␥␤共␪兲

冏

␦^I␣共x,t^␦2兲␦^ln˜I␥^Z共x⬘^,t⬘兲

冏

I_␣,I˜

␣=0

. 共12兲 III. FIELD THEORY IN AN OSCILLATING REFERENCE

FRAME A. Amplitude equation

The frequency␻0and the phase␪can be eliminated from Eq. 共2兲 by a time-dependent variable transformation of the formY⬅e^i␻0tZ,H⬅e^i␻0t⌳⁻¹e^i␪F, and␨⬅e^i␻0t␩^{, whereYde-} notes the oscillation amplitude,His a forcing amplitude, and

␨is a transformed noise that has the same correlators as ␩^. This procedure leads to the amplitude equation

⳵tY= −rY+共c+ica兲⌬Y−共u+iua兲兩Y兩²Y+H+␨^. 共13兲 Defining two real fields␾␣byY=␾1+i␾2, Eq.共13兲reads

⳵t␾␣= −R_␣␤␾␤−U_␣␤␾␤␾␥␾␥+H_␣+␨␣, 共14兲 whereH=H₁+iH₂.

The correlation and response functionsG_␣␤=具␾␣␾␤典cand

␥_␣␤=具␾_␣␾^˜_␤典c of the fields ␾_␣ are related to the physical correlation and response functions by

C_␣␤共x,t兲=⍀_␣␴共−␻0t兲G_␴␤共x,t兲,

␹␣␤共x,t兲=⌳⁻¹⍀_␣␴共␪⁻␻0t兲␥␴␤共x,t兲. 共15兲 Similar time-dependent transformations exist between higher-order correlation and response functions of the physi- cal fields and those calculated in the oscillating reference frame.

B. Analogy with an equilibrium critical point in a particular case

For the particular case where ca= 0 and ua= 0, Eq. 共14兲 becomes identical to the model A dynamics of a real Ginzburg-Landau field theory with anO共2兲symmetry of the order parameter关3兴. The critical behavior of this theory at thermodynamic equilibrium has been extensively studied 关33–35兴. This leads, in this particular case, to a formal anal- ogy between an equilibrium phase transition and a Hopf bi- furcation. The amplitudeY plays the role of the order param- eter of the transition. The disordered phase with 具Y典= 0 corresponds to noisy oscillators that are not in synchrony, while nonzero order 具Y典 implies the existence of a global phase and amplitude of synchronous oscillations with fre- quency␻0. The correlation lengths and times of the equilib- rium field theory correspond to lengths and times over which oscillators are in synchrony. The correlation and response functions C_␣␤ and ␹␣␤ for this case can be obtained from those of the equilibrium field theory by using Eq.共15兲. Since the O共2兲 symmetric theory at thermodynamic equilibrium obeys an FD relation, a generic relation between the corre- lation and response functionsC_␣␤and␹␣␤appears.

C. Generating functional

The functionsG_␣␤ and␥_␣␤, as well as higher-order cor- relation and response functions, can be formally calculated using field-theoretical techniques共see, e.g.,关2,13兴兲. The gen- erating functional of the theory is given by

Z关˜J

␣,J_␣兴=

冕

^{D关␾␣兴D关−i␾˜␣兴}

⫻exp

再

^{S关␾˜␣,␾␣兴+}

^冕

^{ddxdt关J˜␣␾˜␣+J␣␾␣兴}

冎

^,

共16兲 where we have introduced the Martin-Siggia-Rose response field␾^˜_␣关30兴. The associated action reads

S关␾^˜_␣^,␾_␣兴=

冕

^{ddxdt兵D␾˜␣␾˜␣−␾˜␣关⳵t␾␣+R␣␤␾␤兴}

−U_␣␤␾^˜␣␾␤␾␥␾␥其. 共17兲 Correlation and response functions are given by

G_␣␤共x−x⬘^,t−t⬘^兲=

冏

␦^J␣共x,t兲^␦2␦^lnJ␤^Z共x⬘^,t⬘^兲

冏

J_␣,J˜

␣=0

, 共18兲

␥_␣␤共x−x⬘^,t−t⬘^兲=

冏

␦^J␣共x,t兲^␦2␦^ln˜J␤^Z共x⬘^,t⬘兲

冏

J_␣,J˜

␣=0

. 共19兲

The effective action of the theory is defined by

⌫关⌽˜

␣,⌽_␣兴=

冕

^{ddxdt关˜J␣⌽˜␣+J␣⌽␣兴− lnZ关˜J␣,J␣兴,}

共20兲 with

⌽_␣共x,t兲= ␦^lnZ

␦^J␣共x,t兲 and ⌽˜

␣共x,t兲= ␦^lnZ

␦^˜J␣共x,t兲. 共21兲 In order to perform calculations perturbatively, we split the action into a harmonic or “Gaussian” part, and a quartic or “interaction” part asS=S0+Sint.S0is given by

S0=

冕

k

„␾^˜_␣共k兲兵D␾^˜_␣共−k兲−关i␻␦␣␤+R_␣␤共−k兲兴␾_␤共−k兲其…

= −1 2

冕

k

␾_គ␣^t共k兲A=␣␤共−k兲␾_គ␤共−k兲, 共22兲

where

␾_គ␣=

冉

^␾˜␾^␣_␣

冊

^{, ␾គ␣t =共␾˜␣,␾␣兲, 共23兲}

A=␣␤共−k兲=

冉

⁻ⁱ␻␦^{− 2D}␤␣+^␦R^␣␤_␤␣共k兲 ^{i␻␦␣␤+}0^{R␣␤共−k兲}

冊

^,

共24兲 andR_␣␤共k兲=共r+cq²兲␦␣␤+caq²␧_␣␤. In these expressions and in the following, we label k=共q,␻兲, 兰k=兰_q,␻ denotes 兰关d^dq/共2␲兲^d兴共d␻^{/ 2}␲兲, and we use the following convention for Fourier-transforms:

f共x,t兲=

冕

_q,␻^{f共q,␻兲ei共q·x−␻t兲=}

冕

k

f共k兲e^ik·x. 共25兲 The interaction term of the action takes the form

Sint= −

冕

_兵k_i其

U_␣␤␾^˜␣共k1兲␾␤共k2兲␾␥共k3兲␾␥共k4兲

⫻共2␲兲^d+1␦^共d+1兲

冉 ^兺

^{i ki}

冊

^{. 共26兲}

Graphic representations of the basic diagrams of the pertur- bation theory are given in Appendix A.

IV. MEAN-FIELD THEORY

Dimensional analysis reveals that for d⬎4 mean-field theory applies. In this case, the mean-field approximation allows us to calculate valid asymptotic expressions for the effective action of the theory, as well as the two-point corre- lation and response functions. In the framework of the Janssen–De Dominicis formalism for dynamic field-

theoretical models, this approximation consists in substitut- ing the saddle-node value of the path integral共16兲to the full functional generatorZ. We obtain

Z^mf关J˜

␣,J_␣兴= exp

再

^{S关␾˜␣mf,␾␣mf兴+}

^冕

^{ddxdt关J˜␣␾˜␣mf+J␣␾␣mf兴}

冎

^,

共27兲 whereS is given by Eq. 共17兲 and␾_␣^mfand ␾^˜_␣^mfsatisfy the stationarity conditions at the saddle node

␦S

␦␾^˜␣

= −˜J

␣ and ␦S

␦␾␣= −J_␣, 共28兲 which give the mean-field dynamic equations

J_␣= −⳵t␾^˜_␣^mf+R_␤␣␾^˜_␤^mf+U_␤␣␾^˜_␤^mf␾_␥^mf␾_␥^mf+ 2U_␤␴␾_␴^mf␾_␣^mf␾^˜_␤^mf,

˜J

␣= − 2D␾^˜_␣^mf+关⳵t␾_␣^mf+R_␣␤␾_␤^mf+U_␣␤␾_␤^mf␾_␥^mf␾_␥^mf兴.

共29兲 The correlation and response functions can be obtained most easily by first eliminating the nonphysical field␾^˜_␣^{mfand writ-} ing a mean-field equation for␾_␣^mfonly,

2DJ_␣=关−⳵t␦␣␤+R_␤␣+U_␤␣␾_␥^mf␾_␥^{mf+ 2U}_␤␴␾_␴^mf␾_␣^mf兴

⫻关⳵t␾_␤^mf+R␤␴␾_␴^mf+U␤␴␾_␴^mf␾_␥^mf␾_␥^mf−˜J␤兴. 共30兲 The mean-field generating functional for the field ␾_␣^mf obeying Eq. 共30兲 is obtained from Eq.共27兲 by eliminating the field␾^˜_␣^{mf. It reads}

Z^mf关J˜

␣,J_␣兴= exp

再

^−4D1

^冕

^{ddxdt关共⳵t␾␣mf−F␣关␾␤mf兴−˜J␣兲2}

+J_␣␾_␣^mf兴

冎

^{, 共31兲}

where

F_␣关␾_␤兴= −R_␣␤␾_␤−U_␣␤␾_␤␾_␥␾_␥. 共32兲 Note that forJ_␣= 0 the stationary condition of the generating functional共31兲leads to

⳵t␾_␣^{mf= −R}_␣␤␾_␤^mf−U_␣␤␾_␤^mf␾_␥^mf␾_␥^mf+˜J_␣^, 共33兲 which is compatible with Eq.共30兲.

The linear-response and correlation functions are obtained as

␥_␣␤^mf共x−x⬘^;t−t⬘^兲=

冏

␦^␦␾˜J␤^␣mf共x^共x,t兲⬘^,t⬘兲

冏

J_␣,J˜

␣⬅0

,

G_␣␤^mf共x−x⬘^;t−t⬘兲=

冏

␦^␦␾J␤^␣mf共x^共x,t兲⬘^,t⬘^兲

冏

J_␣,J˜

␣⬅0

. 共34兲 Calculating these functions and applying the time-dependent transformations共15兲, we obtain the physical correlation and response functions in mean-field theory,

␹_␣␤^mf共q,␻兲= 1

⌳⌬兵关共−i␻⁺R兲cos␪⁺⍀0sin␪兴␦␣␤

+关共−i␻⁺R兲sin␪−⍀0cos␪兴␧_␣␤其

C_␣␤^mf共q,␻兲= 2D

兩⌬兩²

冉

^{␻2− 2i+⍀}␻⁰²⍀⁺0^R2 ␻^22i+␻⍀^⍀₀²+⁰R²

冊

^{, 共35兲}

whereR=r+cq²,⍀0=␻0+caq², and⌬=共−i␻⁺R兲²+⍀₀². The diagonal elements of these matrices are given by

␹11mf共q,␻兲= 1

2⌳

冋

^{R−i共e␻i␪−⍀0兲+R−i共e␻−i␪+⍀0兲}

册

^,

C₁₁^mf共q,␻兲= D

R²+共␻⁻⍀0兲²+ D

R²+共␻⁺⍀0兲², 共36兲 and the nondiagonal elements by

␹12mf共q,␻兲= i

2⌳

冋

^{R−i共e␻i␪−⍀}0兲− e^−i␪ R−i共␻⁺⍀0兲

册

^,

C₁₂^mf共q,␻兲= iD

R²+共␻⁻⍀0兲²− iD

R²+共␻⁺⍀0兲². 共37兲 Note finally that for r= 0 and for the critical mode 共q

=0,␻= 0兲, the response function of the system is nonlinear even at small amplitudes. We have indeed in this case

␦具兩Y共0,0兲兩典^mf⬀兩H兩^1/3. 共38兲

V. WILSON’S RENORMALIZATION SCHEME Ford⬍4, mean-field theory breaks down and another ap- proach is necessary to investigate the critical behaviors of the theory. We apply perturbative renormalization group 共RG兲 methods using an⑀expansion near the upper critical dimen- siond_c= 4共d= 4 −⑀兲 关7,36,37兴. We present here the RG struc- ture of the theory within Wilson’s momentum shell RG scheme adapted to the renormalization of the complex Ginzburg-Landau field theory. This scheme has the advan- tage to be conceptually transparent and to provide a clear physical interpretation to the calculations. However, for cal- culations beyond one-loop order, this technique is less suited than the Callan-Symanzik RG scheme. The adaptation of the latter to the complex Ginzburg-Landau theory is presented in Sec. VI.

A. Renormalized fields

The renormalization procedure within Wilson’s scheme is performed as follows. We start from a dynamic functional of the theory with a small distance cutoff⌳in the integrals over wave vectors. This cutoff corresponds to an underlying lat- tice of mesh size a⯝2␲^/⌳. We interpret this as a micro- scopic theory with an action of the form共8兲, and associated quantities are labeled with a superscript “0.” We calculate the effective action in an oscillating reference frame at a given scale⌳/b, whereb=e^l is a dilatation coefficient larger than

1. This reference frame is defined such that, described in terms of effective or renormalized parameters, the effective action has the structure 共17兲. The renormalized quantities that satisfy this requirement can be expressed as

q=bq⁰, ␻^=b2Z␻共b兲␻^0,

␾_␣共q,t兲=b^{−共d+2兲/2}

冑

Z_␾共b兲Z_␻共b兲⍀_␣␤„␻^ˆ0共b兲t…␺_␤⁰共q⁰,t⁰兲,

␾^˜_␣共q,t兲=b^{−共d−2兲/2}

冑

^Z_␾^˜共b兲Z_␻共b兲⍀_␣␤„␪共b兲+␻^ˆ0共b兲t…␺^˜_␤⁰共q⁰,t⁰兲. 共39兲 Here we have introduced scale-dependent Z factors for the renormalization of the two dynamic fields关Z_␾共b兲andZ_␾^˜共b兲兴 and for the frequencies关Z_␻共b兲兴. In addition to the usual RG transformations and scale dilatation q=bq^共0兲, the complex Ginzburg-Landau theory requires us to perform time- dependent transformations between bare and renormalized fields. Indeed, the effective theory is described in a reference frame that oscillates with effective frequency ␻^ˆ0共b兲 and phase␪共b兲relative to the reference frame of the bare theory.

The definition of the fields ␾␣ and ␾^˜␣ takes this relative rotation into account by terms involving the rotation matrix

⍀_␣␤ 关Eq. 共10兲兴. The scale-dependent oscillating reference frame represents a key element of the RG procedure for os- cillating systems. For the microscopic theory, ␻^ˆ0共1兲=␻00,

␪共1兲= 0 and the fields ␺_␣^{0 and} ␺^˜_␣⁰ coincide with the fields introduced in Sec. II.

B. Renormalization group flow

In order to determine the behavior of the effective param- eters under renormalization, we determine their variations with respect to a small changes of the dilatation␦^b/b共or␦l兲.

Integrating over the momentum shell of wave vectors in the interval关⌳/e^␦l,⌳兴, we obtain an effective action that, before rescaling, reads

S^共1兲关␾^˜_␣^,␾_␣兴=

冕

q

⌳/e^␦l

冕

_␻^{兵共D+␦D兲␾˜␣␾˜␣}

−␾^˜␣关i共1 +␦␭1兲␻⁺共r+␦r兲+共c+␦c兲q²兴␾␣

−␾^˜_␣关i␦␭2␻⁺␦␻^ˆ0+共ca+␦^ca兲q²兴␧_␣␤␾_␤其

−

冕

k₁k₂k₃

⌳/e^␦l

共U_␣␤+␦^U␣␤兲␾^˜␣␾␤␾␥␾␥. 共40兲

Here, we have introduced variations of the parameters under this procedure, which can be calculated perturbatively from the “one-particle irreducible”共1-PI兲diagrams of the theory.

Other terms are either forbidden by symmetry properties of the theory or irrelevant in the infrared共IR兲 limit. Two new terms appear, which were absent from Eq.共17兲 and which correspond to the coefficients␦␭2and␦␻^ˆ0. They reflect the renormalization of frequency and phase. These terms are ab- sorbed by time-dependent variable transformations of the fields,

␾_␣ⁱ共q,t兲=⍀_␣␤共␦␻^ˆ0

it兲␾␤共q,t兲,

␾^˜_␣ⁱ共q,t兲=⍀_␣␤共␦␻^ˆ0it+␦␪兲␾␤共q,t兲, 共41兲 where

␦␪= arctan关−␦␭2共1 +␦␭1兲⁻¹兴,

␦␻^ˆ0i=␦␻^ˆ0cos␦␪⁺共r+␦^r兲sin␦␪^. 共42兲 Rewriting the effective action in terms of the new fields

␾_␣ⁱ and ␾^˜_␣ⁱ requires a redefinition of the changes of all pa- rameters. The so defined parameter changes are label by a superscript “i” in the following. We finally rescale all lengths byq⬘^=e␦lq, and introduce three Z factors at the scalel+␦^l, such that the effective action retains its form共17兲under an RG step. Furthermore, we impose that D共l+␦l兲=D共l兲 and c共l+␦l兲=c共l兲, i.e., that the coefficientscandD remain con- stants under renormalization. As a result, the variations of the parameters under a small renormalization step are given by

c_a共l+␦^l兲=关c_a共l兲+␦^cai兴共1 +␦^ci/c兲⁻¹, r共l+␦l兲=e^2␦l关r共l兲+␦^ri兴共1 +␦^ci/c兲⁻¹,

U_␣␤共l+␦^l兲=e^{共4−d兲␦l}关U_␣␤共l兲+␦^U␣␤i 兴共1 +␦^ci/c兲⁻²共1 +␦^Di/D兲

⫻共1 +␦␭ⁱ兲⁻¹,

Z_␻共l+␦l兲=Z_␻共l兲共1 +␦^ci/c兲⁻¹共1 +␦␭ⁱ兲, Z_␾共l+␦l兲=Z共l兲共1 +␦^ci/c兲³共1 +␦^Di/D兲⁻¹共1 +␦␭ⁱ兲⁻¹,

Z_␾^˜共l+␦^l兲=Z˜共l兲共1 +␦^ci/c兲共1 +␦^Di/D兲共1 +␦␭ⁱ兲⁻¹, 共43兲 where

␦^Di=␦^D, ␦␭ⁱ=共C− 1兲+␦␭1C−␦␭2S,

␦^ri=r共C− 1兲+␦^rC−共␻0+␦␻0兲S,

␦^ci=c共C− 1兲+␦^cC−共ca+␦^ca兲S,

␦^cai =c_a共C− 1兲+␦^caC+共c+␦^c兲S,

␦^U␣␤i =U_␣␴共⍀_␴␤共␦␪兲−I_␴␤兲+␦^U␣␴⍀_␴␤共␦␪兲. 共44兲 Here, C and S denote cos␦␪ ^{and sin}␦␪, respectively. The evolution of the effective frequency and phase that enter the transformations共39兲is given by

␪共l+␦^l兲=␪共l兲+␦␪^,

␻^ˆ0共l+␦l兲=e^2␦l关␻^ˆ0共l兲+␦␻^ˆ0i兴共1 +␦^ci/c兲⁻¹共1 +␦␭ⁱ兲.

共45兲

C. Correlation and response functions

Using the RG transformations of the effective parameters, as well the correlation and response functions of the renor-

malized fields␾␣ and␾^˜␣, we can write expressions for the physical correlation and response functions, where the trans- formations共39兲have been used,

C_␣␤共q,t兲=b²Z_␾共b兲⁻¹Z_␻共b兲⁻²⍀_␣␴„−␻0共b兲t…

⫻G_␴␤^R „bq,b⁻²Z_␻共b兲⁻¹t…,

␹_␣␤共q,t兲=关Z_␾共b兲Z_␾^˜共b兲兴^−1/2Z_␻共b兲⁻²⍀_␣␴„␪共b兲−␻0共b兲t…

⫻␥_␴␤^R „bq,b⁻²Z_␻共b兲⁻¹t…. 共46兲 Here, the superscript “R” indicates that the functions have to be calculated using the renormalized set of parameters. The scale-dependent renormalized frequency is given by

␻0共b兲=b⁻²Z_␻共b兲⁻¹␻^ˆ0共b兲, 共47兲 and ␪共b兲 describes the scale-dependent phase lag between external forcing and response of the system.

At the fixed point of the RG, the theory is scale invariant and, consequently, theZ factors exhibit simple scaling rela- tions as a function ofb,

Z_␻共b兲=b^z−2, Z_␾共b兲=b^{−2共z−2兲+␩},

Z_␾^˜共b兲=b^{−2共z−2兲+␩˜}. 共48兲 These relations define three independent critical exponents of the theory:z,␩^{, and}␩^˜. A further critical exponent␯ ^{is asso-} ciated with the positive eigenvalue of the linearized RG equations around the fixed point.

D. Renormalization of the time and independent critical exponents

In dynamic RG procedures, there is in general a freedom to choose some parameters constant while others are renor- malized in a nontrivial way. In the system described here, we choose to renormalize the time共i.e., the frequency coordinate

␻ in Fourier space兲 and to keep the parameters c and D invariant under renormalization. In other cases, different choices are commonly used. For example, renormalizing the dynamic model A with O共2兲 symmetry at thermodynamic equilibrium, the coefficient D is usually chosen to change under renormalization, while time is simply rescaled关34,35兴.

This model corresponds to the particular case of the ampli- tude equation of the complex Ginzburg-Landau theory where bothcaanduaare equal to zero. It is described by a dynam- ics of the form

⳵t␾_␣= −D␦H

␦␾␣+␨␣, 共49兲 and relaxes towards a thermodynamic equilibrium. Note that since this model satisfies an FD relation, the noise strengthD appears as the mobility coefficient in the dynamics. Equation 共49兲 shows that both choices, renormalizing the time or renormalizing D, are equivalent. The complex Ginzburg- Landau theory discussed here does not obey an FD relation.

Therefore, a factorization of the coefficientDas it is done in Eq.共49兲would be artificial in this case, and would require a

redefinition of all other parameters. Without this factoriza- tion, the structure of the theory requires one to renormalize the time if the coefficientcis kept constant.

Note also that the absence of an FD relation in the theory changes the structure of the RG equations as compared to an equilibriumO共2兲 model. Indeed, the FD relation imposes a constraint on the renormalized quantities of the equilibrium theory, which can be written as

Z_␻

冑

^Z␾Z_␾−1˜ ⬅1, 共50兲 and which implies the following relation between the critical exponents关13兴:

z= 2 +共␩^˜−␩兲/2. 共51兲 In the nonequilibrium case considered here, such a constraint does not exist and four truly independent critical exponents are present in the theory as compared with three in theO共2兲 dynamic model A.

E. Results to one-loop order

To one-loop order in perturbation theory, theZfactors, as well as the phase factor ␪ and the parameter ca, are not renormalized. We define the following reduced parameters:

¯r=r/c⌳², ␻^¯0=␻0/c⌳², ¯c_a=ca/c, u¯=u/共c⌳²兲², ¯u_a=ua/共c⌳²兲²,

D= 4D⌳^d

共4␲兲^d/2⌫共d/2兲, 共52兲 to express the RG flow in d= 4 −⑀. It is given by three coupled equations

dr¯

dl= 2r¯+ 2D ¯u 1 +¯r, du¯

dl =⑀¯u−D

冋

^{共¯u2共1 +−¯u¯ra2兲共兲关¯1 +ca2+¯r共兲1 ++ 2u¯r¯u兲¯2a兴¯ca+共1 +4u¯¯2r兲2}

册

^,

du¯_a

dl =⑀^¯ua+D ¯c_a

1 +¯r

冋

^{共u¯2共1 +−¯u¯ra2兲共1 +兲关c¯a2+¯r共1 +兲+ 2u¯r¯u兲¯2a兴¯ca}

册

− 6D ¯uu¯_a

共1 +¯r兲², 共53兲

and a fourth one associated with the renormalization of the oscillation frequency,

d␻^¯0

dl = 2D ¯u_a

1 +¯re^−2l. 共54兲 This last equation has to be integrated after the system共53兲 has been solved.

Since the coefficient¯c_a is not renormalized to first order, we find one infrared stable fixed point for each of its values.

It is given by

¯r^*= − ⑀

5, u¯^*= ⑀

5D, and ¯u_a^*=¯c_a ⑀

5D. 共55兲 Writing¯r=¯r^*+␦^r,¯u=u¯*+␦^{u, and¯u}a=¯u_a^*+␦^ua, the linearized RG equations at the fixed point are given by

d

dl

冢

^{␦␦␦ruua}

冣

⁼

冢

^{2共1 −00⑀/5兲 − 4c2D−¯a⑀⑀/5 −00⑀/5}

冣 ^冢

^{␦␦␦uura}

^冣

^.

共56兲 The RG flow of the theory is three-dimensional. We show in Fig. 1 the qualitative RG flow projected on the plane共r,u兲.

Finally, the critical exponents to the one-loop order read

␯⁼¹ 2+ ⑀

10, z= 2, ␩^{= 0,} ␩^{˜= 0.} 共57兲 To one-loop order, the threeZ factors and the parameter c_aare not renormalized. Therefore, these calculations are in- sufficient to determine the fixed-point value ofc_aas well as three of the four independent critical exponents of the theory.

Calculations to two-loop order are necessary to obtain the full RG structure and the critical properties. Wilson’s mo- mentum shell RG scheme is not well suited to perform such calculations and it is technically far more convenient to per- form these using a Callan-Symanzik RG scheme, adapted to the renormalization of critical oscillators. This is discussed in the next section.

VI. CALLAN-SYMANZIK RG SCHEME

The Callan-Symanzik RG scheme avoids the introduction of a cutoff in the momentum space, which is responsible for making the evaluation of multiple integrals technically diffi- cult in Wilson’s scheme. In its absence, the calculation of two-loop and higher-order Feynman integrals is easier. Here we present the general structure of Callan-Symanzik’s RG scheme adapted to the study of coupled oscillators. We dis- cuss the RG flow to two-loop order in perturbation theory and show that the Callan-Symanzik RG scheme described here is consistent with the momentum shell procedure pre- sented before.

FIG. 1. Qualitative representation of RG flow to one-loop order in perturbation theory, projected on the plane共r,u兲and for a space dimensiond⬍4.