Exact results and open questions in first principle functional RG

Annals of Physics 325 (2010) 49–150

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Exact results and open questions in first principle functional RG Pierre Le Doussal LPTENS CNRS UMR 8549 24, Rue Lhomond, 75231 Paris Cedex 05, France

a r t i c l e

i n f o

Article history: Received 8 May 2009 Accepted 19 October 2009 Available online 24 October 2009 Keywords: Disordered systems Glasses Renormalization group Pinning Turbulence Burgers equation

a b s t r a c t Some aspects of the functional RG (FRG) approach to pinned elastic manifolds (of internal dimension d) at finite temperature T > 0 are reviewed and reexamined in this much expanded version of Le Doussal (2006) [67]. The particle limit d = 0 provides a test for the theory: there the FRG is equivalent to the decaying Burgers equation, with viscosity m  T-both being formally irrelevant. An outstanding question in FRG, i.e. how temperature regularizes the otherwise singular flow of T = 0 FRG, maps to the viscous layer regularization of inertial range Burgers turbulence (i.e. to the construction of the inviscid limit). Analogy between Kolmogorov scaling and FRG cumulant scaling is discussed. First, multi-loop FRG corrections are examined and the direct loop expansion at T > 0 is shown to fail already in d = 0, a hierarchy of ERG equations being then required (introduced in Balents and Le Doussal (2005) [36]). Next we prove that the FRG function R(u) and higher cumulants defined from the field theory can be obtained for any d from moments of a renormalized potential defined in an sliding harmonic well. This allows to measure the fixed point function R(u) in numerics and experiments. In d = 0 the beta function (of the inviscid limit) is obtained from first principles to four loop. For Sinai model (uncorrelated Burgers initial velocities) the ERG hierarchy can be solved and the exact function R(u) is obtained. Connections to exact solutions for the statistics of shocks in Burgers and to ballistic aggregation are detailed. A relation is established between the size distribution of shocks and the one for droplets. A droplet solution to the ERG functional hierarchy is found for any d, and the form of R(u) in the thermal boundary layer is related to droplet probabilities. These being known for the d = 0 Sinai model the

E-mail addresses: [email protected], [email protected] 0003-4916/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2009.10.010

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function R(u) is obtained there at any T. Consistency of the  ¼ 4  d expansion in one and two loop FRG is studied from first principles, and connected to shock and droplet relations which could be tested in numerics. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction 1.1. Overview Success of field theory and renormalization methods for pure systems usually stems from being able to identify a few relevant operators, usually from symmetry, and control them in some e.g. dimensional expansion in order to obtain universal quantities [1] (independent of short scale details, i.e. a continuum renormalizable limit). Extending these methods to systems with quenched disorder presents new challenges. Often those exhibit strong disorder regimes, or glass phases, signaled by a flow away from weak coupling of some operators associated to disorder. However, the question of what is really a relevant quantity in a random system is quite delicate. Often these fast growing operators represent high moments of the probability distribution of some observables over samples. Thus they may be associated with far tails of these distributions, i.e. to ultra rare events, and may not be that important in the end (if their feedback into typical events is negligible). A better strategy is then to focus on the RG flow of probability distributions which contain information about typical events. This is what functional RG methods aim to do, and why they are needed in disordered systems. 2D disordered Coulomb gases (DCG) for instance exhibit a frozen phase which at first appears non-perturbative, since the average charge fugacity grows with the scale. The RG flow for the typical fugacity, however, becomes different from the average one in that phase, and turns out to be manageable [2–4] through control of the probability distribution. The same mechanism is at play in related 2D fermion models with quenched disorder and leads to freezing transitions [5–7]. Another example of a strong disorder phase which can be controlled by RG methods arises in one dimensional quantum and classical random chains. There, a (stochastic) real space RG (RSRG) which also decimates in energy space is well adapted to the structure of the strong disorder fixed point (FP). It requires consideration of probability distributions and yields a host of exact results [8–11]. An outstanding question is how general this is, whether other systems can be treated in that way, and whether a unified (functional) RG approach to strong disorder problems can be found. Here we focus on pinned elastic objects, one of the simplest class of systems which form glass phases where temperature (thermal fluctuations) is formally irrelevant. We investigate some properties of the functional RG method to describe these systems, and the issues to be solved. The standard model involves an elastic manifold of internal dimension d parameterized by a displacement field ux ). The energy in a given sample (i.e. one realization noted ux  uðxÞ (which may have N-components ~ of Vðu; xÞ) is given by:

HV ½u ¼

Z

d

d x

 1 ðrx uÞ2 þ m2 u2x þ Vðux ; xÞ 2

ð1Þ

where the random potential lives in a d þ N dimensional space. Its distribution can be chosen gaussian with second cumulant:

Vðu; xÞVðu0 ; x0 Þ ¼ dd ðx  x0 ÞRðu  u0 Þ

ð2Þ

and Vðu; xÞ ¼ 0 where : describes average over samples. Being the bare disorder one usually denotes this function as R0 ðuÞ, while R(u) denotes the renormalized, or coarse grained disorder, precisely defined in this paper. However, in this introduction section we will not keep the distinction. A small confining parabolic potential (the mass term) is added for convenience and provides an infrared cutoff at large length scale Lm  1=m. We do not review here the numerous numerical and analytical studies of this problem, nor the many applications to physical systems (see e.g. Refs. [12–16] for references).

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Pedagogical notes about the FRG method, following the plan of this introduction can be found in [17,18]. An introductory review is given in [19,20] where many references can be found. Here we just recall a few important facts. There are three main classes depending on whether the function R(u) is (i) periodic, then it describes periodic objects such as lattices with substrate impurities, (ii) short range, typically describing Ising domain walls with random bonds, and (iii) long range, typically Ising domain walls with random field disorder. The minimal energy ground state configuration (unique for finite size and continuous distribution), here denoted u1 ðxÞ  u1x , is known to be rough with roughness exponent u1 ðxÞ  xf . This is mostly from numerics and from very few exact [21,22,24] or rigorous [23,25] results, i.e. f ¼ 2=3 for the directed polymer d ¼ N ¼ 1. Universality classes are expected to depend on N; d and the symmetries and boundary conditions on R(u) (mentioned above). The minimum energy E ¼ HV ½u1  fluctuates wildly from sample to sample with a width dE  Lh , with h ¼ d  2 þ 2f (a relation guaranteed by the statistical translational invariance of the disorder, the so-called STS symmetry [26,27]). Here and below L is system size. On a qualitative level, since the manifold sees an energy landscape with roughness Lh , equilibrium at non-zero temperature T > 0 does not alter the T = 0 picture, i.e. the manifold remains localized near u1 ðxÞ. The proper rescaled temperature T L  TLh flows to zero, and the system is in a glass phase controlled by a zero temperature FP (whenever h > 0). It is important to stress that the long range case (iii) has a non-trivial and quite useful d = 0 limit, the so-called toy model, one instance being the famous Sinai model of a particle in a Brownian landscape (which is the d = 0 limit of random field disorder [28]). Temperature is irrelevant, however, even at very low T there are some accidental quasi-degeneracies where the energy difference between two (or more) low lying configurations u1 ðxÞ and u2 ðxÞ happen to be of order T. Then the (normalized) Gibbs measure eHV ½u=T =Z is splitted between two (or more) states. How rarely it happens is not settled yet. The droplet picture (or scenario) assumes [29–31] that quasi-degeneracies of two states differing only below scale L happen more and more rarely, with probability pL  TLh as L grows large. It also assumes a kind of statistical independence of these rare events so that quasi-degeneracies between more than two states are negligibly rare. By contrast, the many pure states picture (based example on the replica symmetry breaking (RSB) saddle point for infinite N [32]) finds that these quasi-degeneracies (within T) always occur (i.e. at any T > 0). For manifolds, numerical studies [33] seem to disfavor the many pure state picture at least for small N and d. The marginal case h ¼ 0 is of particular interest, and may be intermediate [3] (the DCG mentioned above falls in that class). In any case, even within the droplet scenario these rare events produce large effects. The exact identity (from STS) for thermal (i.e. connected) correlation (in Fourier):

huq uq i  huq ihuq i ¼

T q2 þ m2

ð3Þ

where h:i denotes thermal average over the Gibbs measure in a given sample, suggests that although the typical du ¼ u  hui is small, dutyp  Oð1Þ, du can be large, du  Lf , with probability TLh , yielding a net result after disorder (i.e. sample) averaging ðduÞ2n  TL2ðn1Þfþ2d . Thus certain correlation functions (thermal correlations) are dominated by rare events. As emphasized in [34–36] a challenge for the field theory is to be able to describe both (i) typical events, i.e. zero temperature correlations encoding the full ground state statistics of u1 ðxÞ (believed, surely not yet proved, to be critical and universal), and (ii) significant rare events, i.e. the ones which would lead to universal behaviour in thermal correlations. An important question is whether these are two decoupled sectors or how much mutual feedback they enjoy. These correlations can be computed from the replicated partition function and action, R Z p ¼ Dua eS½u ; a ¼ 1; . . . ; p, with:

S½u ¼

Z Z 1X 1 X d d d xðrx ua Þ2 þ m2 u2ax  2 d xRðuax  ubx Þ T a 2T ab

ð4Þ

and the fields ua ðxÞ  uax have been introduced (not to be confused with the u1x introduced above). Various thermal and disorder averages are encoded through averages with distinct replicas n huna11 . . . uakk iS ¼ huin1 :huink and here and below the p ¼ 0 limit is implicit (space indices suppressed). The disorder average couples the replica, and their interaction is precisely R(u). This is a convenient

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tool to perform perturbative calculations, although equivalent to direct perturbation theory in given sample, averaging later. It was observed long ago that weak disorder becomes relevant, and the manifold rough, for d 6 dc ¼ 4. The bare disorder being smooth, perturbation theory can be performed in the derivatives Rð2nÞ ð0Þ. If only R00 ð0Þ were non-zero, i.e. keeping only the quadratic part of S½u in (4), one obtains (upon replica inversion):

D

E uaq ubq ¼

T 1 dab þ R00 ð0Þ q2 þ m2 ðq2 þ m2 Þ2

ð5Þ

i.e. dc ¼ 4 appears through the 1=q4 propagator and u  xfL with fL ¼ ð4  dÞ=2, the so-called Larkin model [37]. Since simple power counting at the Larkin fixed point (FP) shows that the coefficient of the u2n interaction, i.e. Rð2nÞ ð0Þ grows with scale as L4dþð2n4Þf , one expects that adding all these n > 1 non-linearities would yield flow away from the Larkin FP and produce a non-trivial f. Amazingly, and quite distinctly from more standard field theory (FT), one finds that at T = 0 the corrections due to these non-linearities all cancel in any (smooth) observable, i.e. (5) is exact. This is the celebrated dimensional reduction (DR) property [38] which also occurs in more complicated, random field models [39,40]. It is in general a (all-order) perturbative statement [41], assuming that R(u) is analytic at u ¼ 0. It was shown by Fisher [45] that already at the one loop level the functional RG equation for the function R(u) at T = 0 yields flow outside the space of the analytic functions. Fixed point functions with a linear cusp, R00 ðuÞ  juj, were found [45,13]. They are uniformly OðÞ with  ¼ 4  d and thus yield non-trivial f ¼ 0ðÞ, evading DR. The physics of the cusp was argued [46] to be related to existence of many metastable states (see below). One loop FRG has been applied to numerous models and physical systems for two decades, including the depinning transition [47,48], moving systems [13], quantum systems and correlated disorder [49–51]. Amazingly then and for almost 15 years, it was not been extended to higher loop, nor its consistency (even at one loop) really checked! Given its usefulness, this was an uncomfortable situation. Claims that it could not yield a consistent  expansion beyond leading order [45] may have discouraged efforts. However, these were based on a toy model for which we now know that FRG works to all orders (see below). It is true, however, that handling multiple minima intuitively appear as a non-perturbative problem, so it remains to be understood precisely how it can ever be controlled. 1.2. Review of our previous work In a series of works we attempted to make progress [52,55–58,53,54]. First we identified the problem. Because of the cusp, R000 ð0þ Þ ¼ R000 ð0 Þ–0, ambiguities arise in T = 0 calculations of the beta function at two loop level, and even already in the one loop corrections to correlations. Schematically there is no obvious way to interpret a R000 ð0Þ in a graph, or, phrased otherwise, perturbation theory requires evaluations of derivatives at u ¼ 0, while only the left and right derivatives are known. Thus it appears that some additional information about the physics of the system is necessary to continue the calculation. Furthermore it is crucial information since DR suggests, and calculations confirm, that all the physics is contained in these singularities. We have followed two routes in parallel. The first one is heuristics. One physical requirement is the existence of a continuum limit, i.e. a renormalizable field theory (RFT). Since one loop counter terms are found to be unambiguous, this requirement strongly points to a simple prescription [55] to lift ambiguities at two loop in the case N ¼ 1. It then yields a candidate field theory with nice properties: renormalizable to two loop, preserving the linear cusp, and yielding reasonable predictions for f to next order in . It also suggests more powerful ‘‘prescriptions” to lift ambiguities and produces a three loop RFT [57]. This route is more delicate for N-component manifolds and random field sigma models, but has also been attempted there [58–60]. These theories are in a sense the most natural candidates RFT. Another route is to attempt to construct the field theory from first principles. At zero temperature, this was possible in two cases: depinning and large N. The depinning transition is continuously related

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53

to statics, since the system remains pinned in presence of an additional force f until the threshold is reached f ¼ fc (in fact, to one loop [47,48] the depinning FP (and beta function) is identical to the statics one). However, at f ¼ fcþ , and for N ¼ 1, the famous Middleton theorem [61] shows that ot uxt > 0 (in the stationary state, or at any time if the initial condition satisfies it). With this additional information, one can compute all graphs without ambiguities (within the dynamical formalism) and check that the theory is indeed renormalizable to two loop [56,52]. This provides an instructive and controlled higher loop extension of the theory. Since the same theorem indicates also a unique state, one may wonder whether N ¼ 1 depinning is, in a sense, simpler, the challenge in the statics being to treat multiple quasi-degenerate minima. Progress in the statics was possible in the large N limit, the second ‘‘solvable” case. There, it was shown [62] that the FRG yields the contribution of the most distant states in the RSB solution of Mézard and Parisi [32]. Further aspects of this connection were elucidated more recently [63]. It still remains to be understood though whether this is not a rather peculiar limit (i.e. whether large but finite N be different from infinite N). The most natural idea for a first principle understanding of FRG is to perform it at non-zero temperature T > 0. Then one finds (in some cases) that the effective action remains smooth and there are no ambiguities. The price to pay is that it is far more complicated to implement since one must keep track of an irrelevant variable, the temperature. On the other hand, the physics is then more accessible. Let us summarize our recent results in that direction. First we have argued for the existence of a ‘‘thermal boundary layer” (TBL) in the effective action [49,64,34] which describes how the temperature rounds the cusp, e.g. in R00 ðuÞ, on scales u  T L . For fixed u ¼ Oð1Þ, R(u) converges to the non-analytic fixed point as L ! 1, but for u  T L , R(u) is an analytic function (which is novel and is not encoded in the T = 0 fixed point). We have shown [34,36] that not just the second cumulant, but all cumulants possess TBL analytic scaling forms and derived the FRG equations which couple all of them. This was done within the Wilson one loop approach. To be more systematic we have used exact RG schemes (ERG) [34,36] extending previous studies [53,54,65]. In particular we could analyze in a non-perturbative manner the exact FRG equation for the useful limit d = 0. Second, we have shown that these TBL forms reproduce the scaling expected for correlations from the droplet picture [53,36]. The TBL thus appears to encode for droplet probabilities. The ultimate goal is to understand how a critical theory emerges in the large system size limit L ! 1, i.e. how, starting from a non-zero bare temperature, the limit T L ¼ TLh ! 0 can produce an asymptotic ‘‘zero temperature theory” with all ambiguities resolved. It requires solving the so-called matching problem: one wants to connect information about e.g. R(u) for u  Oð1Þ (which is what the  expansion a priori computes), to derivatives of R(u) at u ¼ 0, needed to compute correlations, which a priori requires knowledge of the function within the TBL u  T L . We have progressed towards that goal in the toy model d = 0. By considering partial boundary layers, i.e. how renormalized disorder cumulants behave when multiple points are brought within u  T L , we have been able to show how to derive a beta function for the T = 0 theory. Extensions to higher d and N have been attempted but are not yet conclusive [36,66]. 1.3. Aim of the present paper and outline The aim of the present paper is to investigate further the first principle approach to FRG at zero and non-zero temperature. It is partially a review, as far as explaining what has been already achieved. We also derive some new results, clarify some points from our previous works and detail some connections to other problems and models such as the decaying Burgers equation and ballistic aggregation. Some of the results have already appeared in [67]. The present paper is a much expanded version of Ref. [67], giving all detailed derivations, and presenting all the tools, which we hope will be useful. Since it took much time to be completed, some companion works, sometimes using some of the results presented here, have already appeared, for instance the measurement of R(u) using the method proposed here and in [67] was successfully performed in [68]. Its extension to the dynamics was presented in [69,71,70]. Interesting developments about shocks and avalanches, quite complementary of the present work, have appeared in [72,63] The outline of the paper is as follows. We start by showing (Section 2) what are the difficulties encountered when trying to develop renormalization at T > 0 along the standard FT method, i.e. within the loop expansion. It is illustrated

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in the text on the d = 0 model (up to four loop), the d > 0 case (up to three loop) is summarized in Appendix A. It is a pedagogical way of showing why more systematic methods, such as ERG are needed here. In our previous study, Refs. [36,66], the thermal boundary layer scaling was presented as an ‘‘ansatz”. A hierarchy of ERG equations for all cumulant’s scaling functions was derived, and the existence of a solution was assumed. Here we obtain an exact solution for the full hierarchy of these TBL equations. It makes the connection to droplets simple and transparent. This is achieved by first asking whether the function R(u), which was introduced and defined in the FRG somewhat abstractly (from the effective action) could be related to an observable. In particular, what precise physical information is contained in the derivatives Rð2nÞ ð0Þ? In our previous work, Refs. [53,36], we had shown how some low order thermal correlations are related to some of these derivatives, as well as to derivatives of higher cumulants. Here, and in Ref. [67], we obtain the full solution of the problem which turns out to be (a posteriori) very simple. For that purpose we reexamine (Section 3), in any dimension, various generating functions for correlations, the effective action C½u and the functional W½j. We prove that they take similar dual forms and that they define the same second cumulant functional R½u, while they differ in higher cumulants: R½u simply encodes the two point correlation of a renormalized potential. One way to define this potential is to add a quadratic well, i.e. an external harmonic potential, with a center which can be translated. It can thus be measured in numerical simulations. Its definition in d = 0 is reminiscent of the ‘‘toy RG” model (with only two degrees of freedom) introduced heuristically in [46], with somewhat different interpretations. Here the correspondence with the field theoretic definition is demonstrated in d = 0 and we obtain the proper generalization to any d. As a result, the renormalized force is found to satisfy a decaying functional Burgers equation. With this knowledge we then go back to the d = 0 toy model (Section 4). We write the ERG hierarchy (Section 4.1) and then work out the full TBL solution corresponding to droplets (Section 4.2). The matching property can then be checked explicitly. Next we derive the correct beta function in d = 0 up to four loop with all ambiguities removed (Section 4.3): the method was sketched in our previous work, Ref. [36], but not fully explicited. In the case of the Sinai model (Brownian landscape, random field disorder) using our previous result on the toy model statistics (obtained through the strong disorder RSRG [28]) we obtain the exact result for the function R(u) (Section 4.4), in terms of multiple integrals of Airy functions, and implicit forms for higher cumulants as well as full and partial boundary layer functions. In Sections 4.5 and 4.6 we discuss the connection between FRG and Burgers turbulence. In d = 0 the renormalized force in the FRG satisfies the usual N-dimensional Burgers equation. Although it is a simplified version of the Navier–Stokes equation they share some common properties (see e.g. Refs. [73–76] for reviews). At large Reynolds number (small viscosity m) both exhibit (i) a dissipative regime at small scale where viscosity dominates, and (ii) an inertial range at larger scales with (multi)scaling and intermittency (where viscosity can formally be set to zero). Understanding how the two regimes connect is an outstanding problem in (decaying or stirred) turbulence (resp. Burgulence). Here the FRG exactly describes ‘‘decaying Burgers” with viscosity m ¼ T=2, where the renormalization scale plays the role of time. Although this connection is not new, it has not been pushed too far. Here we study in detail the mapping between Burgers and the FRG and find an exact relation between shocks and droplet size distributions. In particular we emphasize the general correspondence:

 R00 ð0Þ  v ðxÞ2 T 0000 R ð0Þ  mðrv ðxÞÞ2 ¼  2

ð6Þ ð7Þ

where v ðxÞ is the velocity field in Burgers. In the second line the r.h.s. has a finite limit as m ! 0 called the dissipative anomaly: also present in Navier–Stokes, in Burgers it is due to shocks. The (equivalent) finite limit of the l.h.s. implies the existence of a thermal boundary layer in the FRG. It is related to droplets if shocks are dilute. The issue of the construction of the inviscid limit m ! 0 in Burgers equation was recently addressed using distributions [77,78]. We discuss the relations to our work

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on the existence of an ambiguity free T L ! 0 critical limit in the FRG. Indeed, the above-mentioned matching problem in the FRG maps onto the question of how the inertial range in Burgers turbulence matches onto the dissipative scale. In particular, the celebrated Kolmogorov scaling in the inertial range:

1 S111 ð0; 0; uÞ  u $ ðv ðxÞ  v ð0ÞÞ3  x 2

ð8Þ

corresponds to the non-analytic third cumulant behaviour in the critical zero temperature theory, related via the FRG hierarchy to the cusp in R00 ðuÞ:

R000 ð0Þ  v ð0Þrv ð0Þ

ð9Þ

These relations hold for any d = 0 model, and can be fully explicited in the case of the Sinai (random field) landscape d ¼ 0; N ¼ 1. In that case many results about shock statistics are known, some since Burgers [79,80]. Here we derive a more general result for the joint distribution of the renormalized potentials and forces at several points. We can then relate it to the droplet solution to the exact FRG hierarchy mentioned above, and prove matching in that case. The dynamics of the shocks of also of high interest and given exactly by a ballistic aggregation model which can also be solved exactly [81]. This can in turn be interpreted as a Markov property in the merging of droplets which holds in that case as we detail in Section 4.6. These relations can be pushed quite far in d = 0 and we hope they will help progress in d > 0. While the d = 0 pinning problem maps onto decaying Burgers, one should mention that d ¼ 1 pinning problem (the so-called directed polymer) has connections to noisy Burgers. Although the mapping is quite different, there are some common issues. In particular, the shocks and inviscid limit construction are not expected to be too different. Attempts to solve a hierarchy similar to the FRG was done in [82]. Finally, we extend our analysis to higher d in Section 5. The corresponding ERG equation now relate second (and higher) cumulant functionals (R½u is the second cumulant functional and R(u) its local part). Again one identifies a ‘‘zero temperature” region u ¼ Oð1Þ and a thermal boundary layer e L , in the functionals. This hierarchy appears formidable, and a previous attempt to solve (TBL), u  T it in an expansion in powers of R was not successful [66]. Here, however, we obtain an exact solution to all orders within the TBL. It is based on (and inspired by) a simple droplet scenario. We then discuss the consistency and closure of the ERG hierarchy at T L ! 0 and address the ambiguity issue. We identify the assumptions underlying the  expansion in terms of continuity properties of the force functional R00 ½u. Under these assumptions we perform the one and two loop derivation, from first principle, of the anomalous terms in the FRG beta function.

2. Finite temperature FRG: preliminaries The aim of this Section is to examine the direct approach to the FRG at finite temperature, i.e. computing the beta function for the function R(u) in a loop expansion, and understand how it fails. It is an instructive exercise which allows to introduce a few basic facts, and to motivate the use of more systematic exact RG (ERG) methods. Since this method is more economical to use than ERG, it is worth checking thoroughly. 2.1. Finite temperature beta function Let us consider model (4) calling R0 ðuÞ the bare disorder appearing there, and consider perturbation theory in R0 ðuÞ. R0 ðuÞ plays the role of the coupling constant of the theory, as g 0 /4 in the /4 theory. The graphical rules have been described in [55]. The interaction is represented by a splitted (i.e. P 2 double) vertex with two free replica indices cd R0 ðuc  ud Þ=2T . Each free propagator line Tdab g q ; g q ¼ 1=ðq2 þ m2 Þ, gives a factor of T and identifies replica indices. Thus a graph with p connected components corrects a p-replica term. Each line drawn from a vertex results in a derivative of the vertex

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with respect to u. Vertices connected to a single replica component are called saturated, i.e. evaluated at u ¼ 0. One then computes the renormalized second cumulant R(u), defined from the effective action at zero momentum. Its detailed definition is given in Section 3. It plays the role of the renormalized coupling g/4 in the /4 theory.1 To compute it one writes all one particle irreducible graphs2 with two replica connected components (since it is a two replica term), in an expansion in a number of loops:

R ¼ R0 þ dð1Þ R0 þ dð2Þ R0 þ dð3Þ R0 þ   

ð10Þ

Including only the one loop diagrams one finds:

dð1Þ R0 ¼ TJ 1 R000 þ J 2

  1  00 2 R0  R000 ð0ÞR000 2

ð11Þ

(here and below the u-dependence is often implicit, and primes denote derivatives) while the two loop corrections read [55]:

dð2Þ R0 ¼

 000 2  00    1 2 0000 2 1  000 2 00 00 00 0000 R0  R000 ð0Þ IA T R0 J 1 þ T R0 J 3 þ T R0000 0 ðR0  R0 ð0ÞÞ  R0 R0 ð0Þ J 2 J 1 þ R0 2 2 2 1  00 þ R0000 R  R000 ð0Þ J 22 2 0 0

ð12Þ

with the integrals:

Jn ¼

Z k1 þþkn ¼0

g k1 :g kn ;

IA ¼

Z k1 þk2 þk3 ¼0

g k1 g k2 g 2k3

ð13Þ

The three loop corrections at T > 0 are given in Appendix A (at T = 0 they were computed in [57]). Note with p ¼ 0; . . . ; n. The loop expansion is thus a double that n-loop corrections are of the form T p Rnþ1p 0 expansion in T and R0 treated on the same footing (both are considered ‘‘small” of the same order). The standard method to extract the RG beta function is to first compute mom R at fixed bare disorder R0. This results in similar expressions as (10)–(12) where the integrals, which are m-dependent, are differentiated. Next one reexpresses R0 as a function of R, i.e. one inverts (10) order by order in R0, treating R0 and T to be of the same order, i.e. one writes (schematically) R0 ¼ R  TR000 þ  00 2  2 R0 þ    ¼ R  TR00 þ R00 þ   . Inserting in mom R one finally obtains the beta function:

mom RjR0 ¼ b½R; T

ð14Þ

which is also a polynomial expansion of the form T p Rnþ1p . Here STS guarantees that there are no corrections to the one replica part of the effective action, hence to T, which can thus be treated as a constant number. Note that in this calculation we have taken both R0 and R analytic at u ¼ 0, which is natural since we work at T > 0. If one sets T to zero in the result, one can check order by order in R that it yields a function b½R; T ¼ 0 whose coefficients are finite (no poles in  ¼ 4  d). This would be the zero temperature beta function for an analytic R(u). Its finiteness would mean that the theory is renormalizable at T = 0, i.e. a continuum limit exists. Unfortunately, it is not the correct zero temperature beta function, since the assumption of R(u) being analytic is not self-consistent at T = 0 (it develops a cusp in finite renormalization time). Dealing with the resulting ambiguities directly at T = 0 was studied in [55], but it is not our aim in this Section. We want to keep T > 0 and follow the RG flow: since temperature is irrelevant, we hope that it may lead us to the correct, ambiguity-free beta function. 1 As in massless /4 , one could set m ¼ 0 and define instead renormalization conditions at non-zero momentum. Closing the FRG then becomes quite difficult. We found it more convenient to add the mass m and define the renormalized disorder from the zero momentum limit. 2 That is such that the set of vertices cannot be disconnected by cutting a line.

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2.2. Toy model We will now continue in d = 0, for illustration, simplicity, and also because there we know the answer by other means (see below). Hence the model describes a particle in a N ¼ 1 dimensional random potential (i.e. on a line):

Z

1 2 2 m u þ VðuÞ 2 X X 1 1 Sðfua gÞ ¼ m2 u2a  2 Rðua  ub Þ 2T a 2T ab

ZV ¼

due

1T HV ðuÞ

HV ðuÞ ¼

;

ð15Þ ð16Þ

Although it is a simple integral, it is still a non-trivial model. It exhibits a glass phase when correlations of the potential grow with u, ðVðuÞ  Vð0ÞÞ2 ¼ 2ðRð0Þ  RðuÞÞ  juj2a with a > 0. The position u1 of the minimum of the energy HV ðuÞ fluctuates from sample to sample as u1  mf and HV ðu1 Þ  mh and one has a ¼ h=f and h ¼ 2ðf  1Þ. The case a ¼ 1=2 corresponds to a Brownian landscape and to the d = 0 limit of the random field universality class for the manifold ðf ¼ ð4  dÞ=3Þ. We stress that no bifurcation (such as a lower critical dimension) is expected to occur between d = 4 and d = 0 so the toy model should be continuously related to the RF manifold problem. In particular if the latter has a well defined finite beta function, it should have a good limit in d = 0. It is this limit that we are studying here (for arbitrary a). We now compute b½R; T, using that J n ¼ m2n and IA ¼ m8 in d = 0 and write the flow Eq. (14). To e obtain a fixed point it is more convenient to replace R in (14) by the rescaled disorder R:

RðuÞ ¼

1 4f e Rðumf Þ m 4

ð17Þ

One easily sees that this does not change the beta function, apart from an additional linear rescale ¼ 2Tmh : ing term, and replacing T with the rescaled temperature T h i e e 0 ðuÞ þ Te R e 00 ðuÞ þ 1 R e 00 ð0Þ R e 00 ðuÞ þ b e e e e 00 ðuÞ2  R mom RðuÞ þ fu R ¼ ð  4fÞ RðuÞ ð18Þ 2loop R; T ðuÞ þ    2 e flows to zero as Here of course  ¼ 4; f ¼ 2=ð2  aÞ; h ¼ 2a=ð2  aÞ and the rescaled temperature T m ! 0:

Te ¼ 2Tmh !m!0 0

ð19Þ

and the main problem is to understand the limit, as m ! 0 of (18). Before studying higher loops let us recall the physics of the one loop truncation (thus setting all bn>1loop to zero). 2.2.1. One loop e e  ðuÞ the non-analytic fixed point solution of the For a fixed u ¼ Oð1Þ, as m ! 0 one has RðuÞ !R 1loop naive zero temperature equation:3

e e 0 ðuÞ þ 0 ¼ ð  4fÞ RðuÞ þ fu R

1 e 00 2 e 00 e 00 R ðuÞ  R ð0Þ R ðuÞ 2

ð20Þ

which exhibits a non-analytic expansion in juj:

e 00 ðuÞ ¼ R e 00 ð0þ Þ þ R e 000 ð0þ Þjuj þ Oðu2 Þ; R

u ¼ Oð1Þ

ð21Þ

with the relation:

e 00 ð0þ Þ ¼ R e 000 ð0þ Þ2 ð  2fÞ R 1loop 1loop þ

ð22Þ

obtained from taking u ! 0 in the second derivative of (20). e However, for any non-zero m, RðuÞ is analytic and rounded in the thermal boundary layer (TBL) ree , with a scaling form: gion u  T 3

Where

e can be scaled away setting f ¼ f1

e1. e ¼ R and R

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

    1 e 00 e e RðuÞ ¼ Rð0Þ þ u2 R ð0Þ þ Te 3 r u= Te þ O Te 4 ; 2

  u ¼ O Te

ð23Þ

Consistency is checked by plugging this form in the one loop truncation of (18). Leading terms are of   e 2 (the term mom R ¼ O T e 3 and only the quadratic piece in the rescaling term contributes) and order T one finds that the TBL function r(x) satisfies

e 00 ð0Þx2 þ r 00 ðxÞ þ 1 ðr 00 ðxÞ  r00 ð0ÞÞ2 ¼ r00 ð0Þ ð  2fÞ R 2 yielding r00 ðxÞ  r00 ð0Þ ¼

ð24Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 00 ð0Þx2  1. Using the relation (22) one sees that the large x 1  ð  2fÞ R

behaviour of r00 ðxÞ exactly matches the cusp of the T = 0 fixed point. We also see that:

e 00 ð0Þ ¼ R e 00 ð0þ Þ R

ð25Þ

e alone) yields a simple consistent answer Thus this one loop truncation of the FRG equation (for R (which has been used in several studies at non-zero temperature [64]). Let us now see if this carries to higher loop. 2.2.2. Two loop To two loop we find:

h i   e 000 ðuÞ2  1 Te R e 0000 ð0Þ R e 000 ðuÞ2 R e Te ðuÞ ¼ 1 Te R e 00 ðuÞ  R e 00 ðuÞ þ 1 R e 00 ð0Þ b2loop R; 8 4 4

ð26Þ

and to check whether the one loop analysis carries through, we should study this equation both in the e . However, one must first ask whether the same so-called outer region u ¼ Oð1Þ and in the TBL u  T TBL scaling to holds to two loop. Let us make an important observation. One can show the exact result, displayed here both in rescaled and unrescaled version:

 mom R00 ð0Þ ¼ 2TRð4Þ ð0Þ e 00 ð0Þ ¼ ð  2fÞ R e 00 ð0Þ þ Te R e ð4Þ ð0Þ  mom R

ð27Þ ð28Þ

We emphasizes that it holds to all orders and yields a check on the expressions (18), (26), (35) and (42) – a rather non-trivial one as it is satisfied for each combination of R derivatives corresponding to each power of T. To explain its origin let us recall the physical content of the quantities which appear in (28). First one has:

hu2a i ¼ hui2 ¼

e 00 ð0Þ R00 ð0Þ  R ¼ 4 m 4m2f

ð29Þ

e 00 ð0Þ must flows to a well defined limit. Then a quantity expected to be continuous as T ! 0, hence R e . To leading order in temperature, the relation (28) strongly constrains TBL scaling, i.e. Rð4Þ ð0Þ  1= T (28) connects the sample to sample fluctuations u21 of the absolute minimum (i.e. a zero temperature quantity), to the sample to sample fluctuations of the thermal width vs ¼ hðu  huiÞ2 i of the Gibbs measure in a given sample. Indeed one has:

v2s  vs 2 ¼

e ð4Þ ð0Þ T 2 Rð4Þ ð0Þ Te 2 R ¼ 8 m 4m4f

ð30Þ

where vs is a ‘‘droplet quantity”, i.e. it is Oð1Þ in a typical sample, but can be large, vs  m2f , with e ¼ 2Tmh in the rare samples where almost degenerate minima exist. There is an infiprobability p  T nite set of relations such as (28), consequence of STS and ERG. We will recall and discuss them later. We can now come back to the two loop beta function. The first observation about the two loop contribution (26) is that if we evaluate it in the TBL region, e a catastrophy happens. Indeed, using TBL scaling (23) we find that all terms are as  T e. i.e. for u  T e 2 , thus the two loop terms are huge, But we should remember that the one loop terms in (18) are  T e compared to the one loop ones. One can check that no cancellation occurs. uniformly of order 1= T These would be unlikely anyway since the situation gets worse at three and higher loops, each new

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e . The direct loop expansion thus does not appear to describe the TBL and we see loop yields a factor 1= T the need for the ERG method. The ERG equations introduced in Ref. [65] and analyzed in [34,36] allow to understand what goes wrong in the above procedure, which is subtle. We explain it here schematically, for more details about the TBL structure see Ref. [36] and Section 4. The first ERG equation states that the sum of all loop corrections (two and higher) for R(u) is exactly S110 ð0; 0; uÞ, i.e. a second derivative of the (renormalized) third cumulant Sðu1 ; u2 ; u3 Þ of the disorder.  2 Next, S itself obeys an ERG equation of the type (schematically) oS ¼ TS00 þ T R00 þ R003 þ    where the    contains a feedback from the fourth cumulant (and the hierarchy goes on). To get the above loop expansion one truncates the ERG equations in powers of R (which is fine) and in powers of T.  2 For instance, the two loop term (26) can be obtained solving only oS ¼ T R00 þ R003 , and computing 00 the ensuing feedback of S in the R equation. To three loop the term TS (and others) will be added perturbatively to this equation, and so on. Unfortunately, this double expansion in T and R incorrect in the TBL. One can show that in the TBL equation for R the term oS is negligible compared to TS00 (very much like the term oR is also negligible in the TBL for R). Thus to obtain the correct result in the TBL one must  2 rather equate TS00 with the feeding terms T R00 þ R003 þ   , which is the opposite of what is done here! It then amounts to a non-trivial resummation of the above loop expansion, necessary to recover consistency with TBL scaling. One may wonder whether a resummation of all orders in T, to a fixed order in R may provide a meaningful result in the TBL. This question is examined in Appendix B of [94]. e ). At this stage we must renounce to use this direct loop expansion method in the TBL (i.e. for u  T The next question is then whether it can be used in the outer region u  Oð1Þ, i.e. whether there is a good limit as m ! 0 for fixed u. That question is more subtle. In fact to two loop it works! (at least for d = 0). Consider (26) for a u ¼ Oð1Þ and m ! 0. The first term in the r.h.s. flows to zero. The only problematic one is the second term. But we know that it has a finite limit since, from (28)

e 00 ð0Þ; Te Rð4Þ ð0Þ ! ð  2fÞ R ðalwaysÞ þ 2 000 e ¼ R ð0 Þ ; ðto one loopÞ

ð31Þ ð32Þ

Thus it seems that the two loop contribution flows to a well defined limit:

b2loop ðuÞ !

 1  e 000 2 e 000 þ 2  e 00 e 00 ð0Þ R ðuÞ  R ð0 Þ R ðuÞ  R 4

ð33Þ

where we have inserted a one loop relation into a two loop one. This is allowed, to this order, if the end result is an expansion in powers of R. This turns out to coincide with the result from the ‘‘correct” method (see Section 4.3). It obeys the requirement that the cusp remains linear, the constraint of ‘‘no supercusp”: expanding the r.h.s. in juj the term linear in juj cancels (i.e. R0 ð0þ Þ – 0). 2.2.3. Three loops and beyond Embolded by this success, we compute the three loop contribution:

 2  2   h i e 0000 ð0Þ þ 1 Te R e Te ðuÞ ¼ 1 Te 2 R e 0000 R e 0000  2 R e 0000  5 R e 0000 e 000 e 0000 ð0Þ þ 1 Te R00 R b3loop R; R 48 8 4 16 1 00 2  e 0000 2 3  e 000 4 1 00  e 000 2 e 0000 R R þ þ R R ðR Þ R þ 16 32 4

ð34Þ ð35Þ

where here and below we often use the shorthand notation:

e 00 ðuÞ  R e 00 ð0Þ R00 ðuÞ ¼ R

ð36Þ

We see that again it yields a well defined limit in the outer region u ¼ Oð1Þ. Let us define the limit:

  e 0000 ð0Þ ¼ r ð4Þ ð0Þ þ O Te Te R

ð37Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

  e we get: and keeping only terms with a finite limit as m ! 0 (discarding terms which are O T

    1 e 00 2 e 00 e 00 1  e 000 2  e 00 e 00  1 ð4Þ e 00 R R  R ð0Þ  r ð0Þ R  R ð0Þ R þ R 2 4 4 1  e 00 e 00 2  e 0000 2 3  e 000 4 1  e 00 e 00  e 000 2 e 0000 5 ð4Þ  e 000 2 R  R ð0Þ R R R  r ð0Þ R þ þ þ R  R ð0Þ R 16 32 4 32

e ¼ ð  4fÞ R e þ fu R e0 þ oR

ð38Þ

The question is now to fix the value of the number r ð4Þ ð0Þ. One requirement is the absence of supere 0 ð0þ Þ ¼ 0. This yields: cusp, i.e. mom R

e 000 ð0þ Þ2 R e 0000 ð0þ Þ þ OðR4 Þ e 000 ð0þ Þ2 þ 5 R r ð4Þ ð0Þ ¼ R 4

ð39Þ

e 00 ð0þ Þ (in the outer region) Although it looks reasonable, there is something puzzling. The flow of R reads, to two loop:

e 00 ð0þ Þ ¼ ð  2fÞ R e 00 ð0þ Þ þ R e 000 ð0þ Þ2 þ oR

5 e 000 þ 2 e 0000 þ 1 R ð0 Þ R ð0 Þ  rð4Þ ð0ÞR0000 ð0þ Þ 4 4

ð40Þ

which can be compared with the exact identity (28):

e 00 ð0Þ ¼ ð  2fÞ R e 00 ð0Þ þ r ð4Þ ð0Þ oR

ð41Þ

e 00 ð0Þ ¼ R e 00 ð0 Þ at two loop it yields Thus if we want to enforce (25), i.e. the continuity relation R e 000 ð0þ Þ2 þ R e 000 ð0þ Þ2 R e 0000 ð0þ Þ and is incompatible with the absence of supercusp at three loop. rð4Þ ð0Þ ¼ R Thus this procedure has produced a nice limit at three loop, but this limit turns out to be incorrect. The correct beta function is derived below in Section 4.3 and is incompatible with (38) for any value of rð4Þ ð0Þ. The reason why this procedure fails has to do with the existence of partial boundary layer forms, discussed in Section 4.3 and in Appendix B of [94]. However, we cannot exclude that a clever way may exist to resum consistently, equivalent to what is done with the ERG. Thus, to close this section on a challenge, we display the result for the four loop contribution to the beta function: þ

 2 h i e Te ðuÞ ¼ 1 Te 3 R e ð5Þ  1 Te 3 Rð6Þ ð0ÞR0000  1 Te 2 R e 0000 ð0Þ2 R e 0000 b4loop R; 384 192 64 1 e 2 e 0000  e 0000 2 1 e 2  e 0000 3 1 e 2 e 0000 e 000 e ð5Þ T R ð0Þ R T R T R ð0Þ R R  þ  32 64 48  2  2   5 e 2 e 000 e 0000 e ð5Þ 1 e 2 e 00 e 00 e 000 R e ð6Þ ð0Þ e ð5Þ  1 Te 2 R T R R R þ T R  R ð0Þ R þ 96 96 64 3 e e 0000 2  e 000 2 13 e e 0000  e 000 2 e 0000 33 e  e 000 2  e 0000 2 T R ð0Þ R T R ð0Þ R T R  R þ R  64 64 128         2 3 5 e e 0000 e 00  R e 00  R e 00 ð0Þ R e 0000 þ 1 Te R e 00 ð0Þ R e 0000 T R ð0Þ R  64 16 1 e  e 000 3 e ð5Þ 3 e  e 00 e 00  e 000 e 0000 e ð5Þ T R T R  R ð0Þ R R R R þ þ 16 16  2  4   2 1 e e 00 e 00 e ð5Þ þ 5 R e 0000 e 000 R þ R T R  R ð0Þ 64 16 9  e 00 e 00  e 000 2  e 0000 2 1  e 00 e 00 2  e 0000 3 þ þ R R R  R ð0Þ R R  R ð0Þ 16 16       3 2 1 e 00 e 00 e 00  R e 00 ð0Þ R e ð5Þ þ 3 R e 000 R e 0000 R e ð5Þ e 000 R R  R ð0Þ R þ 8 16 1  e 00 e 00 3  e ð5Þ 2 R  R ð0Þ R þ 96

ð42Þ

One notes that a new feature arises to four loop. All terms have a nice limit except for the combination:



 2 1 e 2  e 000 2 ~ ð6Þ 3 e e 0000 2  e 000 2 1  ð6Þ e 000 T R T R ð0Þ R r ð0Þ þ 3rð4Þ ð0Þ2 R R ð0Þ   64 64 64 Te

ð43Þ

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61

Thus for the well defined limit in the outer region to exist one must have:

  rð6Þ ð0Þ þ 3rð4Þ ð0Þ2 ¼ c Te þ O Te

ð44Þ

e 0000 ð0Þ. This condition happens to which also implies the cancellation of the anomalous term linear in R be correct for the one loop truncation for the TBL function rðxÞ given above in (24). However, we also know (from ERG see [36]) that the exact equation

rð6Þ ð0Þ þ 3rð4Þ ð0Þ2 þ s114 ð0; 0; 0Þ ¼ 0 ð3Þ

ð45Þ

involves some derivative of the third cumulant in the TBL, which has no good reason to vanish. Thus we have shown in this Section some of the issues and difficulties facing FRG at finite temperature, using d = 0 as a test. The direct loop expansion does show some features which are qualitatively correct, such as the TBL scaling and its matching with zero temperature solution. However, unless it can be extended in a clever way, it appears to fail beyond two loop (in d = 0) as a quantitative method. Of course we can be certain of that only if we know a correct method. Before we get to a more promising approach we need to understand better the physical meaning of the tools used in the FRG, which is the aim of the next Section. 3. Basic tools and functionals In this section, we analyze the physical information contained in the generating functionals in the replica formulation. We recall the definition of the renormalized disorder in term of the effective action. Then we show that this definition is equivalent to a much more physical one directly related to an observable. For pedagogical purpose we start by establishing the correspondence on the d = 0 model. Then it is extended to higher d and N. 3.1. Renormalized disorder in d = 0 3.1.1. Connected correlations There are two basic generating functions in the replica formulation. The first one is:

WðjÞ ¼ ln ZðjÞ ! Z Y X dua exp ja ua  SðuÞ ZðjÞ ¼ a

SðuÞ ¼

ð46Þ ð47Þ

a

1 X 2 2 1 X m ua  2 R0 ðua  ub Þ 2T a 2T ab

ð48Þ

where R0 ðuÞ is the second cumulant of the bare disorder. Here and below we often use for simplicity the same notation for a function of p replica variables, e.g. WðjÞ  Wðfja gÞ, and functions of a single real variable (i.e. HV ðuÞ below). W(j) is the generating function, via polynomial expansion, of the connected correlations:

WðjÞ ¼ Wð0Þ þ

1X 1 X Gab ja jb þ Gabcd ja jb jc jd þ    2 ab 4! abcd

ð49Þ

where Ga1 ...an ¼ hua1 . . . uan icS where c here means connected with respect to the replica measure S, e.g.:

T R00 ð0Þ dab  2 m m4 ¼ hua ub uc ud i  ðGab Gcd þ 2permÞ

Gab ¼ hua ub i ¼

ð50Þ

Gabcd

ð51Þ

(sixth order correlations are studied in Appendix B). The last equality in the first formula anticipates on the definition of R(u) given below. Let us examine the physical content of WðjÞ which, at least within its polynomial expansion, seems clear. There are a priori two distinct two-point correlation

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

hu2a i ¼ hu2 i and hua ub i ¼ hui2 , where in these type of formula we will assume distinct replicas, a – b. Similarly4 there are a priori five distinct four-point correlations, each related (for p ¼ 0) to a particular   combination of thermal and disorder averages in the usual way: u4a ¼ hu4 i, u3a ub ¼ hu3 ihui,  2 2  ua ub ¼ hu2 ihu2 i, u2a ub uc ¼ hu2 ihui2 and hua ub uc ud i ¼ hui4 . More generally one expects N n distinct P Q k 1 is the boson partition function. elements Ga1 ...an , where n N n zn ¼ 1 k¼1 ð1  z Þ In fact there is less. This is because of the STS symmetry, namely that Wðja þ ~jÞ ¼ Wðja Þþ P T ~ j a ja þ p 2mT 2 ~j2 for any (replica independent) ~j, as can be seen from shifting the integration in m2 (48). One easily sees that it implies (for arbitrary p):

X

Gab ¼

a

T ; m2

X

Gabcd ¼ 0



ð52Þ

a

To order un , these imply N n1 STS relations linking the N n variables. For n ¼ 4 there are five variables and three STS relations, which leaves two independent correlations. For n ¼ 6 one finds 11  7 = 4 independent correlations. More insight into the physical meaning of these relations will be given below. For now they are an infinite set of constraints on the authorized form for the correlations. 3.1.2. Effective action and renormalized disorder The second important generating function is the effective action defined as the Legendre transform. Define:

oja WðjÞ ¼ ua

CðuÞ ¼ ja ua  WðjÞ

ð53Þ

Note that here we are a priori not trying to minimize anything, we can construct this function e.g. perturbatively. As we will see below it is somehow a formal definition. It also admits a polynomial expansion:5

CðuÞ ¼ Cð0Þ þ

1X 1 X Cab ua ub þ Cabcd ua ub uc ud þ    2 ab 4! abcd

ð54Þ

with:

m2 R00 ð0Þ dab þ 2 T T 2 4 m ¼ Gabcd T

Cab ¼ G1 ab ¼

ð55Þ

Cabcd

ð56Þ

and so on (see Appendix B). Note that we somehow assume here that both C and W admit a polynomial expansion, i.e. that they are smooth. This should pose no problem at non-zero temperature.6 At low T this smoothness region may be limited to a boundary layer u  T around u ¼ 0 (see below). The more general approach presented below does not rely on smoothness and can be used directly to handle the T = 0 problem.7 Let us point out that the effective action CðuÞ is useful in the field theory and renormalization because it is the generating function of one particle irreducible graphs in perturbation theory (of the 4 Technically replica symmetry is assumed here. This can be enforced by restricting to a system with a finite number of degree of freedom, and use continuous distributions of disorder to avoid exact degeneracies. It turns out that the formalism and results of this Section can be extended to situations with spontaneous replica symmetry [63]. The reason is that replica symmetry is broken explicitly when defining the effective action. 5 Note that the constants C½0 ¼ W½0 ¼ pln Zð0Þ are proportional to p and to the quenched free energy and play little role in the following. 6 If necessary, assuming also a finite number of degree of freedom. 7 Note that in the replica formulation T appears in all formula, even if in the end some observables have a well defined T = 0 limit. It is not too surprising since an infinitesimal temperature is necessary to select the true minimum among all the extrema which satisfy the zero force condition. The definition of the T = 0 problem, i.e. minimization of the energy, is to take first the limit T ¼ 0þ at fixed size (finite number of degree of freedom) then take the thermodynamic limit.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

63

interaction, here R0 ðuÞ). It is often used to define renormalized vertices and renormalization conditions. Physically, CðuÞ contains all the coarse grained information. Since it sums all fluctuations (all loops) the connected correlation functions are then obtained as the sum of all connected tree graphs constructed from the vertices of CðuÞ. ~Þ ¼ The STS symmetry also constrains the form of CðuÞ. From (53) above one finds Cðua þ u P 2 ~ a ua þ p m2T2 u ~ 2 and on the polynomial expansion it again implies Cðua Þ þ mT u

X

Cab ¼

a

m2 ; T

X

Cabcd ¼ 0



ð57Þ

a

again valid for any p. The discussion of the number of independent components within the polynomial expansion is thus identical to the one above. More global constraints can also be deduced from STS. If one assumes that CðuÞ can be expanded in number of replica sums, then the STS implies the form:

CðuÞ ¼ C0 þ

m2 X 2 1 X 1 X 1 X u  Rðua  ub Þ  Sðua ; ub ; uc Þ  Qðua ; ub ; uc ; ud Þ þ    3 2T a a 2T 2 ab 3!T abc 4!T 4 abcd ð58Þ

where C0 is a constant (see below) and the functions R, S, Q and so on can be thought of ‘‘renormalized disorder cumulants”, second, third and fourth, respectively. More generally they will be denoted SðnÞ with Sð3Þ ¼ S; Sð4Þ ¼ Q ; . . .. The nth cumulant then correspond to the n replica terms, and always come with the factors T n . Since they are defined from CðuÞ we call them C-cumulants. The STS imply that they satisfy Sðu1 þ v ; u2 þ v ; u3 þ v Þ ¼ Sðu1 ; u2 ; u3 Þ as if they were true cumulants of some ‘‘renormalized” statistically invariant random potential.8 Indeed since these C-cumulants are usually the output of the FRG, it would be quite useful to relate them to observables. In particular one wants to know to which physical quantity the derivatives Rð2nÞ ð0Þ correspond to. It should be possible to answer in principle since, as mentioned above, the correlations can be obtained as tree graphs from C vertices. The Legendre transform being involutive one can also express each C vertex as a sum of tree graphs constructed from W(j), i.e. connected correlations. One can do it systematically in the polynomial expansion. To lowest order, u2 , the meaning of R00 ð0Þ is simple and transparent from (55, 50). To order u4 , it is also clear from (56, 51), but needs more calculation as one also needs to expand (58) to order u4 :

Cabcd ¼ 

1 T4

F4 

1 T2

Rð4Þ ð0Þððdab dcd þ 2permÞ  ðdabc þ 3permÞÞ

ð59Þ

thus it contains not only Rð4Þ ð0Þ but also the fourth cumulant F 4 ¼ Q 1111 ð0; 0; 0; 0Þ (this notation means one derivative with respect to each argument, see below). All (connected) four points correlations can be expressed from only these two quantities, one finds:

 T 2 F4 ¼ u3a ub c ¼ 8 Rð4Þ ð0Þ þ 8 m m  2 2 T2 F4 ua ub c ¼ 8 Rð4Þ ð0Þ þ 8 m m  2 F4 ua ub uc c ¼ hua ub uc ud ic ¼ 8 m 

u4a



c

ð60Þ

Thus Rð4Þ ð0Þ corresponds to various combinations (one of them was given in (30) above) which are all thermal correlations. As can be seen from (60) the zero temperature information (i.e. u41 where u1 here denotes the position of the absolute minimum, see below) is strictly contained in the fourth cumulant F 4 (it can be interpreted as describing a renormalized random force in a non-gaussian Larkin model). One can go on and the order u6 is given in Appendix B. It contains Rð6Þ ð0Þ together with three other derivatives of higher cumulants. This is the approach that was studied in Ref. [36] and is summarized here. Unfortunately, this polynomial route does not lead too far because the complexity increases very 8

It is not clear whether there are, if yes it would be interesting to find its probability measure.

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

fast with the order. We now turn to a more powerful approach. It will give us not only the Rð2pÞ ð0Þ which contain information about thermal excitations, but also the zero temperature part R(u). Before doing so let us note that the formulae (48), (49), 52,53,54 and (58) as well as the STS transformations for W and C given in the text all hold for any number of replica p. Similarly for the formulae (61), (63) and (64) of the next section. The resulting definitions for the cumulant functions R, S, etc. are such that they are independent of p, as will be obvious from the next section. By contrast the explicit forms in (55), (59) and (60) and in the first formula in (51) have been given for p ¼ 0.9 3.1.3. Back to the W(j) functional We go back to W(j) and ask what is the physical information contained there beyond the polynomial expansion. One has:

eWðjÞ ¼ ZðjÞ ¼

YZ

dua eja ua 

a

HV ðuÞ ¼ R

HV ðua Þ T

¼

Y

heja ua iHV Z pV

ð61Þ

a

1 2 2 m u þ VðuÞ 2 HV ðuÞ=T

ð62Þ p

where Z V ¼ due and at p ¼ 0 the Zð0Þ term in the first formula can be set to unity. At low temperature, expansion at small j at fixed T contains information about the minimum of HV ðuÞ and thermal flips between quasi-degenerate minima whenever they exist. There is much more information in W(j), hence in CðuÞ. Indeed if one instead rescale j ¼ J=T the particle sees a fixed force as T ! 0 and hence the position of the minimum is shifted. Thus W(j) and CðuÞ also contain, in an essential way10 information about correlations of minima shifted by a force. We will thus define j ¼ m2 v =T so that v has the same dimension as u. To economize notation we will write Wðv Þ  WðjÞ. The important observation is that the STS constraints are exactly the same on W(v) and CðuÞ, i.e. (52) and (57) have an identical form, thus if W(v) has an expansion in replica sums the STS implies the form:

m2 X 2 1 Xb 1 Xb va þ 2 Rðv a  v b Þ þ Sðv a ; v b ; v c Þ 2T a 2T ab 3!T 3 abc 1 Xb þ Q ðv a ; v b ; v c ; v d Þ þ    4!T 4 abcd

Wðv Þ ¼ W 0 þ

ð63Þ

where b Sðv a þ w; v b þ w; v c þ wÞ ¼ b Sðv a ; v b ; v c Þ, etc. and W 0 a constant (see below). The general term b ; . . .. The relation between C(u) S ð3Þ ¼ b S; b S ð4Þ ¼ Q in the above expansion will be denoted b S ðnÞ with b and W(v) is now very symmetric:

CðuÞ þ Wðv Þ ¼

m2 X v a ua T a

ð64Þ

b and so on, and the C-cumulants, R, S, b b and there is a duality between the W-cumulants, denoted R; S; Q Q, etc. b b It is easy to find which random potential has R; S, as its cumulants and corresponds to W(v). For each realization of the random potential V(u) one defines the (W-)renormalized potential b ðV; v Þ in a given sample as: b ðv Þ  V V



Z

1b 1 1 2 2 ðv Þ ¼ du exp  exp  V m u þ Vðu þ v Þ T T 2

Z 1 1 2 ¼ du exp  m ðu  v Þ2 þ VðuÞ T 2

ð65Þ

  2 1 For arbitrary p one has Cab ¼ Adab þ B with A ¼ mT þ pR00 ð0Þ and B ¼ R00 ð0Þ, and Gab ¼ 1A dab þ 1p AþpB  A1 . A too naive p ¼ 0, T = 0 limit seems to fail: keeping only the contribution of the absolute minimum u1 , and its distribution P P E .D P E D R eu1 a ja , independent of a. Pðu1 Þ, would yield ZðjÞ ¼ du1 Pðu1 Þeu1 a ja . The LT condition then yield ua ¼ u ¼ u1 eu1 a ja P P   P P a a 2 n ^ 2 Evaluating C½u ¼ u formally as the LT of W j ¼ a ja gives C½u ¼ u ¼ u =ð2u Þ þ nP3 cn u where cn are some combinations of 9

10

connected correlations. However, C should contain 1=T n factors, and there are none to be found here. Presumably this Legendre transform is ill-defined as hinted by the fact that it cannot be evaluated away from the point ua ¼ u.

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

then one sees that:

eWðv Þ ¼ Zðj ¼ m2 v =TÞ ¼

Z Y

1m2 T

dua e

P a

v a ua 1T ð12m2 u2a þVðua ÞÞ

a

¼

Z Y

e

1 m2 2 a 2T

v

Z

1T ð12m2 ðua v a Þ2 þVðua ÞÞ

dua e

a

¼

Y

1 m2 2T

e

P a

v 2a 1T b V ðv a Þ

ð66Þ

a

Averaging over disorder (i.e. over V) reproduces the above expansion (63) in terms of cumulants, i.e. connected moments, provided: c

b ðv 1 Þ V b ðv 2 Þ ¼ Rð b v 1  v 2Þ V

ð67Þ

c

b ðv 1 Þ V b ðv 2 Þ:: V b ðv n Þ ¼ ð1Þn b V S ðnÞ ðv 1 ; . . . ; v n Þ

ð68Þ

b ðv Þ  V b ðV; v Þ induced by the Of course the overline here denote averaging over the measure on the V bare measure (of cumulants R0, etc.) on the VðuÞ. Note that the formula (66) can be written for any b b number of replica and together with (65) shows quite clearly [83] that the functions R; S in (63) and, through Legendre transforms R, S, etc. are independent of the number of replica p. Finally, note that b ðv Þ ¼ V b ð0Þ :¼ F V ¼ Tln Z V ¼ TW 0 =p independent of v from STS (i.e. translational invariance of the V measure on V) and equal to the averaged free energy. More generally, since Wðua ¼ 0Þ ¼ c 2 3 b Sð0; 0; 0Þ þ    the cumulants of the free energy are given by F n ¼ ðÞn b S ðnÞ W 0 þ p 2 Rð0Þ þ p3b 2T

V

6T

ð0; . . . ; 0Þ. Anticipating a bit,11 Eq. (64) implies that Cð0Þ ¼ Wð0Þ hence C0 ¼ pln Z V , and also c

F nV ¼ ðÞn SðnÞ ð0; . . . ; 0Þ. Thus the information contained in W(v) is exactly the statistics of the W-renormalized random pob ðv Þ. We can now perform explicitly the Legendre transform (64), i.e. relate (63) and (58). This tential V is done in Appendix C, and we only quote the result. The most interesting property that we find is that the second C-cumulant is the same as the second W-cumulant, i.e. one has:

b vÞ Rðv Þ ¼ Rð

ð69Þ

the two functions are the same! Hence from now on we will use the same symbol, i.e. note R(v) both. This is remarquable, since, as we will see these two functions obey quite different RG equations. The difference of course appears at the level of third and higher cumulants. One finds that:

 1  0 R ðuab ÞR0 ðuac Þ þ R0 ðuba ÞR0 ðubc Þ þ R0 ðuca ÞR0 ðucb Þ Q ðuabcd Þ m2

00  0 0 0 0 b ðuabcd Þ þ 6 sym ¼Q abcd R ðuab ÞðR ðuac Þ  R ðubc ÞÞðR ðuad Þ  R ðubd ÞÞ m4 h i 12  2 symabcd b S 100 ðuabc ÞR0 ðuad Þ m

Sðua ; ub ; uc Þ ¼ b Sðua ; ub ; uc Þ 

ð70Þ

where symabcd is 1/4! times the sum of all permutations of abcd and here and below:

uab :¼ ua  ub

ð71Þ

uabc :¼ ua ; ub ; uc

ð72Þ

and so on. We recall the notations used in this paper everywhere for derivatives: p Snmp ðuabc Þ ¼ onua om ub ouc Sðuabc Þ

ð73Þ

and so on. A simple graphical interpretation allows to recover from tree diagrams the combinatoric S ðnÞ . factors in the SðnÞ cumulants from the b This solves, in d = 0, the question of finding an operational way to compute R(u) and higher cumulants and relate it to observable. It is thus: 11 Using the Legendre Transform presented in Appendix C, it relies on v a ðu ¼ 0Þ ¼ 0 which is a consequence of R0 ð0Þ ¼ 0; S0 ð0; 0; 0Þ ¼ 0; . . . always true at non-zero T (at T = 0 it is equivalent to the absence of super-cusp).

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P. Le Doussal / Annals of Physics 325 (2010) 49–150 c

b ðuÞ V b ðu0 Þ ¼ Rðu  u0 Þ V

ð74Þ

b ðuÞ, defined from (65), has a number of nice properties. At where the ‘‘renormalized random potential” V T = 0 it is, up to a constant piece, the Legendre transform (with a true minimization) of VðuÞ, namely:

b ðv Þ ¼ min HV;v ðuÞ; V u

HV;v ðuÞ ¼

1 2 m ðu  v Þ2 þ VðuÞ 2

ð75Þ

The role of the mass is important as one minimizes in presence of an harmonic well centered on point v. Ve ðv Þ is the resulting free energy. As v is moved the absolute minimum will also move and it will result in a non-trivial renormalized energy landscape. At T = 0, in situations discussed below, these moves become discrete jumps. At any temperature, the derivative (minus the renormalized force) satisfies:

b 0 ðv Þ ¼ m2 ðv  hui Þ V v Z 2 1 1 1 2 du ueT ð2m ðuv Þ þVðuÞÞ huiv  huiHV;v ¼ Z V;v

ð76Þ ð77Þ

in terms of the thermal averaged position in a given sample. At T = 0 it exhibits jump discontinuities at some discrete set of values v ¼ v s , so-called shocks (the above formula (76) still holds then for left and right derivatives at the shock positions v s ). These result in a non-analyticity in the force correlator R00 ðuÞ computed from (74). One expects the switch between minima to be abrupt at T = 0 and smooth at finite T as the Gibbs measure gradually shifts from one minimum to the other as v increases. This results in a thermal boundary layer in R(u). These issues are studied quantitatively in the following Sections. The random potential satisfies a Kardar–Parisi–Zhang (KPZ) [91] type of FRG equation:

b¼ mom V

T 2b 1  b 2 ov V  2 ov V 2 m m

ð78Þ

b ¼ V for m ¼ 1. Its derivative yields the decaying Burgers equation (in dimenwith initial condition V sion N). It is indeed well known that shock singularities appear in this equation. There is no noise term and the equation describes the merging and coarsening of shocks. Their physical connection to the FRG hierarchy for cumulants is studied in the coming Sections. Note that the reverse problem, often b ðv Þ at fixed m is fixed, satisfies the same known as Polchinski RG, i.e. how to evolve VðuÞ so that the V (reversed) equation. Note that a version of the present renormalized potential was proposed in d = 0 in Ref. [46], as a ‘‘toy RG” for the random manifold problem. The interpretation was different, in terms of elimination of fast modes: the role of m, which is here the infrared cutoff, was played by K, the UV cutoff. Iterative minimization was discussed and somewhat qualitative arguments were given as to relevance to the b ðv Þ has a precise content: it yields the second cumulant defined FRG. Here we have established that V from the replicated effective action. We have also shown how to obtain the higher cumulants. In Appendix we check explicitly, on the formulas established previously that the (sixth) derivatives of R(u) at u ¼ 0 agree with formulas obtained by the polynomial method, explained above. The main advantage of the present analysis is that it is now easily extended to any d (and N). 3.2. Renormalized disorder functionals, any d 3.2.1. Functionals and their relations To generalize the previous section we consider the model, defined on a discrete d-dimensional lattice:

X 1 X 1 g ux uy þ Vðux ; xÞ 2 xy xy x  1 X 1 a a 1 X  a S½u ¼ g u u  R0 ux  ubx 2T xya xy x y 2T 2 xab

HV ½u ¼

ð79Þ ð80Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Respectively, the energy functional in a given sample and the replicated action functional. We consider for now g 1 xy an arbitrary matrix (d = 0 is recovered suppressing space indices and setting 2 g 1 xy ¼ m ). The disorder is chosen to be uncorrelated from site to site on the lattice, i.e. c Vðu; xÞVðu0 ; x0 Þ ¼ dxx0 R0 ðuÞ, so that (80) enjoys the exact STS property at the level of the lattice model. R d P A continuum model can also be considered x ! d x and here we denote uax  ua ðxÞ. The W½j functional is defined as:

W½j ¼ ln

Z Y

a

Pa

jx uax S½u

dux e xa

ð81Þ

ax

 c The connected correlations Gax11 :x...an p ¼ uax11 . . . uaxnn S are generated upon polynomial multilocal expansion:

W½j ¼ W½0 þ

1 X ab a b 1 X abcd a b c d G j j þ G j j j j þ  2 xyab xy x y 4! xyztabcd xyzt x y z t a

It satisfies the STS identity W½fjx þ jðxÞg ¼ W

ð82Þ

 a  P a P P jx þ T xy g xy jðyÞ a jx þ p T2 xy g xy jðxÞjðyÞ which is ob-

tained performing the shift uax ! uax þ /x with TjðxÞ ¼ g 1 xy /y . This implies:

X

X

Gab xy ¼ Tg xy ;

a

Gabcd xyzt ¼ 0;



ð83Þ

a

bacd In the latter we have used the simultaneous permutation symmetry Gabcd xyzt ¼ Gyxzt , etc. The proper change of variable is now:

T

X

a

g xy jy ¼ v ax

ð84Þ

y

so that

v has the same dimension as u. Then:

W½v  ¼ W½0 þ

1 X ab a b 1 X abcd a b c d G v v þ G v v v v þ  2 xy xy x y 4! xyzt xyzt x y z t

ð85Þ

where: 2 Gab xy ¼ T

X

1 ab g 1 xx0 g yy0 Gx0 y0

ð86Þ

x0 y0

X

4 Gabcd xyzt ¼ T

1 1 1 abcd g 1 xx0 g yy0 g zz0 g tt 0 Gx0 y0 z0 t 0

ð87Þ

x0 y0 z0 t 0

and so on. Thus one still has:

X

1 1 Gab g xy ; xy ¼ T

a

X

Gabcd xyzt ¼ 0

ð88Þ

a

which is exactly the symmetry obeyed by the polynomial expansion of C. Because of STS one thus has:

W½v  ¼ W 0 þ

1 X 1 a a 1 X b ab 1 Xb a b c R½v  þ g v v þ S½v ; v ; v  þ    2T axy xy x y 2T 2 ab 3!T 3 abc

ð89Þ

(for any p, W 0 ¼ pF V =T being a constant proportional to the averaged free energy-see discussion in preb v ab  is a two replica functional which only depends on the field v ab  v a  v b with vious section) where R½ x

x

x

a, b given. It can itself be decomposed into a local part, the usual function Rðv ax  v bx Þ, bilocal and higher. This will be discussed in Section 5. Similarly the third cumulant functional b S½v a ; v b ; v c  only depends on with a, b, c given. It satisfies statistical translational invariance the fields v a;b;c x b S½fv ax þ /x ; v bx þ /x ; v cx þ /x g ¼ b S½v a ; v b ; v c . This form is dual to the one for its Legendre transform CðuÞ:

1 X 1 a a 1 X 1 X a b c g u u  R½uab   S½u ; u ; u  þ    2T axy xy x y 2T 2 ab 3!T 3 abc X a 1 X a 1 a C½u ¼ uax jx  W½j ¼ u g v  W½v  T axy x xy y ax

C½u ¼ C0 þ

ð90Þ ð91Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

The calculation of the Legendre transform is performed in Appendix C. One finds the functional identities:

b v ab  ¼ R½v ab  R½

ð92Þ

S½ua ; ub ; uc   3symabc S½ua ; ub ; uc  ¼ b

X xy

ab

g xy

ac

dR½u  dR½u  duax duay

ð93Þ

which generalize the d = 0 result. One also finds C0 ¼ W 0 ¼ pF V =T. b ½fv x g functional in a given We are now in position to define the renormalized random potential V b b S and so on. Some equivalent definitions are: sample whose connected cumulants will reproduce R;

Z Y

X 1 1 X 1 g xy ux uy þ Vðux þ v x ; xÞ T 2 xy x x X 1 X 1 HV;v ½u ¼ g ðux  v x Þðuy  v y Þ þ Vðux ; xÞ 2 xy xy x

1b eT V ½fv x g ¼

dux exp 

!!

¼

Z Y

1

dux eT HV;v ½u

ð94Þ

x

ð95Þ

Thus one has:

b ½fv x g V b V



v 0x

c

¼R



v x  v 0x



ð96Þ P

note that this is now true as a functional. The simpler form x Vðux ; xÞ which defines the bare disorder is of course not preserved under coarse graining as higher multilocal components develop. Thus in general the functional R½fv x g is not local. However, one can still define a function Rðv Þ from the local part of the renormalized random potential functional. This is sketched below and detailed in Section 5. There are again some nice properties. One can also define the renormalized force functional, which satisfies:

F x ½v  ¼

b ½fv z g dV ¼ g 1 xy ðv y  huy iHV;v Þ dv x

ð97Þ

The renormalized potential in a given sample obeys a RG functional equation as g xy is varied (its variation is noted og xy ), also called Polchinski equation in the ERG context:

b g ½v  ¼ oZ

" # b g ½v  T d2 Z ; tr og 2 dv dv

b g ½v  ¼ e1T bV ½fv x g Z

ð98Þ

b ½v  satisfies a functional KPZ type [91] equation: hence V

b ½v  ¼ oV

" !# b ½v  d V b ½v  d V b ½v  1 d2 V  tr og T 2 dv dv dv dv

ð99Þ

b ½fv x g ¼ V½fv x g. Its (functional) derivThe ‘‘initial condition” at g ¼ 0 (analogous to m ¼ 1) is again V ative is a functional decaying Burgers equation. One can then also expect ‘‘functional shocks” as a generalization of the standard Burgers equation. Finally, we note that all formula of the present Section are straightforwardly extended to the case of a N-component vector u  ui ; i ¼ 1; . . . ; N for arbitrary N. 3.2.2. How to measure R(u) The present study has opened the way to measuring the FRG fixed point function(al) in numerical simulations (or in experiments). The simplest procedure is to confine the manifold in a harmonic well centered on a given configuration v x . The simplest model is then g 1 ðqÞ ¼ q2 þ m2 in Fourier space. As v x is varied at T = 0 shocks will occur as the manifold switches from one ground state to another. A difference with d = 0 is that these switches occur now on various scales (in x). Very little is known at present on these functional shocks and their statistics. They are reminiscent of avalanches in the driven dynamics but they truly are ‘‘static” shocks where the manifold is at equilibrium (or in the mine ½v x ¼ v  is the imum energy position) for each v. The simplest choice is a uniform v x ¼ v . Then V

P. Le Doussal / Annals of Physics 325 (2010) 49–150

69

ground state (free)-energy, which is proportional to the volume Ld . As shown in Section 5 this allows to measure the local part Rðv Þ. One has simply: c

b ½fv x ¼ v g V b ½fv x ¼ v 0 g ¼ Rðv  v 0 ÞLd V

ð100Þ

which holds at any T. It can also be obtained from the force. At T = 0 is particularly simple. Denote P  ¼ Ld x ux ðv Þ the center of mass poux ðv Þ the minimum energy configuration for a fixed v x ¼ v and u sition. Then:

 ðv ÞÞðv 0  u ðv 0 ÞÞ ¼ Dðv  v 0 ÞLd m4 ðv  u

ð101Þ

00

where from (97) Dðv Þ ¼ R ðv Þ is the local part of the correlator of the pinning force (one has v  u ðv Þ ¼ 0). One expects the manifold to behave as roughly ðL=Lm Þd independent pieces, with Lm  m1 , hence the inverse volume factor Ld in (101) simply expresses the central limit theorem. Hence Dðv Þ should have a limit as L ! 1 proportional to the fluctuations of the force density in a correlation volume Ldm . Since force density scale as m2 u  m2f this in turns suggests that, as a function of e should have a fixed point form as m ! 0. This is indeed what is e ðumf Þ where D m, DðuÞ  mdþ42f D predicted by the FRG (see e.g. Section 2). The above formula (101) is exact, however, for any m and allows in principle to measure in numerics all earlier stages of the RG, e.g. (i) the Larkin mass where DðuÞ suddenly acquires a cusp12 and the Larkin regime for m > mc where f ¼ ð4  dÞ=2, (ii) the convergence to the fixed point, and (iii) crossovers between distinct universality classes. At non-zero temperature one simply replaces ux ðv Þ by the thermal average. The effect of temperature is discussed in the next Section where a ‘‘droplet” solution is obtained. The predicted rounding of the cusp can also be measured in numerics. Finally, it is also possible to measure the non-local part of R½v  using a non-uniform v x . These formula can be generalized to a number of situations. First they extend straightforwardly to any N. Next, they can be modified in the case of a model which does not possess exact STS symmetry, as is often the case in numerical simulations. This extension is discussed in Appendix D. They also allow to study (and define properly) the FRG for chaos discussed in [84]. Considering two copies indexed by 1, 2 seeing slightly different disorders, e.g. V  dW with small d, V and W being two statistically independent random potentials, one can study the cross-correlation: c

b 1 ½v x ¼ v  V b 2 ½v x ¼ v 0  ¼ R12 ðv  v 0 ÞLd V

ð102Þ

The mutual correlation of the two ground states in each copy:

 2 ðv 0 Þ  v 0 Þ ¼ Ld D12 ðv  v 0 Þ  1 ðv Þ  v Þðu m4 ðu

ð103Þ

defines the renormalized pinning force cross-correlator D12 ðuÞ ¼ R0012 ðuÞ. At zero temperature these functions measure the correlations between the shocks in the two copies. Having defined the important functional and observables we now turn to the derivation and analysis of the FRG equations, first in zero dimension. 4. Zero dimension, exact FRG and decaying Burgers turbulence 4.1. ERG for moments and cumulants Here we write the exact RG equations satisfied by the ‘‘renormalized disorder” in d = 0. As was shown in the previous section there are three interesting set of functions. First the W-moments, i.e. b ðv Þ, i.e. the simple averages, and second the W-cumuthe moments of the renormalized potential V b ðv Þ, denoted, respectively: lants, i.e. the connected correlations of V

b ðv 1 Þ: V b ðv n Þ ¼ ð1Þn SðnÞ ðv 1 ; . . . ; v n Þ V c

n bðnÞ

b ðv 1 Þ: V b ðv n Þ ¼ ð1Þ S ðv 1 ; . . . ; v n Þ V

12

A finite mc < þ1 should exist for smooth and bounded bare disorder.

ð104Þ ð105Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Finally, the C-cumulants denote SðnÞ ðv 1 ; . . . ; v n Þ. We now give the ERG equation for each set. We will use the notation v 1;2;...;n ¼ v 1 ; . . . ; v n and often denote [  ] the symmetrization with respect to the all variables (i.e. sum over all permutation divided by number of permutations). There are some relations between the lowest ones [86]:

S¼b S

b ¼ R; R¼R

ð106Þ

b ðv 1234 Þ þ 3sym Q ðv 1234 Þ ¼ Q 1234 Rðv 12 ÞRðv 34 Þ

ð107Þ

together with (70). 4.1.1. ERG for the W-moments Starting from the flow equation:

b¼ mom V

T 2b 1  b 2 ov V  2 ov V 2 m m

ð108Þ

b ðv n Þ one obtains: b ðv 1 Þ: V Computing e.g. mom V

mom SðnÞ ðv 1;2;...;n Þ ¼

i i nT h ðnÞ n h ðnþ1Þ S20:0 ðv 1;2;...;n Þ þ 2 S110:0 ðv 1;1;2...n Þ 2 m m

ð109Þ

The lowest ones are:

2T 00 2 R ðv Þ þ 2 S110 ð0; 0; v Þ m2 m i i 3T h 3 h  mom Sðv abc Þ ¼ 2 S200 ðv abc Þ þ 2 Q 1100 ðv aabc Þ m m i  4T h 4  mom Q ðv abcd Þ ¼ 2 Q 2000 ðv abcd Þ þ 2 P11000 ðv aabcd Þ m m  mom Rðv Þ ¼

ð110Þ ð111Þ ð112Þ

where P is the fifth moment. As will be studied below, this set of equation generate the loop expansion. More precisely, inserting only the R2 part of (107) into (111), solving for S and reporting into (110) yields the order R2 part of the beta function, a priori to all orders in T. To get the two loop contribution one needs to go up to the equation for the fifth cumulant (not written) and so on, so while these equations are very simple (they form a linear system) they are not very economical. Note the b ð0Þ ¼ F V : ERG equation for the flow of the averaged free energy V

mom F V ¼

1 00 R ð0Þ ¼ m2 hui2 m2

ð113Þ

one of the many ERG identities derived in [36] (see Section III.D there). It implies the universal behaviour for the averaged free energy F V  ðA=hÞmh where A ¼ limm!0 m2f hui2 valid at any temperature.13 4.1.2. ERG for the W-cumulants Another way to proceed, equivalent but a bit faster, is to study the cumulants. One starts from the general (Polchinski type) functional ERG equation [85]:

1 d2 W½j dW½j dW½j oW½j ¼  Trog 1 þ 2T djdj dj dj

! ð114Þ

We specialize to d = 0, g ¼ 1=m2 ; j ¼ m2 v =T. One defines:

Wðv Þ ¼

m2 X 2 b v þ Uðv Þ 2T a a

ð115Þ

b with Wð0Þ ¼ Uð0Þ. The ERG for W can then be rewritten: P In dimension d > 0 this generalizes as mom F V ¼ m2 x hux i2  ALd mdh . For short range disorder since h 6 d=2 this always A mdh , but for sufficiently long range disorder such that gives universal corrections to the free energy density f ¼ F V =Ld ¼ f0 þ dh h > d, then the dominant contribution is universal, very much as in d = 0. 13

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

b b X o2 U T X oU b ¼ T mom U þ 2 m a ov a ov a m2 a ov a

!2 ð116Þ

where to obtain the above one should remember that Wðv Þ ¼ Wðj ¼ m2 v =TÞ when differentiating with respect to m. Expanding in replica sums, along (63), this yield the general equation for the cumulants:

i i nT h ðnÞ n h ðnþ1Þ 1 mom b S ðnÞ ðv 1;2;...;n Þ ¼ 2 b S 20:0 ðv 1;2;...;n Þ þ 2 b S 110:0 ðv 1;1;2...n Þ þ 2 m m m E½ðnþ1Þ=2 h i X ðkÞ ðnþ1kÞ ck;n b S 10:0 ðv 1;kþ1;...;n Þ S 10:0 ðv 1;2;...;k Þb

ð117Þ

k¼2 k

k

n! with ck;n ¼ 2 ðk1Þ!ðnkÞ! ¼ 2kC n except for n odd and k ¼ ðn þ 1Þ=2 then ck;n ¼ kC n . The lowest ones are:

2T 00 2 R ðv Þ þ 2 b S 110 ð0; 0; v Þ m2 m h i i 3T 3 hb 6 0  mom b ½R ðv ab ÞR0 ðuac Þ Sðv abc Þ ¼ 2 b S 200 ðv abc Þ þ 2 Q 1100 ðv aabc Þ þ m m m2 h i h i h i b ðv abcd Þ ¼ 4T Q b 2000 ðv abcd Þ þ 4 P b 11000 ðv aabcd Þ þ 24 b  mom Q S 100 ðv abc ÞR0 ðv ad Þ m2 m2 m2  mom Rðv Þ ¼

ð118Þ ð119Þ ð120Þ

These equations are equivalent [86] to the above (though derived very differently) and allow to recover the loop expansion a bit faster recursively. 4.1.3. ERG for the C-cumulants We now turn to the effective action, and define:

CðuÞ ¼

m2 X 2 u  UðuÞ 2T a a

ð121Þ

The general ERG for CðuÞ instead obeys [85]:

!1 T o2 UðuÞ mom UðuÞ ¼ Tr d  2 m ouou

ð122Þ

The expansion in replica sums is more tedious, and a general formula to all orders (for the rescaled cumulants) was derived in Appendix A of [36]. Let us recall the result for the second and third cumulants (here given in the present, i.e. unrescaled, notations):

mom RðuÞ ¼

 2T 00 2 2  R ðuÞ þ 2 S110 ð0; 0; uÞ þ 4 R00 ðuÞ2  2R00 ð0ÞR00 ðuÞ 2 m m m

mom Sðuabc Þ ¼

 12   3T 6T ½S200 ðuabc Þ þ 4 R00 ðuab ÞR00 ðuac Þ þ 4 R00 ðuac ÞS110 ðuaad Þ m2 m m

   3 6  R00 ðuac Þ  R00 ð0Þ S110 ðuacd Þ þ 2 ½Q 1100 ðuaabc Þ þ 6 ð3 R00 ðuab ÞR00 ðuac Þ m m  00  R ðuac ÞR00 ð0Þ  R00 ðuab ÞR00 ðubc ÞR00 ðuca ÞÞ

ð123Þ

ð124Þ

where [  ] denotes again symmetrization. Note that if one is interested only in the dependence in uabc one can replace all unmatched R(u) by RðuÞ  Rð0Þ since this produces only gauge terms for p ¼ 0 (thanks to a cancellation for the last term). But the above equation contains a bit more information (e.g. about free energy cumulants). The advantage of this set of ERG equation is that the one loop beta function can already be read off from (123), the two loop from (124), it is thus well suited to the loop expansion. Obtaining the order R2 to all orders in T still requires (124). In Appendix E we show how the C-ERG equations (123) and (124) can be derived from (118) and (119), being careful with the flow term mom (being unrescaled equations, this term is always important).

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4.2. Exact ‘‘droplet” solution in the TBL region for the ERG hierarchy We now show how one can obtain an exact solution to the full ERG hierarchies displayed in the previous section   2. It is restricted to the thermal boundary layer (TBL) region, defined in Section as e . Via matching, however, it does also provide some zero temperature information. umf ¼ O T The strategy is to first compute the low temperature behaviour for the W-moments of the potential e ðv Þ. Once this is done, one can easily get iteratively all W-cumulants and all C-cumulants. V In this Section, to avoid inflation of notations, we do not work with rescaled quantities (unlike in Section 2). Of course, the statements made below are expected to become exact in the (universal) limit e ¼ 2Tmh ! 0 with displacements scaled appropriately u  mf . These are easily restored m ! 0, i.e. T e , but it is as efficient to use instead a formal expansion afterwards. The true expansion is in powers of T in T at small but fixed m. 4.2.1. Droplets: general considerations Let us study the difference of the renormalized potential at two closeby points, and we define v ¼ T v~ . The TBL region is defined as v~ being constant as T goes to zero. The definition (65) can be written in the form of a thermal average in a given sample:

e

  bV ðv ÞbV ð0Þ



T

2 2 1 ¼ e2Tm v~

R

m2 v~ uT1ð12m2 u2 þVðuÞÞ D 2 E 2 2 due 1 ¼ e2Tm v~ em v~ u R 1T ð12m2 u2 þVðuÞÞ HV ðuÞ due

ð125Þ

~ as T ! 0 (and fixed m) and for LR type disorder ðh > 0Þ in (almost) all samples there is a For fixed v single minimum of HV ðuÞ (we assume continuous disorder distributions) which gives the dominant contribution to this average. The droplet assumption is that one can obtain the behaviour up to order T by considering no more than two quasi-degenerate minima (i.e. wells). Restricting to these two wells one obtains:

  b ðv Þ  V b ð0Þ ¼ 1 T 2 m2 v~ 2  T ln pem2 v~ u1 þ ð1  pÞem2 v~ u2 V 2 eE1 =T 1 p ¼ E =T ¼ e 1 þ eE2 =T 1 þ w

ð126Þ ð127Þ

with w ¼ eE=T with E ¼ E2  E1 P 0. Here we call u1 the absolute minimum, u2 the secondary minimum, and Pðu1 ; u2 ; EÞdu1 du2 dE the joint probability density of position and energy difference (normalized to unity). One usually denotes:

Pðu1 Þ ¼

Z

Pðu1 ; u2 ; EÞ du2 dE

ð128Þ

Dðu1 ; u2 Þ dE ¼ Pðu1 ; u2 ; E ¼ 0Þ dE

ð129Þ

the probability of the position of the absolute minimum, and the (unnormalized) probability density that there are two quasi-degenerate minima within a window dE near E ¼ 0 (this window will be small, of order T, and the function of E there can be assumed to be constant, provided it does not vanish). Since many observables depend only on the relative position y ¼ u12 ¼ u1  u2 we also denote:

DðyÞ ¼

Z

du1 Dðu1 ; u1  yÞ

ð130Þ

The droplet assumption allows to compute any average over disorder as the sum of single well events, and quasi-degenerate rare events, as [36,87]:

Fðw; u1 ; u2 Þ :¼ Fðw; u1 ; u2 Þ ¼

Y Z

i

hOi ðuÞi ¼

Y

ðpOi ðu1 Þ þ ð1  pÞOi ðu2 ÞÞ

i

du1 Pðu1 ÞFð1; u1 Þ þ T

Z

du1 du2 Dðu1 ; u2 Þ

ð131Þ Z 0

1

dw ðFðw; u1 ; u2 Þ  Fð1; u1 ÞÞ w

ð132Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

note that Fð1; u1 ; u2 Þ ¼ Fð1; u1 Þ is independent of u2 . We often use the shorthand notation hOiu1 or hOiP for the first average and hOiu1 u2 or hOiD or hOiui or even hOiy (when it depends only on the difference) for the second (unnormalized) average. One may worry that the lack of normalization of the second average may make predictions weaker, thanks to STS this is not the case. Indeed the STS relation:

T ¼ hu2 i  hui2 ¼ pð1  pÞðu1  u2 Þ2 ¼ T m2

Z

1

0

D E E dw w TD ðu1  u2 Þ2 ¼ ðu1  u2 Þ2 2 ui ui w ð1 þ wÞ 2 ð133Þ

provides a normalization

hðu1  u2 Þ2 iui ¼ hy2 iy ¼

2 m2

ð134Þ R

where here and below hOðyÞiy ¼ dyOðyÞDðyÞ. In practice it is easier to compute moments of the renormalized force, i.e. take a derivative of (127):



 b 0 ðv Þ V X 1 þ wX 2  ¼ T v~  ðu1 X 1 oX1 þ u2 X 2 oX 2 Þ ln  m2 u v~ 2 m 1þw i X i ¼e  u1 X 1 þ u2 wX 2  ¼ T v~  X þ wX  m2 u v~ 1

2

X i ¼e

ð135Þ ð136Þ

i

One can check that the STS also guarantees that the average force is zero:

b 0 ðv Þ V ¼0 m4

ð137Þ

We have used (132) and:



dw u1 X 1 þ u2 wX 2 X1 þ X2  u1 ¼ Tðu2  u1 Þ ln w X þ wX X1 1 2 0   X1 þ X2 1  2 2  T u12 ln ¼ T u12 D m v~ ¼ T v~ 2 X1 D

T

Z

1

ð138Þ ð139Þ

and (134). Here and below we repeatedly use two important symmetry property of the droplet averages:

Dðu1 ; u2 Þ ¼ Dðu2 ; u1 Þ

ð140Þ

Dðu1 ; u2 Þ ¼ Dðu1 ; u2 Þ

ð141Þ

The functions Pðu1 Þ and Dðu1 ; u2 Þ also satisfy two important relations. It is shown in Ref. [36] (Section 4.2 there) that:

P0 ðu1 Þ ¼ m2

Z

du2 u21 Dðu1 ; u2 Þ Z  mom Pðu1 Þ ¼ m2 du2 ðu21  u22 ÞDðu1 ; u2 Þ

ð142Þ ð143Þ

The first equation is a consequence of STS and implies (upon integration) an infinite set of relations between moments. It encodes for the low temperature limit of a subclass of all the STS relations between moments. The second equation is a consequence of the more general one:

mom Pðu1 ; u2 ; EÞ ¼ m2 ðu21  u22 ÞoE Pðu1 ; u2 ; EÞ

ð144Þ

for E > 0, which arises from the facts that (i) the dependence of E ¼ HV ðu1 Þ  HV ðu2 Þ in m arises only from the explicit m dependence (using the minimum condition) and (ii) the dependence of the typical u1 and u2 in m becomes subdominant (compared to mf ) when E > 0 in the universal limit m ! 0. Indeed changes in u1 and u2 come only from switching from one low lying state to another (i.e. at E ¼ 0). Integrating (144) over E yields (143). This point reexplained in Section 4.4.1. Integrating (144) over

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

E yields (143). This equation also implies an infinite set of relations between moments, as discussed in [36] (Sections III.D and IV.B, and especially Appendix C there). Of course (144) neglects an additional feeding term from three wells, and as such is a truncation of an infinite hierarchy of equations. These additional contributions are, however, expected to lead to corrections to higher order in T. 4.2.2. Second moment R(u) in the thermal boundary layer We now compute:





b 0 ðv Þ V b 0 ðtÞ R00 ðv  tÞ V u X þ u2 wX 2 ~t þ u1 Y 1 þ u2 wY 2 j m2 u v~ ~þ 1 1 ¼ ¼ T v T 2 ~Þ i ;Y i ¼em ui t X i ¼e m4 m4 X 1 þ wX 2 Y 1 þ wY 2

ð145Þ

Upon expanding the first term is subdominant of order T 2 and is discarded. The cross term vanish b 0 ðv Þ ¼ 0. Remains to be computed: since, as we have just shown V

u1 X 1 þ u2 wX 2 u1 Y 1 þ u2 wY 2 ¼ hu21 iP þ ThhðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þiui þ OðT 2 Þ X 1 þ wX 2 Y 1 þ wY 2

ð146Þ

¼ hu21 iP þ Thhss ðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þiui þ OðT 2 Þ

ð147Þ



dw u1 X 1 u2 X 2 w u1 Y 1 þ u2 Y 2 w  u21 w X1 þ X2 w Y 1 þ Y 2w

ð148Þ

where we have defined:

hðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þ ¼

Z

1

0

This integral is easily done but is greatly simplified if one uses the symmetries (141). For that purpose one defines:

1 ðhðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þ þ hðX 2 ; Y 2 ; u2 ; X 1 ; Y 1 ; u1 ÞÞ 2 1 hss ðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þ ¼ ðhs ðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þ þ hs ðX 2 ; Y 2 ; u1 ; X 1 ; Y 1 ; u2 ÞÞ 2 hs ðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þ ¼

ð149Þ ð150Þ

A little calculation yields:

hss ðX 1 ; Y 1 ; u1 ; X 2 ; Y 2 ; u2 Þ ¼ Z¼

1 1þZ ðu1  u2 Þ2 ln Z 4 1Z

X1Y 2 2 ¼ em ðu1 u2 Þðv tÞ X2Y 1

ð151Þ ð152Þ

Denoting y ¼ u2  u1 the final result is very simple:

 R00 ðv Þ ¼ R00 ð0Þ þ m4 T y2 F 2 ðm2 yv~ Þ y z

F 2 ðzÞ ¼

ð153Þ 2

4

z z 1 z e þ1 1 z z coth  ¼  ¼  þ Oðz6 Þ 4 2 2 4 ez  1 2 24 1440

ð154Þ

Thus we have now an exact correspondence between the droplet probability of two degenerate minima distant from y, DðyÞ in (130), and the full function R(u) in the TBL: they contain the same information. The question posed in the beginning of this paper is thus finally answered. Each higher derivative Rð2pÞ ð0Þ is proportional to a moment hy2pþ2 iy of DðyÞ. It may come as a surprise that only y ¼ u1  u2 appears in these formulae, since after all the system is in an harmonic well. We will see below that this property extends to the third moment, things change after the fourth. The value of R00 ðu ¼ 0Þ can also obtained more directly as:

hui2 ¼ 



2 E R00 ð0Þ u1 þ u2 w 1D T 2 ¼ hu i þ  u21 ¼ hu21 iP  T ¼ hu21 iP  2 ðu1  u2 Þ2 1 P ui m4 1þw 2 m

It is instructive to display the integrated versions:

ð155Þ

P. Le Doussal / Annals of Physics 325 (2010) 49–150

 R0 ðv Þ ¼ R00 ð0Þv þ m2 T 2 yG2 ðm2 yv~ Þ y  1 Rðv Þ ¼ Rð0Þ þ R00 ð0Þv 2 þ T 3 H2 ðm2 yv~ Þ y 2 p2 z 1 G2 ðzÞ ¼ þ ðz  4 þ 4 lnð1  ez ÞÞ  Li2 ðez Þ 2 12 8 H2 ðzÞ ¼ fð3Þ þ

p2 12

75

ð156Þ ð157Þ ð158Þ

1 5 1 1 1 z  z2 þ z3 þ z2 lnð1  ez Þ  z2 lnð1  ez Þ  zLi2 ðez Þ  zLi2 ðez Þ þ Li3 ðez Þ 4 24 2 2 2

P n k which contain polylogarithm functions Lin ðzÞ ¼ 1 k¼1 z =k . Those would have appeared if we had computed disorder averages of the ln in the potential (127), rather than the force. ~ limit coming from the TBL should match the As was recalled in Section 2 we expect that the large v ~ ! 1: small v behaviour coming from the outer (zero temperature) region. We find that for v

hjyj3 iy R00 ð0Þ  R00 ðv Þ Tm2 1 ~ jhjyj3 iy ¼  jv j   v j m4 2 4 hy2 iy The linear cusp of the zero temperature fixed point is thus beautifully reproduced with the exact result for d = 0: 3 R000 ð0þ Þ 1 hjyj iy ¼ m4 2 hy2 iy

ð159Þ

both sides have dimension of length. Similar relations will be derived in higher d in Section 5. In d = 0 it will be further confirmed below for the Sinai Random Field case from the exact solution for the function R(u) in its outer region. The last question of course is the validity of the droplet calculation. One should see the droplet model as one possible solution of all the STS and ERG constraints on correlations at low temperature. In d = 0 and for LR disorder (i.e. when there is a glass phase) it seems fairly inescapable, although a proof would be welcome (beyond the Sinai case). It could fail in two ways. In the first case, two wells are indeed sufficient to describe low T, and failure would then require very peculiar correlations between E2  E1 , when it is of order T, and well positions (e.g. like eigenvalue repulsion). Or, second case, more than two wells are needed. The latter may happen as h ! 0 since then we know (e.g. for a logarithmic R0 ðuÞ) that many wells are important in the low temperature phase, where some replica symmetry breaking (RSB) phenomenon takes place [3]. We also know that for infinite N in d = 0 these many-well quasi-degeneracies take place. Lastly, let us point out that these results, described here within the disordered model, can also be understood within the Burgers approach. We do not wish to anticipate on this topic until Section 4.5, and pursue here along the FRG path. 4.2.3. Higher moments The droplet calculation of all the higher W-moments  SðnÞ ðv 1...n Þ turns out to be possible along the ~ i fixed. From these, one can of course obtain all the same line in the full TBL, i.e. for all arguments v W-cumulants and C-cumulants. Here we give only the result for the third and fourth moments. The general result and calculation is performed in Appendix F. The third cumulant reads:

b 0 ðv 1 Þ V b 0 ðv 2 Þ V b 0 ðv 3 Þ S111 ðv 1 ; v 2 ; v 3 Þ ¼ b S 111 ðv 1 ; v 2 ; v 3 Þ ¼  V  3  2  6 2 2 ¼ m T y F 3 m v~ 1 y; m v~ 2 y; m v~ 3 y y F 3 ½z1 ; z2 ; z3  ¼





1 2 z1 F½z1  z2 ; z1  z3   þ 2perm 4 3

 

a cosh aþb 2 1 þ eaþb 2 1 2 1 b 1 2      ¼ ¼ coth coth   Fða; bÞ  ¼ 3 ð1  ea Þð1  eb Þ 3 2 sinh 2a sinh 2b 3 2 2 2 6

ð160Þ ð161Þ

ð162Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

where y ¼ u1  u2 . One can check (statistical) translational invariance (STS), i.e. S111 ðv 1 þ v ; v 2 þ v ; v 3 þ v Þ ¼ S111 ðv 1 ; v 2 ; v 3 Þ for any v, S using cothðx þ yÞ ¼ ð1 þ coth x coth yÞ=ðcoth x þ coth yÞ and symabc ð1  abÞð1  acÞ=ða  bÞða  cÞ ¼ 1=3. Since one can shift all the arguments by v 1 , the equivalent information is contained in the function:

F 3 ½0; z2 ; z3  ¼

1 ðz2  2z3  ðz2 þ z3 Þ cosh z3 Þ sinh z2  ðz3  2z2  ðz2 þ z3 Þ cosh z2 Þ sinh z3 12 sinh z2  sinh z3  sinhðz2  z3 Þ ð163Þ

Starting from the fourth moment the structure changes slightly:



 Q 1111 ðv 1 ; v 2 ; v 3 ; v 4 Þ ¼ hu41 iP þ m8 T y4 F 4 m2 yv~ 1 ; m2 yv~ 2 ; m2 yv~ 3 ; m2 yv~ 4 y D

E F 4 m2 yv~ 1 ; m2 yv~ 2 ; m2 yv~ 3 ; m2 yv~ 4 þ OðT 2 Þ þ m8 T u1 u2 y2 e u1 ;y

1 ðz1 F½z1  z2 ; z1  z3 ; z1  z4  þ 3permÞ 4 1 e F 4 ½z1 ; z2 ; z3 ; z4  ¼ ðz1 e F ½z1  z2 ; z1  z3 ; z1  z4  þ 3permÞ 4 1 þ eaþbþc F½a; b; c ¼ a ð1  e Þð1  eb Þð1  ec Þ 2 2 2 e F ða; b; cÞ ¼ 3 þ þ þ 1  ea 1  eb 1  ec F 4 ½z1 ; z2 ; z3 ; z4  ¼

ð164Þ ð165Þ ð166Þ ð167Þ ð168Þ ð169Þ

i.e. the second term involves not only y ¼ u12 but also an explicit u1 , i.e. on the precise nature of the harmonic well. Note that this term resembles a disconnected contribution with one second cumulant and one temperature. The fact that the TBL of R and S do not depend on u1 probably reflects some universality of R and S with respect to infrared cutoff procedure. 4.2.4. Check that the droplet solution obeys the FRG equations The above solution should satisfy the FRG equation (within the TBL). There should thus be differential relations between the functions F n introduced above. Let us examine the derivative of the ERG equation for the second moment (110):

mom R0 ðv Þ ¼ and consider

2T 000 2 R ðv Þ þ 2 S111 ð0; 0; v Þ m2 m

v ¼ T v~

F 3 ½0; 0; z ¼

ð170Þ

~ ¼ Oð1Þ. From (161) the feedback from the third moment involves: with v

3 þ z þ 4zez þ ðz  3Þe2z 12ðez  1Þ

2

¼

z3 þ Oðz5 Þ 360

ð171Þ

Thanks to the exact relation:

F 3 ½0; 0; z þ F 02 ½z ¼

z 12

ð172Þ

where F 2 is the TBL function of the second cumulant (153) one finds that the r.h.s. of (170) simplifies into:

2T 000 2 1 R ðv Þ þ 2 S111 ð0; 0; v Þ ¼ m6 Thy4 iy v~ þ OðT 2 Þ m2 m 6

ð173Þ

a simple (linear) quantity of order OðTÞ in the TBL. The l.h.s. of (170) is a priori of order OðT 2 Þ apart ~ ) thus if FRG is obeyed from the single ‘‘zero temperature” piece containing R00 ð0Þv (since v ¼ T v one should have:

mom R00 ð0Þv ¼

1 6 4 m hy iy v þ OðT 2 Þ 6

ð174Þ

P. Le Doussal / Annals of Physics 325 (2010) 49–150

77

This can be checked using now an exact relation (not dependent on droplet assumption) which is obtained by taking an additional derivative of (170) at u ¼ 0, and noting that the third moment S can ð3Þ ð3Þ only start at order u6 (i.e. S112 ð0; 0; 0Þ ¼ S112 ð0; 0; 0Þ ¼ 0, see e.g. Appendix B of Ref. [36]). Hence:

mom R00 ð0Þ ¼

2T 0000 R ð0Þ m2

ð175Þ 0000

If we now use (153) to evaluate R ð0Þ we find consistency since:

TR0000 ð0Þ ¼ m8 hy4 iy F 002 ð0Þ þ OðTÞ ¼ m8

1 4 hy iy þ OðTÞ 12

ð176Þ

Thus the FRG equation for the second cumulant is verified by the droplet solution in the TBL. Let us now check the ERG equation for the third moment. From (112), taking three derivatives one finds:

 T   mom S111 ðv 1 ; v 2 ; v 3 Þ ¼ 2 o21 þ o22 þ o23 S111 ðv 1 ; v 2 ; v 3 Þ m  2  þ 2 Q 2111 ðv 1 ; v 1 ; v 2 ; v 3 Þ þ Q 2111 ðv 2 ; v 2 ; v 1 ; v 3 Þ þ Q 2111 ðv 3 ; v 3 ; v 1 ; v 2 Þ m

ð177Þ ð178Þ

In the TBL the r.h.s. of this equation should be of order O(1), and one can in fact show that it exactly vanishes. Indeed one can check that the above solutions (161) and (164) satisfies:

  o2z1 þ o2z2 þ o2z3 F 3 ½z1 ; z2 ; z3  þ oz1 F 4 ½z1 ; z1 ; z2 ; z3  þ oz2 F 4 ½z2 ; z2 ; z1 ; z3  þ oz3 F 4 ½z3 ; z3 ; z1 ; z2  ¼ 0

ð179Þ

F 4 ½z1 ; z1 ; z2 ; z3  þ oz2 e F 4 ½z2 ; z2 ; z1 ; z3  þ oz3 e F 4 ½z3 ; z3 ; z1 ; z2  ¼ 0 oz1 e

ð180Þ

In addition, there is no O(1) constant piece from the r.h.s. since S111 ð0; 0; 0Þ ¼ 0. Thus the FRG equation is obeyed in the TBL. To conclude we have obtained the solution of the FRG equation for all cumulant in the TBL. It is parameterized by a single droplet distribution, i.e. a function Dðu1 ; u2 Þ. This function remains arbitrary, i.e. one needs information outside the TBL to determine it. Let us point out that in the terminology of ~ i ¼ Oð1Þ. Cumulants higher than R(u) exhibit addiRef [36] what is obtained here is the full TBL, all v tional intermediate regimes between the full TBL and the outer region where all v i ¼ Oð1Þ. These are the so-called partial BL, where some v ij are of order one, other of order T. We have not obtained the solution for these (except, see below in the Sinai RF case where they can in principle be computed). They will be discussed below in the language of Burgers, where they are associated to correlations between shocks, while the full TBL corresponds to a single shock quantity, rounded by viscosity (i.e. temperature). 4.2.5. Droplet observables in Sinai case (random field) For the RF Sinai case, thanks to the Markovian properties of V(u) (which is a simple Brownian walk in ‘‘time” u) it is possible to obtain analytically the droplet probabilities Pðu1 Þ and Dðu1 ; u2 Þ. This was performed in Ref. [28] and we recall the results. Here we choose the parameters m2 ¼ 1 and ðVðuÞ  Vðu0 ÞÞ2 ¼ 2rju  u0 j with r ¼ 1, the general case is easily restored at the end performing the change of spatial scale u ! ur1=3 m4=3 and energy V ! V r2=3 m2=3 . Let us first recall a more general result obtained in Ref. [28] for the joint probability, denoted14   1 2   Nð2Þ 1 ðu0 ; V 0 ; u1 ; V 1 ; u1 Þdu1 dV 0 dV 1 that the minimum of HðuÞ ¼ VðuÞ þ 2 u on a given interval u 2 ½u0 ; u1  is at position u1 (within du1 ) and that the energy differences with the two edges are V 0 ¼ Hðu0 Þ  Hðu1 Þ and V 1 ¼ Hðu1 Þ  Hðu1 Þ (within dV 0 and dV 1 ). This is illustrated in Fig. 1. This is equivalent (see Fig. 2) to the condition that the Brownian walk VðuÞ must remain above the (inverted) parabola centered on u ¼ 0:

VðuÞ >  14

u2 þ E1 2

It is N ð2Þ 1 ðxL ; uL  u; x; uR  u; xR Þ in notations of Ref. [28].

ð181Þ

78

P. Le Doussal / Annals of Physics 325 (2010) 49–150

H(u)

V0

V1

u u0*

u1

u1*

Fig. 1. A random lansdscape such that the minimum of HðuÞ ¼ VðuÞ þ 12 u2 on a given interval u 2 ½u0 ; u1  is at position u1 and that the energy differences with the two edges are V 0 ¼ Hðu0 Þ  Hðu1 Þ and V 1 ¼ Hðu1 Þ  Hðu1 Þ.

with one contact point at u1 where the equality holds (i.e. the walk is on the parabola), i.e. u2 E1 ¼ Hðu1 Þ ¼ 21 þ Vðu1 Þ, and start and finish at

Vðui Þ  Vðu1 Þ ¼ 

ðui Þ2 u21 þ þ Vi 2 2

ð182Þ

It was found (formula (D.5) in Ref. [28]) that:

   ~ ~     Nð2Þ 1 u0 ; V 0 ; u1 ; V 1 ; u1 ¼ h u1 ; u0 ; V 0 h u1 ; u1 ; V 1

ð183Þ

~ uR ; V R Þ ¼ e121 ðu3 u3R Þþ12uR V R hðuR  u; V R Þ hðu; Z þ1 dk iku AiðaV þ ikbÞ hðu; VÞ ¼ e AiðikbÞ 1 2p

ð184Þ ð185Þ

b ¼ 1=a2 and a ¼ 21=3 and the normalization identity:

1¼

Z

1

dV 1

Z

0

1

dV 0

0

Z

u1

u0

    du1 Nð2Þ 1 u0 ; V 0 ; u1 ; V 1 ; u1

ð186Þ

~ uR ; V R Þ is the From the Markovian property it factors into two blocks, where the right block is hðu; probability that a walk which starts on the centered parabola at time u0 ¼ u remains above it and ends up a vertical distance V R above the parabola at u0 ¼ uR (the left block is its mirror image). These are obtained as ‘‘renormalized bonds”15 in the RSRG method of Ref. [28]. The function Aiðz þ VÞ=AiðzÞ 1=2  eVz thus for V > 0 it decays everywhere. If u > 0 the contour in (185) can be closed on the side ReðzÞ < 0, and gives a strictly positive result. If u < 0 the contour can be closed on the ReðzÞ > 0 side R þ1 and gives zero, i.e. hðu; VÞ ¼ 0 for u < 0. Note that 1 duhðu; VÞ ¼ AiðaVÞ=Aið0Þ and hðu; 0Þ ¼ dðuÞ. ð2Þ From N 1 one can obtain both Pðu1 Þ and Dðu1 ; u2 Þ. Taking the interval ½u0 ; u1  to become the whole real axis   1; þ1½ one obtains the distribution of the position of the absolute minimum:

Pðu1 Þ ¼ g~ðu1 Þg~ðu1 Þ ¼ gðu1 Þgðu1 Þ Z 1 ~ uR ; V R Þ ¼ eu3 =12 gðuÞ g~ðuÞ ¼ lim dV R hðu; uR !1 0 Z þ1 dk eiku gðuÞ ¼ 2 p aAiðikbÞ 1 1 X sk u=b 1 ¼ e for u < 0 0 ab k Ai ðsk Þ The

normalization 0

condition 2

1¼

R þ1

pðBiðzÞ=AiðzÞÞ ¼ 1=AiðzÞ which implies 15

du1 Pðu1 Þ follows from 1 R þ1 dk 1 ¼ 1. Note also 1 2p a2 AiðikbÞ2

hðuR  u; V R Þ ¼ e E 1 ð0; V R ; uR  uÞ in the notations of Ref. [28].

ð187Þ ð188Þ ð189Þ ð190Þ the

Airy

functions

that AiðzÞ  z1=4 e2z

3=2 =3

identity for large

P. Le Doussal / Annals of Physics 325 (2010) 49–150

79

V(u)

V0

V1

u u0*

0 u1

u1*

Fig. 2. The same as Fig. 1 where VðuÞ is plotted, together with the inverted parabola y ¼  12 u2 þ Hðu1 Þ.

jzj. Thus 1=AiðzÞ decays only for jargðzÞj > p=3 and the contour can be closed along negative z only when u < 0 resulting in a sum over the zeroes sk of the Airy function. Each factor g~ðuÞ arises from the probability that a walk which starts on the centered parabola at time u0 ¼ u remains above it for all larger u0 . One has the asymptotic behaviour:

gðuÞ u!þ1 2a3 uea Pðu1 Þ juj!1

6 u3 =3

ð191Þ

2a4 2 6 3 ju1 jea js1 jju1 ja u1 =3 Ai ðjs1 jÞ 0

ð192Þ

The probability of (quasi) degenerate minima (within ) can be obtained by considering two adjacent blocks (see Fig. 3), setting e.g. V 1 ¼ . One defines:

~ 1 ; u2 Þ ¼ oV hðu ~ 1 ; u2 ; V R Þj dðu V R ¼0 R

ð193Þ

such that the quantity:

~ 1 ; u2 Þ ¼ e dðu

1 ðu3 u3 Þ 12 1 2

dðuÞ ¼ oV hðu; VÞjV¼0

dðu2  u1 Þ Z þ1 0 dk iku Ai ðikbÞ ¼a e AiðikbÞ 1 2p

ð194Þ

describes the probability that a walk which starts on the centered parabola at time u0 ¼ u1 remains above it and terminates within  in energy of the parabola again at time u0 ¼ u2 > u1 . Note that the integral in (194) should be taken in the sense that udðuÞ is the Fourier transform of the second derivative of ln AiðzÞ. Using the above asymptotics for Airy functions, this yields that at small u one has:

udðuÞ  u1=2

ð195Þ

as can be obtained from the return probability to the origin of a simple random walk (since on that scale the curvature of the toy model energy landscape does not play any role). The total droplet probability also takes into account the two outer intervals with the net result [28]:

b 1 ; u2 Þhðu2  u1 Þ þ Dðu b 2 ; u1 Þhðu1  u2 Þ Dðu1 ; u2 Þ ¼ Dðu ~ 1 ; u2 Þg~ðu2 Þ ¼ gðu1 Þdðu2  u1 Þgðu2 Þ b 1 ; u2 Þ ¼ g~ðu1 Þdðu Dðu

ð196Þ ð197Þ

It is easy at this stage to restore the dependence in arbitrary m and r. One just needs to replace everywhere here and below:

a ¼ 21=3 r2=3 m2=3 ;

b ¼ 22=3 r1=3 m4=3

ð198Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

H(u)

V0

V2

u0*

u1

u u2*

u2

Fig. 3. The probability of a landscape with two degenerate minima (within ) can be obtained from two adjacent blocks with V 1 ¼ .

which still satisfies a2 b ¼ 1. It was checked in Section IV.D of Ref. [36] that the STS and ERG identities (143) are indeed satisfied16 by these exact results for Pðu1 Þ and Dðu1 ; u2 Þ. The total probability density (130) that there are two quasi-degenerate minima separated by y ¼ u1  u2 thus reads

DðyÞ ¼ dðjyjÞ ¼

þ1

dugðuÞgðu þ jyjÞ ¼ ab

1

Z

a b

Z

þ1 1

dk1 2p

Z

Z

þ1

1 þ1

1

dk1 2p

Z

þ1 1

0

dk2 iðk1 k2 Þjyj Ai ðik1 bÞ e 2p Aiðik1 bÞAiðik2 bÞ2

ð199Þ

0

dk2 iðk1 k2 Þjyj=b Ai ðik1 Þ e 2p Aiðik1 ÞAiðik2 Þ2

ð200Þ

with DðyÞ ¼ DðyÞ and which satisfies the normalization:17

Z

þ1

2

2

dyy DðyÞ ¼ ab ¼

1

2 m2

ð201Þ

The final result for the renormalized disorder correlator in the TBL for the Sinai model is thus:

R00 ðv Þ  R00 ð0Þ ¼ m4 T

Z

þ1

dyDðyÞy2

0



m2 yv~ m2 yv~ 1 coth  2 4 2

ð202Þ

where D(y) is given by (200). Below we obtain R(v) outside the TBL, and we check the matching explicitly between the two regimes. It is useful to recall the (half) generating function [28]:

b DðpÞ ¼

Z

1

dyDðyÞepy ¼ a

Z

0

¼a

Z

i1

i1 i1

i1

dz2 1 2ip Aiðz2 Þ2

Z

i1 i1

0

dz1 Ai ðz1 Þ 1 2ip Aiðz1 Þ bp þ z2  z1

ð203Þ

0

dz2 1 Ai ðz2 þ bpÞ 2ip Aiðz2 Þ2 Aiðz2 þ bpÞ

ð204Þ

The second expression is obtained for bp > 0 by closing the contour on the Reðz1 Þ > 0 side and using Cauchy’s theorem, and is somewhat formal, but allows to obtain18 the moments [28]: 16 2 1 e 1 1 e e For general m, r one thus has Pðu1 Þ ¼ u1 m Pðu1 =um Þ, Dðu1 ; u2 Þ ¼ um Em Dðu1 ; u2 Þ and DðyÞ ¼ um Em Dðy=um Þ where e D e are given by the same expressions as P and D here setting a ¼ b ¼ 1. um ¼ 22=3 r1=3 m4=3 and 21=3 Em ¼ r2=3 m2=3 and P; Hence the check of STS and RG in Section IV.D of Ref. [36] was performed using m ¼ 1; r ¼ 1=4, as may not be clear from the explanation there. 17 For y > 0 the function DðyÞ is the same as the one in Ref. [28]. However, the definitions differ since here y ¼ u1  u2 has arbitrary sign, and E2  E1 > 0, while in Ref. [28] y ¼ ju1  u2 j and E2  E1 has arbitrary sign. The end result for expectation values are of course identical. 18 By derivation with respect to p the conditions for convergence are restored.

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Z

þ1

dyjyjk DðyÞ ¼ 2rðk2Þ=3 mð24kÞ=3 yk

1

yk ¼ ð1Þk 2ð2k1Þ=3

Z

ð205Þ

0 ðkÞ dk Ai ðzÞ 1 ðz ¼ ikÞ 2p AiðzÞ AiðikÞ2

þ1

1

ð206Þ

b One notes also, from formula (124) of [28], an unexpected relation DðpÞ ¼ ad=dEP½0;1½ ðEÞjE¼bp with the probability of the minimum energy E on the half line. One has:

y2 ¼ 1

ð207Þ

y3 ¼ 1:80258

ð208Þ

y4 ¼ 4:21695

ð209Þ

y5 ¼ 11:6187 y6 ¼ 36:0516

ð210Þ ð211Þ

y7 ¼ 122:769

ð212Þ

The droplet distribution immediately allows to obtain the (normalized) distribution p(s) of shock sizes s ¼ uiþ1  ui > 0. This quantity will be defined in Section 4.5.4 where we demonstrate:

pðsÞ ¼ hsiP

2 sDðsÞhðsÞ hs2 iD

ð213Þ

From this we can compute the dimensionless universal ratios, independent of any parameters (m or r):

hskþ1 ihsk1 ihsk i2 ¼ ykþ2 yk y2 k1

ð214Þ

This yields:

hs3 ihsihs2 i2 ¼ y4 y2 y2 3 ¼ 1:2978

ð215Þ

hs4 ihs2 ihs3 i2 ¼ y5 y3 y2 4 ¼ 1:17776

ð216Þ

4

2

2 3

hs ihsi hs i

¼

y5 y22 y3 3

¼ 1:98369

ð217Þ 3=2

Note that the density of small droplets diverge as DðyÞ  y at small y, hence the distribution of shock sizes has a pðsÞ  s1=2 at small s. At large s they both decay with a faster than gaussian stretched exponential tail expðBs3 Þ as can be seen from (199) and (191). 4.3. Ambiguity-free zero temperature beta function to four loop via the ERG In this section we study the C-Exact RG defined in Section 4.1.3, for d = 0 and N ¼ 1. We show how it reproduces the T > 0 beta function of Section 2. We then recall the method introduced in Ref. [36] to e l ! 0, ‘‘zero temperature” limit study the partial boundary layers (PBL) and obtain the correct large l, T of the beta function. Next, we finish here the job started in Ref. [36] to obtain the unambiguous beta function. There, only a simplified version was presented to three and four loop, in which one arbitrarily sets the roughness exponent f to zero. In the two-loop part, f was kept non-trivial, but the version given there was unnecessarily heavy as it contained convolutions which, we show here, can be removed. Let us recall the C-ERG equations for the rescaled cumulants of Section 4.1.3:

RðuÞ ¼

1 4f e Rðumf Þ; m 4

Sð3Þ ¼

1 226f eð3Þ m S ; 8

Sð4Þ ¼

1 348f eð4Þ m S ; 16

Te ¼ 2Tmh

ð218Þ

ui mf being implicit as arguments of all rescaled cumulants and h ¼ d  2 þ 2f ¼ 2   þ 2f. Everywhere here  ¼ 4 (d = 0). We denote ol ¼ mom everywhere ð3Þ e 00 ðuÞ2  R e e 00 ðuÞ þ 1 R e e 00 ð0Þ R e 00 ðuÞ þ e ¼ ð  4f þ fuou Þ RðuÞ ol RðuÞ þ Te l R S 110 ð0; 0; uÞ 2

ð219Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

" # 3 Te l eð3Þ 3 Te l e 00 ð3Þ ð3Þ 00 e e e R ðu13 Þ R ðu23 Þ ol S ðu123 Þ ¼ ð2  2  6f þ fui oui Þ S ðu123 Þ þ S ðu123 Þ þ 2 200 2 h   i ð3Þ ð3Þ e 00 ðu12 Þ R e 00 ðu13 Þ2  c R e 00 ðu23 Þ R e 00 ðu31 Þ e 00 ðu12 Þ e e 00 ðu12 Þ R S 110 ðu113 Þ  e S 110 ðu123 Þ þ c R þ3 R 3 i 3 h ð4Þ ðu1123 Þ S þ e 2 1100 ð220Þ

where c ¼ 3=4 and [  ] denote symmetrization over the three arguments. We have defined e 00 ðuÞ ¼ R00 ðuÞ  R00 ð0Þ and various other notations are given in (72) and (73). The fourth cumulant is R needed only to three loop. Eq. (219) setting e S ð3Þ ¼ 0 already yields the one loop beta function. 4.3.1. Two loop We first show how to recover the two loop contribution to the beta functions at non-zero temperature, displayed in Section 2 and obtained there by a more standard field theoretic method. We recall, as detailed there, that the n-loop contribution is a sum of terms of the form T p Rnþ1p ; p ¼ 0; . . . ; n. To two loop we need the third cumulant Eq. (220) but we can discard the fourth cumulant feeding term in (220) (which contains only TR3 and R4 terms), as well as the R S term, and the TS00 term (since the expansion is also in T), these terms only yield contributions at three loop. Thus only the TR2 and R3 terms remain in the r.h.s. of (220), apart from rescaling. It is then natural to look for the solution under the form:

 h i  e e 00 ðu23 Þ þb R e 00 ðu12 Þ R e 00 ðu13 Þ2  1 R e 00 ðu23 Þ R e 00 ðu31 Þ þOðT 2 R2 ;TR3 ;R4 Þ e 00 ðu13 Þ R e 00 ðu12 Þ R S ð3Þ ðu123 Þ ¼ a Te R 3

ð221Þ where [. . .] means symmetrization. One can compute its flow using (220) to lowest order (i.e. zero loop) and one finds, in schematic notations:

L3 ¼ ol  fui oui  ð2  2  6fÞ



 L3 R00 R00 R00 ¼ ð2 þ Þ R00 R00 R00 þ OðR4 ; TR3 Þ



 L3 Te R00 R00 ¼  Te R00 R00 þ OðR4 ; TR3 Þ e l and e l ¼ ð  2  2fÞ T where we have used ol T   OðR4 ; TR3 Þ ¼ 3ð  2fÞ R00 R00 R00 þ O R4 ; TR3 . This implies

ð222Þ ð223Þ ð224Þ   

 ol  fui oui R00 R00 R00 ¼ 3 R00 R00 ðol  fuou ÞR00 þ

a ¼ 3=8; b ¼ 3c=ð þ 2Þ ¼ 3=8

ð225Þ

Note that f plays no role here, hence it can be first set to zero and later restored in the beta function by global rescaling. In that case, however, one does not deal with a fixed point and it is essential to retain and compute to each number of loop the flow term ol (which provides order by order what is usually called counterterms). The feeding term into (219):

00 i 3 h 00 00 00 00 i 3 h ð3Þ e R R R þ S 110 ð0; 0; uÞ ¼ Te R00 R00 8 8     1 e 000 2 1 e e 00 e 00 ðuÞ  R e 00 ð0Þ R e 0000 ð0Þ þ 1 R e 00 ð0Þ R e 000 ðuÞ2 ¼ Te R ðuÞ  T R ðuÞ  R 8 4 4

ð226Þ ð227Þ

where we use schematic notations, reproduces correctly the two loop contribution (26). As discussed in Section 2.2.2 its large l limit is:

    1  e 00 ð3Þ e e 00 ð0Þ R e 000 ðuÞ2  r ð4Þ ð0Þ þ O Te R ðuÞ  R S 110 ð0; 0; uÞ ¼ 4

ð228Þ

On the other hand, one can evaluate (221) in the outer region u12 ¼ 0ð1Þ ¼ u13 . It has a nice, but e l ! 0. We can now take the limit of the resulting function non-analytic limit as l ! 1, i.e. setting T when arguments become close:   1  e 00 ð3Þ ð3Þ e e 00 ð0Þ R e 000 ðuÞ2  R e 000 ð0þ Þ2 ðuÞ  R ð229Þ S 110 ð0; 0þ ;uÞ :¼ limþ e S 110 ð0; v ;uÞ ¼ R 4 v !0 1 e 000 þ 2 e 0000 þ 2 ¼ R ð0 Þ R ð0 Þu þ Oðjuj3 Þ ð230Þ 2

P. Le Doussal / Annals of Physics 325 (2010) 49–150

83

  e 00 ðv Þ  R00 ð0Þ R0000 ðv Þ vanishes. Of course taking instead uniformly the limit where the limit limv !0þ R

v ! 0 yields the same result. The notation here means that v is taken to zero with v ¼ Oð1Þ, i.e. within the outer region, also called inertial range in Burgers turbulence (see Section 4.5). To the two loop accuracy these two expressions are identical since we have shown in (32) that to one loop e 000 ð0þ Þ2 (in the turbulence context it is the Kolmogorov relation to one loop, see Section r ð4Þ ð0Þ ¼ R 4.5). We will see in the Burgers Section 4.5 below why we expect very generally that: ð3Þ ð3Þ e S 110 ð0; 0; uÞ S 110 ð0; 0þ ; uÞ ¼ e

ð231Þ

holds exactly (to any number of loop). This ‘‘matching” identity was in fact demonstrated in Ref [36] using a systematic analysis of partial boundary layers which we now recall. Consider u13 ¼ Oð1Þ. Examination of the ERG Eq. (220) indicates that there are two regions as u12 ! 0:

  e ~212 /ðu13 Þ þ Te 3l sð21Þ ðu ~ 12 ; u13 Þ þ O Te 4l S 3 ðu123 Þ ¼ Te 2l u e3

S ðu123 Þ ¼

 13 Þ u212 phiðu

3

þ ju12 j wðu13 Þ þ

Oðu412 Þ

~ 12 ¼ Oð1Þ u

u12 ¼ Oð1Þ

ð232Þ ð233Þ

e , the first region being the partial boundary layer (PBL21) and the second the outer ~ 12 ¼ u12 = T with u region. Plugging each form of (233) in (220) one obtains two equations, one valid for PBL21, the other, displayed below, is the outer equation. As discussed in Ref [36] they appear to imply (as an exact rela tion to all orders) that phiðuÞ ¼ /ðuÞ, which we set from now on and is equivalent to (231) since19 ð3Þ e S ð0; 0; uÞ ¼ 2/ðuÞ. The (loop) expansion in powers of R can be set up as: 110

ð3Þ e S 110 ð0; 0; uÞ ¼ 2/ðuÞ ¼ 2ð/0 ðuÞ þ /1 ðuÞ þ   Þ nþ3

where /n ðuÞ is OðR the outer region:

ð234Þ

Þ. At two loop order we can write the equation for /0 ðuÞ from the one for e S ð3Þ in

  e 000 ðuÞ2  R e 000 ð0þ Þ2 ol /0 ðuÞ ¼ ð2  2  4f þ fuou Þ/0 ðuÞ  cR00 ðuÞ R

ð235Þ

 e l ! 0, expanded the last term is the R00 R00 R00 feeding term in (220), the only left to two loop order and T 2 to order u12 . It is by definition the same term as (229) above. Using that:

  e 000 ðuÞ2  R e 000 ð0þ Þ2 ¼ 0ðR4 Þ ðol  3 þ 4f  fuou ÞR00 ðuÞ R

ð236Þ

e one can check that the solufrom the lowest order, rescaling part (i.e. zero loop) of the equation for R, tion obtained by equating 2/0 ðuÞ with the right-hand side of (229) obeys indeed (235). The coefficient 1/4 in (229) comes from the coefficient 2c=ð þ 2Þ ¼ 1=4. Hence we recover the above two loop bewta function and the convolutions in the two loop term of Ref. [36] (Eq. (187)) are indeed unnecessary. Note finally that we would obtain the same Eq. (235) for /0 ðuÞ from the large argument ~ 221 limit of the PBL21 for e S, as shown in Ref. [36]. u 4.3.2. Three loop To obtain the non-zero temperature beta function to three loop from the ERG one needs R4 ; TR3 or T 2 R2 , thus one must now include the equation for the fourth cumulant

h i e 00 ðu12 Þ R e 00 ðu13 Þ R e 00 ðu14 Þ ol e S ð4Þ ðu1234 Þ þ 4c Te R S ð4Þ ðu1234 Þ ¼ ð3  4  8f þ fui oui Þe h e 00 ðu14 Þ þ R e 00 ðu24 ÞÞ  2 R e 00 ðu31 Þ R e 00 ðu14 Þ R e 00 ðu12 Þ R e 00 ðu23 Þ e 00 ðu12 Þ2 R e 00 ðu13 Þð2 R þ 6c0 R  1 e 00 e 00 ðu12 Þ R e 00 ðu23 Þ R e 00 ðu34 Þ þ R ðu41 Þ R ð237Þ 2 with c ¼ 3=4; c0 ¼ 1=2, keeping only the needed terms (the complete one is displayed in Eq. (A15) of Appendix A in Ref. [36]). [. . .] denote symmetrization, here over the four indices. One can solve (237) 19

Note that we have not introduced the factor

v in the definition of u~ thus it can be set to one in Ref. [36].

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

by extending the method of the previous section, plug in (220) and solve again using this time the one e It is somewhat tedious and is summarized in Appendix G.1. One recovers the loop beta function for R. three loop finite temperature beta function (35). We now recall and finish the derivation of the correct three loop ‘‘zero temperature” beta function from the method of Ref. [36]. Let us first write Eq. (220) for e S ð3Þ in the outer region:

h  i ð3Þ e 00 ðu12 Þ 2/ðu13 Þ þ e ol e S ð3Þ ðu123 Þ  3 R S 110 ðu123 Þ S ð3Þ ðu123 Þ ¼ ð2  2  6f þ fui oui Þe   h i e 00 ðu13 Þ2  1 R e 00 ðu23 Þ R e 00 ðu31 Þ  3 /ð211Þ ðu12 ; u13 Þ e 00 ðu12 Þ R e 00 ðu12 Þ R þc R 3

ð238Þ ð239Þ

where the last term is the fourth cumulant feeding. It requires two points taken close together, and involves a function /ð211Þ which generalizes the function / of the third cumulant:

  e ~ 212 /ð211Þ ðu13 ; u14 Þ þ Te 3l sð211Þ ðu ~ 12 ; u13 u14 Þ þ O Te 4l ~ 12 ; u13 ; u14 ; u34 ¼ Oð1Þ ð240Þ S ð4Þ ðu1234 Þ ¼ Te 2l u u e S ð4Þ ðu1234 Þ ¼ u212 /ð211Þ ðu13 ; u14 Þ þ Oðju12 j3 Þ uij ¼ Oð1Þ

ð241Þ

the first region is called PBL211 and the second one is the outer region. The identity of the two functions /ð211Þ is again the continuity statement of the zero temperature limits: ð4Þ ð4Þ e S 110 ðu1 ; u1 ; u3 ; u4 Þ S 110 ðu1 ; uþ1 ; u3 ; u4 Þ ¼ e

ð242Þ

Next one needs to expand (239) itself to order Oðu212 Þ. One finds:

  e 000 ðuÞ2  R e 000 ð0þ Þ2 ðol  ð2  2  4f þ fuou ÞÞ/ðuÞ ¼ cR00 ðuÞ R 00

00

00

0

e 000

e 000

þ

þ ð/ ðuÞ  / ð0ÞÞR ðuÞ þ 3/ ðuÞ R ðuÞ þ 6 R ð0 ÞwðuÞ  gðuÞ

ð243Þ ð244Þ

with u ¼ u13 . On the first line one recognizes the two loop terms, the second line contains the contribution of the R00 S00 terms, involving also the cubic expansion function w in (233), and g represents the fourth cumulant feeding. We now expand each function /, w and g as in (234). We recall from the previous section:

/0 ðuÞ ¼ 

  1 cR00 ðuÞ Re 000 ðuÞ2  Re 000 ð0þ Þ2 2þ

ð245Þ

e we obtain: Expanding (244) to next order in R

e 000 ðuÞ ðol  ð2  2  4f þ fuou ÞÞ/1 ðuÞ ¼ ½ð/000 ðuÞ  /000 ð0ÞÞR00 ðuÞ þ 3/00 ðuÞ R e 000 ð0þ Þw ðuÞ þ g ðuÞ  ol j 2 / ðuÞ þ 6R 0 0 0 R

ð246Þ

where we must take also into account

  h  i 1 c dR00 ðuÞ Re 000 ðuÞ2  Re 000 ð0þ Þ2 þ 2R00 ðuÞ Re 000 ðuÞd Re 000 ðuÞ  Re 000 ð0þ Þd Re 000 ð0þ Þ 2þ e 000 ðuÞ2  R e 000 ð0þ Þ2 þ R00 ðuÞR0000 ðuÞ ð247Þ dR00 ðuÞ ¼ R 000 ð5Þ 00 000 0000 e ðuÞ R e ðuÞ þ R ðuÞR ðuÞ dR ðuÞ ¼ 3 R ð248Þ ol jR2 /0 ðuÞ ¼ 

e 000 ð0þ Þ R e 0000 ð0þ Þ dR000 ð0þ Þ ¼ 3 R

ð249Þ

e i.e. the beta function of R e to order R e 2 (since / was found by neglecting those terms, from the flow ol R, 0 and higher order ones). The two other functions are obtained from:

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

  1 e 000 þ e 000 2 e 000 ð0þ Þ R e 0000 ð0þ Þ ol w0 ðuÞ ¼ ð2  2  3f þ fuou Þw0 ðuÞ þ c R ð0 Þ R ðuÞ þ R00 ðuÞ R 2   2 0 7 e 000 e 000 ð0þ Þ R ðuÞ2  R ol g0 ðuÞ ¼ ð3  4  4f þ fuou Þg0 ðuÞ þ c 2 i   2 e 0000 þ 2 e 0000 00 000 000 e 0000 ðuÞ2 e e þ12R ðuÞ R ðuÞ R ðuÞ  R ð0 Þ R ð0þ Þ þ R00 ðuÞ2 R

ð250Þ

ð251Þ

Proceeding as before one finds:     1 0 7  e 000 2 e 000 þ 2 e 0000 ðuÞ  R e 0000 ð0þ Þ þ R00 ðuÞ2 R e 0000 ðuÞ2 e 000 ðuÞ2 R e 000 ð0þ Þ2 R R ðuÞ  R ð0 Þ þ 12R00 ðuÞ R c 4þ 2   1 1 c Re 000 ð0þ Þ Re 000 ðuÞ2 þ R00 ðuÞ Re 000 ð0þ Þ Re 0000 ð0þ Þ w0 ðuÞ ¼ 2þ 2

g0 ðuÞ ¼

ð252Þ ð253Þ

All terms feeding /1 have the same eigenvalue with respect to the linear operator so we find:

/1 ðuÞ ¼ 

1 e 000 ðuÞ  6 R e 000 ð0þ Þw ðuÞ  ol j 2 / ðuÞÞ ðg ðuÞ þ ½ð/000 ð0Þ  /000 ðuÞÞR00 ðuÞ  3/00 ðuÞ R 0 0 R 2 þ 2 0 ð254Þ

It is instructive to compute each piece separately:

ð/000 ð0Þ  /000 ðuÞÞR00 ðuÞ ¼

   1 cR00 ðuÞ 5 Re 0000 ðuÞ Re 000 ðuÞ2  Re 000 ð0þ Þ2 Re 0000 ðuÞ þ 4 Re 0000 ð0þ Þ 2þ   e 0000 ðuÞ2 þ R e 000 ðuÞ R e ð5Þ ðuÞ þ 2R00 ðuÞ R

    e 000 ðuÞ2 R e 000 ðuÞ2  R e 0000 ðuÞ e 000 ðuÞ ¼ 3 c R e 000 ð0þ Þ2 þ 2R00 ðuÞ R e 000 ðuÞ2 R  3/00 ðuÞ R 2þ   e 000 ð0þ Þ2 R e 000 ðuÞ2 þ R00 ðuÞ R e 0000 ð0þ Þ e 000 ð0þ Þ2 R e 000 ð0þ Þw ðuÞ ¼  6 c 1 R  6R 0 2þ 2

ð255Þ ð256Þ ð257Þ ð258Þ

while the contribution to the correction term is:

h i e ðuÞ ¼ db3loop R

2 ol j 2 / ðuÞ 2 þ 2 R 0   h  i 1 e 000 ðuÞ2  R e 000 ð0þ Þ2 þ 2R00 ðuÞ R e 000 ðuÞd R e 000 ðuÞ  R e 000 ð0þ Þd R e 000 ð0þ Þ dR00 ðuÞ R ¼ 40

Putting all together we obtain the final result for the beta function to three loops displayed in the next subsection. 4.3.3. Final result for the beta function to four loop e l ! 0 beta function to four loop. The Let us now display our final result for the unambiguous T three loop term was obtained by three independent methods. The first two are based on the C-ERG: the first one involves detailed considerations of the partial boundary layers and is described in the previous subsection. The second one uses the continuity structure of the various cumulants and was described in Appendix G of Ref. [36], hence we will not detail it here. It allowed us to obtain the four loop term by going up to the C-ERG equation for the fifth cumulant. The third method is described in Appendix G.3 and uses a closure of the W-moment hierarchy. Being a bit more memory consuming, we could only use it to three loops. In all cases where they can be compared the results for the anomalous terms are found to be non-ambiguous and in agreement. One finds, e 00  R e 00 ð0Þ: up to a constant, with R00 ¼ R

   

1 e 00 2 e 00 e 00 1  e 000 2 e 000 þ 2 00 R R  R ð0Þ R þ  R ð0 Þ R 2 4

2  

2 1 00 2  e 0000 2 3  e 000 2 e 000 þ 2 1 e 000 R e 0000  R e 0000 ð0þ Þ e 000 ð0þ Þ2 R R þ ðR Þ R þ  R ð0 Þ þ R00 R 16 32 4

e ¼ ð  4fÞ R e þ fu R e0 þ  mom R

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1 00 3  e ð5Þ 2 3 00 2 e 000 e 0000 e ð5Þ 1 00  e 000 3 e ð5Þ e 000 þ 3 e ð5Þ þ R R  R ð0 Þ R ð0 Þ þ ðR Þ R R R þ R ðR Þ R 96 16 8

1 00 2  e 0000 3 9 00  e 000 2  e 0000 2 1 000 þ 2  e 0000 2 5 e 000 þ 2 e 0000 þ 2 þ ðR Þ R R R þ R  R ð0 Þ R  R ð0 Þ R ð0 Þ 16 16 6 6

 

  2 5  e 000 2 e 000 þ 2 1 11 2 þ 000 0000 0000 000 0000 e R e ð0þ Þ R e e6 e þ R e ð0 Þ  R e 000 ð0þ Þ2 þ O R R R R  R ð0 Þ þ 16 10 10 þ

ð259Þ The first line are one and 2-loop terms, the second is 3-loop, the last three are 4-loop. Normal terms (i.e. non-vanishing for analytic R(u)) are grouped with anomalous ‘‘counterparts” to show the absence of O(u) term, a strong constraint (linear cusp, no supercusp): these combinations can hardly be guessed beyond 3 loop. This shows the difficulty in constructing the FT, already in d = 0. We emphasize that (259) results from a first principle derivation. This was the main point of this calculation, i.e. to show that it can be done. We expect that a large class of scale invariant ‘‘fixed point models” in d = 0 should be solution of this equation. That includes presumably a line of long range random potentials parameterized by a continuously varying f > 1 (equivalently h > 0). Unfortunately we do not, at this stage, have any such model which would be a perturbative test solution. Hence the interest of this b-function is mostly as a d = 0 limit of the one we hope can be computed in higher d. Note that it would be interesting to derive the corresponding beta function for depinning, since the two beta functions are expected to differ only by anomalous terms. 4.4. Sinai random field landscape: exact solution of the FRG hierarchy Here we describe the solution of the FRG in d = 0 at zero temperature and focus on the Sinai random field case. Modifications at non-zero temperature are discussed in the next Section, where the mapping to the Burgers equation is further analyzed. Here we specialize to N ¼ 1 and whenever we deal with the Sinai RF case we mention it. 4.4.1. Shape of the renormalized energy landscape at T = 0 and shocks At zero temperature one has:

  b ðv Þ ¼ min HV ðu; v Þ ¼ min 1 m2 ðu  v Þ2 þ VðuÞ V u u 2 2 b 0 2 0 m V ðv Þ ¼ v  u1 ðv Þ ¼ m V ðu1 ðv ÞÞ

ð260Þ ð261Þ

where we denote u1 ðv Þ the value of u which realizes the minimum in (260). There is an implicit m dependence everywhere, which plays the role of time in Burgers via t ¼ m2 , and we are interested in the small m regime (large Burgers time t). As can be seen integrating (261) over v if there is no applied force (statics) u1 ðv Þ remains on average near v. Furthermore, the motion of u1 ðv Þ is always forward as v increases. It is either smooth and satisfies:

1 1 þ m2 V 00 ðu1 ðv ÞÞ

ð262Þ

1  mom u1 ðv Þ ¼ ðv  u1 ðv ÞÞov u1 ðv Þ 2

ð263Þ

ov u1 ðv Þ ¼

Since u1 ðv Þ is the minimum of HV ðu; v Þ the denominator is always positive and ov u1 ðv Þ > 0. If the motion is smooth one easily show from (263):

mom ðu1 ðv Þ  v Þ2 ¼

4 ov ðu1 ðv Þ  v Þ3 þ 4ðu1 ðv Þ  v Þ2 3

ð264Þ

which, averaged over disorder produces the famous dimensional reduction result ðu1 ðv Þ  v Þ2  m4 . This cannot be correct, though since there are also discontinuous switches forward to another minimum. This happens at special points v i called shocks where HV ðu; v Þ has two minima. As m decreases

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(RG time increases) the quantity m2 V 00 ðu1 ðv ÞÞ becomes larger and larger. This is because V 00 is governed by the typical curvature of the bare potential which we can take as smooth and of order unity. In the universal asymptotic regime, when u1 ðv Þ is properly scaled by mf , one then finds that in the smooth regions, i.e. in between shocks, u1 ðv Þ is both v-independent and m-independent. The asymptotic motion u1 ðv Þ thus becomes a staircase discontinuous forward motion. This can also be seen if one takes the renormalized (rescaled) landscape to be e.g. the Brownian motion, which has V 00 infinite. We expect it to extend to any of the h > 0 landscape. We now study the asymptotic behaviour. Thus for each (bare) disorder realization V(u) there is a set of successive minima ui and shock positions v i , such that:

b 0 ðv Þ ¼ v  u i m2 V

v i1 < v < v i

ð265Þ

such that uðv Þ ¼ ui in this interval.  At position v ¼ v i minima ui and uiþ1 become degenerate and e 0 ðv i Þ ¼ ui  uiþ1 ¼ V 0 ðuiþ1 Þ  V 0 ðui Þ (at T = 0 V b is the b 0 ðv i þÞ  V switch. The discontinuity is m2 V Legendre transform of VðuÞ). This discontinuity will be rounded at small non-zero T since:

b 0 ðv Þ ¼ v  hui m2 V H V ðv Þ

ð266Þ

and the switch from one minimum to the next will occur smoothly on scale v  T. This is related to the internal structure of the shocks and studied in the next Section. b ðv Þ illustrated in Fig. 4. For a There is a simple and well known geometric construction to obtain V given v, one writes the condition that the landscape V(u) must remain, for all u, above a parabola centered on u = v:

VðuÞ > E 

ðu  v Þ2 2

ð267Þ

with one contact point at u ¼ u1 where the equality holds. The value of E is fixed by the single contact point condition. It corresponds to the apex of the parabola and its value is precisely the minimum of b ðv Þ. HV ðu; v Þ, i.e. E ¼ V This construction is then repeated for increasing values of v. Since u1 ðv Þ does not change, the touching parabola first rotates around this point until, for a given v ¼ v s a second contact point appears. At this point v ¼ v s there is a shock (characterized by u21 ¼ u2  u1 ). The statistics is then the one of degenerate minima in the toy model. There is another, equivalent, useful construction to find the shock positions from the graph of the 2 2 b ðv Þ is the Legendre transform of the function UðuÞ ¼ m2 u2 þ VðuÞ. Since the (convex) function m2 v 2 þ V function UðuÞ, then it should also be the Legendre transform of the convex envelope Uc ðuÞ of UðuÞ. The two functions coincide Uc ðuÞ ¼ UðuÞ on regular points, while they differ on shock intervals þ ui ¼ uðv  i Þ; uiþ1 ¼ uðv i Þ½. Note that this construction, as well the parabola construction generalize easily to Burgers in any dimension N > 1. Finally, note that there is yet another construction for shocks, known as the Maxwell rule [75], which does not seem to admit any known extension to N > 1.

V(u)

E

u u1(v)

v

Fig. 4. Geometrical construction of the renormalized landscape. The parabola yðuÞ ¼  m2 ðu  v Þ2 þ E0 centered on u ¼ v is raised (E0 increased from E0 ¼ 1 to E0 ¼ E) until it touches the curve y ¼ VðuÞ at a single point (for E0 ¼ E) u ¼ u1 ðv Þ, position of b ðv Þ ¼ E is obtained from the maximum of the parabola. the minimum of HV ðu; v Þ. The value at the minimum V 2

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

4.4.2. Statistics for the random field Sinai landscape case: preliminaries We can now go back to the RF Sinai case, i.e. VðuÞ a Brownian walk, where exact results can be obtained using the Markov property. Extending the analysis of Section 4.2.5 for droplet probabilities it is b ðv Þ. When convenient, possible to obtain analytically the full statistics of the renormalized landscape V we use the same choice of parameters m2 ¼ 1 and ðVðuÞ  Vðu0 ÞÞ2 ¼ 2ju  u0 j and emphasize connections between the two set of results. Note that some of our results here are similar to those obtained in the context of Burgers equation [80]. Our method, however, is different (the real space RG of Ref. [28]) b itself, and more general, e.g. we also obtain results about the distribution of renormalized potential V and later compute the explicit form of the function R(u). Single point (i.e. v) correlations of the renormalized landscape are related to the distribution Pðu1 Þ of the position of the minimum in the toy model. From the discussion of Section, the probability that the Brownian walk remains above the parabola centered on v and with one contact point at u1 (see Fig. 4) is:

pðu1 ; v Þdu1 ¼ g~ðv  u1 Þg~ðu1  v Þdu1 ¼ gðv  u1 Þgðu1  v Þdu1 ¼ Pðu1  v Þdu1

ð268Þ

where g(u) and g~ðuÞ are given in (189) and (188), and there is a shift of v due to the position of the parabola. Each factor of g represents the probability that V(u) remains above the parabola on the right, b 0 ðv Þ ¼ v  u1 this gives also the distribution of and on the left, respectively. Since the force is Fðv Þ ¼ V the renormalized force at a single point:

p1 ðF; v ÞdF ¼ PðFÞdF ¼ gðFÞgðFÞdF

ð269Þ

which is normalized to unity. Its moments were computed in Ref. [28]. It is easy to get the energy and force joint distribution in the case where there is no shock between v 1 and v 2 . The contact point u1 remains the same and it is necessary and sufficient that the walk VðuÞ remains above the parabola centered on v 1 to the left of u1 and above the parabola centered on v 2 to the right of u1 . Then it is above both parabola everywhere. From the Markov property and the above result one sees that the measure is:

pðu1 ; v 1 ; v 2 Þ ¼ g~ðv 1  u1 Þg~ðu1  v 2 Þdu1

ð270Þ

This implies that the joint probability that simultaneously (i) there is no shock in the interval ½v 1 ; v 2  b ðv 2 Þ, and (iii) F 1 ¼ V b 0 ðv 1 Þ; F 2 ¼ V b 0 ðv 2 Þ is simply: b ðv 1 Þ  V and (ii) E ¼ V

  3 v 21 1 3 e12ðF 1 F 2 Þ gðF 1 ÞgðF 2 ÞdðF 2  ðF 1 þ v 21 ÞÞd E þ ðF 1 þ F 2 Þ dF 1 dF 2 dE 2

ð271Þ

since one has Fðv 1 Þ ¼ F 1 ¼ v 1  u1 and Fðv 2 Þ ¼ F 2 ¼ v 2  u1 , also u1 ¼ 1=2ðv 1 þ v 2 Þ þ E=v 21 (here and below v 21 ¼ v 2  v 1 ). From this measure one can extract one contribution to the function R(u), which is done below, i.e. the part corresponding to no shock (the other piece is more complicated). One can integrate over the energy and obtain

p0 ðF 1 ; v 1 ; F 2 ; v 2 ÞdF 1 dF 2 ¼ g~ðF 1 Þg~ðF 2 ÞdðF 2  ðF 1 þ v 21 ÞÞdF 1 dF 2

ð272Þ

i.e. the joint probability for F 1 ; F 2 and that there is no shock in the interval v 21 . Integrating further over F 2 and F 1 yields the probability that there is no shock in an interval of length v 21 , which varies between one and zero. There is a direct connection between the shocks and the degenerate minima. The statistics of the shock is described by the droplet probability for a toy model whose parabola is centered at the position of the shock. Let us call v 2 the point where the first shock to the right of v 1 occurs (it is called v 3 in Fig. 5). The walk touches the parabola centered on v 1 at u1 , and is above it to its left, hence a first factor ~ðv 1  u1 Þ. The walk touches the parabola centered at v 2 in two points, at u1 and at u2 , and is above it g in between. Finally, it must remain above the parabola centered at v 2 for all points u0 > u2 (if it was crossing there would be a shock at a smaller v < v 2 ). The total probability is (taking into account for each parabola the shifted position of its center):

~ 1  v 2 ; u2  v 2 Þg~ðu2  v 2 Þdu1 du2 g~ðv 1  u1 Þdðu ð273Þ ~ 1  v 2 ; u2  v 2 Þ was defined in (194) and describes the probability of degenerwhere the function dðu ate minima in the toy model.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

89

V(u)

u u1

u2 v1 v2 v3

Fig. 5. As the position v of the center of the parabola is increased (shift to the right with fixed curvature m2 ), from v 1 to v 2 , the position of the minimum u1 ¼ u1 ðv 1 Þ ¼ u1 ðv 2 Þ does not change: there is no shock between v 1 and v 2 , the parabola effectively rotates around the contact point. The next shock is at v ¼ v s ¼ v 3 when there are two contact points at u ¼ u1 and u ¼ u2 . Increasing v further, the parabola rotates again around u2 until the next shock and so on.

One finally obtains the joint probability of u1 ðv 1 Þ ¼ u1 and that the first shock is at v 2 (within dv 2 ) with the new minimum at u2 :

~ 1  v 2 ; u2  v 2 Þg~ðu2  v 2 Þdu1 du2 u21 dv 21 pns ðu1 ; v 1 ; v 2 ; u2 Þ ¼ g~ðv 1  u1 Þdðu  ~ F  ; F þ g~F þ dðF   F 1  v 21 ÞðF   F þ ÞdF 1 dF  dF þ dv 21 ¼ g~ðF 1 Þd 2 2 2 2 2 2 2 2

ð274Þ

where in the last line we have expressed the probability for the force variables F 1 ¼ v 1  u1 ; þ F 2 ¼ v 2  u1 ; F 2 ¼ v 2  u2 . This is just (273) taking into account that  ¼ u21 dv 2 is the vertical shift at u2 of two parabola passing both through u1 and corresponding to v 2 and v 2 þ dv 2 , respectively.20 From (270) the probability that the first shock is at v 2 (within dv 2 ) is dv 2 ov 2 pðu1 ; v 1 ; v 2 Þ, thus compatibility between the two results requires:

Z

1

~ 1  v 2 ; u2  v 2 Þg~ðu2  v 2 Þu21 ¼ g~0 ðu1  v 2 Þ du2 dðu

ð275Þ

u1

which is exactly the STS relation for the droplets. On the other hand, one can also study the probability density for a shock. There we have just a single parabola at say, v 2 ¼ 0 the position of the shock. Thus we get the one shock distribution function:

~ 1 ; u2 Þg~ðu2 Þu21 du1 du2 ¼ gðu1 Þdðu2  u1 Þgðu2 Þu21 du1 du2 ps ðu1 ; u2 Þdu1 du2 ¼ g~ðu1 Þdðu

ð276Þ

one can check that this result is consistent with the one obtained for q1 ðl; gÞdldg in Ref. [80] in different variables such that dldg ¼ du21 12 dðu22  u21 Þ ¼ 12 u21 du21 dðu2 þ u1 Þ ¼ u21 du1 du2 (u2 > u1 is assumed). One method to describe the statistics of the full landscape is to construct successive shocks, thus to write the probabilities of the force at two points v L and v R and having n shocks in between:

~ 1  v 1 ; u2  v 1 Þdðu ~ 2  v 2 ; u3  v 2 Þ . . . dðun  v n ; unþ1  v n Þg~ðunþ1  v R Þ g~ðv L  u1 Þdðu u21 u32 . . . unþ1;n du1 . . . dunþ1 duL duR dv 1 . . . dv n

ð277Þ

In principle summing this over n should reproduce the probabilities computed in the next section by different methods. 4.4.3. Multipoint statistics for the random field Sinai landscape b ðv i Þ and forces F i ¼ V b 0 ðv i Þ at multiple points. Here we obtain the joint distribution of energies Ei ¼ V We need to impose the condition that the walk VðuÞ remains above all inverted parabolas centered in v i of offset Ei :

VðuÞ >  20

ðu  v i Þ2 þ Ei 2

ð278Þ

The equations of the two parabolas are V ¼ ðu  u1 Þ2  v 2 ðu  u1 Þ and  V ¼ ðu  u1 Þ2  ðv 2 þ dv 2 Þðu  u1 Þ. This means that

 ¼ u21 dv 2 .

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

V(u) E2 E3 E1

V2

V1

u u1 v1 u1*

u2 v2

u2*v3 u3

b ðv i Þ ¼ Ei at points Fig. 6. Construction of the joint probability that the renormalized potentials are V must remain above all (inverted) parabola centered on the v i and of apex Ei .

v i : the random walk VðuÞ

with one contact point at ui where the equality holds (i.e. the walk is on the parabola). This is represented in Fig. 6. We first assume that each interval v iþ1;i contain at least one shock. Neighboring parabolas intersect in:

v iþ1;i

ui  v i ¼

2



Eiþ1;i

ð279Þ

v iþ1;i

where Eiþ1;i ¼ Eiþ1  Ei . One must have ui < ui < uiþ1 (see Fig. 6). The case ui ! ui means that there is only one shock in the interval and it is in v iþ1 (whose parabola has then two contact points ui and uiþ1 ). No shock in the interval corresponds to uiþ1 ¼ ui ¼ ui and is examined separately. The intersection point of two neighboring parabola is at coordinate:

yi ¼

  2 u  v iþ1 ðui  v i Þ2  Ei ¼ i  Eiþ1 2 2

ð280Þ

The random walk at u ¼ ui must be above both parabola thus:

Vðui Þ ¼ yi  V i

ð281Þ

where V i > 0 is the vertical distance, it is also:

V i ¼ Vðui Þ þ

ðui  v i Þ2  Ei 2

ð282Þ

i.e. the difference between the energy at ui and its minimum value (i.e. its value at ui ). The condition that the walk remain above all parabola with a contact point on each is equivalent to the condition that in each interval ½ui1 ; ui  it remains above the corresponding parabola centered in v i with a contact point in each. For a single parabola the corresponding probability was displayed in Section. From the Markovian property, the total probability is thus just a product of the same blocks, each shifted by v i . This gives for the probability in case of n parabolas:

Z 0

1

dV 0

Z 0

1

dV 1   

Z 0

1

 dV n Nð2Þ 1 ð1; V 0 ; u1  v 1 ; V 1 ; u1  v 1 Þ

n1 Y

 Nð2Þ 1 ðui1  v i ; V i1 ; ui

i¼2

  v i ; V i ; ui  v i ÞNð2Þ 1 ðun1  v n ; V n1 ; un  v n ; V n ; þ1Þ

ð283Þ N ð2Þ 1

for n ¼ 2 the central product is just suppressed. The functions were given in (183). Replacing b 0 ðv i Þ ¼ F i (283) becomes the joint probability of the energies ui  v i by (279) and v i  ui ¼ V b ðv i Þ and the forces F i ¼ V b 0 ðv i Þ. Ei ¼ V Using the explicit form for the block given in (183), one finds, after some rearrangements and taking the limits that (283) takes the form:

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P. Le Doussal / Annals of Physics 325 (2010) 49–150





2 v v v þv 2 v v þv 2 v þv  1 v 3 þv 3 þþv 3n;n1  21 u  1 2 2 Þ  32 u  2 2 3 Þ  n;n1 u  n12 n Þ 4 ð n1 4 ð 1 4 ð 2 e 48 21 32 Z 1 Z 1 1 dV 1    dV n1 e2ðV 1 v 21 þV 2 v 32 þþV n1 v n;n1 Þ gðv 1  u1 Þ 0 0 " # n1    Y       h u1  u1 ; V 1 h ui  ui1 ; V i1 h ui  ui ; V i h un  un1 ; V n1 gðun  v n Þ

ð284Þ

i¼2

where the functions hðu; VÞ and gðuÞ are given in (189) and (194) in terms of Airy functions. Upon integration over the V i this gives the joint distribution of the Eiþ1;i and F i , via the replacements:

ui  ui ¼

v iþ1;i 2

uiþ1  ui ¼

Eiþ1;i



v iþ1;i

v iþ1;i

Eiþ1;i

þ

2

þ Fi

ð285Þ

 F iþ1

v iþ1;i

ð286Þ

and v i  ui ¼ F i . Integrating over the forces amounts to integrate over the ui . The distribution of the energies is then:

e

1 48



v 321 þv 332 þþv 3n;n1



E2

Z

E2

E2

4v21 4v32 4vn;n1 21

32

1

dV 1   

n;n1

Z

0

1

dV n1

0

Z

u1

du1

Z

u2

du2   

Z

u1

1

1

dun un1

" # n1   Y     1 e2ðV 1 v 21 þV 2 v 32 þþV n1 v n;n1 Þ gðv 1  u1 Þh u1  u1 ; V 1 h ui  ui1 ; V i1 h ui  ui ; V i

ð287Þ

i¼2

  h un  un1 ; V n1 gðun  v n Þ

which can be simplified using convolutions. From this expression one can, in principle, compute all moments SðnÞ ðv 1 ; . . . ; v n Þ. We now give an explicit expression for n ¼ 2. 4.4.4. Exact formula for the FRG function R(u) We now specialize to n ¼ 2 points v 1 ; v 2 and denote v ¼ v 21 . We obtain the distribution pðEÞ of the b ðv 1 Þ  V b ðv 2 Þ, and from there: energy difference E ¼ E1  E2 ¼ V

2ðRð0Þ  Rðv ÞÞ ¼

Z

1

2

dEE pðEÞ

ð288Þ

1

The part with no shock is obtained by integrating (271) over the forces F 1 and F 2 as: 1

pðnsÞ ðEÞdE ¼ e48v

3  E2 4v

g

E

v





E v dE g   2 v 2 v

v

ð289Þ

The part with shocks can be read from the previous paragraph, we denote V ¼ V 1 and u ðEÞ ¼ u1 ¼ v 1 þ2 v 2 þ vE . One has: v3

E2

pðsÞ ðEÞ ¼ e 48 4v

Z

1

1

dVe2

0 1

¼ e48v

3  E2 4v

Z 0

vV

Z

u ðEÞ

du1

Z

u ðEÞ

1

1

þ1

du2 gðv 1  u1 Þhðu ðEÞ  u1 ; VÞhðu2  u ðEÞ; VÞgðu2  v 2 Þ





1v V v E v E / V; þ ; dVe2 / V;  2 v 2 v

/ðV; v Þ ¼

Z

þ1

duhðu; VÞgðu  v Þ

0

ð290Þ 1 2

One notes that the part with shocks can be put in the same form replacing e v V ! dðv VÞ. The final formula for the distribution of the energy difference in terms of Airy functions is thus: 1

pðEÞ ¼ ðabÞ2 e48v

3 E2 4v

Z

þ1

1

dz1 2p i "

dz2 v ðz1 þz2 Þþ E ðz2 z1 Þ vb e2b 2pi

Z

þ1

1

1 þ v Aiðz1 ÞAiðz2 Þ

R1 0

# vV dVe2 AiðaV þ z1 ÞAiðaV þ z2 Þ Aiðz1 Þ2 Aiðz2 Þ2

ð291Þ

92

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pffiffiffiffi where the integration contour is along the imaginary axis. Note that this form suggests E  v , and this is indeed true at large E. However, at small E the landscape is more regular (random but finite first derivative) and one finds instead E  F v . This implies the absence of ‘‘supercusp”, i.e. R0 ð0þ Þ ¼ 0. Using (288) one finds finally:

Rð0Þ  Rðv Þ ¼

Z Z 2 pffiffiffiffiffiffiffi  1 v 3 þ1 dk1 þ1 pv e 48 b 2p 1 1 # " # i v ðk þk Þðk2 k1 Þ2 " R v v 01 dVe2V AiðaV þ ik1 ÞAiðaV þ ik2 Þ dk2 2ðk2  k1 Þ2 e 2b 1 2 b2 v 1 1 þ 2 Aiðik1 ÞAiðik2 Þ 2p Aiðik1 ÞAiðik2 Þ b v ð292Þ

where we have chosen the contour zi ¼ iki . In this formula the integral over V converges very well and 2 the double integral along the imaginary axis also converges. We recall that ba ¼ 1 and in this Section a ¼ 21=3 . The general case is obtained as Rm;r ðv Þ ¼ m4=3 r4=3 R1;1 ðm4=3 r1=3 v Þ, where Rðv Þ ¼ R1;1 ðv Þ is given by (292). Asymptotics and alternate formula are studied in Appendix H. In the large v limit it is found that:

 1 3 2ðRð0Þ  Rðv ÞÞ  2v þ R1 þ O e48v Z þ1 Z dz1 þ1 dz2 4 ðz2  z1 Þ2 8 R1 ¼ ¼ 2 ðA2;2  A21;2 Þ ¼ 25=3 0:510756 ¼ 1:62155 2 2 2 1 2pi 1 2pi b Aiðz1 Þ Aiðz2 Þ b with Ap;n ¼

R þ1

p du ðiuÞ 1 2p AiðiuÞn

ð293Þ ð294Þ

; A2;2 ¼ 1:06458; A1;2 ¼ 1:25512 (and A0;2 ¼ 1), and b ¼ 22=3 . This shows that

the value of r (here chosen to be unity) is not renormalized, as expected from the long range nature E2 4

1 v one finds, in a rescaled sense, pðEÞ pffiffiffiffiffiffi e v . 4pv ~ðÞd ¼ pðEÞdE and expand in v: In the small v limit, one writes E ¼ v , define p

of the random potential. Indeed, at large

pðÞ ¼ p0 ðÞ þ v p1 ðÞ þ    ; with

R

p0 ðÞ ¼ gðÞgðÞ

ð295Þ

dp1 ðÞ ¼ 0 and where the first correction p1 ðÞ is computed in Appendix H. Using that:

Rðv Þ ¼ v 2

Z

d2 p0 ðÞ þ v 3

Z

d2 p1 ðÞ þ   

ð296Þ

one finds that there is indeed a linear cusp to the force correlator, R00 ðv Þ ¼ R00 ð0Þ  R000 ð0þ Þv þ   , of amplitude:

Z

Z þ1 du 1 1 2 d2 gðÞgðÞ ¼ b o2iu ¼ 1:05423856 AiðiuÞ 1 2p AiðiuÞ Z

Z 1 d4 gðÞgðÞ  3 d2 g 0 ðÞgðÞ  R000 ð0þ Þ ¼ 4  R00 ð0Þ ¼

ð297Þ ð298Þ

This cusp was obtained from the small v ¼ Oð1Þ limit of the zero temperature function Rðv Þ. As dis~ ¼ v =T behaviour from the thermal boundary cussed in previous sections it should match the large v layer, and this provides a check for our droplet formula. The droplet formula, using (159,201,206) predicts:

R000 ð0þ Þ ¼

1 4

Z

1

jyj3 DðyÞ ¼

1

1 y 2 3

ð299Þ

and it is checked in Appendix H that this agrees with (298) both expressions being equal to:

R000 ð0þ Þ ¼

1 8 a2 15

Z

1

1

dk k2 ¼ 0:901289 2p AiðikÞ2

which confirms matching and the exactness of the droplet hypothesis.

ð300Þ

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4.4.5. Exact formula for the FRG function Dðv Þ It is also useful to derive an independent formula for the correlator of the force Dðv Þ ¼ R00 ðv Þ as:

Dðv Þ ¼

Z

þ1

1

dF 1

Z

þ1

dF 2 F 1 F 2 pðF 1 ; F 2 ; v Þ

ð301Þ

1

where pðF 1 ; F 2 ; v Þ is the two point force distribution. It can be obtained from our general formula for the joint distribution of forces and energies (284,285). This yields:

h 3 1 3 pðF 1 ; F 2 ; v Þ ¼gðF 1 ÞgðF 2 ÞdF 1 dF 2 dðF 2  v  F 1 Þe12ðF 1 F 2 Þ þ hðv þ F 1  F 2 Þ  Z 1 Z v þF 1 F 2 2 1 3 1 v uF v dV 1 e2V 1 v due 4ð 1 2Þ hðu; V 1 Þhðv þ F 1  F 2  u; V 1 Þ e48v

ð302Þ

0

0

After some manipulations summarized in Appendix H one obtains:

pffiffiffiffi 1 3 2 Dðv Þ ¼ 2 pv 1=2 b a2 e48v

Z

þ1

dk1 2p

Z

þ1

0

dk2 ðk1 kv 2 Þ2 þiv ðk1 þk2 Þ Ai ðibk1 Þ 2 e 2p Aiðibk1 Þ2 1 1 " # R 1 1 vV 0 dVe2 AiðaV þ ibk1 ÞAiðaV þ ibk2 Þ Ai ðibk2 Þ 1þ 0 2 Aiðibk1 ÞAiðibk2 Þ Aiðibk2 Þ

ð303Þ

Eqs. (292) and (303) are thus the explicit form of the fixed point of the FRG in d = 0 for the random field class. Up to a rescaling they should be a fixed point solution of the d = 0 FRG equation, obtained to four loop in (259), with the value f ¼ 4=3 for the roughness exponent. These functions satisfy all the expected requirements (cusp, large u behaviour, matching, etc.) and confirm the validity of the FRG as a method to handle disordered systems with many metastable states leading to shock singularities. 4.5. Decaying Burgers and FRG, inviscid limit 4.5.1. Generalities Let us now detail the connection between the FRG in d = 0 (and N components) and the decaying Burgers equation for a N-component velocity field uðx; tÞ in N-dimension. We focus on N ¼ 1, some aspects extend to any N. Let us recall that the latter is a simplified version of the Navier–Stokes equation (without pressure) and reads, in standard notations (for N ¼ 1):

ot uðx; tÞ þ uðx; tÞox uðx; tÞ ¼ mo2x uðx; tÞ

ð304Þ

where uðx; tÞ is the velocity field and m is the viscosity. The decaying Burgers problem amounts to solve (304) with an initial data uðx; t ¼ 0Þ ¼ u0 ðxÞ. We are interested in random initial data, a prominent example being u0 ðxÞ gaussian with short range correlations. It corresponds to the random field Sinai problem. More general initial (gaussian) velocity correlations are also studied corresponding to initial N1 n (kinetic) energy spectrum E0 ðkÞ ¼ k u0 ðkÞ  u0 ðkÞ  k at small k (in Fourier), n ¼ 0 being the Sinai case. Of high interest is the inviscid (i.e. large Reynolds number) limit m ! 0. In that limit one recovers (formally) Euler equation whose solutions uðx; tÞ ¼ u0 ðx  tuðx; tÞÞ develop shock singularities at finite time [73]. These singularities are smoothed by a small non-zero m, at some scale called the dissipative length Ld , which must thus be kept small but non-zero, the question of the proper construction of the inviscid limit m ! 0 being an outstanding problem, both for Burgers and Navier–Stokes turbulence. One must also mention the driven Burgers problem, which has an additional term f ðx; tÞ ¼ ox Wðx; tÞ on the r.h.s. of (304) where Wðx; tÞ is usually white noise in time and of correlation scale n in space. Under suitable boundary conditions it is expected to reach a stationary measure (i.e. suitable correlations become time independent). A lot of effort has been devoted to study the statistics of the velocity field in that case, as well as the stationary measure for shocks. It is important to point out that the structure of shocks in stirred or decaying Burgers is believed to be rather universal, the small distance (of order and slightly above Ld ) structure being analogous. The detailed time dependent statistics of these shocks in the inertial range Ld < x depend, however, on the model, with some universality classes.

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In decaying Burgers the evolution is expected to reach an asymptotic (statistically) scale invariant f form uðx; tÞ ¼ t21 uðw ¼ xt f=2 Þ (in law), i.e. there is also a stationary measure upon the corresponding rescaling of lengths and time, while in stirred Burgers no rescaling is necessary. For decaying Burgers the velocity correlations in this stationary measure identify with the FRG fixed point, as discussed below, and the universality classes are parameterized by f. For uncorrelated initial velocities f ¼ 4=3, its random field value, other values of f correspond to the other initial statistics, with f ¼ 4=ð3 þ nÞ (see below). The shocks are constantly merging and the typical scale (distance between shocks) grows as tf=2 . Although the width of an isolated shock grows as Ld  mt, one shows (see below) that because of f merging of shocks the width of the surviving shocks actually grows as L0d  mt 12 . The important dimensionless parameter is the ratio of (surviving) shock width to their separation L0d =t f=2  mt1f . Hence shocks become effectively thinner for h ¼ d  2 þ 2f > 0 (here d = 0), i.e. f > 1, which in the FRG corresponds to an attractive zero temperature fixed point (FP). Equivalently one can define an effective meff flowing to zero as th=2 . This picture holds for n < 1, for which the (kinetic) energy 12 u2 asymptotic decay is EðtÞ  t2þf ¼ t 2ðnþ1Þ=ðnþ3Þ . In the language of FRG it is called long range FP, and in Burgers it is termed as being dominated by the persistence of large eddies [75]. In addition there is also a short range (SR) FP regime, called Kida regime in Burgers turbulence [88,75,77,89], which holds for all n > 1. There the decay is EðtÞ  1=ðtðln tÞ1=2 Þ (for gaussian statistics, see below for generalization) and the scale21 is ðt= ln tÞ1=2 . Since now the shock width grows faster than the typical distance (i.e. temperature is relevant) this regime exists only strictly for m ! 0 before t ! 1, and in that case the limits do not commute, while they do for the long range (LR) class n < 1 ðh > 0Þ. For 2 > n > 1 one still has persistence of large eddies (persistence of tail of FRG function) but still the system flows to the SR (Kida) fixed point.22 This is the so-called Gurbatov phenomenon, which states that the velocity statistics is not fully scale invariant, and finds here a very natural interpretation in terms of the crossover from the LR to the SR FRG fixed point. The value n ¼ 2 corresponds to the Flory value f ¼ 4=5, and at short scale the SR correlator of the random potential is behaves effectively as dðxÞ  1=x, while at large scale it flows to the SR Kida FP. Finally, the analogous of the random periodic class f ¼ 0 correspond to Burgers in a periodic box which converges to a single random shock per period and EðtÞ  t 2 ðn ¼ 1Þ. Note that although the above discussion was for N ¼ 1, the phenomenology of the LR FP holds for any N, but the precise exponent values for the crossover from LR to SR depend on N. 4.5.2. Connection to FRG approach b ðv Þ To be more specific, we now switch to the notations of this paper. The renormalized potential V satisfies a KPZ type [91] equation:

 2 b ¼ m3 om V b ¼ To2 V b b 2ot V v  ov V

ð305Þ

where the time is t ¼ m2 , large time corresponding to small mass and to the universal region. Definb 0 ðv Þ, it obeys the Burgers equation: ing the renormalized force Fðv Þ ¼ V

ot Fðv Þ ¼

T 00 F ðv Þ  Fðv ÞF 0 ðv Þ 2

ð306Þ

here written for N ¼ 1, with the correspondence:

uðxÞ  Fðv Þ; t  m2 T viscous layer  thermal layer m ; 2

ð307Þ ð308Þ

and we will from now on switch freely between the two set of notations for time (inverse mass) and viscosity (temperature). The initial condition is precisely the bare potential (see Section 3.1.3 b m¼þ1 ðv Þ ¼ Vðv Þ, hence the Sinai random potential corresponds to random SR correlated initial force V 21 That is, for a SR correlated gaussian random potential V, with non-singular two point function at short scale lim!0 j ln j1=2 uðx=; j ln j1=2 t=2 Þ exists. 22 The case n ¼ 2 corresponds to the LR Flory value f ¼ 4=5, hence at short scale the SR correlator dðxÞ of the random potential behaves effectively as 1/x, while at large scale it remembers that it is short range and flows to the SR Kida FP.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

95

F(v), the case n ¼ 0 defined above. In the T ! 0þ inviscid limit we recall that tFðv Þ ¼ v  uðv Þ, where u(v) is position of minimum of HV ;v ðuÞ ¼ ðu  v Þ2 =ð2tÞ þ VðuÞ. Hence we expect (equivalently for n < 1 in the large t limit upon rescaling of lengths) shock solutions of the type (see Section 4.4.1 or below):

tF 0 ðv Þ ¼ 1 

X

ðsÞ

u21 dðv  v s Þ

ð309Þ

s ðsÞ

 where u21 ¼ u2  u1 ¼ uðv þ s Þ  uðv s Þ > 0 is the strength of the shock, i.e. the force (velocity) discontiðsÞ 0 þ nuity across it uð0 Þ  uð0þ Þ  Fðv  s Þ  Fðv s Þ ¼ u21 =t. Note that the term F ðv ÞFðv Þ in (306) becomes ill defined in the T = 0 inviscid limit. It does, however, possess a distributional limit, i.e. as a distribution, as discussed below. Note finally that the STS symmetry in the FRG corresponds to the galilean invariance of the decaying Burgers equation, Fðv ; tÞ ! Fðv þ a þ bt; tÞ  b (the stirred Burgers has a larger invariance which also involves the forcing). The FRG approach consists in writing from (306) the coupled RG flow (i.e. time evolution) equations for the moments of the ‘‘Burgers velocity field”:

Fðv 1 Þ::Fðv n Þ ¼ ðÞn S1...1 ðv 1...n Þ ðnÞ

ð310Þ

b . In particular the two point velocity correlation in involving the derivatives of the W-moments of V Burgers corresponds to the second cumulant of the renormalized force: ð2Þ

Fðv 1 ÞFðv 2 Þ ¼ R00 ðv 1  v 2 Þ ¼ S11 ðv 1 ; v 2 Þ

ð311Þ

00

i.e. uðxÞuðyÞ  R ðx  yÞ. They satisfy the following hierarchy of dynamical equations: ðnÞ

ot S1...1 ðv 12...n Þ ¼ n

i h i T h ðnÞ ðnþ1Þ S31:1 ðv 12...n Þ þ n S21:1 ðv 112...n Þ 2

ð312Þ

where [  ] denotes symmetrization with respect to the n arguments v i . These are well defined for T > 0 and can be obtained directly from (306) or by taking derivatives of (109). As detailed in Section b . In the 4.1 one may also study the hierarchy for connected W-cumulants, b S ðnÞ , or C-cumulants SðnÞ of V FRG it is more natural to study hierarchies of correlations of the potential, while in Burgers one usually focus on the force (velocity field). The former are usually less singular: as will be discussed below all ðnþ1Þ terms in (109) have a well defined limit for T = 0 while the term S21:1 ðv 112...n Þ ¼ ð1Þnþ1 F 0 ðv 1 ÞFðv 1 Þ:Fðv n Þ a priori is dominated by shocks since it involves F 0 ðv 1 Þ, as discussed below. These FRG functions can be obtained, whether at zero or non-zero T, from an a priori more fundamental obb 0 ðv i Þ which at the ject, namely the joint probabilities Pn ðv 1 ; F 1 ; . . . ; v n ; F n ÞdF 1 . . . dF n where the F i ¼ V fixed point should take the form:

Pn ðv 1 ; F 1 ; . . . ; v n ; F n Þ ¼ mð2fÞn pn ðmf v 1 ; mð2fÞ F 1 ; . . . ; mf v n ; mð2fÞ F n Þ leading to the moments:

ð1Þn S1...1 ðv 1 ; . . . ; v n Þ ¼ ðnÞ

Z

ð313Þ

dF 1 . . . dF n F 1 . . . F n P n ðv 1 ; F 1 ; . . . ; v n ; F n Þ

 ðnÞ  ¼ mð2fÞns1...1 mf v 1 ; . . . ; mf v n

ð314Þ

In this formula the points are supposed to be all distinct, and ordered. If some are repeated then the corresponding F i are raised to the proper power. Thus there is no more information in the P n than in the SðnÞ provided one includes their values at coinciding points. These are also the only quantities required for the disordered model. For instance, at T = 0 the one point probability of Burgers velocity PðF 1 Þ ¼ P1 ð0; F 1 Þ yields the distribution of the minimum u1 of the toy model. For the Sinai case, f ¼ 4=3 and n ¼ 0 these probabilities can be obtained in closed form (see Section 4.4.2 and Ref. [80]) as they satisfy the Markov property:

pn ðv 1 ; F 1 ; . . . ; v n ; F n Þ ¼ p1 ðF 1 Þ

n1 Y p2 ðv iþ1  v i ; F i ; F iþ1 Þ p1 ðF i Þ i¼1

ð315Þ

In general it is of course quite difficult to solve this hierarchy. Other hierarchies have been studied in  PN  Burgers turbulence, usually for the generating functions Z N ðv i ; ki Þ ¼ e i¼1 ki Fðv i Þ , GN ðF i ; v i Þ ¼

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DQ N

i¼1 hðF i

E  Fðv i ÞÞ , or PN ðv i ; F i Þ ¼ oF 1 . . . oF N GN ðv i ; F i Þ, with some (failed) attempts at ‘‘exact” closure

[82]. Another interesting case is the Kida model. This one is analyzed in Appendix I and it is recalled how to compute the two point force (or in Burgers, velocity) correlator. It also provides an interesting example of a fixed point function R(u) which can be explicitly computed, i.e. formula (I32) providing an example of a short range fixed point in d = 0. 4.5.3. Dissipation, viscous layer and inertial range Let us start with the first equation of the hierarchy and compare the information it carries in Burgers and in the FRG for the disordered model.

ot R00 ðv Þ ¼ TR0000 ðv Þ þ S112 ð0; 0; v Þ

ð316Þ

note that we have used the STS identity 2S211 ð0; 0; v Þ ¼ S112 ð0; 0; v Þ to transform the quantity appearing in (312) which a priori requires knowledge of derivatives in the TBL (and is dominated by shocks) into one which is defined in the outer region and has a T = 0 limit. Eq. (316) is the usual dynamical equation which in Burgers relate two and three point velocity correlations. At v ¼ 0 it yields:

ot R00 ð0Þ ¼ TR0000 ð0Þ

ð317Þ

an important identity encountered before (the Taylor expansion of a third moment can only start as S  v 6 at small v as a consequence of STS). In Burgers it expresses the decay of the energy density EðxÞ ¼ 12 uðxÞ2 on average:

ot E ¼  ¼ mðruÞ2 

T 0000 R ð0Þ 2

ð318Þ

 being the ‘‘dissipation rate”, i.e. the energy dissipated from viscosity small scales in the shocks. Note that here, in decaying burgers this rate is time dependent (i.e. m-dependent, see below). It is well known in Burgers-decaying and stirred, as well as in Navier–Stokes – that this rate has a finite limit as m ! 0 (or T ! 0). This is called the dissipative anomaly. It implies that derivatives of velocity field must become very large at small scales. As was discussed in Section 2.2, the equivalent statement in the FRG, i.e. that TR0000 ð0Þ ¼ 2

ð319Þ

implies the existence of a non-trivial thermal boundary layer (TBL). As detailed below the correlations in the TBL region v  Tt are determined by the fine structure of a shock (i.e. two points separated by v 21  Tt will typically either both be inside a shock or both outside). The dissipation occurs in the viscous layer and the dissipation rate (319) is thus a TBL quantity. Next, as discussed in Section 4.2.4, one can consider (316) for v ¼ Oð1Þ in the limit T ! 0, equivalently Tt  v (the so-called outer region). This corresponds to the inertial range, i.e. scales larger than Ld . On the small v side of this region, setting v ¼ 0þ in (316):

ot R00 ð0þ Þ ¼ S112 ð0; 0; 0þ Þ

ð320Þ

since the term involving T is smaller in that region. Now it appears as a general property that:

R00 ð0Þ ¼ Fð0Þ2 ¼ Fð0ÞFðv ¼ 0þ Þ ¼ R00 ð0þ Þ

ð321Þ

is continuous across the TBL, equivalently there is a well defined limit for the joint probability distribution of the force (velocity field) pT¼0 ðF 1 ; 0; F 2 ; v Þ ! pT¼0 ðF 1 ÞdðF 2  F 1 Þ. This is clear from (309), i.e. the force is discontinuous but remains bounded in one shock, so unless there is a strong accumulation of shocks near a point, the above continuity should hold when averaging over a uniform density of shocks at random positions. Identifying (320) and (316) one finds that:

S112 ð0; 0; 0þ Þ ¼ 2

ð322Þ

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This yields the celebrated Kolmogorov law for the third cumulant in the inertial range:

1 1 S111 ð0; 0; uÞ  u $ ðuðxÞ  uð0ÞÞ3  x 2 12

ð323Þ

Identical coefficients in (318) and (323) are a consequence of matching across the TBL (i.e. viscous layer). Similar relations exist in stirred Burgers (and Navier–Stokes) [73]: there the dissipation rate  is balanced by forcing instead of scale invariant time decay of correlations, but small scale shock properties should be rather similar. Let us give some simple consequences for Burgers of the existence of fixed points in the FRG in 2

n2

associated to an initial (kinetic) d = 0. The correlations of the random potential, V k V k  EðkÞ=k  k energy distribution E0 ðkÞ, are long range for n < 1 (and logarithmic for n ¼ 1). Using the disordered model notations, we know from FRG that there should be a LR fixed point where asymptotically  2 b ðuÞ  V b ð0Þ  ðVðuÞ  Vð0ÞÞ2  2rjuj2h=f with h ¼ 2ðf  1Þ, i.e. renormalized and bare asymptotics V should be the same, which implies that f ¼ 4=ð3 þ nÞ and h ¼ 2ð1  nÞ=ð3 þ nÞ. Hence one finds the law of energy decay:

R00 ð0Þ ¼

1 42f e 00 R ð0Þ; m 4

EðtÞ

1 e 00 1 e 00 ð0Þjt2ðnþ1Þ=ð3þnÞ j R ð0Þjt ð2fÞ ¼ j R 8 8

ð324Þ

where the prefactor is a universal function of r, e.g. for the Sinai case (uncorrelated initial velocities) e 00 ð0Þ ¼ 1:054238r2=3 using (298). Note that for the marginal case n ¼ 1 the disorwe have that  14 R dered model exhibits a freezing transition, hence the Burgers problem will also exhibit an interesting phase transition as a function of m, which can be studied using the results of Ref. [3]. As mentioned above the case n > 1 corresponds to short range disorder, and h < 0. The corresponding T = 0 FRG fixed point is thus unstable to temperature. It is dependent on the tail of the distribution of disorder, and related to extreme value statistics, but for the so-called Gumbel class (which contains the gaussian) it seems fairly universal, up to non-universal logarithmic corrections. This fixed point corresponds for a Gaussian disorder to the so-called Kida law in Burgers turbulence [88]. Its explicit form is recalled in Appendix I where the fixed point function R(u) is given in (I32). Next, the Gurbatov phenomenon occurs when RðuÞ  u1n for 1 < n < 2. In that case there is still a memory of a LR fixed point, which is unstable towards the SR fixed point. However, at any finite t, for 1 < n < 2 the renormalized function b will still decay as RðuÞ  u1n (conservation of tail in FRG, persistence of large eddies in Burgers). What happens is that this algebraic behaviour will hold only in the tail for v > v m where v m grows to infinity as t ¼ m2 ! 1. Such crossover between LR and SR are well known and have been studied within the  expansion (see e.g. [45]). Other quantities studied for Burgers, such as the dimensionless ‘‘velocity flatness” [75], have counterpart in the disordered model:

F ¼ lim

t!1

uðx; tÞ4 uðx; tÞ2

2



u4 u2

ð325Þ

2

and indeed F ¼ 2:83827 was derived [28] for the Sinai case n ¼ 0. 4.5.4. Shocks and droplets As discussed in Section 4.4.1, asymptotic (large time) solutions of the Burgers equation in the T = 0 inviscid limit are expected to be of the form:

tFðv ; tÞ ¼ v  ui

v i1 ðtÞ < v < v i ðtÞ

ð326Þ

where the v i ðtÞ are the positions of the shocks, and the ui the minima ui ¼ uðv ¼ uðv This assumes dilute (i.e. well separated) shocks, i.e. h=f < 1, i.e. f < 2 (faster growing correlations result in a function UðuÞ ¼ VðuÞ þ u2 =2m which differs almost everywhere from its convex envelope). One can then show [73] ballistic motion of shocks:  i Þ

v i ðtÞ ¼ bi t þ

ui þ uiþ1 þ Oð1=tÞ 2

bi being a constant discussed below.

þ i1 Þ.

ð327Þ

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At small T > 0 one expects each shock to be smooth in a layer of size t T. To find its shape one can directly look for a solution of the Burgers Eq. (306) of the form:

Fðv ; tÞ ¼

1 t



v



ui þ uiþ1 uiþ1  ui v  v i ðtÞ  / tT 2 2

ð328Þ

with /ð1Þ ¼ 1 to guarantee the boundary conditions for a single shock (326). One finds:

u þ u  uiþ1  ui 1 i iþ1 þ tot v i  v i /0 þ //0 þ /00 ¼ 0 2 2 2

ð329Þ

u u  The first term vanishes from (327) and one finds the unique solution /ðxÞ ¼ tanh iþ12 i x . This procedure can be pushed to any order in an expansion in t T and to the case of many shocks. In the regime where shocks are thin and dilute, i.e. when their width t T is much smaller than relative distance, the velocity (renormalized force) can be written as: ðsÞ

tFðv ; tÞ ¼ v

ðsÞ

b s  u21 tanh u21 ðv  v s ðtÞÞ D 2 2tT

! ð330Þ

where the shock parameters are denoted, from now on:

b i ¼ ui þ uiþ1 D 2 ðiÞ u21 ¼ uiþ1  ui

ð331Þ ð332Þ

which makes contact with the droplet notations of Section 4.2. To make further connection, let us recall that there we wrote a two well approximation:

  # "  ðv u1 Þ2 ðv u Þ2 1 1 Vðu2 Þþ 2t2 e ðv Þ T ln eT Vðu1 Þþ 2t V þe T ¼



ðv  v s Þ2 V 21 u21 ðv  v s Þ ðv  v s Þ  T ln 2 cosh þC þ 2tT 2t u21

ð333Þ

where C is v independent and V 21 ¼ Vðu2 Þ  Vðu1 Þ. The position of the shock is given by equality of the two terms v s ¼ ðu1 þ u2 Þ=2 þ tV 21 =u21 . This allows to identify the shock velocity bi ¼ V 21 =u21 . Hence shocks with zero velocity correspond to exact degenerate states in the bare disorder potential. Taking a derivative of (333) one recovers (330). Note from (330) that the width of the shock is really L0d ¼ Dv  tT=u21  T=DF. Hence although an individual shock broadens with time as Tt, when there is a collection of (dilute) shocks (such as for random initial conditions with f < 2) they merge upon collision. As a result their typical separation u21 (see below) grows as u21  tf=2 and their width hence grows only as Tt 1f=2 . The dimensionless ratio decays even faster Ld =t f=2  Tt 1f as discussed above. At the same time the shock amplitudes decay with time as DF  u21 =t  t ð1f=2Þ . Let us now use the assumption of a (small) uniform density of shocks q, to compute the equal time velocity correlations at closeby points. Let us first consider:

tðFðv 2 Þ  Fðv 1 ÞÞ ¼ v 21 

u u21  21 tanh 2 2

v 2  v s tT

 tanh

u 21 2

v 1  v s  tT

ð334Þ

This expression holds if there is at most one shock in the common neighborhood of v 1 and v 2 , and we will consider v 21  1=q. We now average over this shock position, with measure qdv s (or R 1=ð2qÞ q2 1=ð2 qÞ dv s to be more specific). This yields:

tFðv 2 Þ  Fðv 1 Þ ¼ v 21 ð1  q

Z

1

du21 pðu21 Þu21 Þ

ð335Þ

0

where pðxÞ denotes the normalized probability that for a given shock the parameter u21 ¼ x. We have   R1 used the identity 1 daðtanh zþa  tanh 2aÞ ¼ 2z, and in averaging (334) we assumed tT q=u21  1 2 (dilute shocks) hence we can push the integration to infinity in the terms containing the tanh. Since

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for statistically translational invariant initial conditions the above average (335) must be zero (by the STS symmetry) it implies:

q

Z

1

du21 pðu21 Þu21 ¼ 1

ð336Þ

0

hence the shock parameter u21 is also the typical distance between shocks. We now compute the single shock contribution to the correlator of the force. Taking the square of (334) and averaging over the shock position, we expand the square and use the above calculation to evaluate the cross terms. Using

Z

þ1

1

h z þ a a i2 da tanh ¼ 16F 2 ðzÞ  tanh 2 2

ð337Þ

where F 2 ðzÞ was introduced in (153), we finally obtain:

R00 ðv 21 Þ  R00 ð0Þ ¼

1 2T ðFðv 2 Þ  Fðv 1 ÞÞ2 ¼ q 2 t

Z

du21 pðu21 Þu21 F 2

u

21

v 21  v 221

tT



2t 2

ð338Þ

The second term can be dropped since it is subdominant in the region v 21  tT=u21  qtT that we are studying. This result, obtained here from Burgers, can be compared to the result (153) obtained from the disordered model. It is consistent, provided

e pðyÞ y DðyÞ R1 ; ¼ R1 0 0 e 0Þ y pðy Þ ðy0 Þ2 Dðy 0

y>0

ð339Þ

0

e using hy2 iy ¼ 2t and (336) we have defined DðyÞ ¼ DðyÞ þ DðyÞ ¼ 2DðyÞ for y ¼ u21 > 0. This relates the shock size distribution pðs ¼ u21 Þ to the droplet size distribution DðyÞ. This relation was analyzed in the Sinai case n ¼ 0; f ¼ 4=3 at the end of Section 4.2.5, where the universal ratios where computed. Although it makes sense that shocks statistics should be generally related to droplets since a shock is nothing but a droplet with exact degeneracy, it remains as a tantalizing question to generalize the relation (339) to higher N and d. 4.5.5. Inviscid limit In Burgers (and Navier–Stokes) one is particularly interested in the inviscid limit m ! 0, equivalent to the limit T ! 0 in the FRG. As discussed above, for h > 0; n < 1, this limit is also the relevant one for the dynamics in Burgers and for the flow of the FRG. An important question is thus whether it is possible to construct directly this inviscid limit without solving the complete TBL or viscous layer problem, but keeping only the minimal information from its structure. Equivalently, in the FRG, whether one can compute directly the T = 0 beta function, all ambiguities resolved. Let us recall the analysis of Ref. [77,78] for N ¼ 1. In the limit m ! 0 the derivative F 0 ðv Þ of the Burgers velocity field (309) becomes a distribution. Using the limit of the single shock profile:

Fðv Þ ¼

v  u^s t

ðsÞ



u21 ðv  v s ðtÞÞ 2t

ð340Þ

with ðxÞ ¼ hðxÞ  hðxÞ, one sees that quantities such as Fðv ÞF 0 ðv Þ are ill defined directly at m ¼ 0: they are not distributions since the test functions for distributions should be infinitely differentiable w.r.t v. However, one can define them as:

Fðv Þn1 F 0 ðv Þ :¼

1 ov ðFðv Þn Þ; n

1 ekFðv Þ Fðv ÞF 0 ðv Þ :¼ ok ov ekFðv Þ k

ð341Þ

In each case the r.h.s. is a perfectly legitimate distribution at m ¼ 0, and the relations are evidently true for any m > 0. One can then verify explicitly using (340) and (327) that:

1 ot Fðv Þ þ Fðv ÞF 0 ðv Þ :¼ ot Fðv Þ þ ov Fðv Þ2 ¼ 0 2 holds in the sense of distributions at

m ¼ 0. It is not a fully trivial statement since by contrast:

ð342Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Fðv Þot Fðv Þ þ Fðv Þ2 F 0 ðv Þ :¼

1 1 ot Fðv Þ2 þ ov Fðv Þ3 –0 2 3

ð343Þ

This is because of the neglected term mFðv ÞF 00 ðv Þ ¼ 12 mo2v Fðv Þ2  12 mF 0 ðv Þ2 . While the second term has a (zero) distributional limit as m ! 0 the second does not due to the dissipative anomaly field: ðsÞ 1 X u21 lim mF ðv Þ ¼ m!0 12 s t 2

0

!3

dðv  v s ðtÞÞ

ð344Þ

as found from direct integration of (330) around the shock. The following general equation was then shown to hold:

!

ðsÞ 1 2 X kbs ku21 dðv  v s ðtÞÞ ot þ kok ov ekFðv Þ ¼  e G k k s 2t

ð345Þ

with GðxÞ ¼ x cosh x  sinh x and bs the shock velocity. This can be checked directly using (330) and taking the limit m ! 0. Expanding in k one obtains all the anomalies, which contain information about the shock form factor, i.e. the distribution of shock sizes and velocities. It turns out that the dissipative anomaly field can be rewritten using left and right shock velocity, which leads to a very simple and elegant form:



1 ot Fðv Þ þ ðFðv þ dÞ þ Fðv  dÞÞF 0 ðv Þ ekFðv Þ ¼ 0 2

ð346Þ

where the limit d ! 0 is implicit and selects at each shock position the left or right velocity. Note that to lowest order in k one recovers indeed (342) since obviously Fðv þ dÞF 0 ðv Þ þ F 0 ðv þ dÞFðv Þ ¼ 2ov ðFðv þ dÞFðv ÞÞ ¼ 2ov Fðv Þ2 in the distribution sense (i.e. integrated with a test function). The form (346) of the inviscid Burgers equation is physically very natural since it describes convection and that the true shock velocity is the half sum of left and right one. Note that these results invalidate the attempts at closure of Ref. [82] since closures necessarily involve non-trivial information about shocks as also discussed for stirred Burgers [92]. Let us now come to the natural T = 0 limit of the FRG hierarchy (312). Given that it comes from a KPZ like [91] equation, it is thus natural to define it to be:

ot Fðv 1 Þ:Fðv n Þ ¼ 

i nh ov n Fðv 1 Þ:Fðv n ÞFðv n Þ 2

ð347Þ

If we assume that all force (i.e. Burgers velocity) correlation functions Fðv 1 Þ:Fðv n Þ are continuous functions of their arguments, which is expected to hold at least for dilute shocks ðf < 2Þ and can be checked explicitly from the exact solution in the case of Sinai landscape f ¼ 1=2 ðn ¼ 0Þ given in Section, then the r.h.s of (347) is well defined. As explained in Appendix G.3 iterative truncation of this hierarchy is one of the several methods used to obtain the T = 0 beta function given in (259). For N ¼ 1 it does not generate any ambiguity, as was checked up to four loop. If we compare with (342) and (346) we can now understand why the T = 0 FRG based on (347) does work. Indeed this proh i ðnÞ cedure is exactly the one performed in that case. The limits of S1:12 ðv 1 ; . . . ; v n ; v n þ 0þ Þ computed in h i the inertial range for m > 0 coincide with the values 12 ov n Fðv 1 Þ:Fðv n ÞFðv n Þ in the direct m ¼ 0 solution. As a last application of the Burgers-FRG correspondence let us note that the shock form factor controls the small distance behaviour of the moments of the velocity difference in the inviscid limit (equivalently for m > 0 in the inertial range). Since the jump across the shock is DF ¼ u21 =t a simple one shock calculation yields:

ðFðv Þ  Fð0ÞÞp  lp v signðv Þpþ1 tp Z 1 X ðsÞ p lp ¼ q du21 up21 pðu21 Þ ¼ ðu21 Þ dðv  v s Þ 0

ð348Þ ð349Þ

s

which can also be expressed in terms of droplet distribution, i.e. lp ¼ hjyjpþ1 i=hy2 i, e.g. consistent with R000 ð0þ Þ  uð0þ Þruð0Þ ¼ l2 =ð2t 2 Þ given above. One can check for instance, using the full TBL form given in section, that (setting t ¼ m2 ¼ 1):

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

ðFð0Þ  Fðv ÞÞ4 ¼ 2Q 1111 ð0;0;0;0Þ  8Q 1111 ð0;0;0; v Þ þ 6Q 1111 ð0;0; v ; v Þ 1 1 ð3v~ yð9cosh ðv~ y=2Þ þ coshð3v~ y=2ÞÞ  27sinh ðv~ y=2Þ  11sinh ð3v~ y=2ÞÞ ¼ Ty4 3 24 sinh ðv~ y=2Þ   1 ¼ Thy8 iy v~ 4 þ O v~ 6 280 1 ¼ hjyj5 iy jv j þ Oðv 2 Þ 2

ð350Þ ð351Þ ð352Þ ð353Þ

~ ¼ v =T, respectively. for small and large v 4.6. Ballistic aggregation of shocks and closures As discussed in Section 4.2.1 there exist an exact RG Eq. (143) relating Pðu1 Þ the probability that in the disordered model the absolute minimum is at u1 and Dðu1 ; u2 Þ the (droplet) probability density that there are two degenerate absolute minima at u1 and u2 . It is physically reasonable since as m is varied the absolute minimum can change by abrupt switch events whose probability is governed by D. An exact solution for this equation (together with the additional STS relation) was given in (187) and (197) for the Sinai RF class ðf ¼ 4=3Þ. In general, however, these equations (FRG and STS) may not be sufficient to fully determine D (and hence P), and one may need another FRG equation for D itself. Such equation would then involve the probability of three degenerate absolute minima, and it is then not clear whether this set of equations would close. We now see that, at least in the Sinai case, one can close this hierarchy. This closure is related to the ballistic aggregation dynamics of shocks. Let us first recall the expected scaling with time t ¼ m2 :

Pt ðu1 Þ ¼ tf=2 Pðu1 t f=2 Þ; Pt ðu1 Þ ¼ g~ t ðu1 Þg~t ðu1 Þ; g~t ðu1 Þ ¼ t

f=4 ~

g ðu1 t

f=2

Þ;

  Dt ðu1 ; u2 Þ ¼ Pt ðu1 ; u2 ; 0Þ ¼ t12f D u1 t f=2 ; u2 t f=2 ~t ðu1 ; u2 Þg~t ðu1 Þ b t ðu1 ; u2 Þ ¼ g~t ðu1 Þd D

ð355Þ

~t ðu1 ; u2 Þ ¼ t13f2 dðu ~ 1 t f=2 ; u1 t f=2 Þ d

ð356Þ

ð354Þ

where in the last two lines we have put the solution in the same form as in the Sinai case, for which ~ 1 ; u2 Þ ¼ e121 u31 121 u32 dðu21 Þ is proportional to the probability that if the absolute minimum is in u1 then dðu there is a second one in u2 . It turns out that its time dependent version satisfies the following remarkable FRG equation:

~ 1 ; u3 Þ ¼ tot dðu



 2 3f ~ f ~ 1 ; u3 Þ ~ 1 ; u3 Þ  3 fu31 u1 þ u3 dðu dðu1 ; u3 Þ  ðu1 ou1 þ u3 ou3 Þdðu 1 2 2 8 2 Z 1 ~ ; u Þu dðu ~ ;u Þ  du2 u21 dðu 1 2 32 2 3 2

ð357Þ

The first line is simple scaling, but the second represents the event of three degenerate minima, which amounts to switch between two valleys (i.e. two sets of two degenerate minima), and ~ 1 ; u2 Þu32 dðu ~ 2 ; u3 Þ is the usual weight as allows to close the equation. The weight factors u21 dðu discussed in Section 4.4.2, see e.g. Eq. (277). (357) is equivalent to the following equation for dðuÞ:

tot dðu31 Þ ¼



1

Z 3f f f 3 1 0 du2 u21 dðu21 Þu32 dðu32 Þ dðu31 Þ  u31 d ðu31 Þ þ u31 dðu31 Þ  2 2 32 2

ð358Þ

and in both equations the term tot vanishes at the fixed point, i.e. when scale invariance holds exactly. For the Sinai case f ¼ 4=3 one explicitly check that it is equivalent to:

2 0 1 3 1 u dðuÞ ¼ ðudðuÞÞ  ðudðuÞÞ dðuÞ  ud ðuÞ þ 3 24 2

ð359Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

where  means convolution, which indeed holds because of the following identity of Airy functions:



0 0 0 000 " 0 0 #2 0 Ai ðzÞ 2 Ai ðzÞ 1 Ai ðzÞ Ai ðzÞ  ¼ þ z AiðzÞ 3 AiðzÞ 6 AiðzÞ AiðzÞ

ð360Þ

R 0 using that dðuÞ ¼ ða=bÞ z ezu=b Ai ðzÞ=AiðzÞ. Three minima degeneracies in the Burgers setting is related to collision of shocks. It is well known that the dynamics of shocks for N ¼ 1 is simple ballistic aggregation. It was studied in Ref. [81] where some exact results where obtained, and we now make contact with notations of that paper. One denotes M ¼ l ¼ u21 and P ¼ g ¼  2t1 ððu1  v Þ2  ðu2  v Þ2 Þ, respectively, the mass and the momentum of a shock at position v. In between collisions the motion of a shock v ðtÞ is ballistic, i.e. the   2  v ðtÞ is constant in time, where the ui are time independent. The shock velocity V ¼ P=M ¼ 1t u1 þu 2 collision process is related to the three well droplet. Let us call ðu1 ; u2 Þ the first shock at v ðtÞ and ðu2 ; u3 Þ the second shock at v 0 ðtÞ > v ðtÞ with u1 < u2 < u3 . Neighboring shocks share a minimum of the random potential u2 . Collision occurs at the time such that v ðtÞ ¼ v 0 ðtÞ and just amounts to erase u2 and can hence be seen as a decimation process. It is characterized by three conservation laws (ballistic aggregation), as detailed in Ref. [81], M ¼ M 1 þ M 2 ; P ¼ P 1 þ P 2 ; ðV 1  V 2 Þt ¼ M2 , which are exactly equivalent to the operation of erasing u2 . Indeed:

M 1 þ M 2 ¼ u21 þ u32 ¼ u31 ¼ M  1 2   1 2 1 2 P1 þ P2 ¼ u  u21 þ u  u22 ¼ u  u21 2t 2 2t 3 2t 3 1 1 1 ðV 1  V 2 Þt ¼  ðu1 þ u2 Þ þ ðu2 þ u3 Þ ¼ ðu3  u1 Þ ¼ M=2 2 2 2

ð361Þ ð362Þ ð363Þ

we set v ¼ 0 the shock collision position, and t is time of the collision. We can now look at formula (122) of Ref. [81] which describes the statistics of the ballistic aggregation process in terms of a probability weight IðM; P; tÞ of aggregating particles. One can then check that

 

~t ðu1 ; u3 Þ ¼ It M ¼ u31 ; P ¼ 1 u31 u1 þ u3  v ðtÞ d t 2

ð364Þ

and that (122), which makes more explicit the conservation laws of the dynamics, is fully equivalent to (357) if one also assumes scale invariance, leading to the same explicit solution in terms of Airy function (working out the jacobian in the collision integral in (122) simply replaces M by M 1 M 2 and recover the measure as in (357). To conclude, one may wonder, as do the authors of Ref. [81], whether similar closing procedure could work for other values of f, an open problem. There seem to be, however, a one to one correspondence between the problem of N ¼ 1 ballistic aggregation and the FRG, i.e. m dependent solution of the disordered model. It would be of high interest to understand what type of aggregation process occur for N > 1 and d > 0 (functional shocks). 4.7. Other solvable models Note that the FRG fixed point function can also be computed for the random periodic model in d = 0. It produces some interesting results. Another case amenable to analytical results is the fully connected model (or the large d limit). Both are studied in Appendix N. 5. Higher dimension We now study FRG for pinned manifolds in d > 0. One outstanding question is whether it can be controlled near d ¼ duc , where duc ¼ 4 for the standard type of elasticity studied here. This being a difficult question we proceed by first trying to extend what we have learned from d = 0 in previous sections. We study the general non-zero T case and examine the important issue whether a T = 0 limit can be constructed, i.e. an ‘‘inviscid” limit for the decaying functional Burgers equation equivalent to the FRG. In Section 3.2 the W½j and C½u functionals and their associated W and C-cumulants (R[u]), . . . where introduced and related. Here we start by giving the ERG equations that they satisfy, and discussing a few other constraints coming from STS. Next we obtain a ‘‘droplet” solution to the functional hierarchy valid in any d. Finally, we discuss in more detail the -expansion.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

103

5.1. ERG equations The derivation of the ERG equation is a simple extension of the d = 0 case presented in Section 4.1. Upon infinitesimal change og of the bare propagator matrix g in (80 and 81) the functional W½j satisfies the standard W-ERG equation:

! 2 1 dW½j dW½j 1 d W½j oW½j ¼  Trog þ 2T djdj dj dj

ð365Þ

P P where og 1 ¼ g 1 ogg 1 and here and below TrM ¼ a trMaa and trM ¼ x M xx . Upon the change of d d variable (84) which implies Tg dv ¼ dj one obtains the W-ERG equation for W½v . Separating the bare part from the interacting one (i.e. due to disorder) in (89):

W½v  ¼

1 X a 1 b c v g v þ W ½v  2T axy x xy y

ð366Þ

it can be written as:

c ½v  ¼ T oW 2

X a

"

c c dW c d2 W dW tr og þ dv a dv a ov a ov a

!# ð367Þ

up to a constant proportional to the number of replicas (the linear term cancels upon the change from j to v, i.e. Wðv Þ ¼ Wðj ¼ g 1 v =TÞ). In a similar fashion the effective action functional C½u:

1 X a 1 b b u g u  C ½u 2T axy x xy y X 1 X b ½u ¼ 1 C R½uab  þ S½uabc  þ    2 2T ab 3!T 3 abc

C½u ¼

ð368Þ ð369Þ

satisfies the C-ERG equation:

b ½v  ¼ oC

" #1 b ½v  1 d2 C Trogg 1 1  Tg 2 dv dv

ð370Þ

c and C b in number of replica up to a constant proportional to the number of replicas. Expanding W sums from (89) and (90) yields ERG equations for the p-replica connected cumulants functionals b b R; S n n P 3 and R, SðnÞ , respectively. These are very similar to the d = 0 ones, except that all functions now become functionals, and all derivatives become functional derivatives. Alternatively one can start from the RG equation obeyed by the renormalized potential in a given sample:

b ½v  ¼ oV

" # b ½v  d V b ½v  d V b ½v  1 d2 V  tr ogðT 2 dv dv dv dv

ð371Þ

from which one easily obtains the linear functional equation for the moments:

" # n X T d2 SðnÞ ½v 1 ; . . . ; v n  oS ½v 1 ; . . . ; v n  ¼ tr og 2 dv i dv i i¼1 " # n d2 Sðnþ1Þ ½v 1 ; v 01 ; v 2 :v n  þ sym1...n tr og j v 01 ¼v 1 2 dv 1 dv 01 ðnÞ

ð372Þ

where, as before:

b ½v 1 : V b ½v n  ¼ ð1Þn SðnÞ ½v 1 ; . . . ; v n  V

ð373Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

The lowest order relates the second moment functional Sð2Þ ½v 1 ; v 2  ¼ R½v 1  v 2  to the third:

"

#    v d2 R½ dS½v 1 ; v 2 ; v  þ tr og oR½v  ¼ Ttr og jv 1 ¼v 2 ¼0 dv dv dv 1 dv 2

ð374Þ

b and b the same equation being valid for the connected cumulants R S, see below. A standard choice is 2 1 2 1 g k ¼ k þ m in Fourier, and o ¼ mom . Then one has og xx0 ¼ 2m2 dxx0 and og ¼ 2m2 g 2 . In d = 0 setting g ¼ m2 one then recovers the equations given in Section 4.1. We now introduce more convenient notations for functional derivatives:

dR½v  ¼ R0x ½v ; dv x

d2 R½v  ¼ R00xy ½v ; dv x dv y

dS½v 1 ; v 2 ; v 3  ¼ S110 xy ½v 1 ; v 2 ; v 3  dv 1 dv 2

ð375Þ

and so on. We now recall the relations between the W-moments (over-bars) W cumulants (hat) and C cumulants, up to the fourth cumulant, derived in Appendix C:

b ¼ R; R¼R

S¼b S

ð376Þ

S½v 123  ¼ S½v 123   g xy ðR0x ½v 12 R0y ½v 13  þ R0x ½v 23 R0y ½v 21  þ R0x ½v 31 R0y ½v 32 Þ

ð377Þ

b ½v 1234  þ R½v 12 R½v 34  þ R½v 13 R½v 24  þ R½v 14 R½v 23  Q ½v 1234  ¼ Q

ð378Þ

0 00 0 b ½v 1234   12sym b100 Q ½v 1234  ¼ Q 1234 g xy S x ½v 123 Ry ½v 14  þ 6sym1234 g xy g zt Rxz ½v 12 ðRy ½v 13 

 R0y ½v 23 ÞðR0t ½v 14   R0t ½v 24 Þ 100 0 00 00 b ½v 1234   12sym ¼Q 1234 g xy Sx ½v 123 Ry ½v 14   6sym1234 g xy g zt Rxz ½v 12 ðRy ½v 13 

 R00y ½v 23 ÞðR00t ½v 14   R00t ½v 24 Þ

ð379Þ

Here and below repeated indices are contracted unless stated otherwise (or the trace notation is used). Using these relations can now write the W-ERG equations for second and third moments (and cumulant) as:

oR½v  ¼ Tog xy R00xy ½v  þ og zz S110 zz0 ½0; 0; v 

ð380Þ

3 3 1100 ð381Þ Tsym123 og xy S200 xy ½v 123  þ sym123 og xy Q xy ½v 1123  2 2 3 3 0 0 b 1100 ¼ Tsym123 og xy S200 xy ½v 123  þ sym123 og xy Q xy ½v 1123  þ 3sym123 og xy Rx ½v 12 Ry ½v 13  ð382Þ 2 2 and the corresponding C-ERG equations as:   oR½v  ¼ Tog xy R00xy ½v  þ og zz0 g xy R00xz ½v R00yz0 ½v   2R00xz ½0R00yz0 ½v  þ og zz0 S110 ð383Þ zz0 ½0; 0; v  oS½v 123  ¼

oS½v 123  ¼

3 00 00 Tsym123 og xy S200 xy ½v 123  þ 3Tsym123 og xy g zt Rxz ½v 12 Ryt ½v 13  2   3 110 1100 þ 6sym123 og xy g zt R00xz ½v 12  S110 yt ½v 113   Syt ½v 123  þ sym123 og xy Q xy ½v 1123  2 

þ 3sym123 og xy g zt g rs R00xz ½v 12 R00yr ½v 12 R00st ½v 13  þ 2R00xz ½v 12 R00st ½v 12 R00yr ½v 13   R00xz ½v 12 R00yr ½v 23 R00st ½v 13 

ð384Þ

where in the last formula we have introduced the notation:

R00xy ½u ¼ R00xy ½u  R00xy ½0

ð385Þ

It is important to note that up to now we have written the W and C-ERG equations at non-zero temperature T > 0, i.e. assuming analyticity of the functional at coinciding points (certainly correct for a finite number of degrees of freedom, i.e. a finite size system). The quantity R00xy ½0 is then well defined.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

105

Solving these functional equations seem hopeless beyond an -expansion. We show, however, in the following that an exact solution can be found in the thermal boundary layer. Before doing so let us give some further definitions and exact constraints on correlation functions. 5.2. Local part and non-local part of the functionals It is useful in the following to note that the functional R½v  can be split unambiguously into a local part and a non-local one:

R½v  ¼

Z

e v Rðv x Þ þ R½

ð386Þ

x

such that the non-local part vanishes for a uniform configuration:

e v z ¼ v g ¼ 0 R½f

ð387Þ

this is in agreement with the definition of the local part R(v) given in previous sections, R½fv z ¼ v g ¼ Ld Rðv Þ. As a consequence:

e 00 ½v  R00xy ½v  ¼ R00 ðv x Þdxy þ R xy Z 00 e R xy ½v jfv z ¼v g ¼ 0

ð388Þ ð389Þ

y

and the second line can also be used to specify the local part, i.e. the function R(v) up to a constant. It was obtained taking two derivatives of (387) and using translational invariance. A similar decomposition exists for all higher moments SðnÞ and SðnÞ functionals. It may sometimes be useful to further split the non-local part in multilocal components, and this is discussed in Appendix M in [94]. 5.3. Correlation functions Here we give relations between two and four point correlations and the renormalized disorder cumulants, as well as some exact relations satified by these correlations. These are extensions of the relations presented in Section in d = 0 apart from subtleties related to space arguments. 5.3.1. Two point functions There are only two distinct two point functions:

Gxy ¼ huax uby i ¼ Gxy ¼ g xx0 g yy0 R00x0 y0 ½0

ð390Þ

e xy ¼ G

ð391Þ

huax uay i

¼ Gxy þ Tg xy

b ¼ RÞ functional at zero. It implies that these derivboth related to the second derivative of the R ð¼ R atives, here R00xy ½0 must be well defined at T > 0 and that the limit:

lim R00xy ½0

ð392Þ

T!0þ

should exist for finite system size and be related to the second moment of the configuration u1 ðxÞ of minimum energy GT¼0 xy ¼ u1 ðxÞu1 ðyÞ. 5.3.2. Four point functions There are five possible four point connected correlations. However, they depend on only two functions: 2 0000 huax uay uaz uat ic ¼ huax uay uaz ubt ic ¼ g xx0 g yy0 g zz0 g tt0 ðQ 1111 x0 y0 z0 t0 ½0  T Rx0 y0 z0 t 0 ½0Þ

ð393Þ

2 0000 huax uay ubz ubt ic ¼ g xx0 g yy0 g zz0 g tt0 ðQ 1111 x0 y0 z0 t 0 ½0 þ T Rx1111 y1111 z1111 t 1111 ½0Þ

ð394Þ

huax uay ubz uct ic

¼

huax uby ucz udt ic

¼

g xx0 g yy0 g zz0 g tt0 Q 1111 x0 y0 z0 t 0 ½0

ð395Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

As for d = 0 this is obtained using that connected correlations are tree graphs from effective action, and P that b Gab oub ¼ Tgoua from the above form of the exact two point function and the STS property. Notation here is again that different replica indices mean distinct replicas. Replica symmetry is assumed.23 Note that these functions have higher permutation symmetry (full with respect to the space points) than can naively be inferred from replica symmetry alone. This is explained below. It is useful to give explicitly also the disconnected parts:

e xy G e xz G e xt G e zt þ G e yt þ G e yz ; huax uay uaz uat idisc ¼ G

e xy Gzt þ G e xz Gyt þ Gxt G e yz huax uay uaz ubt idisc ¼ G

e xy G e zt þ Gxz Gyt þ Gxt Gyz ; huax uay ubz ubt idisc ¼ G

e xy Gzt þ Gxz Gyt þ Gxt Gyz huax uay ubz uct idisc ¼ G

huax uby ucz udt idisc

ð396Þ

¼ Gxy Gzt þ Gxz Gyt þ Gxt Gyz

5.3.3. STS identities One can prove [36] the general STS identity:

T

X c

g xy

  X dO½u ¼ hO½uuxf i y duc f

ð397Þ

for an arbitrary functional O½u. Choosing, respectively, O½u ¼ uay uaz uat ; O½u ¼ uay uaz ubt ; O½u ¼ uay ubz uct one obtains: a a a a a a a b e zt þ g G e e Tðg xy G xz yt þ g xt G yz Þ ¼ hux uy uz ut i  huy uz ut ux i

ð398Þ

e yz Þ ¼ hua ua ua ub i þ hua ua ub ub i  2hua ua ub uc i Tðg xy Gzt þ g xz Gyt þ g xt G x y z t y z x t y z x t Tðg xy Gzt þ g xz Gyt þ g xt Gyz Þ ¼

huax uay ubz uct i

þ

huax uaz uby uct i

þ

huax uat uby ucz i



ð399Þ 3huax uby ucz udt i

ð400Þ

We have used replica symmetry to relabel some indices. It turns out that the l.h.s of these equations (which do not possess full symmetry with respect to spatial indices) are simply the disconnected parts of the r.h.s. Thus these equations are equivalent to the property that the connected parts of the r.h.s. must be zero. One can check that they are indeed obeyed by the above parameterization in terms of R and Q. 5.3.4. ERG identities Similarly one can prove [36] ERG identities directly on correlations:

ohO½ui ¼ 

1 X ðuf og 1 uf ÞO½ui 2T f

ð401Þ

They yield:

 2Tohuax uay i ¼ hðua og 1 ua Þuax uay i  hðua og 1 ua Þubx uby i

ð402Þ

 2Tohuax uby i ¼ hðua og 1 ua Þuax uby i þ hðua og 1 ua Þuay ubx i  2hðua og 1 ua Þubx ucy i

ð403Þ

a where here and below we denote ðua og 1 ua Þ ¼ uaz og 1 zt ut . These equations are not independent, since, subtracting the second to the first yields an identity always true (using the results of the previous paragraph, the connected parts of the r.h.s. cancel and the rest simplifies). One can easily see that either correlation ERG identity is equivalent, using (391) and (395) to the C-ERG identity:

oR00zt ½0 ¼ Tog xy R0000 xyzt ½0

ð404Þ

which generalizes the relation (28) to any d. It can be separately shown from (383) using the fact that 4 a b c c S110 xy ½0; 0; v  starts at small v only as v (indeed the generic term S ¼ ux uy uz ut can be excluded by STS, 6 see the appendix of Ref. [36], so that S starts at u , as in d = 0-and a similar property for S). The relation 1 between (403) and (404) is obtained noting that oR00xy ½0 ¼ oðg 1 xz g yt Gzt Þ and that, from (403): 23

This is automatically satisfied for a finite size system and continuous distribution of disorder.

P. Le Doussal / Annals of Physics 325 (2010) 49–150 1 oGzt ¼ Tog xy g zz0 g tt0 R0000 xyz0 t0 ½0  og x0 y0 ðg x0 z Gy0 t þ g x0 t Gy0 z Þ

107

ð405Þ

Since, as discussed above, the l.h.s. of (404) should have a limit as T ! 0, it again suggests some scaling v  T for the functional thermal boundary layer. In the next section we present a solution of the functional hierarchy which admits such a scaling. Taking the local part of (404) one has:

oR00 ð0Þ ¼ T

Z z

og xy R0000 0xyz ½0

ð406Þ

hence the local part Rðv Þ should exhibit a TBL similar to d = 0, but not identical since the r.h.s. of this equation cannot be expressed in terms of the local part alone. 5.4. A droplet solution to the functional hierarchy We now obtain an exact solution of the full ERG functional hierarchy in the thermal boundary layer region v  T, inspired by the droplet picture. We do not claim that this is necessarily the unique ‘‘correct” solution. The droplet picture serves as a heuristic method to find such a solution, and there are some assumptions, detailed below, which go into this construction. It is quite possible that a more complex solution based on a more complex (and realistic) picture can be constructed in the future. It is already very interesting that an exact solution can be found to this highly non-trivial hierarchy. In viewing the FRG in higher d as a decaying functional Burgers equation, this droplet picture holds in what could be also called a ‘‘dilute functional shocks” scenario. 5.4.1. Structure of droplets Let us first give a qualitative description. Consider a sample of volume Ld and keep the mass m fixed. It is simplest to think (and draw) the case of the directed polymer d ¼ 1, although we consider general d. Let us call u1x the ground state configuration in a given sample (i.e. disorder environment). It is assumed to be unique for continuous disorder probability distributions. We call a droplet a configuration u2x which is close in energy from the ground state, the energy difference being E ¼ HV ½u2   HV ½u1 , i.e. a quasi-degenerate state, where HV ½u is defined in (80). To be qualified as an active droplet E should be of order T, where T is small and fixed. One calls the size (volume) of the droplet the distance (volume) over which it differs from the ground state. There are of course ðiÞ many such droplet configurations, especially of small sizes, and we label them u2x and their energy ðiÞ difference with the ground state E . At any temperature T the Gibbs measure is split between the ground state and the active droplets. The main assumption within the droplet picture is that large droplets are rare. More precisely, the h probability p to find a droplet with energy E (fixed, of order T) of size l < Ld < l þ dl is p  El dl=l. Thus in a sample of size L will most often contain no (active) droplet of size of order L (e.g. of size between k L and L where k < 1 is a fixed number), and rarely will contain one, with probability  TLh . When this occurs, the probability that one of the two quasi-degenerate states contains another droplet of size of the same order is again vanishingly small. Thus there are no droplets within droplets at large scale. This is a very important assumption. For h > 0 it breaks down below some small scale, and for marginal glasses, h ¼ 0, it breaks down at all scales. In that case there is indeed a finite probability that a droplet contains another one of size one order of magnitude less (or a factor k fixed), resulting in a tree-like structure of droplet excitations. One may surmise based on results for Cayley trees [90], that in that case replica symmetry breaking occurs (for h ¼ 0, d = 0 and any N evidence for it was obtained in [3]. In this section, however, we consider h > 0 and assume rare, non-overlapping, droplets. Here we consider a geometry with fixed m and Ld very large (i.e. mL 1). Thus one can consider that the system is roughly cut in N  ðLmÞd independent pieces (and samples) of internal volume md where in each u fluctuates (from sample to sample) of order mf (i.e. the ground state can be assumed to be uncorrelated over internal distances larger that 1/m). The droplets in each piece are thus also uncorrelated (over distances larger than 1/m). We will call ‘‘elementary” droplets the ones in each piece. In a given sample there are few of them, i.e. only a few of the pieces contain an active droplet. In the limit considered here, of small Tmh , their density is small, of order Tmh . For these elementary droplets we denote:

108

P. Le Doussal / Annals of Physics 325 (2010) 49–150 ðiÞ

ðiÞ

u12x ¼ u1x  u2x

ð407Þ d

and consider that this quantity is non-zero only over a region of size m . The elementary droplets in a given sample are assumed to be well separated along the directed polymer (or manifold). At this stage we consider only droplets of volume of order md . We ignore here questions arising from possible accumulation of very small droplets. 5.4.2. Droplet calculation and thermal boundary layer form We now implement these assumptions in a calculation. Extending the d = 0 arguments presented in Section 4.2.1, the renormalized potential can be written:

e



T1 b V ½v b V ½0



¼e

1 2T

P

g 1 xy v x v y

* 1P e

xy

T

g 1 xy ux v y

+

ð408Þ

xy

HV ½u

where HV ½u is defined in (80). For partition function:

P

he xy

~ g 1 xy ux v y

v ¼ T v~ the average in the r.h.s. can be evaluated from the droplet

X v~ g1 u1  e

iHV ½u Z 1 d

ni ¼0;1 ðv~ g ¼ Z 1 d e

1 u

1Þ

P

ðiÞ

ni ðv~ g 1 u12 Þ1T

i

P

ni Ei

i

 Y Ei ~ 1 ðiÞ 1 þ e T ðv g u12 Þ

ð409Þ ð410Þ

i

P P 1 Here Z d ¼ ni ¼0;1 eT i ni Ei is the partition sum of active elementary droplets and we use the notation  1  P v~ g u1 ¼ xy v~ x g1 xy u1y when convenient. Thus one has:

  X   X ~ 1 ðiÞ e ½fv x g  V e ½fv x ¼ 0g ¼ 1 ðv g 1 v Þ  T v~ g 1 u1  V T ln 1 þ wi eðv g u12x Þ þ T lnð1 þ wi Þ 2 i i

ð411Þ with wi ¼ eEi =T . It is convenient to work with the renormalized force:

b 0 ½v  ¼ g 1 T v~ y  u1;y  V x xy ~ðiÞ a

   ðiÞ ¼ exp  v~ g 1 u12

X uðiÞ ~ðiÞ 12;y wi a ~ðiÞ 1 þ wi a

!!

ð412Þ

i

ð413Þ

We now compute disorder averages. For that we need to make minimal assumptions which generalize the d = 0 analysis. One first denotes P½u1  the (functional) probability for the ground state configuration. Next one defines a droplet probability functional D½u1 ; u2 . Contrarily to d = 0 there can be ðiÞ

ðiÞ

several elementary droplet configurations u2 ¼ u1  u12 , thus one writes symbolically D½u1 ; u2  ¼ i P h ðiÞ i Di u1 ; u2 . These functionals can be derived from a more general droplet functional h n oi h i ðiÞ ðiÞ P u1 ; u2 ; EðiÞ . One calls Pi u1 ; u2 ; EðiÞ the single droplet functional (all others droplets variables h i h i ðjÞ ðiÞ ðiÞ u2 ; EðjÞ with j–i integrated out). One has, as in d = 0, Di u1 ; u2 ¼ P i u1 ; u2 ; EðiÞ ¼ 0 . Since elementary droplets are assumed to be independent, upon computing disorder averages, only one active elementary droplet at a time need be considered. Terms involving two simultaneously active elementary droplets give contributions of higher order T 2 . Hence to compute the disorder averaged force we can now use formula (138) and obtain: ðiÞ X uðiÞ 12;y wi a i

1 þ wi aðiÞ

¼T

* X

+   ~ 1 ðiÞ ðiÞ u12;y ln 1 þ eðv g u12 Þ

i

* + T X ðiÞ  1 ðiÞ  ~ u12;y v g u12 ¼ 2 i

ð414Þ

D

ð415Þ

D

¼ T v~ y

ð416Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

P D h ðiÞ iE Here and below the droplet average is defined as hA½u12 iD ¼ i A u12 according to the previous Di line we have assumed, paragraph. We will, however, usually drop the index i on Di . In the second ðiÞ ðiÞ and used the local symmetry u12 ! u12 of the droplet probability distribution, which generalizes (141). In the last line we have used the STS identity:

X ðiÞ ðiÞ hu12;x u12;y iD ¼ 2g xy

ð417Þ

i

which generalizes (134). Since hu1 iP ¼ 0 by parity, one correctly recovers (to lowest order in T) that V 0x ½v  ¼ 0. We can now compute the second cumulant:

b 0 ½v 1  V b 0 ½v 2  ¼g 1 g 1 hu1y u1y i þ T V x1 y1 x2 y2 x1 x2 1 2 P u1y2 

! ðiÞ ðiÞ u12;y2 wi a2 ðiÞ

1 þ wi a2

XZ i

1 0

dwi wi #+

 u1y1 u1y2

*"

ðiÞ

u1y1 

þ D

ðiÞ

u12;y1 wi a1

!

ðiÞ

1 þ wi a1

ðiÞ ðjÞ ðjÞ X uðiÞ 12;y1 wi a1 u12;y2 wj a2 ðiÞ

i–j

1 þ wi a1

ðjÞ

1 þ wj a2

1 A þ OðT 2 Þ ð418Þ

Elementary droplets being independent averages can be decoupled in the last term, which is thus of order T 2 . Since it involves an extensive double sum one could fear that it could contain an additional factor of volume Ld and be of order TLd as compared to the dominant one, hence not negligible. This is not the case, and in fact, up to adding and subtracting the i ¼ j term (which is clearly subdominant), it ~ terms. The algebra is now similar to the case d = 0 and we find: exactly cancels against the T v

b 0 ½v 1  V b 0 ½v 2  ¼ R00 ½v 1  v 2  ¼ R00 ½0 þ Tg 1 g 1 V x1 x2 z1 z2 z1 z2 z1 x1 z2 x2

XD i

ðiÞ

ðiÞ

u12x1 u12x2 F 2



ðiÞ

u12 g 1 v~ 12

E D

ð419Þ ~ 12 ¼ v ~1  v ~ 2 and the function F 2 ðzÞ ¼ 4z coth 2z  12 was defined in Section 4.2.1. Thus the droplet where v picture yields a prediction for the exact second cumulant functional directly related to the droplet probabilities. One can easily infer from (419), or derive independently, that in the functional thermal ~ ¼ v =T  Oð1Þ one has: boundary layer v

R½v  ¼

E X D  1 ðiÞ v x R00xy ½0v y þ T 3 H2 v~ g 1 u12 D 2 i

ð420Þ

where the function H2 ðzÞ was defined in (159). Thus the second cumulant functional predicted from this independent droplet picture has a simple structure. It is an average of a functional which is simply  1  P d ~ x g 1 ~ g u12 ¼ xy v a function of the quantity v xy u12y . It is nicely proportional to L for a uniform conðiÞ figuration, since each term contains a deformation u12 non-zero only within a fixed volume md .  1  R ~ the argument in the function F 2 (or H2 ) is v ~ g u12 ¼ m2 v ~x ¼ v ~ x u12x . For a uniform configuration v For the usual massive propagator the result can be written, in Fourier:



R00q ½v  ðq2 þ

m2 Þ2

  Z  Z 1 1 2 ¼ hu1q u1;q iu1  T u12;q u12;q u12k ðk þ m2 Þv~ k coth u12p ðp2 þ m2 Þv~ p 4 2 p k D ð421Þ

which includes a generalization of (155). The calculation of all higher moments can be performed similarly. Some details are given in Appendix J. We quote here only the result for the third moment:

   1 1 1 1 ~ S111 v 1 Þ; ðu12 g 1 v~ 2 Þ; ðu12 g 1 v~ 3 Þ D z1 z2 z3 ½v 1 ; v 2 ; v 3  ¼ Tg z1 x1 g z2 x2 g z3 x3 u12x1 u12x2 u12x3 F 3 ðu12 g



1 2 F 3 ½z1 ; z2 ; z3  ¼ z 1 F ½ z 1  z 2 ; z1  z 3   þ 2p:c: 4 3 aþb

ð422Þ

1þe where we recall F½a; b ¼ ð1e a Þð1eb Þ. Note that the notion of a TBL in high d may contain some momentum dependence if one chooses v x non-uniform, a property which remains to be investigated.

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

5.4.3. Check that ERG equations are obeyed Since it is quite amazing to obtain a solution to a functional hierarchy, we will check explicitly at least the first equation of the W-ERG hierarchy:

oR½v  ¼ Tog xy R00xy ½v  þ og xy S110 xy ½0; 0; v 

ð423Þ

The two terms which must be added in the r.h.s. have the explicit form, using the droplet solution (419) and (422):

D  E R00xy ½v  ¼ R00xy ½0 þ T ðg 1 u12 Þx ðg 1 u12 Þy F 2 v~ g 1 u12 D D  1

 1 E 2 1 ~ S110 ½0; 0; v  ¼ T g u ðg u Þ G v g u 12 x 12 y 3 12 xy

ð424Þ ð425Þ

D

2

4

z z where one finds G3 ½z ¼ 24  F 2 ½z  1440 þ Oðz6 Þ. Hence the two term cancel and, up to a constant, one is left with a single term, a quadratic term v2, of order T 2 in the TBL. Hence in the TBL the above equation becomes oR½v  ¼ 12 oR00xy ½0v x v y and the only thing left to check is:

oR00zt ½0 ¼

  E X D 1 ðiÞ ðiÞ ðiÞ ðiÞ g 1 u12 ðg 1 u12 Þy ðg 1 u12 Þz g 1 u12 ¼ lim Tog xy R0000 og xy xyzt ½0 T!0 x t D 12 i

ð426Þ

One can check that the second identity holds, as a consequence of (419). Hence the last equation to check is the identity (404) but this one can be proved exact independently of droplets, as consequence of ERG and STS identities, as discussed in the previous section (see also below). Note that Eq. (426) is the generalization of the dissipation rate or anomaly equation in decaying Burgers turbulence. ~ ). Hence if Hence we have found that oR00xy ½v  is constant in the whole TBL (i.e. from v ¼ 0 to large v ~ x then the functional second derivative should be continmatching holds between small v x and large v uous in the whole range. It is useful to give the local version of (426):

oR00 ð0Þ ¼

*

2 +   Z X  1 4 d ðiÞ ðiÞ ðiÞ g 1 u12 g 1 u12 u12t m L og xy x y 12 t i

ð427Þ

D

¼

1 6 d m L 6

* X Z x

i

ðiÞ

ðiÞ

u12x u12x

Z t

ðiÞ

u12t

 P R

2 + ¼ D

1 4 m 3

i

R 2 ðiÞ ðiÞ ðiÞ u u u x 12x 12x t 12t  2  P R ðiÞ dxu12 ðxÞ i

 D

ð428Þ

D 2

00 2 where in the second equality we have restricted to the form g 1 k ¼ k þ m and o ¼ mom . Again R ð0Þ þ 00 is continuous and equal to R ð0 Þ.

5.4.4. Matching and relations between shocks and droplets ~ ! 1. In d = 0, for Rðv Þ, this nicely matches to the small v limit of the outWe now study the limit v ~ 13 and v ~ 23 go to infinity, it matches the er region v ¼ Oð1Þ. For higher moments, such as Sðv 123 Þ, as v partial boundary layer (here PBL21). Let us examine what happens here. ~ . From (419) we find for large v ~: ~x ¼ v Let us consider first a uniform v

*Z 3 +  1 00 1 d 6 3 X  ðiÞ  ðxÞ R ð0Þv 2 þ L m jv j dxu 12   2 24 i D    P R 3 ðiÞ i  dxu12 ðxÞ 1 1 4 3 D  ¼ R00 ð0Þv 2 þ m jv j 2  R P 2 12 ðiÞ dxu12 ðxÞ i

Rðv Þ ¼

ð429Þ

ð430Þ

D

where we have used the normalization of the droplet measure given by the STS symmetry:

* X Z i

ðiÞ dxu12 ðxÞ

2 + D

¼ 2Ld m2

ð431Þ

P. Le Doussal / Annals of Physics 325 (2010) 49–150

111

Hence we obtain the cusp as the following droplet average:

 3  P R  ðiÞ ðxÞ dxu   12 i 1 4 þ 000 D  R ð0 Þ ¼ m 2  R P 2 ðiÞ dxu12 ðxÞ i

ð432Þ

D

On the other hand, within some assumptions, one can also show that [72]:

R000 ð0þ Þ ¼

1 4 hs2 iP m 2 hsiP

ð433Þ

R where s ¼ dxu12 ðxÞ are the shock sizes. This is compatible with a relation between shock and droplet size distributions which generalizes the d = 0 relation (339), with, in d dimension the droplet size R ðiÞ y ! Y ¼ dxu12 ðxÞ.  2 ~ as the term ~ ¼ Oð1Þ limit of (423). From (425) it behaves as O v Indeed let us consider the large v 0000 TR ½v  becomes negligible in that limit, and the coefficient should equal the r.h.s. of (426). Indeed in the TBL, from (422):

lim S112 xyzt ½0; 0; v  ¼

v~ !1

  1 X  1 ðiÞ   1 ðiÞ   1 ðiÞ   1 ðiÞ  g u12 g u12 g u12 g u12 x y z t D 12 i

ð434Þ

On the other hand, matching, i.e. considering the small v ¼ Oð1Þ limit of (423) and making the usual assumption that the term TR0000 can be neglected in that region, requires: þ 112 og xy S112 xyzt ½0; 0; 0  ¼ lim og xy Sxyzt ½0; 0; v 

v~ !1

ð435Þ

a generalization of the anomaly equation in Burgers turbulence (matching of dissipation rate from dissipation range to inertial range). Since one can also show that the third moment of the shock size distribution is given by [72]:

Z xzt

þ S112 0xzt ½0; 0; 0  ¼

1 6 hs3 iP m 6 hsiP

ð436Þ

the continuity (435) implies that the same quantity is related to the fourth moment of droplet sizes, hence it is again compatible with the generalization of the droplet-shock relation (339). It would be quite interesting to investigate further the relation between droplets and shocks, in particular in the non-local, momentum-dependent aspects. While a shock always correspond to a droplet (with exact degeneracy), it is not fully clear how the reverse works, i.e. given a droplet at low T, does it correspond to a unique shock nearby in phase space, and which v x then to choose to find this underlying shock. 5.4.5. STS and ERG droplet identities Let us close this Section by mentioning another useful check of the droplet solution. One can compute using droplets all four point correlations given in Section 5.3 and check that all STS and ERG identities are indeed obeyed. This is performed in Appendix K. Here let us just mention that the STS identities can be encoded in the following functional RG equation which relate P½u1  and D½u1 ; u2 :



dP½u1  ¼ du1x

Z

Du2 g 1 xy ðu1y  u2y ÞD½u1 ; u2 

ð437Þ

which, upon integration, generate an infinite set of identities between correlation functions, the first one being the famous hu12;x u12;y iD ¼ 2g xy . The ERG identities yield:

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

oP½u1  ¼ 

1 2

Z

Du2 ððu1 og 1 u1 Þ  ðu2 og 1 u2 ÞÞD½u1 ; u2 

ð438Þ

These are the functional generalization of the equations given in d = 0.24 It is interesting to present these equations since one expects, if the  expansion makes sense, that P½u1  be nearly gaussian near d ¼ 4, and it may inspire other approaches to this expansion (in particular one wonders what simplification occurs then, if any, in the shape of D½u1 ; u2 ). 5.5. T = 0 limit and -expansion 5.5.1. T = 0 limit and continuity properties The ERG Eqs. (380) and (383) in Section 5.1 were derived for analytic moment and cumulant functions, and as such they are always exact for T > 0. The zero temperature limit T ¼ 0þ (alternatively the small m limit Tmh ! 0) of these equations necessitates a careful analysis. We will follow closely what was learned in d = 0. Consider the W-ERG Eq. (380) and naively set T = 0. One is left with the hierarchy:

oR½v  ¼ og zz0 S110 zz0 ½0; 0; v  3 oS½v 123  ¼ sym123 og xy Q 1100 xy ½v 1123  2

ð439Þ ð440Þ

and so on. The questions are (i) what is the meaning of the functions evaluated at coinciding arguments on the right-hand side (ii) can this hierarchy be closed (iii) can it be solved. The last question is about the  expansion and is examined in the next section. To answer (i), we note that these equations are expected to be correct for small T > 0 and v in the outer region, i.e. v ¼ Oð1Þ T, all v ij ¼ 0ð1Þ T for i – j. Indeed in that region one may assume that the terms proportional to T in (380) are negligible (they become of the same order only in the TBL region v  T; v ij  T discussed in the previous section). Of course this is an assumption but it is supported by the analysis of the previous section. It extends the d = 0 analysis to the functional. As a consequence the meaning of the derivatives appearing in the r.h.s. of (439) is:

lim S110 zz0 ½0; 0; v 

ð441Þ

T!0

and similarly for all members of the hierarchy. These are perfectly well defined quantities, but defined in the TBL, since the derivatives are exactly at coinciding arguments. The next question (ii) then is can the hierarchy be closed, i.e. is there really a hierarchy valid at T ¼ 0þ ? Or, in the language of Burgers turbulence, is there a hierarchy defined solely in the inertial range (by ‘‘closed” here we do not mean truncated-this is the business of the  expansion, we mean defined only in terms of quantities involving the outer region itself). For this one needs continuity properties, as was the case also in d = 0. Since here we deal with a functional we may need to distinguish: (a) the strong continuity property:

lim Sðn;T¼0Þ1100:0 ½v 1234...n  exists for arbitrary xy

v 2 !v 1

v 21x ! 0

ð442Þ

and ¼ lim Sðn;TÞ1100:0 ½v 1134...n  xy T!0

ð443Þ

(b) the weak continuity25 property:

24 Note that these equations, as their d = 0 counterpart, where found on the basis of all analytic correlation functions. If they hold they should also imply relations for averages of non-analytic observables (such as juj). 25 Note that we have termed these properties strong and weak for pure convenience and in any relation to a customary sense in which these terms are used in mathematics.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

lim Sðn;T¼0Þ1100:0 ½v 1234...n  exists for xy

v 12 !0

v 21x ¼ v 12 ! 0

113

ð444Þ

and ¼ lim Sðn;TÞ1100:0 ½v 1134...n  xy

ð445Þ

T!0

Hence the limit may exist uniformly for arbitrary argument, or for a spatially uniform argument. Of course (a) implies (b) and is a stronger property. For n ¼ 2 it reads:

lim R00T¼0;xy ½v  ¼ lim R00T;xy ½0 strong continuity

v x !0

lim

v x ¼v !0

T!0

R00T¼0;xy ½

v ¼

lim R00T;xy ½0 T!0

weak continuity

ð446Þ ð447Þ

These continuity properties are necessary for the ERG equation to be globally closed, i.e. to relate quantities computable in the outer region, hence defining a meaningful T ¼ 0þ W-ERG hierarchy. The continuity of the second derivative of Eq. (448) is necessary for the calculation of the two point correlation function ux uy . In both cases, weak continuity is sufficient (b) but we believe that in fact strong continuity (a) usually holds. Indeed one has, directly at T = 0:

ðv 1x  ux ½v 1 Þðv 2y  uy ½v 2 Þ ¼ g xz g yt R00zt ½v 12 

ð448Þ

where ux ½v  is the ground state configuration of the interface in an harmonic well centered on v  fv x g. Although we cannot prove it, it is hard to imagine from the picture of shocks that the left-hand side is not a continuous function with a unique limit as v 12x ! 0, at least for a finite number of degree of freedom. This is basically the same argument that in d = 0 (see Section 4.5.5) that b 0 ½v  and ux ½v  have jumps at discrete locations as v is varied, here in the space of configurations. UnV x less there is some accumulation of shocks it seems unlikely that continuity in arbitrary moments of b 0 ½v  and ux ½v  should fail. This implies that all W-moment functionals Sðn;T¼0Þ11:1 ½v 1234...n  should be V xy x continuous in their arguments and equal to the T ¼ 0þ limit of the same function in the TBL at exactly coinciding points. What about the C-ERG hierarchy? From the definition (377) one finds:

   110 000 0 0 00 00 00 00 S110 zt ½v 123  ¼Szt ½v 123  þ g xy Rxzt ½v 12  Ry ½v 23   Ry ½v 13   Rxt ½v 12 Ryz ½v 13   Rxz ½v 21 Ryt ½v 23   þR00xz ½v 31 R00yt ½v 32  ð449Þ thus to recover the first C-ERG equation in (383) setting T = 0 one needs the regularity condition:

   0 0 lim g xy R000 ¼0 xzt ½v 12  Ry ½v 23   Ry ½v 13 

v 12 !0

ð450Þ

since it vanishes by parity in the TBL, and continuity of R00xy ½v . Hence for the first C-ERG equation to be valid for all v  fv x g weak continuity of R00 as well as the weak version of (450) should be sufficient. There are very similar continuity, or regularity conditions to be satisfied for the second C-ERG equation in (383) to be valid. Hence if we make the minimal assumption of weak continuity and regularity we should be able to use the C-ERG equation directly at T = 0. 5.5.2.  Expansion Let us now investigate the  ¼ 4  d expansion using the C-ERG hierarchy. A version using the W-ERG hierarchy is given in Appendix L. Let consider the C-ERG hierarchy and set T = 0. The two lowest order equations read:

oR½v  ¼ og 0zz g xy ðR00xz ½v R00yz0 ½v   2R00xz ½0R00yz0 ½v Þ þ og zz0 S110 zz0 ½0; 0; v  and

ð451Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

  3 110 1100 oS½v 123  ¼ 6sym123 og xy g zt R00xz ½v 12  S110 yt ½v 113   Syt ½v 123  þ sym123 Q xy ½v 1123  2   þ 3sym123 og xy g zt g rs R00xz ½v 12 R00yr ½v 12 R00st ½v 13  þ 2R00xz ½v 12 R00st ½v 12 R00yr ½v 13   R00xz ½v 12 R00yr ½v 23 R00st ½v 13  ð452Þ

The spirit of the  expansion is that there is a solution of the ERG hierarchy which has the following e v , of the functional structure. Let us recall the definition of the local part R(v), and non-local parts R½ R[v] as defined in (387) and (389). The  expansion states that:

e v   Oð2 Þ R½

Rðv Þ  OðÞ; S½v 123   Oð Þ; 3

ð453Þ

Q  Oð4 Þ

and so on SðnÞ  Oðn Þ for n P 3, these statements being valid for the properly rescaled fixed point forms (see details below). Examination of the structure of the hierarchy shows superficially compatibility with this counting. We must now check that it works and is unambiguous. Here we will distinguish again the strong  expansion in which the counting (453) is obeyed for any v ¼ fv x g and the weak  expansion in which the counting (453) is obeyed for uniform configuration v or infinitesimally close to it, i.e. by all derivatives evaluated at a uniform configuration. Note that the weak version is sufficient to compute all correlations of the ux field at zero temperature (or at the fixed point) hence it is perfectly respectable. We now study the  expansion. We focus on the hard question of ambiguities and anomalous terms, as in d = 0. Once these are understood the rest, i.e. rescaling and derivation of the FRG equation is easy and similar to what was done in Ref. [55], hence we will not detail that part. Our aim here is to give a first principle derivation of the anomalous terms, or at the very least specify clearly what the assumptions are, which was not done in Ref [55]. There a candidate field theory was proposed, based on some global consistency. 5.5.3. One loop: order OðÞ Following the strategy (453) outlined above, to study R½v  to one loop, i.e. to lowest order in , we can discard the S term in (451) and obtain:

oR½v  ¼ og zz0 g xy ðR00xz ½v R00yz0 ½v   2R00xz ½0R00yz0 ½v Þ þ Oð3 Þ

ð454Þ

e v   Oð2 Þ. Of course these assumptions Let us insert the decomposition (389) and assume that R½ (453) must be checked a posteriori self-consistently. One finds:

oR½v  ¼ og xy g xy R00 ðv x ÞR00 ðv y Þ þ Oð3 Þ

ð455Þ

The local part gives:

oRðv Þ ¼

Z

og y g y R00 ðv Þ2 þ Oð3 Þ

ð456Þ

y

R R 2 Upon evaluation of y og y g y ¼ 12 oJ 2 with J 2 ¼ k ðk þ m2 Þ2 and rescaling this yields the standard one loop FRG equation and fixed point for the rescaled R of order OðÞ. It is important to note that R J 2 ¼ y g 2y  m = has a pole in  which disappear as the derivative oJ 2 is finite in d ¼ 4, producing a finite b-function as it should for a renormalizable theory. Using the fixed point value of R00 ð0Þ and weak continuity yields the T = 0 correlation function. Its general expression is:

e 00 ½0 ux uy ¼ g xz g yt R00yt ½0 ¼ g xz g yz R00 ð0Þ  g xz g yt R yt

ð457Þ

and the knowledge of the local part R00 ð0Þ only gives it at q ¼ 0, i.e. the center of mass fluctuations, as R u u ¼ R00 ð0Þ=m4 . y x y Let us now study the equation for the non-local part which would result from (454).

e v  ¼ ðog g  dxy og g ÞR00 ðv x ÞR00 ðv y Þ þ Oð3 Þ o R½ xy xy z z

ð458Þ

P. Le Doussal / Annals of Physics 325 (2010) 49–150

115

which integrates into:

e v  ¼ 1 ðg 2  dxy g 2 ÞR00 ðv x ÞR00 ðv y Þ þ Oð3 Þ R½ z 2 xy

ð459Þ

here (456) has been used, and the fact that the term oR terms should produce only higher order terms in . The corresponding formula for the second derivative is:

e 00 ½v  ¼ ðg 2  dts g 2 ÞR000 ðv t ÞR000 ðv s Þ þ dst ðg 2  dty g 2 ÞR0000 ðv t ÞR00 ðv y Þ þ Oð3 Þ R ts ts z ty z

ð460Þ

and no summation on s, t. Hence for a uniform configuration:

e 00 ½v  ¼ ðg 2  dts g 2 ÞR000 ðv Þ2 þ Oð3 Þ R ts ts z

ð461Þ 000

þ 2

000

 2

as the second term automatically vanishes. Since R ð0 Þ ¼ R ð0 Þ weak continuity then produces an unambiguous limit:

e 00 ½0 ¼ ðg 2  dts g 2 ÞR000 ð0þ Þ2 þ Oð3 Þ R ts ts z

ð462Þ

and yields the T = 0 correlation function at any momentum q through (457):



Z uq uq0 ¼ g 2q R00 ð0Þ þ g k ðg qþk  g k Þ R000 ð0þ Þ2 þ Oð3 Þ

ð463Þ

k

which is the result displayed in [55] where some (rough) discussion of the ambiguities was also given. R Note the important fact that g 2ts is in Fourier k g k g qþk  1= and that the divergence is removed by the R local part (counterterm), i.e. only the difference g 2ts  dts k g 2k is finite, and produces a finite result in 26 d ¼ 4 for the correlation. There is, however, a feature of the result for the non-local part which at first sight may appear disturbing. Since R00 ðv Þ  jv j at small v Eq. (459) appears incompatible with the existence of an unambige 00 ½0 for the non-local part to the lowest order in . More precisely, the limit of uous second derivative R xy 00 e R xy ½v 12  as v 12 ! 0 is unambiguous and equal to (462) only for non-crossing configurations v 1x and v 2x , i.e. such that v 12x remains of the same sign for all x. Since these are interface configurations ðN ¼ 1Þ it also mean they are partially ordered, a concept familiar from depinning [56]. Hence there seem to be two mutually incompatible properties: (i) strong epsilon expansion (ii) strong continuity. At least one of them must fail. Faced with this dilemma we prefer to assume that strong continuity holds, as argued in the previous section. Then it means that formula (459) and (460) cannot be correct in the counting in  if evaluated for non-uniform configurations v x , more precisely for configurations with at least one sign change. In fact one can check that even as v x ! 0 in a subspace of configurations with sign changes at fixed positions, the first term in the r.h.s of (460) retains a complicated momentum structure which may indeed be incompatible with the epsilon counting, while the second term exhibits delta function singularities at the points where v t vanishes (since R0000 ðv Þ ¼ 2dðv ÞR000 ð0þ Þ þ R0000 ð0þ Þ as can be checked by integrating over a small region containing v ¼ 0). Note that this does not mean a failure of the  expansion since as we found above to one loop the weak epsilon expansion is perfectly fine and is sufficient to give the correlation functions of ux . It just means that the T = 0 functional hierarchy can be easily solved using the usual  counting only in a neighborhood of uniform configurations v x ¼ v þ dv x . This neighborhood can be infinitesimal, such that all derivatives of arbitrary order taken at a uniform configuration will satisfy the usual  expansion counting, or one may try to extend it to partially ordered configurations v 1 . . . v p with no intersections. In Appendix M of Ref. [94] exact equations are written using the multilocal expansion, and confirm that the weak epsilon expansion does work to one loop. We check below that it appears to work also to two loop. A non-trivial question is to understand exactly how the formula (459) and (460) fail for non-uniform configurations. There must clearly be contributions of the same order in  coming from higher order terms. There are then two possibilities. Either (454) remains true and corrections only come from the non-local part itself, or it fails and corrections also come from the third moment term S in (451). 26

As d ! 4 g 2x becomes proportional to a a delta function dx to leading order.

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One reason for which we are confident that indeed (459) fails for (sign changing) non-uniform configurations is that it contradicts the TBL solution which was shown in the previous sections to be an exact (and fully non-perturbative in ) solution of the hierarchy (order by order in T). More precisely ~ limit of (426) and (423) is clearly incompatible with the naive small v limit of (459) for the large v (sign changing) non-uniform configurations. Hence terms of the same order at small v must be hiding in the neglected terms of (451). 5.5.4. Droplet relations to order OðÞ It is interesting to note that if indeed R000 ð0þ Þ is of order , then from (432) and (431) one expects that:

Z

ðiÞ

dxu12 ðxÞ  OðÞ;



X 1 h1iD  O 2 i

ð464Þ



R ðiÞ and the typical value dxu12 ðxÞ  mdþf with the TBL variable for a uniform R . ðiÞ e. dxu12 ðxÞ T  ðv mf Þ= T

v

being z ¼ m2 v

In addition to the local relation (432), the assumption that (459) is correct for a uniform v, i.e. that (461) holds implies a few relations involving droplets which should be valid to leading order in . Matching the flow of R00xy ½0 inside and outside the TBL implies (for x–y:

2og xy g xy R000 ð0þ Þ2 ¼

1 hðu12 og 1 u12 Þðg 1 u12 Þx ðg 1 u12 Þy iD þ Oð3 Þ 12

ð465Þ

to be valid to leading order in , not only the amplitude, but also the spatial dependence-the local part ~ reproduces the relation (432). Matching of the term proportional to Oðjv jÞ from (461) and the large v behaviour from the non-local part of (419) yields:

2ðog x g x  dðxÞ

Z

og y g y ÞR000 ð0þ ÞR0000 ð0þ Þ

ð466Þ

y

 

Z Z   m2 ¼ ðg 1 u12 Þ0 ðg 1 u12 Þx  dðxÞ ðg 1 u12 Þ0 ðg 1 u12 Þy  u12z  þ Oð3 Þ 4 y z D

ð467Þ

a relation which has no local analogous. Note that testing these relations would be as much a check of the structure of the droplet solution (after all, we have considered only the simplest minded one), than a check of the weak epsilon expansion. 5.5.5. One loop: third moment Let us now compute the third moment to lowest order in the  expansion. We need to solve to lowest order, keeping the dominant term in (452):   oS½v 123  ¼ 3sym123 og xy g zt g rs R00xz ½v 12 R00yr ½v 12 R00st ½v 13  þ 2R00xz ½v 12 R00st ½v 12 R00yr ½v 13   R00xz ½v 12 R00yr ½v 23 R00st ½v 13  The solution is:

  S½v 123  ¼ sym123 g xy g zt g rs 3R00xz ½v 12 R00yr ½v 12 R00st ½v 13   R00xz ½v 12 R00yr ½v 23 R00st ½v 13  þ Oð4 Þ Here again we have assumed that oR is higher order. The leading behaviour is expected to be:

  S½v 123  ¼ sym123 g xy g xt g yt 3R00 ðv 12x ÞR00 ðv 12y ÞR00 ðv 13t Þ  R00 ðv 12x ÞR00 ðv 23y ÞR00 ðv 13t Þ þ Oð4 Þ

ð468Þ

with validity a priori either for v x near a uniform configuration (weak epsilon expansion) or any v x (strong epsilon expansion). As discussed below only the first one presumably holds. Since S ¼ S one can compute, from (468): 1 1 1 S111 zz0 z00 ½v 123  ¼ g zs g z0 s0 g z00 s00 ðv 1s  us ½v 1 Þðv 2s0  us0 ½v 2 Þðv 3s00  us00 ½v 3 Þ

for

v 1; v 2

and

v 3 partially ordered (weak epsilon expansion).

ð469Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

5.5.6. Two loop: order Oð2 Þ To go to next order in  we must now study:

oR½v  ¼ og zz0 g xy R00xz ½v R00yz0 ½v  þ og zz0 S110 zz0 ½0; 0; v 

ð470Þ

e 00 ½v R00 ðv z0 Þ þ og zz0 S110 ¼ og xy g xy R ðv x ÞR ðv y Þ þ 2og zz0 g xz0 R xz zz0 ½0; 0; v  00

00

ð471Þ

at this stage there is no approximation. The second term is the feeding from the non-local part of the second moment. We will now insert (459), but we need:

e 00 ½v   R e 00 ½v ¼ 0 e 00 ½v  ¼ R R xz xz xz

ð472Þ

we need to use only the weak continuity property:

e 00 ½v ¼ 0 ¼ lim R e 00 ½fv y g ¼ v  R xz xz þ

ð473Þ

v !0

One then obtains the contribution of the non-local term:

   e 00 ½v R00 ðv z0 Þ ¼ 2og zz0 g xz0 g 2  dxz g 2 R000 ðv x ÞR000 ðv z Þ  R000 ð0þ Þ2 R00 ðv z0 Þ 2og zz0 g xz0 R xz xz t   þ 2og xz0 g xz0 g 2xy  dxy g 2t R0000 ðv x ÞR00 ðv y ÞR00 ðv z0 Þ

ð474Þ

For the feeding from the third moment we use the result (468). It is shown in Appendix M that:

    S½v 123  ¼ v 12z v 12y g xy g yz g zx R000 ðv 12y ÞR000 ðv 12z ÞR00 ðv 13x Þ  R000 ðv 13z ÞR000 ðv 13y ÞR00 ðv 13x Þ þ O v 312 ð475Þ where we have used that R ðv Þ ¼ R ðv Þv þ Oðv Þ. Hence: 00

000

2

000 000 00 000 000 00 S110 zz0 ½v 123  ¼ 2g xz0 g z0 z g zx ðR ðv 12z0 ÞR ðv 12z ÞR ðv 13x Þ  R ðv 13z ÞR ðv 13z0 ÞR ðv 13x ÞÞ þ Oðv 12 Þ

ð476Þ We use again the weak continuity: 110 og zz0 S110 zz0 ½0; 0; v  ¼ limþ Szz0 ½v 123 jv 13x ¼v x ;v 12x ¼w

ð477Þ

w!0

¼ 2og zz0 g xz0 g z0 z g zx ðR000 ð0þ Þ2  R000 ðv z ÞR000 ðv z0 ÞÞR00 ðv x Þ

ð478Þ

Putting all together we find the two loop FRG equation for the functional:

oR½v  ¼ og xy g xy R00 ðv x ÞR00 ðv y Þ þ þ

2og zz0 g xz0 ðg 2xz 2og xz0 g xz0 ðg 2xy

ð479Þ

 dxz g 2t ÞðR000 ð x ÞR000 ð z Þ  R000 ð0þ Þ2 ÞR00 ð z0 Þ  dxy g 2t ÞR0000 ð x ÞR00 ð y ÞR00 ð z0 Þ

v v

v v

v

v

 2og zz0 g xz0 g z0 z g zx ðR000 ð0þ Þ2  R000 ðv z ÞR000 ðv z0 ÞÞR00 ðv x Þ

ð480Þ

This equation is certainly valid near a uniform configuration, i.e. within the weak epsilon expansion. Presumably it is valid again for non-sign changing v x , although we have not checked it in details. As explained above, for it to be correct for arbitrary v x one needs to discard strong continuity, which we are not ready to do. Let us now consider a uniform configuration v x ¼ v . One then gets the two loop FRG equation for the local part:

oRðv Þ ¼ og x g x R00 ðv ÞR00 ðv Þ þ ½2og z0 g x0 ðg 2xz  dxz g 2t Þ þ 2og z0 g x0 g 0z g zx ðR000 ðv Þ2  R000 ð0þ Þ2 ÞR00 ðv Þ ð481Þ which can also be written as:

oRðv Þ ¼



1 1 oJ 2 R00 ðv ÞR00 ðv Þ þ o IA  J 22 ðR000 ðv Þ2  R000 ð0þ Þ2 ÞR00 ðv Þ 2 2

ð482Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

R R with J 2 ¼ x g 2x and IA ¼ xz g 20z g 0x g xz . This has now a very standard form for the T = 0 FRG equation, and having carefully justified the anomalous terms we refer to [55] for the further simple steps leading to the rescaled two loop FRG equation. From the equation above we could also compute the two loop correction to the correlation function, a task left for the future. Acknowledgements My understanding of FRG has been greatly enhanced by vigorous, ongoing and multiple collaborations with Kay Wiese whom I warmly thanks. I am greatly indebted to collaborations with Leon Balents on the subtleties related to finite temperature aspects, part of that work being reviewed here. The ERG techniques where constructed with P. Chauve and G. Schehr whom I also thank. I am also greatly indebted to T. Giamarchi and C. Monthus, as well as A. Rosso, W. Krauth, A. Middleton and C. Marchetti for numerous discussions and collaborations on related topics.

Appendix A. Finite temperature beta function to three loop in any dimension Using the notations of the text one finds the following corrections to the disorder to three loop:

dð3Þ R0 ¼

1 3 ð6Þ T J 1 R0 þ T 2 6





 2 1 2 7 1 0000 000 ð5Þ  J 21 J 2 þ J1 J2 þ J 4 R0000 J 4 R0000 0 0 ð0ÞR0 þ J 1 J 3 R0 R0 2 24 12



   2 0000  000 2 0000  ð6Þ 1 1 ð6Þ R  þ I J þ J 21 J 2 R000  R000 ð0Þ R0  R000 R0 ð0Þ þ T ð2Ir þ IA J 1 Þ R000 I R0 R0 ð0Þ r A 1 0 0 2 2    00  0000 2 2 00 0000 0000 2 00 J 1 J 22 R000  R000 ð0Þ R0000 0 ð0ÞR0  I2A R0 ð0Þ R0 þ ðJ 1 J 2 þ I2A Þ R0  R0 ð0Þ R0    2 ð6Þ  4 1 2  00 1 ð5Þ 00 þð2IA J 1 þ J 2 J 3 Þ R000  R000 ð0Þ R000 R þ J R  R ð0Þ R J þ ðIi þ Im Þ R000 1 2 0 0 0 0 0 0 2 2

 2 0000  00  2 1 3  0000 2  00 00 þ ð4Il þ Io þ IA J 2 Þ R000 R R  R ð0Þ þ I þ R0  R000 ð0Þ J R0 j 0 0 0 0 2 2  2 3 ð6Þ 1 3  00 ð5Þ 00 þ 2IA J 2 R000  R000 ð0Þ R000 0 R0  J 2 R0  R0 ð0Þ R0 6 ðA1Þ where some integrals are defined in (13), the others are:

Im ¼ Il ¼

Z 1234

Z

1234

Ij ¼

Z

1234

Ir ¼

Z

1234

g 213 g 14 g 23 g 224 ;

Ii ¼

Z 1234

g 12 g 213 g 14 g 23 g 24 ;

Io ¼

g 212 g 13 g 14 g 23 g 24 ;

Ik ¼

g 212 g 213 g 23 ;

I2A ¼

g 12 g 13 g 14 g 23 g 24 g 34

Z

1234

Z 1234

Z

1234

g 212 g 213 g 24 g 34 g 312 g 13 g 34 g 24

g 312 g 13 g 23

R R where 1234 denotes the real space integral over the four points X1 x1 ;x2 ;x3 ;x4 , X the volume of the system. The corresponding diagrams are represented in Ref. (one and two loop) and [57] (three loop at T = 0). e one obtains, via the proceThe last line contains finite T integrals. Denoting o ¼ mom , and R ¼ m R, dure described in the text, the beta function:

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        2 2 e 00  R e þ oJ T R e ¼ R e 00 þ m oJ 1 R e 00 ð0Þ R e 00 þ m 1 oJ  J oJ T R e 0000 ð0Þ R e 00 e 000  m J oJ T R oR 1 2 2 1 2 1 2 2 3    2   2  e 00  R e 0000  1 m oJ T 2 R e 0000 ð0Þ R e 000 e 00 ð0Þ þ m 7 oJ  J oJ T 2 R e 0000 R þ m2 ½oIA  J 2 oJ 2  R 3 1 4 24 4 12    2  2 1 e 0000 e 000 þ m2 ½2oIr þ ð5J 2  4IA ÞoJ  3J oJ  Te R e 000 R e 0000 ð0Þ R þ m2  oIr þ ðJ 22  IA ÞoJ 1 T R 1 2 3 2 2   2 e 00 e 00  R e 00 ð0Þ R e 0000 þ m2 ½oI2A þ 3J 2 oJ þ J oJ T R e 0000 ð0Þ2 R þ m2 ½oI2A  2IA oJ 1  J3 oJ 2 T R 3 2 2 1 h i 4 2  2 1

 00 e 000 e R e 0000 þ m3 oIi þ oIm  2J oIA þ J 2 oJ e 00 ð0Þ R R þ m3 oIj  2IA oJ 2 R 2 2 2 2 h i  2 e 00  R e 0000 e 00 ð0Þ R e 000 R þ m3 4oIl þ oIo  6J 2 oIA  4IA oJ 2 þ 5J 22 oJ 2 R ðA2Þ and the flow of the second derivative is now: h i  2 e R e 0000 ð0Þ e 0000 ð0Þ þ 6 eI A  eJ 2 T e 00 ð0Þ ¼ð  2fÞ R e 00 ð0Þ þ 2eJ 2 Te R oR 2   h i  3 1 e 0000 ð0Þ e 0000 ð0Þ R e ð6Þ ð0Þ þ 3m2 oIr  4IA oJ þ 5J 2 oJ  2J oJ T R þ m oJ 4  2J 3 oJ 1 T 2 R 1 2 3 2 1 2 2

2

2

ðA3Þ

2

One can now use special relations such as: oJ 1 ¼ 2m J 2 ; oJ 3 ¼ 6m IA ; oJ 4 ¼ 8m I2A ; oI2A ¼ 2m ð2Ik þ 3Ij Þ; oIr ¼ 2m2 ð4Il þ Io Þ or scaling relations m oJ 2 ¼ eJ 2 ; m2 oIA ¼ 2eI A ; m3 oIl ¼ 3eI l and similar for any of the zero temperature three loop integrals (the tilde denotes scaled integrals with m ¼ 1). For d < 2 one has Jn ¼ m2n4ðn1ÞeJ n and thus oJ n ¼ ððn  1Þ þ 4  2nÞJ n . This yields, upon introducing further e ¼ md2þ2f T: rescaling of u by mf and R by m4f , and defining here T

    2 e 00  R e 00 þ eJ 2 1 R e ¼ ð  4fÞ R e þ fu R e 0 þ 2eJ 2 Te R e 00 ð0Þ R e 00 oR ðA4Þ 2   2  2  e 0000 ð0Þ R e 00  R e 000  2eJ 2 Te R e 00 þ ½2eI A  eJ 2  R e 000 e 00 ð0Þ R þ ½3eI A  2eJ 22  Te R 2 2     2 7 e 0000  2 eI 2A Te 2 R e 0000 ð0Þ R e 0000 þ eI 2A  2eJ 2eJ 3 Te 2 R 3 3  2  2 e 0000 e 0000 ð0Þ R e 000 þ ½16eI l þ 4eI o þ 10eJ 3  26eJ 2eI A  Te R e 000 R þ ½4eI l  eI o þ 2eJ 32  2eJ 2eI A  Te R 2   2 e 00 e 00  R e 00 ð0Þ R e 0000 ½4eI k  6eI j þ 6eJ 3 þ eJ 2eJ 3  Te R e 0000 ð0Þ2 R þ ½4eI k þ 6eI j  4eI AeJ 2  eJ 2eJ 3  Te R 2  4  2  2 1 e 0000 þ ½3eI i þ 3eI m  4eJ 2eI A þ eJ 3  R e 00  R e 00 ð0Þ e 000 R þ ½3eI j  2eI AeJ 2  R 2 2 h i  2 e 00  R e 0000 e 00 ð0Þ R e 000 R þ  12eI l þ 3eI o  16eJ 2eI A þ 5eJ 32 R

ðA5Þ One must be careful that the rescaled integrals eI 2A ; eJ 3 are still functions of K=m for d > 10=3 and d > 8=3, respectively, while all other integrals have a well defined UV limit. All combinations of rescaled integrals which appear as coefficients of the T = 0 terms are finite when multiplied by , which is a consequence of (formal) renormalizability of the T = 0 theory. The combinations entering the T = 0 part of the three loop term is found to be identical to the one obtained in Ref. [57], where these integrals are computed. Here we have in addition non-zero temperature terms up to three loop. In d = 0, e!T e =2, R e ! R=4 e upon the rescaling T and setting all rescaled integrals to unity, this equation yields the results given in the text. The flow of the second derivative becomes, for d < 2: 

2  h i  e 00 ð0Þ ¼ ð  2fÞ R e 00 ð0Þ þ 2eJ 2 Te R e ð6Þ ð0Þ e 0000 ð0Þ þ 3   2 eJ 4  4eJ 3eJ 2 Te 2 R e 0000 ð0Þ þ 6 eI A  eJ 2 Te R e 0000 ð0Þ R oR 2 2 3 h i  e 0000 ð0Þ ðA6Þ þ 3 8eI l þ 2eI o  8eI AeJ 2 þ 10eJ 32 þ 4ð1  ÞeJ 2eJ 3 Te R

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Remarkably all but the first two terms vanish in d = 0, when all rescaled integrals are unity. As discussed in the text this is a consequence of an exact identity which is local only in d = 0. It holds also in d > 0, see (404), but involves also non-local contributions of the R½v  functional, producing the extra terms above. One can then analyze this beta function (A5) as in the text. The one loop equation (first line) is identical (up to rescaling) to the d = 0 case and the analysis is identical, with a well defined  expansion. To one loop one has a well defined TBL as for d = 0, namely:

e 00 ð0Þ ¼ eJ 2 R e 0000 ð0Þ ! ð  2fÞ R e 000 ð0þ Þ2 2eJ 2 Te R

ðA7Þ

Now, however, one is not guaranteed that the TBL holds without modification beyond one loop. Indeed e for each additional loop. This is not necinserting TBL scaling in (A6) yields additional power of 1= T essarily a failure of TBL scaling, as we know already from the analysis in d = 0 that in the TBL this loop expansion fails. If TBL still holds, resummations of all loops higher than one in (A6) should yield a result Oð1Þ. Assuming that this is the case one can again examine whether (A5) is finite in the outer region u ¼  Oð1Þ. 2 Now trouble already starts at two loop (second line). In the outer region u ¼ Oð1Þ the e R e 000 flows to zero and the other term has a finite limit if one uses the one loop result term T e 00 ð0Þ ¼ eJ 2 R eR e 0000 ð0Þ ! ð  2fÞ R e 000 ð0þ Þ2 , namely: 2eJ 2 T

e 0000 ð0Þ R e 000 ð0þ Þ2 R e 00 e 00 ! eJ 2 R 2eJ 22 Te R 2

ðA8Þ

Unfortunately cancels the supercusp from the T = 0 two loop term only for d = 0. Worse, close h this limit i to d ¼ 4,  2eI A  J 22 has a finite limit, while the coefficient of (A8) diverges as 1=. Let us close by indicating the calculation of the correlation function to two loop at non-zero temperature. We obtain:

huqa uq b i ¼ ¼

1

h

m2 Þ2

ðq2 þ 1 ðq2

þ

m2 Þ2

h

2 R000 ð0Þ þ ðdð1Þ R0 Þ00 ð0Þ þ ðdð2Þ R0 Þ00 ð0Þ þ TR0000 0 ð0Þ ½J 3 ðqÞ  J 3 ð0Þ

e 00 ð0Þ þ Tm2 ½J ðqÞ  J ð0Þ R e 0000 ð0Þ2 m R 3 3

i

i

ðA9Þ

e 00 ð0Þ contains all q ¼ 0 contributions. The lowest non-trivial q dependent diagram thus occurs where R at two loop and contained in J 3 . Appendix B. Higher correlations in d = 0 Here we examine the polynomial expansion to sixth order. In particular we check explicitly that b ð6Þ ð0Þ ¼ Rð6Þ ð0Þ. Since six point correlations have been examined in detail in Appendix C.2. of [36] R we only give material not presented there. The six point connected correlation reads:

Gabcdef ¼ hua ub uc ud ue uf i  15½hua ub ihuc ud ue uf i þ 30hua ub ihuc ud ihue uf i ¼ hua ub uc ud ue uf i  15½hua ub ihuc ud ue uf ic   15½hua ub ihuc ud ihue uf i

ðB1Þ ðB2Þ

where [  ] denote full symmetrization upon the p indices. It is related to the C vertices through:

2 6 2 7 m m Gabcdef þ 10 ½Gabch Gdefh  T T 6 7 T T ¼ C þ 10 ½Cabch Cdefh  abcdef m2 m2

Cabcdef ¼ 

ðB3Þ

Gabcdef

ðB4Þ

since Legendre transform is involutive C is also made of tree graphs of W, and there is a symmetry between these formulae. In principle there is a G1 ab on each external leg, but thanks to the above STS property, only the Kronecker delta part remains (R00 ð0Þ disappears). In Appendix C.2 of [36] the following parameterization of the C vertex was used:

P. Le Doussal / Annals of Physics 325 (2010) 49–150

Cabcdef ¼  

1 T2

1 T6

m6 þ

6! 4!4T 4

q6 ð8½dabc   6½dab dcd Þ 

6! 3!8T 3

s6 ð½dabcd   2½dabc dde  þ ½dab dcd def 

Rð6Þ ð0Þððdabcde þ 5permÞ þ ðdabcd def þ 14permÞ  ðdabc ddef þ 9permÞÞ

121

ðB5Þ ðB6Þ

where s6 ; q6 and m6 are derivatives at zero of, respectively, third, fourth and sixth cumulants given there. In all above formula replica indices are arbitrary (they can be equal). This form is imposed by STS since there are 11 a priori distinct elements Caaaaaa ; Caaaaab ; . . . (distinct indices), etc. (same for P the G) and the general STS relation h Cabcdeh ¼ 0 (arbitrary indices) implies 7 relations. It thus remains 4 independent variables at order u6 . In Appendix C.2. of [36] all 11 distinct connected correlations where given as a function of these parameters (formulae (C.15)–(C.25) in the condmat version). We have checked that these relations are correct. Taking linear combinations of those one can hope to relate Rð6Þ ð0Þ to some simple observable. This is not that simple, and in particular (see below) it involves correlations with more that two distinct replica indices. Relation if any is indirect.27 b ð6Þ ð0Þ which does have a simple We now follow the route of the present paper and first compute R physical meaning from the renormalized potential. Formula (76) yields, introducing two sets of replicas with n1 and n2 separately going to zero:

* e 00 ðv 1  v 2 Þ ¼ m4 R

m2 T



ðua1  v 1 Þðua2  v 2 Þe

v1

P a1

ua1 þv 2

P a2

ua2

+ ðB7Þ

where h:i is the standard replica average. Indices a1 are always distinct from indices a2 (and ai belongs to ai ). The above expression is clearly a function of v 1  v 2 as can be verified by performing the shift uc ! uc þ v 2 . Setting v 2 ¼ 0 and v 1 ¼ v and Taylor expanding one finds:

b 00 ðv Þ ¼ hua ua i þ m4 R 1 2

1 X m2p v p T p p! p¼2

* ua1

X

!p ua1

+ ua2

ðB8Þ

a1

where a term proportional to n1 has been dropped. Here one has to be careful with the constraints. It yields for the sixth derivative: 4 b ð6Þ

m R ð0Þ ¼

m8

*

T4

ua1

X

!4 ua1

+ ua2

ðB9Þ

a1

To perform the calculation the safest method is to add variables one by one, recursively (n1 ¼ 0 being implicit everywhere):

ua ua ub uc ud ¼ ðu2a  ua ub Þub uc ud ¼ ðu3a  u2a ub  2u2a ub þ 2ua ub uc Þuc ud ¼

ðu3a



3u2a ub

þ 2ua ub uc Þuc ud

ðB10Þ ðB11Þ

and so on. the final result is:

b ð6Þ ð0Þ ¼  m4 R

m8 

hu5a ub i  5hu4a ub uc i  10hu3a u2b uc i T4  þ 20hu3a ub uc ud i þ 30hu2a u2b uc ud i  60hu2a ub uc ud ue i þ 24hua ub uc ud ue uf i

ðB12Þ ðB13Þ

Now let us give here explicitly the disconnected pieces not given in Appendix C.2. of [36]:

 

u5a ub

disc

u4a ub uc

 2    ¼ 15 u2a hua ub i þ 5hua ub i u4a c þ 10 u2a u3a ub c

disc

  2   ¼ 12 u2a hua ub i2 þ 3 u2a hua ub i þ hua ub i u4a c þ 8hua ub i u3a uc c   þ 6 u2a u2a ub uc c

ðB14Þ

ðB15Þ

27 The quantity s6 was found to be related to third moment of susceptibility. Because of vanishing of D3 to leading order it means that in effect r 6 is related to third moment of susceptibility.

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u3a ub uc ud

disc

   ¼ 9 u2a hua ub i2 þ 6hua ub i3 þ 3hua ub i u3a ub c þ 9hua ub i u2a ub uc c  2 þ 3 ua hua ub uc ud ic

ðB16Þ

hu2a u2b uc ud idisc ¼ hu2a i2 hua ub i þ 4hu2a ihua ub i2 þ 10hua ub i3 þ hua ub ihu2a u2b ic þ 2hu2a ihu2a ub uc ic þ 8hua ub ihu2a ub uc ic þ 4hua ub ihua ub uc ud ic

ðB17Þ

hu2a ub uc ud ue idisc ¼ 3hu2a ihua ub i2 þ 12hua ub i3 þ 6hua ub ihu2a ub uc ic þ 8hua ub ihua ub uc ud ic þ hu2a ihua ub uc ud ic

ðB18Þ

3

hua ub uc ud ue uf idisc ¼ 15hua ub i þ 15hua ub ihua ub uc ud ic

ðB19Þ

Amazingly, a tedious calculation shows that in the above combination all disconnected parts cancel (as is already the case for the fourth cumulant). One finally gets:

e ð6Þ ð0Þ ¼  m4 R

m8

ðhu5a ub ic  5hu4a ub uc ic  10hu3a u2b uc ic ðB20Þ T4 þ 20hu3a ub uc ud ic þ 30hu2a u2b uc ud ic  60hu2a ub uc ud ue ic þ 24hua ub uc ud ue uf ic Þ ðB21Þ ¼ m4 Rð6Þ ð0Þ

ðB22Þ

where the last line results from evaluating the combination using formulae (C.15)–(C.25) of Appendix C.2. of [36]. All contributions of m6 ; s6 ; q4 ; Rð4Þ ð0Þ2 cancel in this calculation. This tedious and non-trivb ¼ R should convince the reader of the amazing efficiency of the Legendre ial (partial) check that R transform method given in the text. Furthermore formula (B8) provides a general method to express b (and thus of R) at zero to one linear combination of correlation functions (others any derivative of R can be generated by adding all STS identities). Appendix C. Legendre transform: relations between moments and C-moments We perform the Legendre transform in any dimension, d = 0 is recovered setting g ¼ 1=m2 and discarding space indices. We start from the definition of the Legendre transform (91):

1 X a 1 a u g v  W½v  T axy x xy y X dC½u ¼T g xy duay y

C½u ¼

ðC1Þ

v ax

ðC2Þ x

C½u where the second line is the condition ja ¼ ddu together with (84). Taking the derivative in the a x C-cumulant expansion (90), (C2) can be rewritten:

v ax ¼ uax 

1X dR½uac  1 X dS½ua ; uc ; ud  1 X dQ ½ua ; uc ; ud ; ue  g  2 g xy  3 g xy a T yc xy duay du duay 2T ycd 6T ycde y

þ 4replica sums

ðC3Þ

where we have used the symmetry of the cumulants. One can also rewrite (C2) by rearranging the quadratic term and inserting the W-cumulant expansion:

1 X 1 a a 1 X 1 1 X b ab R½v  g u u  g ðv  uÞax ðv  uÞay  2 2T axy xy x y 2T axy xy 2T ab 1 Xb a b c 1 Xb a b c d  3 S½v ; v ; v   Q ½v ; v ; v ; v  þ 5replica sums 6T abc 4!T 4 abcd

C½u ¼ C0 þ

ðC4Þ

We can now insert (C3) into (C4), perform Taylor expansion in the cumulants. Remarkably, since in that operation the number of free replica sums can only increase, it allows to identify simple relations between cumulants by comparing with:

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C½u ¼ C0 þ

1 X 1 a a 1 X 1 X a b c 1 X g xy ux uy  2 R½uab   3 S½u ; u ; u   Q½v a ; v b ; v c ; v d  2T axy 2T ab 6T abc 4!T 4 abcd

þ 5replica sums The calculation yields the formula (93) in the text as well as:

X

dR½uab  dS½ua ; uc ; ud  duax duay xy X d2 R½uab  dR½uac  dR½ubc  dR½uad  dR½ubd 

g yt g    6symabcd duax duby xz duaz dubz duat dubt xyzt

b ½ua ; ub ; uc ; ud   12sym Q ½ua ; ub ; uc ; ud  ¼ Q abcd

g xy

ðC5Þ ðC6Þ

and using (93) one finds that the same formula holds with S ! b S and the 6 replaced by +6. Appendix D. Generating functionals and Legendre transform in general, non-STS case Since many models do not posses an exact STS symmetry we give here a derivation valid in the general case. The only assumption is that the expansion in number of replicas is valid. Of course the final formula is not as nice as in the STS case, and we hence stop at the level of second cumulant. Q One defines as usual the functional W½j ¼ ln Z½j ¼ ln pa¼1 Z V ½ja  where:

Z V ½j ¼

Z

R

1T HV ½uþ

Due

x

jx ux

1

¼ eT F V ½j

ðD1Þ

where F V ½j is the free energy in presence of sources, e.g. an external force jx ¼ fx =T. One writes the general expansion:

W½j ¼ W½0 þ

1Xb 1 Xb 1 Xb U½ja  þ 2 R½ja ; jb  þ 3 S½ja ; jb ; jc  þ    T a 2T ab 6T abc

ðD2Þ

in terms of fully symmetric functionals. Note that the number of replica p is arbitrary here [83]. One shows that:

b  F V ½j ¼ U½j

ðD3Þ

c

b ;j  F V ½j1 F V ½j2  ¼ R½j 1 2

ðD4Þ

c

 F V ½j1 F V ½j2 F V ½j3  ¼ b S½j1 ; j2 ; j3 

ðD5Þ 



P

n c

P P ð1Þ It is clear from the definition W½j ¼ ln exp  1T a F V ½ja  ¼ 1 . It can also be a F V ½ja  n¼1 n!T n checked by inserting groups of values ja ¼ ji for ni replicas and expanding in multinomial powers of ni . These function(al)s can be simply measured by applying a force to the system. For instance, at T = 0, denoting Ef the ground state energy in one realization of disorder in presence of a force one has:

b   Ef ¼ U½f c b 1 ; f2  Ef Ef ¼ R½f 1

2

n

ðD6Þ ðD7Þ

In some cases, STS is restored at large scales (such a a manifold on a lattice) and these formula can be used to check that. In that case it should depend only on f1  f2 . Of course in that case a non-STS local part such as non-STS gradient terms may remain, and are typically irrelevant. R a Let us now perform the Lengendre transform C½u þ W½j ¼ x uax jx with u ¼ W 0 ½j. This condition yields:

uxa ¼

1 b0 1 X b0 1 X b0 U x ½ja  þ 2 R x ½ja ; jb  þ 3 S x ½ja ; jb ; jc  þ    T T b 2T bc

ðD8Þ

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where derivatives here are with respect to the first variable. Now define j ½u such that:

ux ¼

1 b0 0 U ½j ½u T x

ðD9Þ a

0

In the absence of the terms involving replica sums in (D8) the solution would be j ¼ j ½ua . In general a 0 a a one writes j ¼ j ½ua  þ dj where dj contains at least one replica sum. One rewrite (D8) as:

uax ¼

1 b0 0 a 1 b 00 0 a a 1 X b 0 a b U ½j ½u  þ U R x ½j ; j  þ 2replica sums þ    ½j ½u djy þ 2 T x T xy T b

ðD10Þ

The two first terms cancel one gets:

1 b 00 0 a 1 X b 0 0 a 0 b a R z ½j ½u ; j ½u  þ 2replica sums þ    djy ¼  U ½j ½u yz T b

ðD11Þ

We can now express the effective action:

C½u ¼

X

C0 ½ua  þ

a

XZ a

x

a

uax djx 

1 b0 0 a a 1 Xb 0 a 0 b U x ½j ½u djx  2 R½j ½u ; j ½u  þ 3replica sums þ    T 2T ab ðD12Þ

C0 ½u ¼

Z

0

ux jx ½u 

x

1b 0 U½j ½u T

ðD13Þ

2

0

where the terms Oðdj Þ yield three sums. A cancellation occurs from the definition of j giving the final result:

C½u ¼

X

C0 ½ua  

a

1 Xb 0 a 0 b R½j ½u ; j ½u  þ 3replica sums þ    2T 2 ab

ðD14Þ

Thus one has:

b 0 ½ua ; j0 ½ub  R½ua ; ub  ¼ R½j 0

ðD15Þ

0

where j ½u ¼ f ½u=T with:

F 0Vx ½f 0 ½u ¼ ux

ðD16Þ

note that f0 ½u is a non-random quantity depending only on the average energy. 0 Finally, note that if j ½0 ¼ 0 then:

C0 ½0 ¼ F V =T;

R½0; 0 ¼ F 2V

c

ðD17Þ

c

and S½0; 0; 0 ¼ b S½0; 0; 0 ¼ F 3V , checked in the STS case. In general one can write: 2

b ¼ T jGj þ U b i ½j U½j 2 d2 F V ½f  G¼ df df G ¼ g; for STS

ðD18Þ ðD19Þ ðD20Þ

b i ¼ 0 in the latter case. Thus in general: with U

ux ¼ v 0x ½u þ

v 0 ¼ TGj

1 b0 0 U ½j ½u T i

0

in the STS case

ðD21Þ ðD22Þ

v 0 ½u ¼ u; j0 ½u ¼ g 1 u=T.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

125

Appendix E. From C-FRG to W-FRG We start with the second cumulant W-ERG:

mom Rðv Þ ¼

2T 00 2 R ðv Þ þ 2 b S 110 ð0; 0; v Þ m2 m

ðE1Þ

and substitute using (70):

 00  1  0 0 00 00 00 00 b S 110 ðv abc Þ ¼ S110 ðv abc Þ þ 2 R000 ab ðRac  Rbc Þ  Rab Rbc  Rac Rab  Rcb m

ðE2Þ

and we denote R0ab ¼ Rðv ab Þ, etc. The limit v ab ! 0 is always unambiguous for T > 0 (and here also unambiguous at T = 0 provided R000 is bounded which we assume from now on). It yields the second cumulant C-ERG (123) in the text. Let us now consider the third cumulant W-ERG:

i i 3T h 3 hb 6 0 0  mom b R R Sðv abc Þ ¼ 2 b S 200 ðv abc Þ þ 2 Q 1100 ðv aabc Þ þ m m m2 ab ac

ðE3Þ

Let us write, using (70):

 6  3  mom Sðv abc Þ ¼ mom b Sðv abc Þ  2 ðmom Þ R0ab R0ac  2 R0ab R0ac m m i 3 i

 3 h 3T h b 1100 ðv aabc Þ ¼ 2 b S 200 ðv abc Þ  2 ðmom Þ R0ab R0ac þ 2 Q m m m i 0 i 3

 3 h 3T 9T h b 1100 ðv aabc Þ ¼ 2 ½S200 ðv abc Þ þ 4 o2a R0ab Rac  2 ðmom Þ R0ab R0ac þ 2 Q m m m m

ðE4Þ ðE5Þ ðE6Þ

One has:

2 0 0  2 00 00  4 000 0  oa Rab Rac ¼ Rab Rac þ Rab Rac 3 3

ðE7Þ



 4T     4 8  0 0 S111 ðuaab ÞR0ac þ 2 R00ab  R00 ð0Þ R000 mom R0ab R0ac ¼ 2 ðmom ÞR0ab R0ac ¼ 2 R000 ab Rac  ab Rac m m2 m ðE8Þ Thus we obtain:

mom Sðv abc Þ ¼

i  12  3T 6T 3 hb ½S200 ðv abc Þ þ 4 R00ab R00ac þ 4 S111 ðv aab ÞR0ac þ 2 Q 1100 ðv aabc Þ m2 m m m   24  0  4 R00ab  R00 ð0Þ R000 ab Rac m

One can check that:

h

 i 1 1 1 oa ob S100 ðv abc ÞR0ad a¼b ¼ ½R0ad S111 ðv acc Þ þ ½R00ac S110 ðv aad Þ  ½ðR00ac  R00 ð0ÞÞS110 ðv acd Þ 3 3 3

using that S111 is odd and using S200 þ S110 þ S101 ¼ 0. Thus one gets:

the

STS

relations

ðE9Þ

S111 ðu; u; v Þ þ 2S210 ðu; u; v Þ ¼ 0 and

 12      3T 6T ½S200 ðv abc Þ þ 4 R00ab R00ac þ 4 R00ac S110 ðuaad Þ  R00ac  R00 ð0Þ S110 ðuacd Þ ðE10Þ m2 m m  36 h

   i  3 24  0 þ 2 ½Q 1100 ðv aabc Þ  4 R00ab  R00 ð0Þ R000 ðE11Þ oa ob R00ac R0ab  R0cb R0ad a¼b ab Rac þ m m m4

 mom Sðv abc Þ ¼

Performing the last calculation cancellations occur and one gets Eq. (124) in the text. Note that the R0 R0 b can be guessed from the W-ERG equation, just solving for part of S  b S as well as the R0 S0 part of Q  Q explicit mass dependence in the last term (i.e. the non-linear term). Finally, note that the above was derived using R000 ð0Þ ¼ 0, i.e. at T > 0. There are in fact several terms formally proportional to R000 ð0Þ both

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

in (E9) and in (E11). One finds that these all cancel by parity if one replaces the contractions ja¼b by jua ¼ub þd and keep d infinitesimal until the end (i.e. the result is unambiguous). Appendix F. Two-well droplet solution for any moment F.1. Third and fourth moments We start with the two-well calculation of the third moment:

b 0 ðv Þ V b 0 ðtÞ V b 0 ðzÞ V S111 ðv ; z; tÞ ¼ m6 m6





u1 X 1 þ u2 wX 2 u1 Y 1 þ u2 wY 2 u1 Z 1 þ u2 wZ 2 ¼ T v~ þ T~t þ T ~z þ X 1 þ wX 2 Y 1 þ wY 2 Z 1 þ wZ 2    ¼ T u21 P v~ þ ~t þ ~z þ T hhss ðX 1 ; Y 1 ; Z 1 ; u1 ; X 2 ; Y 2 ; Z 2 ; u2 Þiui þ OðT 2 Þ

ðF1Þ ðF2Þ ðF3Þ

2 2 ~ 2 where we have defined X i ¼ em ui v~ ; Y i ¼ em ui t ; Z i ¼ em ui ~z and:

hðX 1 ; Y 1 ; Z 1 ; u1 ; X 2 ; Y 2 ; Z 2 ; u2 Þ ¼

Z 0

1

dw u1 X 1 þ u2 wX 2 u1 Y 1 þ u2 wY 2 u1 Z 1 þ u2 wZ 2 w X 1 þ wX 2 Y 1 þ wY 2 Z 1 þ wZ 2

ðF4Þ

together with the symmetrized expression hss as in (150). A tedious calculation then yields:

  1 hss ðX 1 ; Y 1 ; Z 1 ; u1 ; X 2 ; Y 2 ; Z 2 ; u2 Þ ¼ m2 u1 u2 ðu1  u2 Þ2 v~ þ ~t þ ~z 2 " 1 2 X 22 Y 1 Z 1 þ X 21 Y 2 Z 2 4 þ m ðu1  u2 Þ v~ 4 ðX 2 Y 1  X 1 Y 2 ÞðX 2 Z 1  X 1 Z 2 Þ þ

ðF5Þ ðF6Þ #

Y 22 X 1 Z 1 þ Y 21 X 2 Z 2 Z 22 X 1 Y 1 þ Z 21 X 2 Y 2 ~t þ ~z ðX 2 Y 1  X 1 Y 2 ÞðY 2 Z 1 þ Y 1 Z 2 Þ ðY 2 Z 1 þ Y 1 Z 2 ÞðX 2 Z 1 þ X 1 Z 2 Þ

ðF7Þ

~!v ~ þ a; ~z ! ~z þ a; ~t ! ~t þ a, the change in hss simplifies drasHere we can check that upon the shift v tically into:

am2



3 1 u1 u2 ðu1  u2 Þ2 þ ðu1  u2 Þ4 2 2

ðF8Þ

Expanding this yields am2 hu41  u31 u2 iui which cancels exactly the change from the first term 3ahu21 iP R due to the STS relation P0 ðu1 Þ ¼ du2 ðu1  u2 ÞDðu1 ; u2 Þ. The final result is thus invariant under trans2 lation. The first term Thu1 iP can hence be combined with the second and third. Defining y ¼ u1  u2 , upon a few transformations one obtains the result given in the text. The fourth moment can be computed in the same way: e 0 ðv Þ V e 0 ðuÞ V e 0 ðtÞ V e 0 ðzÞ Q 1111 ðv ; u; z; tÞ V ¼ m8 m8





u1 X 1 þ u2 wX 2 u Y þ u2 wY 2 u1 Z 1 þ u2 wZ 2 u1 T 1 þ u2 wT 2 ~þ 1 1 ¼ ðT v~ þ T ~z þ T~t þ Þ T u X 1 þ wX 2 Y 1 þ wY 2 Z 1 þ wZ 2 T 1 þ wT 2 ¼ hu41 i þ Thhss ðX 1 ; Y 1 ; Z 1 ; T 1 ; u1 ; X 2 ; Y 2 ; Z 2 ; T 2 ; u2 Þiui þ OðT 2 Þ

ðF9Þ ðF10Þ ðF11Þ

2 2 2 2 ~ with X i ¼ em ui v~ ; Y i ¼ em ui u~ ; Z i ¼ em ui ~z ; T i ¼ em ui t . Let us define:

hðX 1 ; Y 1 ; Z 1 ; T 1 ; u1 ; X 2 ; Y 2 ; Z 2 ; T 2 ; u2 Þ

Z 1 dw u1 X 1 þ u2 wX 2 u1 Y 1 þ u2 wY 2 u1 Z 1 þ u2 wZ 2 u1 T 1 þ u2 wT 2 ¼  u41 w X 1 þ wX 2 Y 1 þ wY 2 Z 1 þ wZ 2 T 1 þ wT 2 0

ðF12Þ

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127

Implementing the two symmetries a nice mathematica calculation yields:

hss ¼



2T 2 X 1 2X 2 Y 1 2X 2 Z 1 þ þ lnðX 1 =X 2 Þ þ p:c: T 2 X 1 þ T 1 X 2 X 2 Y 1  X 1 Y 2 X 2 Z 1  X 1 Z 2 # X 32 Y 1 Z 1 T 1 þ X 31 Y 2 Z 2 T 2 lnðX 1 =X 2 Þ þ p:c: ðT 2 X 1 þ T 1 X 2 ÞðX 2 Y 1  X 1 Y 2 ÞðX 2 Z 1  X 1 Z 2 Þ

1 u1 u2 u221 4 "

1 þ u421 4

 3 þ

ðF13Þ ðF14Þ

One gets finally the result given in the text. One can check the STS symmetry (under a common shift of all the variables) noting that symu;v ;z;t F½v  u; v  t; v  z ¼ 0, both for circular permutations, and full permutations, and one also finds that the second term is invariant by the shift of all variables. One finds also the small argument behaviour:      

   11 T m8 Q 1111 ðv 1 ; v 2 ; v 3 ; v 4 Þ  u41 P ¼  12hu1 u2 y2 iy þ O m2 u1 u2 y4 y v 2i þ hy4 iy þ O m2 y6 y v 2i 4 3 ðF15Þ

all those functions are always even in y, as has been imposed using the symmetries. F.2. Solution for any moment We can write any odd moment as: ðnÞ

S1...1 ðv 1 ; . . . ; v n Þ ¼ Tm2n hun1 1 iP

n X

!

v~ i

þ An ½v~ i  þ OðT 2 Þ

ðF16Þ

i¼1

and every even one: ðnÞ S1...1 ðv 1 ; . . . ; v n Þ ¼ m2n hun1 iP þ An ½v~ i  þ OðT 2 Þ

ðF17Þ

where: n Y u2 w þ u1 ai  un1 ¼ Thh½v i ; ai ; u1 ; u2 iui w þ ai i¼1 ! Z 1 n dw Y u2 w þ u1 ai 2n n ~ h½v i ; ai ; u1 ; u2  ¼ m  u1 w i¼1 w þ ai 0

An ½v~ i  ¼ m2n

ðF18Þ ðF19Þ

~ i Þ; y ¼ u1  u2 . To simplify this expression it is convenient to first subtract un2 inand ai ¼ expðm2 yv stead of un1 , then perform the Mathematica command Apart[,w] to extract the poles. Dividing by w and applying again the command Apart[,w], one gets a term ðun2  un1 Þ=w which can be discarded if the initial substraction is un1 . The final result is remarkably simple:

h½v~ i ; ai ; u1 ; u2  ¼ m2n

Z

1

dw 0

n X i¼1

u21 Y u2 ai  u1 aj w þ ai j–i ai  aj

ðF20Þ

Now, before integrating we first symmetrize under the first symmetry ai ! 1=ai ; u1 $ u2 . Noting that R1 u a u a each fraction 2 ai a1 j is invariant, one ends up to compute 0 dw½ðw þ ai Þ1  ðw þ 1=ai Þ1  ¼  ln ai i j thus one gets simply:

hs ½v~ i ; ai ; u1 ; u2  ¼

n Y u2 ai  u1 aj y 2n X ln ai m 2 ai  aj j–i i¼1

ðF21Þ

Hence one gets:

An ½v~ i  ¼

*

+ n X Y 1 2nþ2 1 2 ~ m T y vi u1  y 2 ~ ~ 2 1  em yðv j v i Þ j–i i¼1

u1 ;y

ðF22Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

The last step is the symmetrization ui ! ui . It yields: * "

#+ n X Y Y 1 2nþ2 1 1 n1 2 An ½v~ i  ¼ m T y v~ i u1  y u1  y þ ð1Þ 2 2 ~ ~ ~ ~ 4 1  em yðv j v i Þ 1  em yðv j v i Þ i¼1 j–i j–i u

1 ;y

ðF23Þ The highest order term in y is:

~ Ahigh n ½v i  ¼

* + n X

  1 2nþ2 T ynþ1 v~ i F m2 y v~ i  v~ j m 4 i¼1

ðF24Þ

u1 ;y

F½fzij g ¼ ð1Þ

Y n1 j–i

1 þ 1  ezji

Y j–i

P

zij

1 1þe j Q zij ¼ 1e ð1  ezij Þ

ðF25Þ

j–i

where zij ¼ zi  zj , a result consistent with the highest order term in y given in the text for the third and fourth moments. To get the other terms given in the text one must use, for the third moment, 2 hu21 iP ¼ m3 hyu31 iy;u1 from the STS identity (143) which yields hu21 iP þ 12 hy2 u21 iy;u1  12 hy3 u1 iy;u1 ¼  16 hy4 iy , and, for the fourth moment, the identity e F ða; b; cÞ ¼ 3symabc ð1  ea Þð1  eb Þ1  ð1  ea Þð1  eb Þ1 e where F is defined in the text. F.3. Generalization to any N The above formula can be generalized to any N. We drop the tilde subscript. We need:

An ½~ v i a1 ;...;an ¼ m2n

n n Y u2;ai w þ u1;ai ai Y  u1;ai ¼ Thh½~ v i ; ai ; ~u1 ; ~u2 a1 ;...;an i~ui w þ ai i¼1 i¼1

ðF26Þ

where ai ¼ expðm2~ v i Þ, ~y ¼ ~u1  ~u2 . Again using Apart one finds: y~

h½v i ; ai ; u1 ; u2 a1 ;...;an ¼ m2n

Z

1

dw

n X

0

i¼1

yai Y u2;aj ai  u1;aj aj w þ ai j–i ai  aj

ðF27Þ

Integration and symmetrization yields:

hs ½v i ; ai ; u1 ; u2 a1 ;...;an ¼

n Y u2;aj ai  u1;aj aj 1 2n X yai ln ai m 2 ai  aj j–i i¼1

ðF28Þ

Using the symmetrization ~ ui ! ~ ui . It yields:

An ½v i a1 ;...;an ¼

* "

n Y 1 2nþ2 X 1 T yai~ vi u1;aj  yaj m y~ 2 4 1  em ~y~v ji j–i i¼1 #+

Y 1 þð1Þn1 u1;aj  yaj 2 1  em ~y~v ji j–i ~ ~

ðF29Þ

u 1 ;y

The highest order term is:

Ahigh n ½v i a1 ;...;an ¼

* + n X 1 2nþ2 ~ y~ T ya1 :yan v i F½m2~y  ~ v ij  m 4 i¼1

ðF30Þ

~ y u1 ;~

where the function F is defined in (F25) and does not depend on N. For the second moment one obtains, restoring the temperature in the TBL variable:

   R00ab ð~ v Þ ¼ R00ab ð0Þ þ Tm4 ya yb F 2 m2~y  ~ v =T ~y

ðF31Þ

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129

Appendix G. Three loop beta function in zero dimension G.1. Three loop at T > 0 by the C-ERG method We start from Eq. (237) in the text. From this equation we expect a solution for e S ð4Þ of a form similar to (221), schematically:

    e e 00 R e 00 R e 00 þ b0 R e 00 R e 00 R e 00 R e 00 S ð4Þ ¼ a0 Te R

ðG1Þ

en R e p in the equation for e To be systematic, note that any feeding term of the form c T S ðqÞ yields the term aqnp c Te n Re p in its solution with the explicit form for the coefficients:

1=aqnp ¼ 2n þ 2q  4 þ ðp þ 1  qÞ ¼ 2n þ 4p  2q

ðG2Þ

Hence one has a0 ¼ 4ca413 ¼ 1=2 and b0 ¼ 6c0 a404 ¼ 3=8 in (G1), and this also reproduces (225). We now S ð3;0Þ þ e S ð3;1Þ with, schematically: write e S ð3Þ ¼ e

3  e 00 e 00  3  e 00 e 00 e 00  e R R R R þ S ð3;0Þ ¼ Te R 8 8

ðG3Þ

The equation for Sð3;1Þ is, schematically:

3  e 00 e 00  3  e 00 e 00 e 00  3 e eð3;0Þ00  R  ðol  2 þ 2Þe S ð3;1Þ ¼  Te 2d R 3d R R R þ T S 8 8 2    ð3;0Þ e 00 e þ3 R S ð3;0Þ00 S 110 ðu113 Þ  e 3 h e 00 e 00 e 00 00 i 9 h e 00 e 00 e 00 e 00 00 i R R R R þ R R þ Te R 4 16

ðG4Þ ðG5Þ

ð3Þ This leads to the various contributions to e S 110 ð0; 0; uÞ:

  9 h e 00 e 00 e 00 e 00 00 i00 7  e 000 4 3 00  e 000 2 e 0000 1 00 2  e 0000 2 R R R R R R þ R R ¼ þ R R 160 160 20 80   3 e h e 00 e 00 e 00 00 i00 1 e  e 000 2 e 0000 1 e  e 000 2 e 0000 ¼ T R R  T R R ð0Þ T R R R 32 32 32   h   i    00 00 00 00 00 3 e eð3;0Þ00 3 3 1 e 000 2 e 0000 3 e  e 000 2 e 0000 3 e 0000 2 00 R R R ð0Þ R R  T R R ð0Þ  Te R ! Te a313 ¼ Te R T S 2 2 8 8 32 32 1 e  e 0000 2 00 1 e e 000 e ð5Þ 00 R þ TR R R þ T R 32 16 3 e ð3;0Þ00 3 e 2 3 3 h 00 00 00 00 i 1 e 2  e 0000 2 1 e 2 e 0000 e 0000 1 e 2 e 000 e ð5Þ 1 e 2 00 e ð6Þ T ðS Þ ! T a22 R R  T R ð0Þ R þ T R R  T R R ð0Þ ¼ T R 2 2 8 48 12 24 24  00 00 i 1 2 0000 3 e e 00 e 00 3 3 e 2 h 0000 0000 e 0000  1 Te 2 R e ð0Þ R e 000 R e ð5Þ þ 1 Te 2 R00 R e ð6Þ ð0Þ  T ðd R R Þ !  a22 T R  R ð0Þ R ¼ Te R 4 4 24 24 24 h     i     00 2 2 3 1 e 0000 e 00 Þ !  3 a3 Te e 000 e 00 R e 0000 R e 000  3 Te R R0002 þ R00 R0000 R00 ð0Þ R ¼ Te R  Te ðd R 4 4 13 32 32 3 e 0000 2 00 1 e 00 e 0000 e 0000 1 e 00 e 000 e ð5Þ þ Te R ð0Þ R þ T R R ð0Þ R  T R R R 32 32 32 h   2 1  2  00 i 1 9 e 00 e 00 e 00 9 e 0000 ð0Þ R e 000 R e 0000  1 Te R00 R e 000 R e ð5Þ e 000  Te R  ðd R R R Þ !  a313 Te R0000  R0000 ð0Þ R00 R00 ¼ Te R 8 8 32 32 16 h     i     00 9 e 00 e 00 e 00 9 1 e 000 4 7 00 e 000 2 e 0000 1 002 e 000 e ð5Þ R R Þ !  a304 R  R R R  ðd R R0002 þ R00 R0000 R00 R00  R R ¼ R 8 8 40 40 20    9 h i  00 ð3;0Þ e 00 e S 110 ðu113 Þ  e 3 R S ð3;0Þ00 ! a313 Te R00 ðR00 R00 Þ000  R00 ðR00 R00 Þ00 8 3 e 0000  e 000 2 3 e e 0000  e 000 2 ¼  Te R ð0Þ R þ TR R 32 32 1 e 00  e 0000 2 1 e 00 e 0000 e 0000 1 e 00 e 000 e ð5Þ þ TR R  T R R ð0Þ R þ T R R R 32 32 32    9 h  00  00 00 i ð3;0Þ e 00 e S 110 ðu113 Þ  e 3 R S ð3;0Þ00 ! a304 R00 R00 R00 R00 0  R00 R00 R00 R00 8 3  e 000 4 11 00  e 000 2 e 0000 ¼ R R þ R R 40 40 1 00 2  e 0000 2 1 002 e 000 e ð5Þ þ R þ R R R R 20 20

ðG6Þ ðG7Þ ðG8Þ ðG9Þ ðG10Þ ðG11Þ ðG12Þ ðG13Þ

ðG14Þ

ðG15Þ

ðG16Þ

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We note that each of these terms has a vanishing second derivative at zero (they all come from a true   e 00 ¼ ðol þ ÞR00 ¼ T e R e 0000  R e 0000 ð0Þ þ third2 cumulant, and third cumulants start as u6 ). Above, d R  2 3 e 000 þ R00 R e 0000 þ OðTR ; R Þ. Remarkably, the sum of all these terms, after multiple cancellations, gives R the T > 0 beta function to three loop derived by other means in Section 2. It is worth pointing out where each anomalous term originates from:   3 e eð3;0Þ00  3 e 3 3  e 00 e 00 e 00 00 00 3 e 0000  e 000 2 3 e e 0000 2 00 T S R R R ¼  Te R ð0Þ R  T R ð0Þ R ! T a13 2 2 8 32 32        00 00 3 e eð3;0Þ00 3 3 1 e 00 R e 00 e ð6Þ ð0Þ ¼  Te 2 R00 R R ! Te 2 a322 T S 2 2 8 24 3 e e 00 e 00 3 3 e 2 h e 0000 e 0000  e 00 00 i 1 e 2 00 e ð6Þ R  R ð0Þ R  T ðd R R Þ !  a22 T ¼ T R R ð0Þ 4 4 24  00 i 1  2 3 3 e e 00 e 00 3 3 e h 0002 00 0000 00 e 0000 ð0Þ R e 0000 ð0Þ2 R00 þ 1 Te R00 R e 0000 ð0Þ R e 000 þ Te R e 0000  T ðd R R Þ !  a13 T R þR R R ¼ Te R 4 4 32 32 32    9   00  00 00   ð3;0Þ 3 e 0000  e 000 2 1 e 00 e 0000 e 0000 e 00 R e 00 R e 00  R e 00 R e 00 e 00 R S 110 ðu113 Þ  e ¼  Te R 3 R00 e S ð3;0Þ00 ! a313 Te R ð0Þ R  T R R ð0Þ R 0 8 32 32    2  00 00 00 i 1 3 e h e 00 e 00 e 00 00 i00 1 e  e 000 2 e 0000 9 e 00 e 00 e 00 9 3 e h 0000 0000 e 000 e 0000 ð0Þ R ¼ T R R ð0Þ  ðd R R R Þ !  a13 T R  R ð0Þ R R ¼ Te R T R R R 32 32 8 8 32

ðG17Þ ðG18Þ ðG19Þ

ðG20Þ

e terms arise in the first four terms, while the last three are more regular (with Cancellations of 1= T e 00 cancelling). The cancellation of divergent terms occurs thus between term linear in R ee ee S ð3Þ00 in the unrescaled equation, and presumably between T S ð3Þ00 and fe S ð3Þ þ fue S ð3Þ0 in the oe S ð3Þ and T e equation. rescaled equation using the R G.2. Three loop at T = 0 by the C-ERG method Let us now recompute the T = 0 terms from the previous appendix, taking all limits at 0þ : 9 hh 00 00 00 00 00 i00 i 7  e 000 4 3 00  ~ 000 2 e 0000 1 00 2  e 0000 2 ¼ R R R R R þ R R R þ R R 160 160 20 80 3 e 000 þþ 2  e 000 2 11 e 000 þ 2  e 000 2 3 00 e 000 þþ 2 e 0000 þ 3 e 000 ð0þ Þ2 R e 0000 ð0þ Þ  R ð0 Þ R R ð0 Þ R  þ R R ð0 Þ R ð0 Þ  R00 R 160 160 80 16 h  00 i 9 9 1  e 000 4 7 00  e 000 2 e 0000 1 00 2 e 000 e ð5Þ ¼  ðdR00 R00 R00 Þ !  a304 R R0002 þ R00 R0000 R00 R00  R R R  R R R 8 8 40 40 20 3  e 000 2 e 000 þ 2 7 00 e 000 þ 2 e 0000 þ 1 00 e 000 þ 2 e 0000 þ R R ð0 Þ þ R R ð0 Þ R ð0 Þ þ R R ð0 Þ R 40 40 80   9   00  00 00 3  e 000 4 11 00  e 000 2 e 0000 ð3Þ R ¼ þ R R S 0;110 ðu113 Þ  S000 ! a304 ½~ R00 R00 R00 R00 0  R00 R00 R00 R00 3 R00 e R 8 40 40   2 1 e 0000 þ 1 R002 R e 000 R e ð5Þ þ R002 R 20 20 1 e 000 þþ 2  e 000 2 1 e 000 þ 2  e 000 2 1 00 e 000 þþ 2 e 0000 þ 3 e 000 ð0þ Þ2 R e 0000 ð0þ Þ  1 R00 R e 000 ð0þ Þ2 R e 0000  R ð0 Þ R  R ð0 Þ R þ R R ð0 Þ R ð0 Þ  R00 R 40 8 20 10 40

ðG22Þ ðG23Þ ðG24Þ ðG25Þ ðG26Þ ðG27Þ

Where we have assumed that R00 ð0þ Þ ¼ 0. We find that each of these expressions has no super-cusp by itself. When two limits have to be taken the one taken first is noted 0þþ (for instance this gives the feeding of the fourth cumulant into the third). This is PLB31. It has no ambiguity. Then the second limit 0þ is taken, this is PBL21 or PBL211. One easily sees that there is no supercusp independently for 0þ and 0þþ . The final result for the beta function to three loop using this procedure is:

    1 e 00 2 e 00 e 00 1  e 000 2  e 00 e 00  1 e 000 þ 2 e 00 R R R  R ð0Þ  R ð0 Þ R  R ð0Þ R þ 2 4 4 1  e 00 e 00 2  e 0000 2 3  e 000 4 1  e 00 e 00  e 000 2 e 0000 þ þ þ R R R  R ð0Þ R R  R ð0Þ R 16 32 4    5 e 000 þ 2 e 000 2 5 e 000 þ 2 e 0000 þ e 00 7 e 000 þþ 2 e 000 2 R ð0 Þ R R ð0 Þ R ð0 Þ R  R ð0 Þ R   32 16 160 7 e 000 þþ 2 e 0000 þ e 00 R ð0 Þ R ð0 Þ R þ 80

e ¼ ð  4fÞ R e þ fu R e0 þ oR

ðG28Þ

Thus if one sets the anomalous 0þþ terms to zero one reproduces the beta function obtained from the three loop finite T R equation using the value of r ð4Þ ð0Þ which cancels the super-cusp. Indeed the 0þþ

P. Le Doussal / Annals of Physics 325 (2010) 49–150

131

contribution has no supercusp by itself. Next, if one adds the 0þþ and 0þ contributions, one still does not obtain the full correct answer, derived in the next Section. As explained there an additional term, called a1 there, is set to zero here, see below for further explanations. G.3. Three loop beta function via W-ERG The simplest method to program in Mathematica, although quite memory consuming, is to use instead the W-functional. One uses the W-ERG Eq. (109) on the moments, directly at T = 0. It has the symbolic form:

 00 ot SðnÞ ¼ an Sðnþ1Þ

ðG29Þ

with an ¼ n=2, where in ðÞ00 denote two derivatives and their arguments set equal at the end, and symmetrization in n arguments is implicit. Similar notations were used in Appendix G of Ref. [36] for a different calculation, using instead the C-ERG equations. We recall that t ¼ m2 . One looks for a solution of the form:

ot R ¼

1 X

t 2q1 dq R

ðG30Þ

q¼1

SðnÞ ¼

1 X

t2p SðnÞ p ½R

n even

ðG31Þ

p¼0

SðnÞ ¼

1 X

t2pþ1 SðnÞ p ½R

n odd

ðG32Þ

p¼0

The recursion relations read: ðnÞ

ðnþ1Þ 00

ð2k þ 1ÞSk ¼ an ðSk

Þ 

k X

ðnÞ

Skq ½dq R; R

n odd

ðG33Þ

q¼1 ðnÞ

ðnþ1Þ 00

ð2k þ 2ÞSkþ1 ¼ an ðSk

Þ 

kþ1 X

ðnÞ

Skþ1q ½dq R; R

n even

ðG34Þ

q¼1 ðnÞ The notation SðnÞ p ½dq R; R just means that one performs the derivative ot Sp ½R and replace each of the p resulting ot R factors by dq R. These are what are usually called counterterms and here they appear automatically in the calculation. Each dq R is called the q-loop contribution to the beta function and is homogeneous of order OðRq Þ. The calculation of d1 R, i.e. the one loop contribution to the beta function, can be performed as follows:

ð4Þ

S0 ¼ 3½RR  00 ð3Þ ð4Þ S0 ¼ a3 S0  00 ð3Þ d1 R ¼ S0

ðG35Þ ðG36Þ ðG37Þ

To obtain d2 R, the two loop contribution to the beta function, one needs some one loop counterterms, hence d1 R. One does it as follows. ð6Þ

S0 ¼ 15½RRR  00 ð5Þ ð6Þ S0 ¼ a5 S0  00 ð4Þ ð5Þ ð4Þ 2S1 ¼ a4 S0  S0 ðd1 R; RÞ  00 ð3Þ ð4Þ ð3Þ 3S1 ¼ a3 S1  S0 ðd1 R; RÞ  00 ð3Þ d2 R ¼ S1

ðG38Þ ðG39Þ ðG40Þ ðG41Þ ðG42Þ

132

P. Le Doussal / Annals of Physics 325 (2010) 49–150

Hence to a given number of loop, q, one performs the gaussian (Wick) truncation on the 2q þ 2 moment, and from there solves the RG equations for all lower moments. It exactly amounts to solve the FRG hierarchy to a given order in powers of t, as shown above. Note that to get d2 R one could instead truncate the fifth moment into a product of the form Sð3Þ R and use the one-loop formula to evaluate the third moment. One can check that it yields the same result. Note that it is convenient to to use R i.e. set R00 ð0Þ ¼ 0 in the calculation, since this is the object which appear in the loop corrections, but then one should not forget when computing the beta function for R, and the related counterterms, to e 00 ð0Þ, with a1 ¼ 1. write oR ¼ oR  a1 u2 =2o R The result of this calculation, performed using Mathematica, reads to three loop, in the rescaled version, with a1 ¼ 1:

     2   2 e 00  R e 000 e 000 ð0þ Þ2 R e 00  R e 00 e ¼ ð  4fÞ R e þ fu R e0 þ 1 R e 00 ð0Þ R e 00 þ 1 R e 00 ð0Þ  1 R R oR 2 4 4 1  e 00 e 00 2  e 0000 2 3  e 000 4 1  e 00 e 00  e 000 2 e 0000 R  R ð0Þ R R  R ð0Þ R R R þ þ þ 16 32 4   3 e 000 þ 2 e 000 2 1 e 000 þ 2 e 0000 þ e 00 R ð0 Þ R   R ð0 Þ R ð0 Þ R 16 4

ðG43Þ

While keeping arbitrary a1 yields changes only in the last line as:

þ



a1 1 e 000 þ 2  e 000 2 a1 9 e 000 þ 2 e 0000 þ e 00 R ð0 Þ R R ð0 Þ R ð0 Þ R   þ 80 5 40 40

and one can check that the result of the previous section, Eq. (G28) corresponds to setting a1 ¼ 0. This is not too surprising since they were derived from a T > 0 procedure, where no non-trivial countere 00 ð0Þ arises (i.e. a1 ¼ 0) if one justs set T = 0. One must indeed properly take into account terms to R e the ensuing non-analyticity of the function RðuÞ, as is done here. The corresponding modification in the calculation of the previous section does indeed lead to the correct value a1 ¼ 1 and the correct three loop beta function (G43). Note that no ambiguity ever appear in this iterative procedure. That there should be no ambiguity in each Eq. (G29) is clear. The r.h.s. involves derivatives such that, e.g. for n ¼ 4, e ðv 3 Þ V e ðv 4 Þ. Despite the presence of shocks, the product Fðv 1 ÞFðv 2 Þ Q 1100 ðv 1 ; v 2 ; v 3 ; v 4 Þ ¼ Fðv 1 ÞFðv 2 Þ V in any correlation involving forces or anything smoother at points distinct from v1, v2 is a continuous function and the coinciding point limit v 1 ! v 2 can be taken unambiguously. This is further discussed in the text. What is less obvious is that this remains true order by order in the loop expansion. For N ¼ 1, we have checked explicitly that it does, up to four loop, but we believe it holds to all orders. Appendix H. Properties of the fixed point functions for the Sinai landscape H.1. Asymptotics of R(v) The large

v limit can be studied using the asymptotics (shown in the appendix of Ref. [80]):

Bðx; z1 ; z2 Þ ¼

Z

1

0

ðz1 z2 Þ2 1 x3 x eyz Aiðy þ z1 ÞAiðy þ z2 Þ pffiffiffiffiffiffi e122ðz1 þz2 Þ 4x 2 px

Inserting this approximation which becomes exact at large

1 E2 pðEÞ pffiffiffiffiffiffiffiffiffiffi e4v 4pv

Z

þ1 1

dz1 2pi

Z

þ1 1

ðH1Þ

v yields:

dz2 bEv ðz2 z1 Þ 21 ðz2 z1 Þ2 1 b v e 2 2pi Aiðz1 Þ Aiðz2 Þ2

ðH2Þ

which is always normalized to unity (the part with no shock decays rapidly at large v) and yields in R þ1 1 1 E2 1 e4v since 1 dz ¼ 1. We can now use the result of the appendix the large v limit, pðEÞ ¼ pffiffiffiffiffiffi 2pi Aiðz1 Þ2 4pv of Ref. [80] for the integral:

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Bðx; z1 ; z2 Þ ¼

Z



1

eyx Aiðy þ z1 ÞAiðy þ z2 Þ ¼ B1 ðx; z1 ; z2 Þ 1 

0

Z

1

dy

x

gðy; z1 ; z2 Þ B1 ðy; z1 ; z2 Þ



1 B1 ðx; z1 ; z2 Þ ¼ pffiffiffiffiffiffi e 2 px

y 1 0 0 Aiðz1 ÞAiðz2 Þ  ðAi ðz1 ÞAiðz2 Þ þ Aiðz1 ÞAi ðz2 ÞÞ gðy; z1 ; z2 Þ ¼ 4 4 1 1 0 0 0 0 þ ð2Ai ðz1 ÞAi ðz2 Þ  ðz1 þ z2 ÞAiðz1 ÞAiðz2 ÞÞ  2 ðz2  z1 ÞðAi ðz1 ÞAiðz2 Þ  Aiðz1 ÞAi ðz2 ÞÞ 4y 4y 2 x3  x ðz þz Þðz1 z2 Þ 2 12 2 1 4x

This yields:

ðH4Þ ðH5Þ ðH6Þ

!

" Z 1 dz2 4 1 gðy; z1 ; z2 Þ 2 2v þ 2 ðz2  z1 Þ ðH7Þ 1  dy B1 ðy; z1 ; z2 Þ Aiðz1 Þ2 Aiðz2 Þ2 v =ð2aÞ 1 1 2pi b  pffiffiffiffiffiffiffi 1 3 v ðz þz Þþðz2 z1 Þ2 ðH8Þ ðabÞ2 2 pv e48v e2b 1 2 b2 v

2ðRð0Þ  Rðv ÞÞ ¼ þ

ðH3Þ

1

Z

þ1

dz1 2pi

Z

þ1

v Aiðz1 ÞAiðz2 Þ where for large v only the 1 contributes yielding the formula given in the text. One can also show the alternative formula:

" Z du þ1 dw iv u w22 2 1  pffiffiffiffi   pffiffiffiffi  2ðRð0Þ  Rðv ÞÞ ¼  a 2v pe e b e b ow Ai iu þ i v w2 Ai iu  i v w2 1 2p 1 2p R1 vV  pffiffiffiffi   pffiffiffiffi # dVe2 Ai aV þ iu þ i v w2 Ai aV þ iu  i v w2 ðH9Þ þv 0  pffiffiffiffi 2  pffiffiffiffi 2 Ai iu þ i v w2 Ai iu  i v w2 2

H.2. Expansion at small

pffiffiffiffi

1 v3 48

Z

þ1

v

The distribution of rescaled energy can be written:

 v g   2 2 R1 vW Z Z 2 dz1 dz2 v ðz1 þz2 Þþðz2 z1 Þ 0 dWe2a AiðW þ z1 ÞAiðW þ z2 Þ 2 v  1 v 3 v  4 b e 48 e2b þ ðabÞ a 2pi 2pi Aiðz1 Þ2 Aiðz2 Þ2

1 3 2 pðÞ ¼ e48v v 4 g

with





v

ðH10Þ ðH11Þ

 ¼ E=v . We use the notations of the text. Hence one has:

R1   Z dWAiðW þ z1 ÞAiðW þ z2 Þ dz1 dz2 ðz2 z1 Þ 1 1 1 1 ðz1 þ z2 Þ  2 þ 0 eb Aiðz1 ÞAiðz2 Þ Aiðz1 ÞAiðz2 Þ 2b 4 a 2pi 2pi Z Z dk dk 1 2 iðk2 k1 Þ ¼ a2 e 2p 2p R1   dWAiðW þ ibk1 ÞAiðW þ ibk2 Þ 1 i 1 1 ðk1 þ k2 Þ  2 þ 0 Aiðibk1 ÞAiðibk2 Þ Aiðibk1 ÞAiðibk2 Þ 2 4 a

p1 ðÞ ¼ ðabÞ2

Z

Thus:

b 1 ð lÞ ¼ P

 Z dk2 1 i deil p1 ðÞ ¼ a2 ð2k2 þ lÞ 2p Aiðibk2 þ iblÞAiðibk2 Þ 2 R1  Z dWAiðW þ ibk2 þ iblÞAiðW þ ibk2 Þ 1 1 dk2 1 þ a2 o2l þ 0 ðH12Þ Aiðibk2 ÞAiðibk2 þ iblÞ a 4 2p Aiðibk2 þ iblÞAiðibk2 Þ Z

Z dk2 1 ¼ a2 2p Aiðibk2 þ ibl=2ÞAiðibk2  ibl=2Þ R1   1 0 dWAiðW þ ibk2 þ ibl=2ÞAiðW þ ibk2  ibl=2Þ ik2 þ  a Aiðibk2 þ ibl=2ÞAiðibk2  ibl=2Þ Z 1 dk 1 2 þ a2 o2l 4 2p Aiðibk2 þ ibl=2ÞAiðibk2  ibl=2Þ

ðH13Þ

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The function f ðzÞ ¼ Aiðz þ z1 ÞAiðz þ z2 Þ satisfies the differential equation:

f 0000 ðzÞ  ð4z þ 2z1 þ 2z2 Þf 00 ðzÞ  6f 0 ðzÞ þ ðz1  z2 Þ2 f ðzÞ ¼ 0

ðH14Þ

This yields:

Z

0

1

dzAiðz þ z1 ÞAiðz þ z2 Þ ¼

Z0 1

0

Aiðz2 ÞAi ðz1 Þ  Aiðz1 ÞAi ðz2 Þ z2  z1

ðH15Þ

dzAiðz þ z1 Þ2 ¼ z1 Aiðz1 Þ2  Ai ðz1 Þ2 0

ðH16Þ

0

Let us denote

p11 ðÞ ¼ ðabÞ2

1 a

¼ ðabÞ2

1 a

Z Z

dz1 2pi dz1 2pi

Z Z

dz2 ðz2 z1 Þ eb 2pi

R1 0

dWAiðW þ z1 ÞAiðW þ z2 Þ Aiðz1 Þ2 Aiðz2 Þ2 0

ðH17Þ

0

dz2 ðz2 z1 Þ Aiðz2 ÞAi ðz1 Þ  Aiðz1 ÞAi ðz2 Þ eb 2pi ðz2  z1 ÞAiðz1 Þ2 Aiðz2 Þ2

ðH18Þ

One finds:

o p11 ðÞ ¼ 2a1 b p10 ðÞ ¼ 



2

4

2

gðÞgðÞ ¼ gðÞgðÞ

ðH19Þ

1 gðÞgðÞ  ðg 0 ðÞgðÞ þ gðÞg 0 ðÞÞ 2

ðH20Þ

Let us first check normalization

3 4

Z

d2 gðÞgðÞ ¼

Z

R

dp1 ðÞ ¼ 0. We need to show:

dg 0 ðÞgðÞ

ðH21Þ

One has:

3 4 Z

" # 2 Z 0 3b dk ibk Ai ðibkÞ2 þ2 d gðÞgðÞ ¼ 4 a2 2p AiðibkÞ2 AiðibkÞ4 Z 1 dk ibk dg 0 ðÞgðÞ ¼ 2 2p AiðibkÞ2 ba

Z

2

ðH22Þ ðH23Þ

These two quantity are indeed equal, from the identity:

3

Z

Ai ðzÞ2 0

dz

AiðzÞ

4

¼

Z

dz

z AiðzÞ2

ðH24Þ

R 0 R 0 which can be checked by integration by part of z Ai ðzÞ2 =AiðzÞ4 ¼  12 z ðAi ðzÞ=AiðzÞÞð1=AiðzÞ2 Þ0 . This implies the expansion and derivatives given in the text. H.3. Check of matching Writing:

Z 1 1 ez=b ab z AiðzÞ Z 0 a Ai ðzÞ dðyÞ ¼ ez=b b z AiðzÞ

gðÞ ¼

ðH25Þ ðH26Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

with

R z

¼

Z

Z Z Z with:

R

dz=ð2piÞ. One finds:

d4 gðÞðÞ ¼

4 J a2 1

d2 g 0 ðÞgðÞ ¼

ðH27Þ

1 J a2 2

ðH28Þ

dudygðuÞy3 dðyÞgðy þ uÞ ¼

2 J a2 3

ðH29Þ

dudygðuÞy2 dðyÞgðy þ uÞ ¼ 1

ðH30Þ

0000 1 1 7 ¼ 5I1  28I2 þ 24I3 ¼ I1 AiðzÞ AiðzÞ 15 z

00 Z z 1 1 J2 ¼ ¼ I1  2I2 ¼ I1 AiðzÞ AiðzÞ 3 z Z 0 000 Ai ðzÞ 1 8 J3 ¼  ¼ 2I1  8I2 þ 6I3 ¼ I1 AiðzÞ 15 AiðzÞ2 z J1 ¼

Z

ðH31Þ ðH32Þ ðH33Þ

where we have defined:

I1 ¼

Z z

I2 ¼

Z z

I3 ¼

Z z

z2

¼ 3I2

AiðzÞ2

zAi ðzÞ2 0

¼

AiðzÞ4

ðH34Þ

5 I3 3

ðH35Þ

Ai ðzÞ4 0

ðH36Þ

AiðzÞ6

the above relations being derived from considering the total derivatives which integrate to zero. The droplets predict:



0  0 3 0  0 ðzÞ 1 ; AiAiðzÞ and AiðzÞ 5 2

zAi0 ðzÞ AiðzÞ3

Z 1 1  R000 ð0þ Þ ¼  2 du dy gðuÞy3 dðyÞgðy þ uÞ ¼  2 J 3 4 a Z

Z 1 1 d4 gðÞgðÞ  3 d2 g 0 ðÞgðÞ ¼ 2 ðJ 1  3J 2 Þ  R000 ð0þ Þ ¼ 4 a

ðH37Þ ðH38Þ

8 We can see that these results agree J 1  3J 2 ¼ J 3 ¼  15 I1 . The final result is:

R000 ð0þ Þ ¼

1 8 a2 15

Z

z2

ðH39Þ

AiðzÞ2

z

H.4. Calculation of DðuÞ Let us start from the formula given in the text. The first part gives, writing F 1 ¼ F  v =2; F 2 ¼ F þ v =2: 1

Dns ðv Þ ¼ e48v

3

Z

þ1

1

1

¼ a2 e48v

3

Z



v2  v  v v 2 dF F 2  g F  g F  e4F 4 2 2 þ1

1

dk1 2p

Z

þ1

1

dk2 iv ðk1 þk2 Þ 1 e2 Aiðibk1 ÞAiðibk2 Þ 2p

ðH40Þ Z

þ1

1



v 2 iðk1 k2 ÞFv4F 2 dF F 2  e 4 ðH41Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

¼

Z Z 1 pffiffiffiffi 5=2 2  1 v 3 þ1 dk1 þ1 pv a e 48 2 2p 1 1  dk2 iv ðk1 þk2 Þðk1 kv 2 Þ2  1 e2 8v  v 4  16ðk1  k2 Þ2 Aiðibk1 ÞAiðibk2 Þ 2p

ðH42Þ

Using the same trick as below it can also be written:

pffiffiffiffi 1 3 2 Dns ðv Þ ¼ 2 pv 1=2 a2 b e48v

Z

þ1

dk1 2p

1

Z

0

þ1

0

dk2 iv ðk1 þk2 Þðk1 kv 2 Þ2 Ai ðibk1 ÞAi ðibk2 Þ e2 2p Aiðibk1 Þ2 Aiðibk2 Þ2

1

ðH43Þ

The part with shocks is: 1

pðF 1 ; F 2 ; v Þ ¼ gðF 1 ÞgðF 2 ÞdF 1 dF 2 hðv þ F 1  F 2 Þe48v

Z

3

þ1

1

Z v þF 1 F 2

v4 ðuF 1 v2 Þ2 ik2 ðv þF 1 F 2 Þ iðk1 k2 Þu

due

e

e

0

dk1 2p

Z

þ1

dk2 2p

1

Z

1

1V

dVe2

v

0

AiðaV þ ibk1 ÞAiðaV þ ibk2 Þ Aiðibk1 ÞAiðibk2 Þ

ðH44Þ

To compute D the best is to go back directly to:

Z

dF 1 dF 2 g s ðF 1 ; F 2 ; v Þ ¼



Z

1

1

Z

1

du1 du2 du1 e48v 

3 v ðu v 1 þv 2 Þ2 2 4 1

ðH45Þ

vV

dVe2 ðv 1  u1 Þgðv 1  u1 Þhðu1  u1 ; V 1 Þhðu2  u1 ; VÞðv 2  u2 Þgðu2  v 2 Þ 0 Z Z 1 2 1v V  1 v 3 v wv ¼  dwe 48 4ð 2Þ dVe2 wðw; VÞwðv  w; VÞ

ðH46Þ

0

(put bounds) where we have defined:

Z

wðw; VÞ ¼

1

duðu  wÞgðu  wÞhðu; V 1 Þ ¼ 

Z

0

1

with w ¼ u1  v 1 . One has ugðuÞ ¼  1

2

Ds ðv Þ ¼ b a2 e48v

3

Z

0 dk iku bAi ðibkÞ e 1 2p aAiðibkÞ2

v wv dwe 4ð 2Þ

0

2

Z

þ1

dk1 2p

Z

þ1

1

dk2 ik2 v iðk1 k2 Þw e e 2p

Z

1

1

dVe2

vV

0

3

1

3

Z

þ1 1

dk1 2p

0

ðH48Þ

Aiðibk2 Þ3

Aiðibk1 Þ

2

ðH47Þ

0

Ai ðibk1 ÞAiðaV þ ibk1 Þ Ai ðibk2 ÞAiðaV þ ibk2 Þ

¼ b a2 e48v

0

dk ikw bAi ðibkÞAiðaV þ ibkÞ e 2p aAiðibkÞ3

R þ1

1



þ1

Z

þ1

1 0

dk2 pffiffiffiffi 1=2 ðk1 kv 2 Þ2 þiv ðk1 þk2 Þ 2 2 pv e 2p

Z

1

vV

0

Ai ðibk1 ÞAiðaV þ ibk1 Þ Ai ðibk2 ÞAiðaV þ ibk2 Þ Aiðibk1 Þ3

1

dVe2

Aiðibk2 Þ3

ðH49Þ

Putting everything together one finds the formula given in the text. Appendix I. Short range random potential and Kida Burgers turbulence Consider the d = 0 toy model

HV;v ðuÞ ¼

m2 ðu  v Þ2 þ VðuÞ 2

ðI1Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

where VðuÞ has short range correlations. This is the problem studied, in the context of Burgers turbulence, by Kida [88] and is by now standard. A nice calculation using replica, and interpretation in terms of extremal statistics was given in [89]. The calculation of the one point function below is a slight generalization of the one in Ref. [89]. We recall it here and extend to the two point function and calculation of R(u) for this problem. We also derive the logarithmic corrections for disorder distributions which fall in the Gumbel class. One method to define the model in the continuum is via a Poisson point process. See Ref. [78] for the solution in that case. We also use below some of the arguments given there. Here we will rather discretize u to integers and consider the small m limit where a continuum limit exists. We consider that VðuÞ are i.i.d random variables and call P0 ðVÞ the on site probability distribution and define:

P< ðVÞ ¼

Z

V

P0 ðWÞdW ¼ eAðVÞ

ðI2Þ

1

R1 and P> ðxÞ ¼ x P0 ðVÞdV ¼ 1  P< ðxÞ. The minima statistics are controlled by the tail of PðVÞ for small V. We treat here all cases in the so-called Gumbel class, and give explicit application to the case AðVÞ ¼ BjVjd  ln C for large negative V, the subcase d ¼ 1 being PðVÞ ¼ BCeBV as V ! 1. I.1. One point function Let us first consider the problem for T = 0, i.e. the inviscid limit m ¼ 0 and define b ðv Þ ¼ minu HV;v ðuÞ. Consider first the distribution of the absolute minimum. It is equivalent to study V v ¼ 0. Then the probability Pðu1 ; V 1 Þ that the absolute minimum is at u1 with energy 2 H ¼ E1 ¼ m2 u21 þ V 1 is given by:

Z

1 m2 2 m2 2 u1  u P> ðV 1 þ m2 ðu21  u2 ÞÞ PðV 1 Þ exp  duP< V 1 þ 2 2 2 u–u1 Z

1 1 2 1 2 ~ eAðV 1 þ2u~1 2u~ Þ ¼ A0 ðV 1 ÞeAðV 1 Þ exp  du m

Pðu1 ; V 1 Þ ¼ PðV 1 Þ

Y

ðI3Þ ðI4Þ

assuming lnð1  P < Þ P< which can be checked a posteriori to hold for the bulk of the resulting dis~ 1 =m and u ¼ u ~ =m. In the limit m ! 0 the bulk of the probability is tribution. We have defined u1 ¼ u around V 1 ¼ V m such that:

eAðV

m

Þ

¼ m;

AðV m Þ ¼ lnð1=mÞ 1=d

m

hence V ¼ ðlnð1=mÞ=BÞ . Expanding AðV 1 Þ around V ~ one finds: performing the gaussian integral over u

Pðu1 ; V 1 Þ mam eam ðV 1 V

m

Þ

ðI5Þ m

0

m

to linear order, defining am ¼ A ðV Þ and

  m 1~2 1 exp eam ðV 1 V þ2u1 Þþ2 lnð2p=am Þ

ðI6Þ

1

with am d lnð1=mÞ=jV m j ¼ d lnð1=mÞ1d B1=d . Hence one finds that the dependence in energy is of Gumbell type from extremal statistics of short range correlated variables, while the one point distribution for the minimum, obtained by integration over V 1 is a simple Gaussian (in a rescaled sense as m ! 0):

rffiffiffiffiffiffiffiffiffiffiffiffi m2 am 1m2 am u2 1 e 2 Pðu1 Þ

2p 1 2d

ðI7Þ

1d 2d

hence u1  d1=2 B m1 ðln m1 Þ . This corresponds to f ¼ 1 with logarithmic corrections. This is also the one point distribution of the Burgers velocity field since uðxÞ $ Fðv Þ ¼ ðv  uðv Þ=tÞ has the same distribution as u1 =t with t ¼ m2 . From this we get the kinetic energy:

EðtÞ ¼

1d 1 1 1 m2 uðx; tÞ2 ¼ 2 hu21 iP ¼ ¼ 21=d d1 B d t1 ðln tÞ d 2 2am 2t

and one finds EðtÞ  1=ðtðln tÞ1=2 Þ for the gaussian case d ¼ 2.

ðI8Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

It is also interesting to note that the joint distribution of the renormalized potential b 0 ð0Þ ¼ m2 u1 , at the same point is juste b ð0Þ ¼ V 1 þ 1 m2 u2 and its derivative, the force F ¼ V E¼V 1 2 the product:

PðE; FÞ ¼ Q e ðE  V m ÞQ f ðFÞ am E

ðI9Þ

am E

Q e ðEÞ ¼ am e expðe Þ rffiffiffiffiffiffiffiffiffiffiffiffiffi am am F 2 =ð2m2 Þ Q f ðFÞ ¼ e 2pm2

ðI10Þ ðI11Þ

of a Gumbel by a Gaussian, hence that these two variables are statistically independent. I.1.1. Two point function Let us now consider the two point probability P tot v 1 ;v 2 ðu1 ; V 1 ; u2 ; V 2 Þ. It has two contributions:

Ptot v 1 ;v 2 ðu1 ; V 1 ; u2 ; V 2 Þ ¼ dðu2  u1 ÞdðV 2  V 1 ÞP v 1 ;v 2 ðu1 ; V 1 Þ þ P v 1 ;v 2 ðu1 ; V 1 ; u2 ; V 2 Þ

ðI12Þ

according to whether there are no shock, or at least one shock in the interval ½v 1 ; v 2 . We start with the second contribution, i.e. we assume that there are shocks. Then one wants the two conditions to be simultaneously fulfilled:

m2 m2 ðu1  v 1 Þ2  ðu  v 1 Þ2 ; 2 2 m2 m2 VðuÞ P V 2 þ ðu2  v 2 Þ2  ðu  v 2 Þ2 ; 2 2

VðuÞ P V 1 þ

equality in u1

ðI13Þ

equality in u2

ðI14Þ

One defines the intersection of the two parabola (the construction is similar to Section 4.4.3):

u ¼

v1 þ v2 2





m2 v 1 þ v 2 E21 2 2 ððu  2 V þ  v Þ  ðu  v Þ Þ ¼ 21 2 2 1 1 m2 v 21 2 2 m v 21 1

ðI15Þ

b ðv 2 Þ  V b ðv 1 Þ must satisfy One must have u1 < u < u2 , hence the energy difference E21 ¼ V 1 < v þ v  u . This is assuming there are shocks in the interval ½v 1 ; v 2 . The  12 v 21 þ v 2  u2 < vE21 21 1 1 2 21 case u1 ¼ u < u2 corresponds to one shock in the interval at v ¼ v 2 . When there are shocks between the points then: P v 1 ;v 2 ðu1 ; V 1 ; u2 ; V 2 Þ A0 ðV 1 ÞeAðV 1 Þ A0 ðV 2 ÞeAðV 2 Þ ! Z u~ Z þ1 ~ AðV 1 þ1ðu~1 v~ 1 Þ2 1ðu~v~ 1 Þ2 Þ ~ AðV 2 þ1ðu~2 v~ 2 Þ2 1ðu~v~ 2 Þ2 Þ du du 2 2 2 2 e e exp   m ~ 1 m u m

m2 a2m eam ðV 1 þV 2 2V Þ Z Z u~ 2 1 2 m 1 ~ eam ðV 1 V þ2ðu~1 v~ 1 Þ 2ðu~v~ 1 Þ Þ  exp  du

þ1

am

~e du





~ 2 e V 2 V m þ12 u v2

2

~ v~ 2 Þ2 12ðu

!

~ u

1

ðI16Þ

~ i ¼ mui and v ~ i ¼ mv i . Since u depends only on V 21 ¼ V 2  V 1 one can inteAs above we have defined u grate over ðV 1 þ V 2 Þ=2  V m which gives: ~

Pv 1 ;v 2 ðu1 ; u2 ; V 21 Þ ¼ hR

~ u 1

m2 am i2 v Þ R þ1 ~ am ð12V 21 þ12ðu~2 v 2 Þ2 12ðu~v~ 2 Þ2 Þ þ u~ due

~ 1  1 Þ2 1ðu ~  ~ 1 Þ2 am 12V 21 þ12ðu 2

~e ð du

v

ðI17Þ

This yields the joint distribution, where we have restored the mass dependence:

Pv 1 ;v 2 ðu1 ; u2 ; E21 Þ ¼ m2 a2m pma1=2 v m

 21

1=2 ma1=2 m ðu1  v 1 Þ; mam ðu2  v 2 Þ; am E21



2

1 w2 þw2 þ1v 2 þ E e 2ð 1 2 Þ 4 v 2 v E v pv ðw1 ; w2 ; EÞ ¼  h   w2 < <  w1     2 2 v 2 U v2  vE þ U v2 þ vE Þ 2 =2

where UðwÞ ¼ ew

Rw

1

z2 =2

dze

¼

R þ1 0

z2 =2þwz

dze

which satisfies U0 ðwÞ ¼ 1 þ wUðwÞ.

ðI18Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

The case with no shock corresponds to u1 ¼ u ¼ u2 . From the above we get:

Pv 1 ;v 2 ðu1 ; V 1 Þ mam eam ðV 1 V

m

Þ

am ðV 1 V m Þ

exp e

"Z

~1 u

~e du

~ 1 v~ 1 Þ2 1ðu ~ v~ 1 Þ2 Þ am ð12ðu 2

þ

1

Z

þ1

~e du

am

 1 2

~ 1 e v2 u

2

~ v~ 2 Þ2 12ðu

 #!

~1 u

ðI19Þ Through integration over the Gumbell function of V 1 we get:

Pv 1 ;v 2 ðu1 Þ ¼ ma1=2 m pv

1=2 21 mam

 1=2  1=2 mam ðu1  v 1 Þ; mam ðu1  v 2 Þ

1 pv ðw1 ; w2 Þ ¼ Uðw1 Þ þ Uðw2 Þ

ðI20Þ ðI21Þ

Adding the two contributions, and setting for now am and m to unity (to be restored below) we find b ðv 1 Þ, and the forces b ðv 2 Þ  V the joint distribution for the energy difference E ¼ V b 0 ðv i Þ ¼ m2 ðv i  ui Þ; i ¼ 1; 2, with v ¼ v 2  v 1 , as: F i ¼ Fðv i Þ ¼ V

  v 1 PðF 1 ; F 2 ; EÞ ¼ dðF 2  ðF 1 þ v ÞÞd E  ðF 1 þ F 2 Þ 2 UðF 1 Þ þ UðF 2 Þ 2

1ðF 2 þF 2 Þþ1v 2 þ E e 2 1 2 4 v2 v E v h  < < þ  þ F þ F   2 2 2 v 2 1 U vE ÞþU vþE v

2

2

ðI22Þ ðI23Þ

v

1=2 To restore m dependence one replaces F i ! a1=2 m F i =m; E ! am E and v ! v mam , and correcting as needed for the probability measure to remain normalized to one. The two point distribution of the forces is obtained by integration over E:

PðF 1 ; F 2 Þ ¼ dðF 2  ðF 1 þ v ÞÞ 2

2

þ v e2ðF 1 þF 2 Þþ4v 1

1 2

Z

v þF

1

UðF 1 Þ þ UðF 2 Þ 2

e d    2 hðF 1  F 2 þ v Þ v v  2þF 2 U 2   þ U v2 þ  2

1

ðI24Þ ðI25Þ

it also yields the two point distribution for Burgers turbulence through the replacement: 1=2

F i ! at t 1=2 uðxi Þ 1=2 1=2 ðx2 t t

v!a

 x1 Þ

ðI26Þ ðI27Þ

with at ¼ am¼1=pffit . Hence the internal scale is  ðt=at Þ1=2 . Since the width of a shock is 1  Tt=u21  Tt=ðt=at Þ1=2 the dimensionless ratio is T at  Tðln tÞ1d . Note that for d < 1 temperature is in effect irrelevant. The case d ¼ 1 is related to the REM. To compute R(u) we instead integrate over F 1 and F 2 gives (being careful with the jacobian factor 1=v in the first term) and obtain the distribution of scaled energy difference  ¼ E=v :

    U v   U v2 þ  1 v þv  2   2 U 2 þU 2þ U v2   þ U v2 þ          U0 v2   U v2 þ  þ U v2   U0 v2 þ  ¼ ¼ o Hð; v Þ

v  v 2 U 2 þU 2þ   U v2 þ   v  Hð; v Þ ¼ v U 2 þU 2þ

PðÞ ¼

v



ðI28Þ ðI29Þ ðI30Þ

pffiffiffiffiffiffi 2 using that wUðwÞ ¼ U0 ðwÞ  1. Since UðwÞ  1=w vanishes as w ! 1 and diverges at 2pew =2 for w ! 1, one has Hð1Þ ¼ 0 and Hðþ1Þ ¼ 1 hence PðÞ is correctly normalized. We now obtain:

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

1 Rð0Þ  Rðv Þ ¼ v 2 2

Z

þ1

d pðÞ ¼ 2v 2

2

Z

þ1

0

1

  U v2      d v U 2   þ U v2 þ 

1 2 1 v  pffiffiffiffi v 3 þ 0:0272494v 4  0:00114373v 5 þ Oðv 6 Þ 2 3 p

¼

ðI31Þ ðI32Þ

where integration by parts and symmetries have been used. One shows, using symmetries and the differential equation for UðwÞ, that the integrand can be rewritten:

 v

Hð; v Þ ¼ ov ln U

2

 v  1 v  1     þU þ þ þ  þ v 2 2 2 U 2   þ U v2 þ 

ðI33Þ

where the linear term exactly cancels the bevavior of the first term at  ! þ1. In the end one shows that:

Z

þ1

1    d v U 2   þ U v2 þ  p ffiffiffi ffi  v 2 = p þ 0:108998v 3  0:00571863v 4 þ Oðv 5 Þ

 R0 ð v Þ ¼ 2 v

ðI34Þ

0

¼v

ðI35Þ

which leads to Kida result for the two point force correlator, i.e. the velocity correlator DðuÞ ¼ R00 ðuÞ. Here we have obtained also Rð0Þ since Rð1 ¼ 0Þ ¼ 0 one gets for large v:

Rð0Þ ¼ lim 2v 2

Z

v !1

1

d

0

1 p2 ¼ v  1þe 6

ðI36Þ

Restoring all m factors one sees that

Rðv Þ ¼ a2 m Rs

pffiffiffiffiffiffi  am mv

ðI37Þ

where the scaled R, noted here Rs ðv Þ is given by formula (I32). It should in principle satisfy the FRG equation with f ¼ 1 (and h ¼ 0) to leading order in m ! 0. Note that it should be the short range solution for this value of f, while the long range one (with the same value of f) corresponds to logarithmic disorder and was studied in Appendix ??. As a result the force correlator satisfies:

Z

1

Dðv Þdv ¼ 0

ðI38Þ

0

and indeed one can see in Fig. 10 or Ref [88] what was probably (unknowingly) the first FRG fixed point correlator (before the FRG was invented) with a nice cusp, and is reminiscent of the results for manifolds in random bond disorder measured recently in [68]. Consider now the small v limit of the part of the two point force probability which contains at least one shock (setting t to unity), from formula (I25) one obtains: 1

2

2

Pshock ðF 1 ; F 2 Þ v e2ðF 1 þF 2 Þ ðF 1  F 2 ÞhðF 1  F 2 Þ

ðI39Þ

Since as v ! 0 one can safely assume that there will be only one shock in the interval one finds that the joint distribution of shock sizes s ¼ uðv þ Þ  uðv  Þ ¼ F 1  F 2 > 0 (for t ¼ 1) and positions ^ ¼ 12 ðuðv þÞ þ uðv  ÞÞ is simply: u 2 s 2 ^ Þ ¼ pffiffiffiffi es =4 eðu^v Þ hðsÞ Pðs; u 2 p

ðI40Þ

^  v is the velocity of the shock in Burgers, and the center position of the normalized to unity, where u ^  v are independent. From the droplet shock relation droplet/shock, and the two variables s and u (213) we find the droplet density distribution function:

1 2 DðyÞ ¼ pffiffiffiffi ey =4 2 p

ðI41Þ

R1 normalized by 1 y2 DðyÞ ¼ 2. We will refrain from giving here results for the other classes of disorder (Weibul and Frechet) but these are easily analyzed along the same lines. For the depinning this was done very recently in [70].

P. Le Doussal / Annals of Physics 325 (2010) 49–150

141

Appendix J. Two-well droplet calculation in higher dimension We start from:

e ½fv x g  V e ½fv x ¼ 0g ¼ 1 V 2

P 1 P v~ x

Xi ¼ e

X

g 1 xy v x v y  T lnðpX 1 þ ð1  pÞX 2 Þ

g xy ui;y

x

ðJ1Þ

xy

ðJ2Þ

y

with p ¼ 1=ð1 þ wÞ. As before calculation is simpler if one considers the force:

F ½v x ¼



e ½fv z g dV u1;y X 1 þ u2;y X 2 w 1 ~ ¼ g 1 T v~ y  xy ðT v y  huy iÞ ¼ g xy dv x X 1 þ wX 2

ðJ3Þ

One defines the moments: 1 1 S11:1 x1 :xn ¼ F ½v 1 x1 :F ½v n xn ¼ g x1 z1 :g xn zn C n ½v 1 ; . . . ; v n z1 ;...;zn

u1;xi ai þ u2;xi w ai þ w i¼1 * + Y ¼ u1;xi þ A2n ½v 1 ; . . . ; v n x1 ;...;xn

C n ½v 1 ; . . . ; v n x1 ;...;xn ¼ C 2n ½v 1 ; . . . ; v n x1 ;...;xn

ðJ4Þ

n Y

i

C 2nþ1 ½v i z1 ;...;zn ¼ A2nþ1 ½v i z1 ;...;zn

ðJ5Þ ðJ6Þ

u1

0 * + 1 Y X A  T@ v~ i;z u1;z

ðJ7Þ

j

i

i

j–i

u1

with ai ¼ X 1;i =X 2;i . The calculation is similar to Appendix F. The result is:

An ½v i x1 ;...;xn

* "

n Y 1 X 1 1 ¼ T Y xi ðYg v i Þ u1;xj  Y xj 1 4 i¼1 1  eðYg v ji Þ j–i #+

Y 1 þð1Þn1 u1;xj  Y xj ðYg 1 v ji Þ 1  e j–i u ;Y

ðJ8Þ ðJ9Þ

1

1

where Y x ¼ u1;x  u2;x and ðYg v Þ ¼ u1;x $ u2;x . The highest order term is:

Ahigh n ½ i x1 ;...;xn

v

P

1 xy Y x g xy

v y.

The two symmetries used are ui;x ! ui;x and

* + n h i X 1 1 1 ¼ T Y x1 . . . Y xn ðYg v i ÞF n ðYg v ij Þ 4 i¼1

ðJ10Þ

u1 ;Y

where the function F n was given in (F25). For the second and third moments we obtain the result given in the text. Appendix K. STS and ERG identities for correlations in higher d and droplets K.1. Functional form of STS and ERG identities The general STS identity is:

T

X dO½u c

duxc

¼ g 1 xy

X hO½uuyf i f

ðK1Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Specializing to observables O½u depending only on a single replica one finds:

T



   dO½u ¼ g 1 xy hO½uuy i  hO½uihuy i dux

ðK2Þ

In the zero T limit it yields:

  dO½u1  ¼ g 1 xy hðu1x  u2x ÞO½u1 iD½u1 ;u2  du1x P½u1 

ðK3Þ

which yields the functional equation given in the text. Similarly the ERG equation is:

ohO½ui ¼ 

1 X ðuf og 1 uf ÞO½ui 2T f

ðK4Þ

Specializing to g independent observables depending on a single replica one gets the functional equation given in the text. K.2. Droplet calculation of correlation functions In the case of independent well separated droplets one evaluates the thermal and disorder averages as follows. The mean of any observable is:

  P ðiÞ ðiÞ ðjÞ Aðu1 Þ þ wi A u1 þ u21 þ i–j wi wj Aðu1 þ u21 þ u21 Þ þ    P P 1 þ i wi þ i–j wi wj þ   

ðK5Þ

Since any disorder average containing more than one distinct wi will be higher order in T upon computing the above average one finds:

hO½uui  hO½uihui ¼

X



wi 2

i

ð1 þ wi Þ

   ðiÞ ðiÞ u1  u2 þ OðT 2 Þ Oðu1 Þ  O u2

ðK6Þ

1  21 1 21 ^ x  12 u21 ^ ^ We now define u1x ¼ u x , u2x ¼ ux þ 2 ux , then one has hux i ¼ ux þ 2  p ux . Only correla^ and an even number of u21 are non-zero and satisfy the symmetries. tions with an even number of u This yields for the two point functions:

Gxy ¼ hu1x u1y i 

T X D ðiÞ ðiÞ E u21x u21y 2 i

ðK7Þ

e xy ¼ hu1x u1y i G

ðK8Þ R0000 xyzt ½0

Q 1111 xyzt ½0

We now want to compute the four point functions. We need to relate and to droplets since all correlations can be obtained from them. We will also check the consistency, i.e. that, as for d = 0 all three STS relations amount to one droplet identity, and similarly that the two ERG relations yield only one droplet identity. The only four point correlation which are non-zero in the droplet cal^ y u21z u21t i and hu21x u21y u21z u21t i. We now show that they are related. ^x u culation are hu Consider the identity (from the above):

  huax uay ubz uct i  huax uby ucz udt i  hux uy i  hux ihuy i huz ihut i ¼ Tg xy Gzt

ðK9Þ

Let us compute within droplets:

^z u ^t i þ Tpð1  pÞ ðhux uy i  hux ihuy iÞhuz ihut i ¼ Tpð1  pÞhu21x u21y u



2 1  p hu21x u21y u21z u21t i 2 ðK10Þ

This yields the relation:

^z u ^t i þ hu21x u21y u

1 hu21x u21y u21z u21t i ¼ 2g xy Gzt 12

ðK11Þ

the right-hand side being presumably the disconnected part of the first correlation on the l.h.s.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

143

The droplet calculation of correlations yields:

huax uay uaz uat i ¼ hu1x u1y u1z u1t i

ðK12Þ

1 1 ^ 21x u ^ 21y i huax uay uaz ubt i ¼ hu1x u1y u1z u1t i  Thu21x u21y u21z u21t i  Tðhu21z u21t u 8 2 ^ 21z u ^ 21y i þ hu21y u21t u ^ 21x u ^ 21z iÞ þ hu21x u21t u ¼ hu1x u1y u1z u1t i  Tg zt Gxy  Tg yt Gxz  Tg xt Gyz

ðK13Þ

1 ^ 21x u ^ 21t i þ hu21x u21z u ^ 21y u ^21t i huax uay ubz ubt i ¼ hu1x u1y u1z u1t i  Tðhu21y u21z u 2 ^ 21x u ^21z i þ hu21x u21t u ^ 21y u ^ 21z iÞ þ hu21y u21t u 1 ¼ hu1x u1y u1z u1t i þ Thu21x u21y u21z u21t i  Tðg yz Gxt þ g xz Gyt þ g yt Gxz þ g xt Gyz Þ 6

ðK14Þ

1 1 ^t u ^x i þ 4permÞ huax uay ubz uct i ¼ hu1x u1y u1z u1t i  Thu21x u21y u21z u21t i  Tðhu21y u21z u 8 2 1 1 ^t u ^x i þ 5permÞ ¼ hu1x u1y u1z u1t i  Thu21x u21y u21z u21t i  Tðhu21y u21z u 8 2 ^z u ^t i  hu21x u21y u ¼ hu1x u1y u1z u1t i þ

1 Thu21x u21y u21z u21t i  Tðg xy Gzt þ 5permÞ þ Tg xy Gzt 12

1 1 ^z u ^t i þ 5permÞ huax uby ucz udt i ¼ hu1x u1y u1z u1t i  Thu21x u21y u21z u21t i  Tðhu21x u21y u 6 2 1 ¼ hu1x u1y u1z u1t i þ Thu21x u21y u21z u21t i  Tðg xy Gzt þ 5permÞ 12

ðK15Þ

ðK16Þ

where in the last steps we use the above equality. Thus everything is consistent with:

1 hu21x u21y u21z u21t i 12 1111 2 0000 g xx0 g yy0 g zz0 g tt0 ðQ x0 y0 z0 t0 ½0  T Rx0 y0 z0 t0 ½0Þ ¼ hu1x u1y u1z u1t ic g xx0 g yy0 g zz0 g tt0 TR0000 x0 y0 z0 t 0 ½0 ¼

ðK17Þ ðK18Þ

and we must check whether these identities are compatible with our result for these cumulants. Finally, the ERG equation yields, using the above:

oGzt ¼

1 hðu21 og 1 u21 Þu21z u21t i  og 1 x0 y0 ðg x0 z Gy0 t þ g x0 t Gy0 z Þ 12

ðK19Þ

consistent with the equations given in the text. Appendix L. W-ERG To one loop we need:

oR½v  ¼ og zz0 e S 110 zz0 ½0; 0; v  3 oS½v 123  ¼ sym123 og xy Q 1100 xy ½v 1123  2

ðL1Þ ðL2Þ

We write the cumulant truncation:

Q ½v 1234  ¼ R½v 12 R½v 34  þ R½v 13 R½v 24  þ R½v 14 R½v 23 

v

Q 1100 xy ½ 1234 

¼

R00xy ½ 12 R½ 34 

v

v

þ

R0x ½ 13 R0y ½ 24 

v

v

þ

ðL3Þ

R0x ½ 14 R0y ½ 23 

v

v

ðL4Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

Hence cumulant truncation+weak (resp strong) continuity of R00xy ½v 12  implies weak (resp strong) continuity of Q to one loop with the result: 0 0 og xy Q 1100 xy ½v 1123  ¼ 2og xy Rx ½v 12 Ry ½v 13 

ðL5Þ

up to the gauge term R00xy ½0R½v 23 . Hence:

oS½v 123  ¼ og xy ðR0x ½v 12 R0y ½v 13  þ R0x ½v 21 R0y ½v 23  þ R0x ½v 31 R0y ½v 32 Þ

ðL6Þ

To this order it integrates into:

S½v 123  ¼ g xy ðR0x ½v 12 R0y ½v 13  þ R0x ½v 21 R0y ½v 23  þ R0x ½v 31 R0y ½v 32 Þ

ðL7Þ

since oR is higher order. Let us now evaluate:

  000 0 0 00 00 00 00 00 00 S110 zt ½v 123  ¼ g xy Rxzt ½v 12 ðRy ½v 23   Ry ½v 13 Þ  Rxt ½v 12 Ryz ½v 13   Rxz ½v 21 Ryt ½v 23  þ Rxz ½v 31 Ryt ½v 32  ðL8Þ Assuming that: 0 0 lim g xy ðR000 xzt ½v 12 ðRy ½v 23   Ry ½v 13 Þ ¼ 0

v 12 !0

ðL9Þ

then one obtains the equations in the text. One has then strong/weak continuity for S if R satisfies the same. Appendix M. Details of two loop C-ERG calculation in  expansion Let us denote M12 ¼ R00yr ½v 12  and rewrite Eq. (468) in the text as:

1 A 2 A ¼ tr½gM12 gM12 ðgM 13 þ gM23 Þ þ tr½gM 23 gM 23 ðgM12 þ gM 13 Þ

ðM2Þ

þ tr½gM13 gM 13 ðgM 12 þ gM23 Þ  2tr½gM 12 gM 23 gM13 

ðM3Þ

S½v 123  ¼

ðM1Þ

On can rewrite:

A ¼ A 1 þ A2 þ A3

ðM4Þ

A1 ¼ tr½gM 12 gM 12 ðgM13 þ gM 23 Þ A2 ¼ tr½gM 12 gðM 13  M 23 ÞgðM13  M 23 Þ 1 A3 ¼ tr½gðM 13 þ M23 ÞgðM 13 þ M23 ÞgðM 13 þ M 23 Þ 3

ðM5Þ ðM6Þ ðM7Þ

up to gauge terms, where we have used cyclic properties of the trace, as well as identity under transposition and that all matrices are symmetric. The second term is of order v 312 . To perform the expansion in the last term we use symmetric expansion of a three replica functional as explained in Appendix of [36]. Let f ½v 13 ; v 23  the symmetric functional. One can either expand:

  1 v 12x v 12y fxy02 ½v 13 ; v 13  þ O v 312 2   1 20 ½v 23 ; v 23  þ O v 312 f ½v 23 þ v 12 ; v 23  ¼ f ½v 23 ; v 23  þ v 12x fx10 ½v 23 ; v 23  þ v 12x v 12y fxy 2   20 11 ¼ f ½v 23 ; v 23  þ v 12x ðfx10 ½v 13 ; v 13   v 12y fxy ½v 13 ; v 13  þ fxy ½v 13 ; v 13 

f ½v 13 ; v 13  v 12  ¼ f ½v 13 ; v 13   v 12x fx01 ½v 13 ; v 13  þ

þ

  1 v 12x v 12y fxy20 ½v 13 ; v 13  þ O v 312 2

performing the half sum and discarding the zero-th order terms which are gauge, one gets:

ðM8Þ ðM9Þ

ðM10Þ

P. Le Doussal / Annals of Physics 325 (2010) 49–150

f ½v 13 ; v 23  ¼ 

  1 v 12x v 12y fxy11 ½v 13 ; v 13  þ O v 312 2

145

ðM11Þ

We obtain:

 3  000 00 A3 ¼ 2v 12x v 12y tr½gR000 x ½v 13 gRy ½v 13 gR ½v 13  þ O v 12

ðM12Þ

The first term gives:

  A1 ¼ 2tr½gR00 ½v 12 gR00 ½v 12 gR00 ½v 13  þ O v 312

ðM13Þ

replacing R00xy ½v 12  ¼ dxy Rðv 12x Þ it becomes:

A1 ¼ 2

Z

g xy R00 ðv 12y Þg yz R00 ðv 12z Þg zx R00 ðv 13x Þ

ðM14Þ

xyz

Appendix N. Periodic case and mean field limit Here we give some partial results about two solvable cases. The random periodic class appears simpler in any d, and in d = 0 it is useful as it provides an exact solution of the FRG hierarchy. It then raises questions about shocks which can be answered in another solvable limit, large d or the fully connected model. N.1. Random periodic class in d = 0 b ðv Þ is also Consider in d = 0, for N ¼ 1, a random potential VðuÞ periodic of period one. Then V b ðv Þ ¼ minu ðVðuÞ þ ðu  v Þ2 =ð2tÞÞ. It periodic of period one. Consider the T = 0 problem such that V is easy to compute the T = 0 fixed point of the FRG in the limit t ¼ m2 ! 1. Denote u1 the absolute minimum of VðuÞ on a period, say 1=2 6 u1 < 1=2. Since for infinite t the curvature of the quadratic well goes to zero, u1 ðv Þ ¼ u1 for u1  1=2 < v < u1 þ 1=2, u1 ðv Þ ¼ u1 þ 1 for u1 þ 1=2 < v < u1 þ 3=2, etc. Hence there is a single shock in the unit cell [0,1[, at position 0 6 v s ¼ u1 þ 1=2 < 1, and on the ðiÞ real axis there is a periodic array of shocks at v s ¼ u1 þ ð2i þ 1Þ=2. The force tFðv Þ ¼ v  u1 ðv Þ is then for 0 6 v < 1:

1 tFðv Þ ¼ v  v s þ ; 2 1 tFðv Þ ¼ v  v s  ; 2

v < vs

ðN1Þ

v > vs

ðN2Þ

The velocity of the shock is zero, because of the constraint that the integral of the force over a period is R1 zero. Note that this solution indeed satisfies 0 dv e F ðv Þ ¼ 0. Because of statistical translational invariance u1 is uniformly distributed in [1/2, 1/2[ and so is v s in [0, 1[. One then obtains the second and third moments:

t 2 R00 ðv Þ ¼ t 2 Fð0ÞFðv Þ ¼

Z 0

1

dv s



1 1 1 1 v  v s  ðv  v s Þ ¼  v ð1  v Þ  vs 2 2 12 2

ðN3Þ

t3 S111 ðv 1 ; v 2 ; v 3 Þ ¼ t3 Fðv 1 ÞFðv 2 ÞFðv 3 Þ ¼

1 ð1 þ v 1 þ v 2  2v 3 Þð1 þ 2v 1  v 2  v 3 Þðv 1  2v 2 þ v 3 Þ 6

ðN4Þ

for 0 6 v 6 1 and for 0 6 v 1 < v 2 < v 3 6 1, all other cases can be obtained by symmetry or periodicity. We can now compute the limits:

1 v ð1  2v Þð1  v Þ 6 1 S 112 ð0; 0þ ; v Þ ¼  v ð1  v Þ t3 e 6

t3 e S 111 ð0; 0þ ; v Þ ¼

ðN5Þ ðN6Þ

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P. Le Doussal / Annals of Physics 325 (2010) 49–150

The first FRG equation reads:

ot R00 ðv Þ ¼ TR0000 ðv Þ þ S112 ð0; 0; v Þ

ðN7Þ

and one easily checks from the above that its T = 0 version is obeyed (i.e. setting T = 0 and replacing 1 e S 112 ð0; 0þ ; v Þ). One also checks that hu21 i ¼ R00 ð0Þt 2 ¼ 12 since for v ¼ 0 the minimum S 112 ð0; 0; v Þ ! e is necessarily in the interval [1/2, 1/2[ and essentially uniformly distributed (note that it crosses over to hu21 i  tT at for large t T-see below). Expressions for higher moments become rapidly complicated, but some generating functions can be computed, e.g. (setting t ¼ 1):



ew1 Fðv 1 Þþw2 Fðv 2 Þ ¼

 1 1 e2ðw1 þw2 Þ ew1 ð1v Þþw2  ew1 ð1v Þ þ ew1 þv w2  ev w2 w1 þ w2

ðN8Þ

with 0 < v ¼ v 2  v 1 . Although we will not do it here in details, it is easy to study the case T > 0. A shock is broadened as:

tFðv Þ ¼ v  v s 



1 1 tanh ðv  v s Þ þ OðTtÞ 2 2tT

ðN9Þ

since u21 ¼ 1, and one can redo the previous calculation using this form and get results consistent with the general ones obtained in the text for the shape of the TBL, specializing to a very simple droplet distribution DðyÞ ¼ dðjyj  1Þ. However, this form works only for tT  1. Since temperature is relevant here (with f ¼ 0 and h ¼ 2) the shock width grows as T t, while their spacing is unity, hence at large b time they overlap. Exact solutions are easily written using the diffusion equation for Z ¼ e V =T , one interesting example being the solution:

b ðv Þ ¼ T ln V

nX ¼1 n¼1

2 1 pffiffiffiffiffiffiffiffiffiffiffi eðv v s þnÞ =ð2TtÞ 2pTt

ðN10Þ

which represents a solution of decaying Burgers with an initial condition a periodic set of (zero temperature) shocks at locations v s þ n (this solution arised in the study of SLE on a cylinder [93]. Averaging over v s yields a general solution at any T for the FRG hierarchy, which describes the flow away from the zero T fixed point when temperature is turned to a non-zero value. Finally, one can check S 111 ð0; 0þ ; v Þ the continuity property discussed S 111 ð0; 0; v Þ ¼ e directly on the T > 0 solution that limT!0 e in the text. Although the d = 0 random periodic (RP) model may appear trivial, it is an interesting limit of the RP class in higher d. Apart from temperature becoming relevant for d < 2, we do not expect any bifurcation between d = 4 and d = 0 in the T = 0 fixed point itself. Let us indeed compare with the result from the  ¼ 4  d expansion (to two loop) of Ref. [55]:

DðuÞ ¼ m





K d=2  2  2 uð1  uÞ þ Oð3 Þ þ  þ 6 9 eJ 2 36 54

ðN11Þ

with eJ 2 ¼ 2ð4pÞd=2 Cð3  2dÞ. Note that from the shear fact that it contains only uð1  uÞ the universal number r1 ¼ aD00 ð0þ Þ=jD0 ð0þ Þj ¼ 2 and next, from the fact that it integrates to zero r2 ¼ aD0 ð0þ Þ=jDð0Þj ¼ 6 (where a is the lattice period, a ¼ 1 here). Note that for depinning 2 =54 is changed into 2 =108 and the universal number becomes r 2 ¼ 6 þ 2 þ Oð2 Þ as the constraint that the integral of the force is zero is relaxed. Finally, there is a global amplitude, r ¼ m Dð0Þ=K d=2 , with r ¼ 1=36 þ =54 in the  expansion, which comes not too far from the exact result r ¼ 1=12 in d = 0. So we see that despite its simplicity, this model does not fare too badly as compared to d > 0. N.2. Large d limit, fully connected model The model can be studied in the large d limit of a hypercubic lattice, as was done in Appendix H.2 of Ref. [36]. Here we study a simpler, but closely related, fully connected model. It is simple enough to allow for easy understanding of the results obtained in Ref. [36], seen here under a different perspective (the definition of the renormalized correlators is slightly different there). One starts from:

147

P. Le Doussal / Annals of Physics 325 (2010) 49–150

"

#

1 X ui N i X 1 X HV ½fui g; v ; u0  ¼ K ðui  u0 Þ2 þ ðui  v Þ2 þ V i ðui Þ 2t i i

HV ½fui g; v  ¼ HV fui g; v ; u0 ¼

K with N continuous variables ui ; i ¼ 1; . . . ; N. The first term can also be written 2N is defined by a minimization condition on the first line, one has:

b ðv Þ ¼ min HV ½fui g; v  ¼ min HV ½fui g; v ; u0  V ui

ui ;u0

ðN12Þ ðN13Þ P

ij ðui

 uj Þ2 . Since u0

ðN14Þ

Regrouping terms one easily sees that:

" #

1 Xb v þ 2tKu0 K 2 0 b þ V ðv Þ ¼ N min Vi v ¼ ðu0  v Þ u0 N i 1 þ 2Kt 1 þ 2tK   b i ðv Þ ¼ min 1 þ 2tK ðu  v Þ2 þ V i ðuÞ V u 2t

ðN15Þ ðN16Þ

and note that the force is:

b 0 ðv Þ ¼ Nðv  u0 Þ=t Fðv Þ ¼ V

ðN17Þ

as can be checked comparing derivatives with respect to v and to u0 of the first line. Hence we are back to a d = 0 model for the center of mass u0 , but it is feeling an effective disorder which is an average over Pb 0 a sum of a large number N of independent random potentials Wðv 0 Þ ¼ N1 i V i ðv Þ. Hence it should be small and also almost gaussian in distribution. Note, however, that each of them has shocks if the parameter t=ð1 þ 2 KtÞ > tc corresponding to a Larkin scale, hence the correlator will have a cusp. For t ! 1 this corresponds to a critical K c such that for K < K c the system is in a strong disorder phase with a cusp, while for K > K c there is no cusp in each layer. Since layers are uncorrelated the correlator of the effective force reads:

R00W ðv Þ ¼ W 0 ð0ÞW 0 ðv Þ ¼

1 00 R ðv Þ N lay

ðN18Þ

b 0 ðv Þ assumed identical b 0 ð0Þ V where Rlay ðv Þ is the renormalized correlator of a d = 0 model R00lay ðv Þ ¼  V i i for any i (we have also assumed the V i uncorrelated). Similarly the three point cumulant of W is re2 lated to 1=N S where S is the three point correlator in each layer and we see that indeed the effect of higher correlators is reduced by the N counting. There are various behaviours in this model depending on the way one scales the parameters. The simplest one is to keep t and K fixed and N ! 1 first. Then since disorder is reduced one has b ðv Þ ¼ Wðv Þ and (N18) gives the final result Rðv Þ ¼ RW ðv Þ. In the cusp phase of each layer, u0 ¼ v and V b will exhibit many independent shocks. For the random the resulting potential of the full system V periodic class then one can then use the result (N3) for Rlay . On the other hand, one could consider t ! 1 at fixed N and K. Then if K > K c although there are no shocks in each layer, there are shocks from the center of mass u0 ðv Þ since the curvature of the term u0  v in (N15) goes to zero. These are global shocks quite different from the local shocks which occur independently in each layer in the other limit. In general both occur, with an interesting scaling behaviour as a function of N=t2 . N.3. Periodic case, many shocks Let us close on a remark about the form of the correlator ubiquitously found for the random periodic class in the  expansion and in recent numerics [68] in d ¼ 3; 2, i.e. D00 ðv Þ independent of v (v not integer). Consider n shocks in the interval]0,1]:

Fðv Þ ¼

n X ðv  ui Þhðv i < v < v iþ1 Þ i¼0

ðN19Þ

148

P. Le Doussal / Annals of Physics 325 (2010) 49–150

with v 0 ¼ 0; v nþ1 ¼ 1 and the condition un ¼ u0 þ 1, so that Fð0Þ ¼ Fð1Þ. We set t ¼ 1. Then one has for 0 < v – v0 < 1

F 0 ðv ÞF 0 ðv 0 Þ ¼ 1 þ

n X

ai aj dðv  v i Þdðv 0  v j Þ

ðN20Þ

i–j¼1

P with ai ¼ ui1  ui , hence ni¼1 ai ¼ 1 and we have used that F 0 ðv Þ ¼ 0. For v –v 0 the term i ¼ j can be discarded since it produces only a d function. For two shocks it yields:

F 0 ðv ÞF 0 ðv 0 Þ ¼ 1 þ 2

Z

daað1  aÞPða; v  v 0 Þ

ðN21Þ

where Pða; v 1 ; v 2 Þ ¼ Pða; v 1  v 2 Þ is the probability of shock positions and amplitude. It can only depend on the difference from translational invariance. The form R00 ðv Þ ¼ Dðv Þ  v ð1  v Þ (periodized on the real axis) thus can only occur if the shock distance distribution is uniform, i.e. independently located shocks. In that case the number:

F 0 ðv ÞF 0 ðv 0 Þ ¼ 1 þ 2að1  aÞ ¼ 2a2

ðN22Þ

by symmetry. This is more general one has for n independent shocks:

F 0 ðv ÞF 0 ðv 0 Þ ¼ 1 þ

n X i–j¼1

aiaj ¼ 

X

a2i ¼ na2

ðN23Þ

i

Hence D00 ðv Þ ¼ na2 and we recall that a ¼ 1=n. This model can be generalized in various ways, such as n not fixed, but it illustrates simply what features of the shock distribution lead to the usual form of the random periodic fixed point. References [1] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, Oxford, 1989. [2] David Carpentier, Pierre Le Doussal, Nucl. Phys. B 588 (2000) 565. cond-mat/9908335; David Carpentier, Pierre Le Doussal, Phys. Rev. Lett. 81 (1998) 2558. cond-mat/9802083 (and references cited therein). [3] David Carpentier, Pierre Le Doussal, Phys. Rev. E 63 (2001) 026110. cond-mat/0003281. [4] Pierre Le Doussal, Thierry Giamarchi, Physica C 331 (2000) 233. cond-mat/9810218. [5] H.E. Castillo, C. Chamon, E. Fradkin, P.M. Goldbart, C. Mudry, Phys. Rev. B 56 (1997) 10668. [6] Baruch Horovitz, Pierre Le Doussal, Phys. Rev. B 65 (2002) 125323. cond-mat/0108143. [7] C. Mudry, S. Ryu, A. Furusaki, Phys. Rev. B 67 (2003) 064202. cond-mat/0207723. [8] D.S. Fisher, Phys. Rev. B 50 (1994) 3799; D.S. Fisher, Phys. Rev. B 51 (1995) 6411. [9] Pierre Le Doussal, Cecile Monthus, Daniel S. Fisher, Phys. Rev. E 59 (1999) 4795. cond-mat/9811300. [10] Daniel S. Fisher, Pierre Le Doussal, Cecile Monthus, Phys. Rev. E 64 (2001) 066107. cond-mat/0012290; Daniel S. Fisher, Pierre Le Doussal, Cecile Monthus, Phys. Rev. Lett. 80 (1998) 3539. cond-mat/9710270. [11] F. Igloi, C. Monthus, Phys. Rep. 412 (2005) 277. [12] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [13] T. Giamarchi, P. Le Doussal, Phys. Rev. B 52 (1995) 1242, cond-mat/9705096, in: A.P. Young (Ed.), Spin Glasses and Random Fields, World Scientific, Singapore, 1997. [14] T. Nattermann, S. Scheidl, Adv. Phys. 49 (2000) 607–704. [15] A. Rosso, W. Krauth, Phys. Rev. Lett. 87 (2001) 187002; A. Rosso, W. Krauth, Phys. Rev. B 65 (2001) 012202; A. Rosso, A. Hartmann, W. Krauth, Phys. Rev. E 67 (2003) 021602. [16] L. Roters, S. Lubeck, K.D. Usadel, Phys. Rev. E 66 (2002) 026127; L. Roters, S. Lubeck, K.D. Usadel, Phys. Rev. E 66 (2002) 069901(E). [17] P. Le Doussal, Euro-Summer School on Condensed Matter Theory 2004, lecture notes at http://www.lancs.ac.uk/users/ esqn/windsor04/program.htm. [18] P. Le Doussal, Beg Rohu School of Physics 2008, lecture notes at http://ipht.cea.fr/Meetings/BegRohu2008/notes.html. [19] Kay Joerg Wiese, in: Proceedings of STATPHYS 22, Pramana 64 (2005) 817–827 cond-mat/0511529. [20] K.J. Wiese, P. Le Doussal, Markov Process. Relat. Fields 13 (2007) 777–818. cond-mat/0611346. [21] D.A. Huse, C.L. Henley, D.S. Fisher, Phys. Rev. Lett. 56 (1986) 899. [22] M. Kardar, Nucl. Phys. B 290 (1987) 582; L. Balents, M. Kardar, Phys. Rev. B 49 (1994) 13030; T. Emig, M. Kardar, cond-mat/0101247. [23] J.Z. Imbrie, T. Spencer, J. Stat. Phys. 52 (1988) 609. [24] E. Brunet, B. Derrida, Phys. Rev. E 61 (2000) 6789–6801. cond-mat/0005352; E. Brunet, B. Derrida, Physica A 279 (2000) 398–407. cond-mat/0005355.

P. Le Doussal / Annals of Physics 325 (2010) 49–150

149

[25] K. Johansson, Commun. Math. Phys. 209 (2000) 437–476. math.PR/9910146, math.CO/9903134; Jinho Baik, Percy Deift, Kurt Johansson, J. Am. Math. Soc. 12 (4) (1999) 1119–1178. math.CO/9810105. [26] U. Schulz, J. Villain, E. Brezin, H. Orland, J. Stat. Phys. 51 (1988) 1–27. [27] T. Hwa, D.S. Fisher, Phys. Rev. B 49 (1994) 3136–3154. [28] P. Le Doussal, C. Monthus, Physica A 317 (2003) 140. cond-mat/0204168. [29] W.L. McMillan, J. Phys. C 17 (1984) 3179; A.J. Bray, M.A. Moore, J. Phys. C 17 (1984) L463; A.J. Bray, M.A. Moore, in: van Hemmen, I. Morgenstern (Eds.), Heilderberg Colloquium on Glassy Dynamics and Optimization, Springer, Heidelberg, 1986. [30] D.S. Fisher, D.A. Huse, Phys. Rev. B 38 (1988) 373–386. [31] D.S. Fisher, D.A. Huse, Phys. Rev. B 43 (1991) 10728. [32] M. Mézard, G. Parisi, J. Phys. I (France) 1 (1991) 809. [33] See, e.g. A.A. Middleton, Phys. Rev. B 63 (2001) 60202; T. Hwa, D.S. Fisher, Phys. Rev. B 49 (1994) 3136. [34] L. Balents, P. Le Doussal, Europhys. Lett. 65 (2004) 685. [35] L. Balents, P. Le Doussal, Phys. Rev. E 69 (2004) 061107. cond-mat/0312338. [36] L. Balents, P. Le Doussal, Ann. Phys. 315 (2005) 213. cond-mat/0408048. [37] A.I. Larkin, Y.N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [38] K.B. Efetov, A.I. Larkin, Sov. Phys. JETP 45 (1977) 1236. [39] A. Aharony, Y. Imry, S.K. Ma, Phys. Rev. Lett. 37 (1976) 1364–1367. [40] G. Grinstein, Phys. Rev. Lett. 37 (1976) 944–947. [41] Although it can also be made a non-perturbative statement [42,43] one easily sees that the computed average is over all energy extrema with a weighting factor ð1ÞN where N is the number of unstable directions (see e.g. [44]). Hence, it cannot select low energy states. [42] G. Parisi, N. Sourlas, Phys. Rev. Lett. 46 (1981) 871; G. Parisi, N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. [43] J.L. Cardy, Phys. Lett. 125 (1983) 470. [44] See e.g. J. Kurchan, cond-mat/0209399. [45] D.S. Fisher, Phys. Rev. Lett. 56 (1986) 1964; L. Balents, D.S. Fisher, Phys. Rev. B 48 (1993) 5959. [46] L. Balents, J.P. Bouchaud, M. Mézard, J. Phys. I 6 (1996) 1007–1020. [47] T. Nattermann, S. Stepanow, L.H. Tang, H. Leschhorn, J. Phys. II Fr. 2 (1992) 1483–1488. [48] O. Narayan, D.S. Fisher, Phys. Rev. B 48 (1993) 7030–7042. [49] L. Balents, Europhys. Lett. 24 (1993) 489. [50] T. Giamarchi, P. Le Doussal, E. Orignac, Phys. Rev. B 64 (2001) 245119. cond-mat/0104583. [51] Pascal Chauve, Pierre Le Doussal, Thierry Giamarchi, Phys. Rev. B 61 (2000) R11906. cond-mat/9910197. [52] P. Chauve, P. Le Doussal, K.J. Wiese, Phys. Rev. Lett. 86 (2001) 1785. cond-mat/0006056. [53] P. Chauve, P. Le Doussal, Phys. Rev. E 64 (2001) 051102. [54] S. Scheidl, Y. Dincer, 2000, cond-mat/0006048. [55] P. Le Doussal, K.J. Wiese, P. Chauve, Phys. Rev. E 69 (2004) 026112. cond-mat/0304614. [56] P. Le Doussal, K.J. Wiese, P. Chauve, Phys. Rev. B 66 (2002) 174201. [57] P. Le Doussal, K.J. Wiese, in preparation. [58] Pierre Le Doussal, Kay Joerg Wiese, Phys. Rev. E 72 (2005) 035101. cond-mat/0501315. [59] Pierre Le Doussal, Kay Joerg Wiese, Phys. Rev. Lett. 96 (2006) 197202. cond-mat/0510344; Pierre Le Doussal, Kay Joerg Wiese, Phys. Rev. Lett. 98 (2007) 269704. [60] D.E. Feldman, Phys. Rev. Lett. 88 (2002) 177202; G. Tarjus, M. Tissier, Phys. Rev. Lett. 93 (2004) 267008; G. Tarjus, M. Tissier, Phys. Rev. Lett. 96 (2006) 087202; G. Tarjus, M. Tissier, Phys. Rev. B 74 (2006) 214419. [61] A.A. Middleton, Phys. Rev. Lett. 68 (1992) 670; For a complete proof of Middleton’s theorem, cf.C. Baesens, R.S. MacKay, Nonlinearity 11 (1998) 949. [62] P. Le Doussal, K.J. Wiese, cond-mat/0109204.; P. Le Doussal, K.J. Wiese, Phys. Rev. Lett. 89 (2002) 125702. cond-mat/0305634; P. Le Doussal, K.J. Wiese, Phys. Rev. B 68 (2003) 174202. cond-mat/0406297. [63] P. Le Doussal, M.M. Muller, K.J. Wiese, Phys. Rev. B 77 (2008) 064203. [64] P. Chauve, Thierry Giamarchi, Pierre Le Doussal, Europhys. Lett. 44 (1998) 110. cond-mat/9804190; P. Chauve, Thierry Giamarchi, Pierre Le Doussal, Phys. Rev. B 62 (2000) 6241. cond-mat/0002299. [65] G. Schehr, P. Le Doussal, Phys. Rev. E 68 (2003) 046101. [66] L. Balents, P. Le Doussal, unpublished. [67] P. Le Doussal, Europhys. Lett. 76 (2006) 457. [68] A.A. Middleton, P. Le Doussal, K.J. Wiese, Phys. Rev. Lett. 98 (2007) 155701. cond-mat/0606160. [69] P. Le Doussal, K.J. Wiese, EPL 77 (2007) 66001. cond-mat/0610525. [70] P. Le Doussal, K.J. Wiese, 0808.3217. [71] A. Rosso, P. Le Doussal, K.J. Wiese, Phys. Rev. B 75 (2007) 220201. cond-mat/0610821. [72] P. Le Doussal, A. Middleton, K. Wiese, 0803.1142; P. Le Doussal, K. Wiese, Phys. Rev. E 79 (2009) 051106. [73] For review, see e.g. D. Bernard, cond-mat/0007106. [74] U. Frisch, J. Bec, nlin.CD/0012033, in: M. Lesieur, A. Yaglom, F. David (Eds.), New Trends in Turbulence, Springer EDPSciences, 2001, pp. 341–383. [75] For review, see J. Bec, K. Khanin, 2007, 0704.1611. [76] J.P. Bouchaud, M. Mezard, G. Parisi, cond-mat/9503144.

150 [77] [78] [79] [80] [81] [82] [83]

P. Le Doussal / Annals of Physics 325 (2010) 49–150 D. Bernard, K. Gawedzki, chao-dyn/9805002. M. Bauer, D. Bernard, J. Phys. A 3 (1999) 5179–5199. chao-dyn/9812018. J.M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974. L. Frachebourg, Ph.A. Martin, cond-mat/9905056. L. Frachebourg, Ph.A. Martin, J. Piasecki, cond-mat/9911346. A.M. Polyakov, hep-th/9506189. Note that there was a degree of arbitrary in the definition of these functions (a gauge invariance). One can perform for b ; j  þ T/½j  þ T/½j ; b ; j  ! R½j instance the simultaneous changes (in the more general notations of Appendix D): R½j a

b

a

b

a

b

b  ! U½j b   p/½j . This gauge choice is fixed by the sample dependent definition of V b chosen here. U½j a a a [84] P. Le Doussal, Phys. Rev. Lett. 96 (2006) 235702. cond-mat/0505679. [85] See e.g. [65] for review. [86] All relations given in Section 4.1 are valid for any number of replica p, in the sense discussed in the previous section, except (107) which is valid only ‘‘for p ¼ 0”. The correct equations must take into account the non-zero value of Vðv Þ ¼ F V b v Þ ¼ Rðv Þ  F V 2 , b (the free energy) and read Rðv Þ ¼ Rð Sðv 123 Þ ¼ Sðv 123 Þ þ F V ðRðv 12 Þ þ Rðv 23 Þ þ Rðv 31 ÞÞÞ  2F V 3 and h i



 b ðv 1234 Þ ¼ Q ðv 1234 Þ  3 Rðv 12 ÞRðv 34 Þ þ 4F V Sðv 123 Þ þ 12F V 2 Rðv 12 Þ  6F V 4 . For p ¼ 0 the additional terms are ‘‘gauge” Q terms83 and hence have been ignored not to burden the text. However, these terms are important if one wants, e.g. to [87] [88] [89] [90] [91] [92]

[93] [94]

compute the flow the free energy cumulants (the b S ðnÞ ð0; . . . ; 0Þ). They are also import. C. Monthus, P. Le Doussal, Eur. Phys. J. B 41 (2004) 535. cond-mat/0407289. S. Kida, J. Fluid Mech. 93 (1979) 337. J.P. Bouchaud, M. Mezard, cond-mat/9707047. B. Derrida, H. Spohn, J. Stat. Phys. 51 (1988) 817. M. Kardar, G. Parisi, Y.C. Zhang, Phys. Rev. Lett. 56 (1986) 889. E.W.E. Vanden Eijnden, Phys. Fluids 11 (1999) 21492153. chao-dyn/9901006; E.W.E. Vanden Eijnden, Phys. Fluids 12 (2000) 149154. chao-dyn/9901029; E.W.E. Vanden Eijnden, Commun. Pure Appl. Math. 53 (2000) 852901. chao-dyn/9904028. C. Hagendorf, P. Le Doussal, 0803.3249. P. Le Doussal, 0809.1192.