Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
TTheory and Experiments for Disordered ElasticManifolds, Depinning, Avalanches, and Sandpiles
Kay J ¨org Wiese
Laboratoire de physique, D´epartement de physique de l’ENS, ´Ecole normale sup´erieure,UPMC Univ. Paris 06, CNRS, PSL Research University, 75005 Paris, France.
Abstract.
Domain walls in magnets, vortex lattices in superconductors, contact lines atdepinning, and many other systems can be modelled as an elastic system subject to quencheddisorder. Its field theory possesses a well-controlled perturbative expansion around its uppercritical dimension. Contrary to standard field theory, the renormalization group flow involves afunction, the disorder correlator ∆( w ) , therefore termed the functional renormalization group(FRG). ∆( w ) is a physical observable, the auto-correlation function of the centre of mass ofthe elastic manifold. In this review, we give a pedagogical introduction into its phenomenologyand techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics).Building on these techniques, avalanche observables are accessible: distributions of size,duration, and velocity, as well as the spatial and temporal shape. Various equivalences betweendisordered elastic manifolds, and sandpile models exist: an elastic string driven at a point andthe Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves andAbelian sandpiles or loop-erased random walks. Each of these mappings requires specifictechniques, which we develop, including modelling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata.Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional KPZ terms. On the other hand, KPZ without disordercan be mapped back to disordered elastic manifolds, either on the directed polymer for itssteady state, or a single particle for its decay. Other topics covered are the relation betweenfunctional RG and replica symmetry breaking, and random field magnets. Emphasis is givento numerical and experimental tests of the theory. Anisotropic depinning with its relation to directed percolation, explained in section 5.7.VERSION 1.0.: PLEASE REPORT MISPRINTS OR MISSING REFERENCES TO THE AUTHOR a r X i v : . [ c ond - m a t . d i s - nn ] F e b ONTENTS Contents1 Disordered Elastic Manifolds: Phenomenology 4 ∆( u ) and the cusp in simulations . . . . . . 172.12 Beyond 1-loop order . . . . . . . . . . . . 172.13 Stability of the fixed point . . . . . . . . . 192.14 Thermal rounding of the cusp . . . . . . . . 202.15 Disorder chaos . . . . . . . . . . . . . . . 222.16 Finite N . . . . . . . . . . . . . . . . . . . 222.17 Large N . . . . . . . . . . . . . . . . . . . 232.18 Corrections at order /N . . . . . . . . . . 232.19 Relation to Replica Symmetry Breaking(RSB) . . . . . . . . . . . . . . . . . . . . 242.20 Droplet picture . . . . . . . . . . . . . . . 252.21 Kida model . . . . . . . . . . . . . . . . . 262.22 Sinai model . . . . . . . . . . . . . . . . . 272.23 Random-energy model (REM) . . . . . . . 282.24 Complex disorder and localization . . . . . 292.25 Bragg glass and vortex glass . . . . . . . . 302.26 Bosons and fermions in d = 2 , bosonization 302.27 Sine-Gordon model, Kosterlitz-Thoulesstransition . . . . . . . . . . . . . . . . . . 312.28 Random-phase sine-Gordon model . . . . . 322.29 Multifractality . . . . . . . . . . . . . . . . 342.30 Simulations in equilibrium . . . . . . . . . 352.31 Experiments in equilibrium . . . . . . . . . 35 d = 2 ) . . . 533.21 Experiments on thin magnetic films ( d = 1 ) 543.22 Hysteresis . . . . . . . . . . . . . . . . . . 553.23 Inertia, and a large-deviation function . . . 553.24 Plasticity . . . . . . . . . . . . . . . . . . 563.25 Depinning of vortex lines or charge-densitywaves, columnar defects, and non-potentiality 563.26 Other universal distributions . . . . . . . . 56 Sandpile Models, and Anisotropic Depinning 72 ζ dep d =1 = 5 / . . . . 745.5 Manna model . . . . . . . . . . . . . . . . 745.6 Hyperuniformity . . . . . . . . . . . . . . 755.7 A cellular automaton for fluid invasion, andanharmonic depinning . . . . . . . . . . . 755.8 Brief summary of directed percolation . . . 765.9 Fluid invasion fronts from directed percola-tion . . . . . . . . . . . . . . . . . . . . . 775.10 Anharmonic depinning and FRG . . . . . . 775.11 Other models in the same universality class 785.12 Quenched KPZ with a reversed sign for thenon-linearity . . . . . . . . . . . . . . . . . 785.13 Experiments for directed Percolation andquenched KPZ . . . . . . . . . . . . . . . 79 β -function for KPZ . . . . . . . . 897.9 Anisotropic KPZ . . . . . . . . . . . . . . 917.10 KPZ with spatially correlated noise . . . . . 917.11 An upper critical dimension for KPZ? . . . 917.12 The KPZ equation in dimension d = 1 . . . 917.13 KPZ, polynuclear growth, Tracey-Widomand Baik-Rains distributions . . . . . . . . 927.14 Models in the KPZ universality class, andexperimental realizations . . . . . . . . . . 937.15 From Burgers’ turbulence to Navier-Stokesturbulence? . . . . . . . . . . . . . . . . . 93 φ -theorywith two fermions and one boson . . . . . . 978.7 Supersymmetry: A critical discussion . . . 978.8 Mapping loop-erased random walks onto φ -theory with two fermions and one boson 978.9 Other models equivalent to loop-erasedrandom walks, and CDWs . . . . . . . . . 1008.10 Conformal field theory for critical curves . 101
10 Appendix: Basic Tools 105
Theory and Experiments forDisordered Elastic Manifolds,Depinning, Avalanches, andSandpiles
Foreword
This review grew out of lectures the author gave in theICTP master program at ENS Paris. While the beginning ofeach section is elementary, later parts are sometimes morespecialized and can be skipped at first reading. Beginnerswishing to enter the subject are encouraged to start readingsections 1.1 (introduction), 2.1-2.12 (equilibrium/statics),and 3.1-3.4 (depinning/dynamics). An introduction toavalanches is given in sections 4.1-4.3, 4.5-4.10. Theremaining sections are more specialized: Sandpile modelsand anisotropic depinning are treated in sections 5 and 6.An introduction to the KPZ equation and its relation todisordered elastic systems is given in section 7. Section heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
48 discusses links between a class of theories encompassingloop-erased random walks, charge density waves, Abeliansandpiles, and n-component φ theory with n = − , linkedby supersymmetry techniques. Further developments andideas are collected in section 9. The appendix 10 containsuseful basic tools.
1. Disordered Elastic Manifolds: Phenomenology
Statistical mechanics is by now a rather mature branch ofphysics. For pure systems like a ferromagnet, it allowsone to calculate with precision details as the behavior ofthe specific heat on approaching the Curie-point. We knowthat it diverges as a function of the distance in temperatureto the Curie-temperature, we know that this divergence hasthe form of a power-law, we can calculate the exponent,and we can do this with at least 3 digits of accuracy usingthe perturbative renormalization group [1, 2, 3, 4, 5, 6,7, 8], and even more precisely with the newly developedconformal bootstrap [9, 10, 11]. Best of all, these findingsare in excellent agreement with the most precise simulations[12, 13, 14], and experiments [15]. This is a true successstory of statistical mechanics. On the other hand, in natureno system is really pure, i.e. without at least some disorder(“dirt”). As experiments (and theory) seem to suggest, alittle bit of disorder does not change much. Otherwiseexperiments on the specific heat of Helium would not soextraordinarily well confirm theoretical predictions. Butwhat happens for strong disorder? By this we mean thatdisorder dominates over entropy, so effectively the systemis at zero temperature. Then already the question: “Whatis the ground-state?” is no longer simple. This goeshand in hand with the appearance of metastable states .States, which in energy are close to the ground-state,but which in configuration-space may be far apart. Anyrelaxational dynamics will take an enormous time to findthe correct ground-state, and may fail altogether, as canbe seen in computer-simulations as well as in experiments,particularly in glasses [17]. This means that our wayof thinking, taught in the treatment of pure systems, hasto be adapted to account for disorder. We will see thatin contrast to pure systems, whose universal large-scaleproperties can be “modeled by few parameters”, disorderedsystems demand to model the whole disorder-correlationfunction (in contrast to its first few moments). We showhow universality nevertheless emerges.Experimental realizations of strongly disordered sys-tems are glasses, or more specifically spin-glasses, vortex-glasses, electron-glasses and structural glasses [18, 19, 20,21, 22, 23, 24, 25, 17]. Furthermore random-field mag-nets [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], Even though there is some tension between values obtained in aspace-shuttle experiment [15] on one side, and simulations [16] and theconformal bootstrap [11] on the other hand. and last not least elastic systems subject to disorder, some-times termed disordered elastic systems or disordered elas-tic manifolds [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,52, 53, 54], on which we focus below.What is our current understanding of disordered elasticsystems? There are a few exact solutions, mostly foridealized or toy systems [55], there are phenomenologicalapproaches (like the droplet-model [56]), and there isa mean-field approximation, involving a method calledreplica-symmetry breaking (RSB) [57]. This methodpredicts the properties of infinitely connected systems, ase.g. the Sherrington-Kirkpatrick (SK) model [58, 59]. Thesolution proposed in 1979 by G. Parisi [60] is parameterizedby a function q ( x ) , where x “lives between replica indices and ”. Today we have a much better understanding ofthis solution [61, 62, 63], and many features can be provenrigorously [64, 65, 66, 67]. The most notable feature is thepresence of an extensive number of ground states arrangedin a hierarchic way (ultrametricity). On the other hand, thissolution is inappropriate for systems in which each degreeof freedom is coupled only to its neighbors, as is e.g. thecase in short-ranged magnetic systems.While the RSB method mentioned above is intellec-tually challenging, and rewarding, the complexity of themethods involved makes intuition difficult, and perform-ing a field theoretic expansion around this mean-field solu-tion has proven too challenging to succeed. Random-fieldmodels, which can be recast in a φ -type theory are seem-ingly more tractable, but still the non-linearity of the φ -interaction makes progress difficult. What one would liketo have is a field theory which in absence of disorder is assimple as possible. The simplest such system certainly isa non-interacting, Gaussian, i.e. free theory, to which onecould then add disorder. Actually, experimental systems ofthis type are abundant: Magnetic domain walls in presenceof disorder a.k.a. Barkhausen noise [68, 69, 70], a contactline wetting a disordered substrate [71], fracture in brittleheterogenous systems [72, 73, 74], or earthquakes [75] aregood examples for elastic systems subject to quenched dis-order. They have a quite different phenomenology frommean-field models, with notably a single ground state. Wewill term this situation equilibrium ; it supposes that if ex-ternal parameters change, they change so slowly that thesystem has enough time to find the ground state.In the opposite limit, notably if there are no thermalfluctuations at all, we can study depinning : Increasing anexternal applied field yields to jumps in the center-of-massof the system (the total magnetization in a magnet). Thesejumps are termed shocks or avalanches. While one canshow that the sequence of avalanches is deterministic givena specific disorder (see below), we are more interested intypical behavior, i.e. an average over disorder. The latteraverage can often be obtained by watching the system for anextended time; one says that the system is self-averaging .In these lectures combined with a review, the author heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Before developing the theory to treat elastic systemsin a disordered environment, let us give some physicalrealizations. The simplest one is an Ising magnet. Imposingboundary conditions with all spins up at the upper and allspins down at the lower boundary (see figure 1), at lowtemperatures, a domain wall separates a region with spinsup from a region with spins down. In a pure system attemperature T = 0 , this domain wall is flat. Disordercan deform the domain wall, making it eventually rough.Figure 1 shows, how the domain wall is described bya displacement field u ( (cid:126)x ) . Two types of disorder arecommon:(i) random-bond disorder (RB), where the bonds betweenneighboring sites are random. On a course-grainedlevel this also represents missing spins. Thecorrelations of the random potential are short-ranged.(ii) random-field disorder (RF), i.e. coupling of the spinsto an external random magnetic field. This disorderis “long-ranged”, as the random potential is the sumover the random fields below the domain wall, i.e.effectively has the statistics of a random walk. Takinga derivative of the potential, one obtains short-rangedcorrelated random forces.Another example is the contact line of a liquid (water,isobutanol, or liquid helium), wetting a rough substrate, seefigure 2. Here, elasticity becomes long-ranged , see Eq. (15)below.A realization with a 2-parameter displacement field (cid:126)u ( x, y, z ) is the deformation of a vortex lattice: the positionof each vortex is deformed from the 3-dimensional vector (cid:126)x = ( x, y, z ) to (cid:126)x + (cid:126)u ( (cid:126)x ) , with (cid:126)u ∈ R (its z -component is ). Irradiating the sample produces line defects. They allowexperimentalists to realise [44](iii) generic long-range (LR) correlated disorder. The mostextreme example are(iv) random forces with the statistics of a random walk .This model, the Brownian force model (BFM) of section 4.5, plays an important role as its center-of-mass motion advances as a single degree of freedom,known as the ABBM or mean-field model (section 4.3),often used to describe avalanches.Another example are charge-density waves, first predictedby Peierls [85]: They can spontaneously form in certainsemiconductor devices, where a uniform charge density isunstable towards a super-lattice in which the underlyinglattice is periodically deformed, and the charge densitybecomes [86, 87, 88, 89]. ρ ( (cid:126)x ) = ρ cos( (cid:126)k(cid:126)x ) . (1)Adding disorder, the latter locally deforms the phase,modifying the charge density to ρ (cid:0) (cid:126)x, u ( (cid:126)x ) (cid:1) = ρ cos (cid:0) (cid:126)k(cid:126)x + 2 πu ( (cid:126)x ) (cid:1) . (2)As the charge density is invariant under u ( (cid:126)x ) → u ( (cid:126)x ) + 1 ,we find another disorder class,(v) random periodic disorder (RP).All these models have in common that they can bedescribed by a displacement field (cid:126)x ∈ R d −→ (cid:126)u ( (cid:126)x ) ∈ R N . (3)For simplicity, we suppress the vector notation whereverpossible, and mostly consider N = 1 . After some initialcoarse-graining, the energy H = H el + H conf + H dis consists of three parts: the elastic energy H el [ u ] = (cid:90) d d x
12 [ ∇ u ( x )] , (4)the confining potential H conf [ u ] = (cid:90) x m (cid:90) x [ u ( x ) − w ] , (5)and the disorder H dis [ u ] = (cid:90) d d x V (cid:0) x, u ( x ) (cid:1) . (6)In order to proceed, we need to specify the correlations ofdisorder. Suppose that fluctuations of u scale as (cid:68) [ u ( x ) − u ( y )] (cid:69) ∼ | x − y | ζ . (7)Notations are such that (cid:104) ... (cid:105) denotes thermal averages,i.e. averages of an observable using the weight e − βH ,properly normalised by the partition function Z = (cid:10) e − β H (cid:11) .Overbars denote the average over disorder. This defines a roughness-exponent ζ . Starting from a disorder correlator V ( x, u ) V ( x (cid:48) , u (cid:48) ) = f ( x − x (cid:48) ) R ( u − u (cid:48) ) (8)and performing a step in the RG-procedure, for eachrescaling by λ in the x -direction one has to rescale by λ ζ in the u -direction. As long as ζ < , this will eventuallyreduce f ( x − x (cid:48) ) to a δ -distribution, whereas the structureof R ( u − u (cid:48) ) may remain visible. We therefore choose asour starting correlations for the disorder V ( x, u ) V ( x (cid:48) , u (cid:48) ) := R ( u − u (cid:48) ) δ d ( x − x (cid:48) ) . (9) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (‘‘random bond’’) defect‘‘random field’’ xu(x) Figure 1.
An Ising magnet at low temperatures forms a domain wall described by a function u ( x ) (right). An experiment on a thin Cobalt film (left)[83]; with kind permission of the authors. Figure 2.
A contact line for the wetting of a disordered substrate by Glycerine [84]. Experimental setup (left). The disorder consists of randomlydeposited islands of Chromium, appearing as bright spots (top right). Temporal evolution of the retreating contact-line (bottom right). Note the differentscales parallel and perpendicular to the contact-line. Pictures courtesy of S. Moulinet, with kind permission. (cid:126)u ( x, y, z )( x, y, z ) Figure 3.
A vorte lattice is described by a deformation of a lattice point ( x, y, z ) to ( x, y, z ) + (cid:126)u ( x, y, z ) . Shown is a cartoon of a single layer,i.e. fixed z . The vortex lines continue perpendicular to the drawing. As we do not consider higher cumulants of the disorder,this implicitly assumes that the distribution of the disorder is Gaussian .There are a couple of useful observables. We alreadymentioned the roughness-exponent ζ . The second is therenormalized (effective) disorder R ( u ) .Noting F ( x, u ) := − ∂ u V ( x, u ) , the correspondingforce-force correlator can be written as (cid:104) F ( x, u ) F ( x (cid:48) , u (cid:48) ) (cid:105) = ∆( u − u (cid:48) ) δ d ( x − x (cid:48) ) . (10)Since (cid:104) F ( x, u ) F ( x (cid:48) , u (cid:48) ) (cid:105) = ∂ u ∂ u (cid:48) V ( x, u ) V ( x (cid:48) , u (cid:48) ) = − R (cid:48)(cid:48) ( u − u (cid:48) ) δ d ( x − x (cid:48) ) , we identify ∆( u ) = − R (cid:48)(cid:48) ( u ) . (11) There are several relevant experimental systems for whichthe elasticity is different from Eq. (4). This mostly happens For the concept of cumulants see e.g. Ref. [90]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles L θ gZ y c x Figure 4.
The coordinate system for a vertical wall.The air/liquid interfacebecomes flat for large x . The height h ( x, y ) is along the z -direction. when the elasticity of a lower-dimensional subsystem ismediated by the surrounding bulk. The simplest suchexample is a contact line [91] in a coffee mug or waterbottle, i.e. the line where coffee, cup and air meet. Alaboratory example is shown in Fig. 2. For fracture thiswas introduced in [92].Consider a liquid with height h ( x, y ) , defined in thehalf-space x ≥ (see Fig. 4). Its elastic energy is surface-tension times surface-area, i.e. H liquidel [ h ] = γ (cid:90) y (cid:90) x> (cid:112) ∇ h ( x, y )] (cid:39) const. + (cid:90) y (cid:90) x> γ ∇ h ( x, y )] (12)We wish to express this as a function of the height u ( y ) := h (0 , y ) on the boundary x = 0 : Note that a minimumenergy configuration satisfies ! = δ H liquidel [ h ] δh ( x, y ) = − γ ∇ h ( x, y ) . (13)This is achieved by the ansatz h ( x, y ) = (cid:90) d k π ˜ u ( k ) e iky −| k | x . (14)At x = 0 this is the standard Fourier transform of the height u ( y ) on the boundary x = 0 . Integrating by parts, theelastic energy as a function of u ( y ) becomes with the helpof Eq. (13) H liquidel [ u ] = (cid:90) y (cid:90) x> γ ∇ h ( x, y )] = γ (cid:20)(cid:90) y (cid:90) x> ∇ (cid:16) h ( x, y ) ∇ h ( x, y ) (cid:17) − h ( x, y ) ∇ h ( x, y ) (cid:21) = − γ (cid:90) y h ( x, y ) ∂ x h ( x, y ) (cid:12)(cid:12)(cid:12) x =0 = γ (cid:90) d k π | k | ˜ u ( k )˜ u ( − k ) (15)In generalization, one can write H α el [ u ] = 12 (cid:90) d d k (2 π ) d | k | α ˜ u ( k )˜ u ( − k ) . (16) For α = 2 , this is equivalent to the local interaction of Eq.(4). For α < , the interaction is non-local in positionspace, H α el [ u ] = A αd (cid:90) d d (cid:126)x (cid:90) d d (cid:126)y [ u ( (cid:126)x ) − u ( (cid:126)y )] | (cid:126)x − (cid:126)y | − d − α , (17 a ) A αd = − α − Γ( d + α ) π d Γ( − α ) . (17 b )For d = α = 1 this yields H α =1el [ h ] = 14 π (cid:90) d x (cid:90) d y [ u ( x ) − u ( y )] | x − y | . (18)Note that for α → , A d ∼ (2 − α ) , reducing the long-rangekernel to the short-range one.Eq. (12) is an approximation, as higher-order termsare neglected. The latter can be generated efficiently [93],and may change the physics of the system [94]. When thecontact angle is different from the inclination of the wall,the elastic energy is further modified [95].The theory we develop below works for arbitrary(positive) α , with α = 2 for standard short-ranged elasticity, and α = 1 for (standard) long-ranged elasticity.Apart from contact lines, long-ranged elasticity appears fora d -dimensional elastic object (a surface), where the elasticinteractions are mediated by a bulk material of higherdimension D > d . Apart from the contact line, the mostimportant examples are the displacement of tectonic platesleading to earthquakes ( d = 2 , D = 3 ) [96, 75, 79] andfracture ( d = 1 , D = 2 or D = 3 ) [73, 74].For magnetic domain walls ( d = 2 ) with dipolarinteractions, the interactions are also long-ranged. Theelastic kernal is given by [97] (page 6357) H el [ u ] = γ (cid:90) d (cid:126)r (cid:90) d (cid:126)r ∂ x u ( (cid:126)r ) ∂ x u ( (cid:126)r ) | (cid:126)r − (cid:126)r | ,(cid:126)r = ( x , y ) , (cid:126)r = ( x , y ) . (19)In Fourier, this reads H el [ u ] = γ π (cid:90) d (cid:126)q ˜ u ( q )˜ u ( − q ) q x | (cid:126)q | . (20) Above, we distinguished four types of disorder, resulting infour different universality classes:(i) Random-Bond disorder (RB): short-range correlatedpotential-potential correlations, i.e. a short-rangecorrelated R ( u ) .(ii) Random-Field disorder (RF): a short-range correlatedforce-force correlator ∆( u ) := − R (cid:48)(cid:48) ( u ) . As thename says, this disorder is relevant for random-fieldsystems where the disorder potential is the sum overall magnetic fields below a domain wall.(iii) Generic long-range correlated disorder (LR): R ( u ) ∼| u | − γ . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles R ( u ) = R ( u + 1) , supposing that u is periodicwith period 1.To get an idea how large the roughness ζ becomes inthese situations, one compares the contributions of elasticenergy and disorder, and demands that they scale in thesame way. This estimate has first been used by Flory[98] for self-avoiding polymers, and is therefore called theFlory estimate . Despite the fact that Flory estimates areconceptually crude, they often give a decent approximation.For RB disorder, this gives for an N -component field u : (cid:82) x u |∇| α u ∼ (cid:82) x √ V V , or L d − α u ∼ L d √ L − d u − N , i.e. u ∼ L ζ with ζ RBFlory = 2 α − d N α → = 4 − d N . (21)For RF disorder ∆( u ) = − R (cid:48)(cid:48) ( u ) is short-ranged, and ζ RFFlory = 2 α − d N α → = 4 − d N . (22)For generic LR correlated disorder ζ LRFlory = 2 α − d γ α → = 4 − d γ . (23)For RP disorder the field u cannot be rescaled or one wouldbreak periodicity, and thus ζ RP = 0 (24)exactly. We will see below in section 2.4 that theseestimates are a decent approximation, and even exact forRF at N = 1 , or for LR disorder. In disordered systems, a particular configuration stronglydepends on the disorder, and therefore general statementsabout a specific configuration are meaningless. What oneneeds to calculate are averages, of the form (“gs” denotesthe ground state) O [ u ] := (cid:10) O [ u ] e −H [ u ] /T (cid:11)(cid:10) e −H [ u ] /T (cid:11) T → −−−→ O [ u gs ] e −H [ u gs ] /T e −H [ u gs ] /T ≡ O [ u gs ] . (25)Note that division by the partition function Z = (cid:10) e −H [ u ] /T (cid:11) itself is crucial. This is particularly pronouncedin the limit of T → , where Z → e −H [ u gs ] /T diverges orvanishes when T → , except if by chance H [ u gs ] = 0 .Thus the denominator can not be replaced by its mean.This is a difficult situation: while integer powers Z n , with n ∈ N could be achieved by using n copies or replicas of For disordered systems this type of argument was employed by Harris[99] and Imry and Ma [100], and the reader will find reference to them aswell. the system, / Z cannot. On the other hand, we observethat, independent of n , O [ u ] = (cid:10) O [ u ] e −H [ u ] /T (cid:11) Z n − Z n . (26)The replica-trick [101, 102] consists in doing thecalculations for arbitrary n . This is in genearal possiblein perturbation theory, as results there are polynomials in n . It may become troublesome for exact solutions (notablyleading to replica-symmetry breaking [57]). Knowing thedependence on n , the idea is to set n → at the end of thecalculation, thus eliminating the denominator, O [ u ] = lim n → (cid:10) O [ u ] e −H [ u ] /T (cid:11) Z n − . (27)Since thermal averages over distinct replicas factorize, wewrite their joint measure as (cid:10) O [ u ] e −H [ u ] /T (cid:11) Z n − = (cid:42) O [ u ] n (cid:89) a =1 e −H [ u a ] /T (cid:43) = (cid:68) O [ u ]e − (cid:80) na =1 H [ u a ] /T (cid:69) = (cid:68) O [ u ] e − T (cid:80) na =1 H el [ u a ]+ H conf [ u a ]+ H dis [ u a ] (cid:69) . (28)Note that in the second equality we have exchanged thermaland disorder averages. We also allowed for differentpositions w of the parabola for the different replicas,denoted w a . Finally, we used that O [ u ] does not dependon the disorder. Since only the last term in the exponentialdepends on V ( x, u ) , and since V ( x, u ) is a Gaussianvariable, e − T (cid:80) na =1 H dis [ u a ] = exp (cid:32) − T (cid:90) x (cid:88) a V ( x, u a ( x )) (cid:33) = exp T (cid:90) x (cid:90) y n (cid:88) a,b =1 V ( x, u a ( x )) V ( y, u b ( y )) = exp T (cid:90) x n (cid:88) a,b =1 R (cid:0) u a ( x ) − u b ( x ) (cid:1) . (29)In the second step we used that V is Gaussian; in the laststep we used the correlator (9).To summarize: to evaluate the expectation of anobservable, we take averages with the help of the replicaHamiltonian or action e −S rep [ u ] S rep [ u ] := 1 T n (cid:88) a =1 (cid:90) x (cid:26)
12 [ ∇ u a ( x )] + m u a ( x ) − w a ] (cid:27) − T (cid:90) x n (cid:88) a,b =1 R (cid:0) u a ( x ) − u b ( x ) (cid:1) . (30) It is not quite clear who “invented” the replica trick. In Ref. [101]P. Brout stresses that ln Z has to be averaged over disorder, not Z .Brout considers a cluster expansion for a quenched disordered system,organizing his expansion in powers of n , equivalent to a cumulantexpansion, or sums over independent replicas , concepts we use below. The partition function for each of these replicas may be different. Theformalism takes this into account. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles /T . Ifthe disorder had a third cumulant, this would appear as atriple replica sum, and a factor , of /T .Let us now turn to perturbation theory. The freepropagator, constructed from the first line of Eq. (30), andindicated by the index “0”, is (first in Fourier, than in realspace) (cid:104) ˜ u a ( − k )˜ u b ( k ) (cid:105) = T δ ab ˜ C ( k ) , (31) (cid:104) u a ( x ) u b ( y ) (cid:105) = T δ ab C ( x − y ) . (32)Noting C ( x − y ) the Fourier transform of ˜ C ( k ) , and S d =2 π d/ / Γ( d/ the area of the d -sphere, we have ˜ C ( k ) = 1 k + m , (33 a ) C ( x ) = (cid:90) d d k (2 π ) d e ikx k + m (cid:39) d − S d | x | − d for x → . (33 b )This allows us to calculate expectation values in the fulltheory. As an example consider (for more explanations seethe calculations around Eq. (51)) (cid:10) [ u ( x ) − w ] (cid:11) w (cid:10) [ u ( z ) − w ] (cid:11) w c ≡ (cid:10) [ u ( x ) − w ][ u ( z ) − w ] (cid:11) S rep = − (cid:90) y C ( x − y ) C ( z − y ) R (cid:48)(cid:48) ( w − w ) + ... (34)Let us clarify the notations: Thirst, (cid:10) [ u ( x ) − w ] (cid:11) w is thethermal average of u ( x ) − w , obtained by evaluating thepath integral for a fixed disorder configuration V , and ata position of the parabola given by w . This procedure isrepeated for (cid:10) [ u ( z ) − w ] (cid:11) w , with the same V . Finally theaverage over the disorder potential V is taken. Accordingto the calculations above, this can be evaluated with thehelp of the replica action S rep [ u ] , represented by (cid:10) [ u ( x ) − w ][ u ( z ) − w ] (cid:11) S rep . The latter is already averagedover disorder. The last line shows the leading order inperturbation theory, dropping terms of order T and higher.Finally, let us integrate this expression over x and z , andmultiply by m /L d . This leads to m L d (cid:90) x,z (cid:10) [ u ( x ) − w ] (cid:11) w (cid:10) [ u ( z ) − w ] (cid:11) w = − R (cid:48)(cid:48) ( w − w ) + ... (35)The combination m [ u ( x ) − w ] is the force acting on point x (a density), its integral over x the total force acting on theinterface. Force correlations are short ranged in x , leadingto the factor of /L d . Note that the thermal 2-point function(32) is absent, as we consider two distinct copies of thesystem. It is an interesting exercise to show that there are noperturbative corrections to Eq. (35) in the limit of T → , as long as one supposes that R ( w ) is an analytic function.Similarly, one shows that in the same limit (cid:104) uuuu (cid:105) c = 0 .This implies that (we set w = w = 0 ) at T = 0 (whicheliminates the thermal fluctuations) (cid:104) ˜ u ( k ) (cid:105) (cid:104) ˜ u ( − k ) (cid:105) = (cid:104) ˜ u ( k )˜ u ( − k ) (cid:105) = ˜ u ( k )˜ u ( − k ) = − R (cid:48)(cid:48) (0)( k + m ) . (36)In the third expression we suppressed the thermal expecta-tion values since at T = 0 only a single ground state sur-vives . Fourier-transforming back to position space yields(with some amplitude A , and in the limit of m → ) (cid:2) u ( x ) − u ( y ) (cid:3) = − R (cid:48)(cid:48) (0) A| x − y | − d . (37)This looks very much like the thermal expectation (32),except that the dimension of space has been shifted by .Further, both theories are seemingly Gaussian, i.e. highercumulants vanish.We have just given a simple version of a beautiful andrather mind-boggling theorem relating disordered systemsto pure ones (i.e. without disorder). The theorem appliesto a large class of systems, even when non-linearities arepresent in the absence of disorder. It is called dimensionalreduction [103, 104, 105]. We formulate it as follows:“Theorem”: A d -dimensional disordered system at zerotemperature is equivalent to all orders in perturbationtheory to a pure system in d − dimensions at finitetemperature. We give in section 8.1 a proof of this theorem usinga supersymmetric field theory introduced in Ref. [32].The proof implicitly assumes that R ( u ) is analytic , thusall derivatives can be taken. The equivalence is ratherpowerful, since the supersymmetric theory knows aboutdifferent replicas, and allows one to calculate even awayfrom the critical point.However, evidence from experiments, simulations,and analytic solutions show that the above “theorem” isactually wrong . A prominent counter-example is the 3-dimensional random-field Ising model at zero temperature[30]; according to the theorem it should be equivalent tothe pure 1-dimensional Ising-model at finite temperature.While it was shown rigorously [30] that the former has anordered phase, the latter is disordered at finite temperature[106]. So what went wrong? Let us stress that there are nomissing diagrams or any such thing, but that the problemis more fundamental: As we will see later, the proof makesthe assumption that R ( u ) is analytic. While this assumptionis correct in the microscopic model, it is not valid at largescales.Nevertheless, the above “theorem” remains importantsince it has a devastating consequence for all perturbativecalculations in the disorder: However clever a procedure For disordered elastic manifolds with continuous disorder, the groundstate is almost surely unique. This is in strong contrast to mean-field spinglasses, where it is highly degenerate, see e.g. [57] heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ζ defined in equation (7),as ζ DR = 4 − d . (38)On the other hand, the directed polymer in dimension d = 1 does not have a roughness exponent of ζ DR = 3 / , but[107] ζ RB d =1 = 23 . (39)Experiments and simulations for disordered elastic mani-folds discussed below in sections 2.30, 2.31, 3.12, 3.13,3.15, 3.16, 3.17, and 3.21 all violate dimensional reduction. To understand the failure of dimensional reduction, letus turn to crucial arguments given by Larkin [108]. Heconsiders a piece of an elastic manifold of size L . Ifthe disorder has correlation length r , and characteristicpotential energy ¯ E , this piece will typically see a potentialenergy of amplitude E dis = ¯ E (cid:18) Lr (cid:19) d . (40)On the other hand, the elastic energy scales as E el = c L d − . (41)These energies are balanced at the Larkin-length L = L c with L c = (cid:18) c ¯ E r d (cid:19) − d . (42)More important than this value is the observation that in allphysically interesting dimensions d < d c = 4 , and at scales L > L c , the disorder energy (40) wins; as a consequencethe manifold is pinned by disorder, whereas on small scalesthe elastic energy dominates. For long-ranged elasticity, thesame argument implies d c = 2 α, and disorder relevant for d < d c . (43)Since the disorder has many minima which are far apartin configurational space but close in energy (metastability),the manifold can be in either of these minima, and localminimum does not imply global minimum. However, theexistence of exactly one minimum is assumed in e.g. theproof of dimensional reduction. (Formally, the field theorysums over all saddle points.)Another important question is the role of temperature.In Eq. (7) we had supposed that u scales with thesystems size as u ∼ L ζ . Demanding that the action (30) be dimensionless, the first term in Eq. (30) scales as L d − ζ /T . This implies that T ∼ a θ , θ = d − ζ, (44)where a is a microscopic cutoff with the dimension of L , tocompensate the factor of L d − ζ . For completeness, wealso give the result for generic LR-elasticity, θ α = d − α + 2 ζ. (45)The thermodynamic limit is obtained by taking L → ∞ .Temperature is thus irrelevant when θ > , which is thecase for d > , and when ζ > even below . As aconsequence, the RG fixed point we are looking for is atzero temperature [109]. The same argument applies to thefree energy F [ u ] = − T ln( Z [ u ]) ∼ (cid:18) La (cid:19) θ . (46)We added u as an argument of F [ u ] , as e.g. in thedirected polymer the partition function is the weight of alltrajectories arriving at u . This is important in section 7.1when considering the KPZ equation.From the second term in Eq. (30) we conclude that the(microscopic) disorder scales as R ∼ a θ − d = a d − ζ . (47)For ζ = 0 , this again implies that d = 4 is the uppercritical dimension. More thorough arguments are presentedin the next section, where we will construct an (cid:15) = 4 − d expansion for the RG flow of R ( u ) .
2. Equilibrium (statics)
In section 1.7, we had seen that 4 is the upper criticaldimension for SR elasticity, which we treat now. As forstandard critical phenomena [1, 2, 3, 4, 5, 6, 7], we nowconstruct an (cid:15) = (4 − d ) -expansion. Taking the dimensionalreduction result (38) in d = 4 dimensions tells us that thefield u is dimensionless in d = 4 . Thus, the width σ = − R (cid:48)(cid:48) (0) of the disorder is not the only relevant couplingat small (cid:15) , but any function of u has the same scalingdimension in the limit of (cid:15) = 0 , and might equivalentlycontribute. The natural conclusion is to follow the fullfunction R ( u ) under renormalization, instead of just itssecond derivative R (cid:48)(cid:48) (0) .Such an RG-treatment is most easily implementedin the replica approach: The n times replicated partitionfunction led after averaging over disorder to a path integralwith weight e −S rep [ u ] with action (30). Perturbation theory The dimension of T invites confusion: arguing that T ∼ L θ with θ > suggests that T is relevant instead of irrelevant. However, T isa parameter which does not change when L is increased, and whether itis relevant or not depends on the ratio of elastic energy E el to temperature T , which scales as ( L/a ) θ , showing that energy wins over temperature,or that temperature is irrelevant (compared to the elastic energy). heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles k flowing through and replicas a and b , see Eqs. (31)–(33 a ), (cid:104) ˜ u a ( k )˜ u b ( k ) (cid:105) = T × a b = T δ ab ˜ C ( k ) . (48)Note that the factor of T is explicit in our graphicalnotation, and not included in the line. The disorder vertexis (we added an index R to R to indicate that this is themicroscopic (bare) disorder) T × xba = 1 T × R (cid:16) u a ( x ) − u b ( x ) (cid:17) . (49)The rules of the game are to find all contributions whichcorrect R , and which survive in the limit of T → . Atleading order, i.e. order R , counting of factors T showsthat we can use at most two correlators, as each contributesa factor of T . On the other hand, (cid:80) a,b R ( u a − u b ) hastwo independent sums over replicas . Thus at order R four independent sums over replicas appear, and in order toreduce them to two, one needs at least two correlators (eachcontributing a δ ab ). Thus, at leading order, only diagramswith two propagators survive.Before writing down these diagrams, we need to seewhat Wick-contractions do on functions of the field. To seethis, remind that a single Wick contraction (indicated bysitting on top of the fields to be contracted) u a ( x ) n u b ( y ) m = nu a ( x ) n − × mu b ( x ) m − × T δ ab C ( x − y ) . (50)Realizing that nu n − = ∂ u u n , we can write the Wickcontraction for an arbitrary function V ( u ) as V (cid:0) u a ( x ) (cid:1) V (cid:0) u b ( y ) (cid:1) = V (cid:48) (cid:0) u a ( x ) (cid:1) × V (cid:48) (cid:0) u b ( y ) (cid:1) × T δ ab C ( x − y ) . (51)Graphically we have at second order for the correction ofdisorder T δR = 12! (cid:34) T xba (cid:35) TT fefe (cid:34) T ydc (cid:35) (52)We have explicitly written all factors: a / from theexpansion of the exponential function exp( −S rep [ u ]) , afactor of / (2 T ) per disorder vertex, and a factor of T per propagator. Using these rules, we obtain two distinctcontributions δR (1) = 12 x yba ba (53) = 12 (cid:90) x R (cid:48)(cid:48) (cid:0) u a ( x ) − u b ( x ) (cid:1) R (cid:48)(cid:48) (cid:0) u a ( y ) − u b ( y ) (cid:1) C ( x − y ) ,δR (2) = x yaa ba (54) = − (cid:90) x R (cid:48)(cid:48) (cid:0) u a ( x ) − u a ( x ) (cid:1) R (cid:48)(cid:48) (cid:0) u a ( y ) − u b ( y ) (cid:1) C ( x − y ) . The concept of sums over independent replicas already appears in thework by P. Brout [101], see footnote 4.
Note that all factors of T have disappeared, and onlytwo replica sums (not written explicitly) remain. Each R ( u a − u b ) has been contracted twice, giving rise to twoderivatives. In the first diagram, since once u a and once u b has been contracted, each R (cid:48)(cid:48) comes with an additionalminus sign; these cancel. In the second diagram, there is aminus sign from the first R (cid:48)(cid:48) , but not from the second; thusthe overall sign is negative.Note that the following diagram also contains twocorrelators (correct counting in powers of temperature), butis not a 2-replica but a 3-replica sum: x yba ca (55)In a renormalization program, we are looking for diver-gences of these diagrams. These divergences are localizedat x = y : indeed the integral over the difference z := y − x ,is in radial coordinates with r = | z | , (cid:15) = 4 − d , and for m → (up to a geometrical prefactor) (cid:90) z C ( z ) ∼ (cid:90) La d rr r d r − d ) = (cid:90) La d rr z − d = 1 (cid:15) ( L (cid:15) − a (cid:15) ) . (56)Note that for (cid:15) → each scale contributes the same: from r = 1 / to r = 1 the same as from r = 1 / to r = 1 / , andagain the same for r = 1 / to r = 1 / . Thus the divergencecomes from small scales, which allows us to approximate R (cid:48)(cid:48) ( u a ( y ) − u b ( y )) ≈ R (cid:48)(cid:48) ( u a ( x ) − u b ( x )) . This is formallyan analysis of the theory via an operator product expansion.For an introduction and applications see [3, 110].Eq. (56) is regularized with cutoffs a and L . It isconvenient to use (cid:15) > (what we need anyway), whichallows us to take a → and L → ∞ while keeping m finite, as the latter appears as the harmonic well introducedin section 2.10. The integral in that limit becomes I := = (cid:90) x − y C ( x − y ) = (cid:90) k k + m ) = m − (cid:15) (cid:15) (cid:15) )(4 π ) d/ . (57)It is the standard 1-loop diagram of massive φ -theory.Setting u = u a ( x ) − u b ( x ) , we obtain for theeffective disorder correlator ˆ R ( u ) at 1-loop order with allcombinatorial factors as given above, ˆ R ( u ) = R ( u ) + (cid:20) R (cid:48)(cid:48) ( u ) − R (cid:48)(cid:48) ( u ) R (cid:48)(cid:48) (0) (cid:21) I + ... (58)We can now study its flow, by taking a derivative w.r.t. m ,and replacing on the r.h.s. R with ˆ R , as given by the aboveequation. This leads to − m ∂∂m ˆ R ( u ) = (cid:20)
12 ˆ R (cid:48)(cid:48) ( u ) − ˆ R (cid:48)(cid:48) ( u ) ˆ R (cid:48)(cid:48) (0) (cid:21) (cid:15)I . (59)This equation still contains the factor of (cid:15)I , which hasboth a scale m − (cid:15) , as a finite amplitude. There are two heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles I itself, defining ∂ (cid:96) ˆ R ( u ) := − ∂∂I ˆ R ( u ) = 12 ˆ R (cid:48)(cid:48) ( u ) − ˆ R (cid:48)(cid:48) ( u ) ˆ R (cid:48)(cid:48) (0) . (60)This is convenient to study the flow numerically.To arrive at a fixed point one needs to rescale both ˆ R and u , in order to make them dimensionless. The field u hasdimension u ∼ L ζ ∼ m − ζ , whereas the dimension of ˆ R can be read off from Eq. (53), namely ˆ R ( u ) ∼ ˆ R (cid:48)(cid:48) ( u ) m − (cid:15) ,equivalent to ˆ R ∼ m (cid:15) − ζ . The dimensionless effectivedisorder ˜ R , as function of the dimensionless field u is thendefined as ˜ R ( u ) := (cid:15)I m ζ ˆ R ( u = u m − ζ ) . (61)Inserting this into Eq. (60), we arrive at ∂ (cid:96) ˜ R ( u ) := − m ∂∂m ˜ R ( u ) (62) = ( (cid:15) − ζ ) ˜ R ( u ) + ζ u ˜ R (cid:48) ( u ) + 12 ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) ( u ) ˜ R (cid:48)(cid:48) (0) . This is the functional RG flow equation for the renormal-ized dimensionless disorder ˜ R ( u ) , first derived in Ref. [111]within the Wilson scheme . We will in general set u → u in the above equation, to simplify notations, and suppressthe tilde as long as this does not lead to confusion. We claim that there are no renormalizations of the quadraticparts of the action which are replicated copies of H [ u ] := H el [ u ] + H conf [ u ] (63)given in Eqs. (4) and (5). This is due to the statistical tiltsymmetry u a ( x ) → u a ( x ) + αx. (64)As the interaction is proportional to R ( u a ( x ) − u b ( x )) , thelatter is invariant under the transformation (64). The changein H [ u ] becomes δ H [ u ] = c (cid:90) d d x (cid:20) ∇ u ( x ) α + 12 α (cid:21) + m (cid:90) d d x (cid:20) u ( x ) αx + 12 α x (cid:21) . (65)To render the presentation clearer, the elastic constant c set to c = 1 in equation (30) has been introduced. Theimportant observation is that all fields u involved are large-scale variables, which are also present in the renormalizedaction, where they change according to H ren [ u ] →H ren [ u ] + δ H ren [ u ] . Since one can either first renormalizeand then tilt, or first tilt and then renormalize, we obtain δ H ren0 [ u ] = δ H bare0 [ u ] . This means that neither the elasticconstant c , nor m change under renormalization. (cid:96) in Eqs. (60) and (62) is different. The RG flow equation (62) is at this order independent of the RGscheme. Universal quantities are scheme-independent to all orders [2, 112,113, 114, 110].
Let us recapitulate the strategy used here: weconsidered k -replica terms, and kept the leading term for T → . A k -replica term is a sum over k replicas,equivalent to a cumulant of order k , and thus comes witha factor of /T k . Factors of n appear e.g. in correlationfunctions, where the limit of n → has to be taken, but thereader should not be surprised to see n absent from otherconsiderations. We now analyze the FRG flow equations (60) and (62). Tosimplify our arguments, we first derive them twice w.r.t. u ,to obtain flow equations for ∆( u ) ≡ − R (cid:48)(cid:48) ( u ) . This yieldsno rescaling: ∂ (cid:96) ˆ∆( u ) = − ∂ u (cid:2) ˆ∆( u ) − ˆ∆(0) (cid:3) , (66)with rescaling: ∂ (cid:96) ˜∆( u ) = ( (cid:15) − ζ ) ˜∆( u ) + ζu ˜∆ (cid:48) ( u ) − ∂ u (cid:2) ˜∆( u ) − ˜∆(0) (cid:3) . (67)For concreteness, consider Eq. (66), and start with ananalytic function, ˆ∆ (cid:96) =0 ( u ) = e − u / . (68)According to our classification, this is microscopically RFdisorder. Since ˆ∆( u ) − ˆ∆(0) grows quadratically in u atsmall u , the r.h.s. of Eq. (66) also grows ∼ u at u = 0 , andboth ˆ∆(0) as well as ˆ∆ (cid:48) (0 + ) do not flow in the beginning.This can be seen on the plots of figure 5.Integrating further, a cusp forms, i.e. ˆ∆ (cid:48)(cid:48) (0) → ∞ ,and as a consequence ˆ∆ (cid:48) (0 + ) becomes non-zero. This isbest seen by taking two more derivatives of Eq. (66), andthen taking the limit of u → , ∂ (cid:96) ˆ∆ (cid:48)(cid:48) (0 + ) = − (cid:48)(cid:48) (0 + ) − (cid:48) (0 + ) ˆ∆ (cid:48)(cid:48)(cid:48) (0 + ) . (69)Since in the beginning ˆ∆ (cid:48) (0 + ) = 0 , only the first termsurvives. Its behavior crucially depends on the sign of ˆ∆ (cid:48)(cid:48) (0) . In the example (68), ˆ∆ (cid:48)(cid:48) (cid:96) =0 (0) < . This is truein general, as can be seen by rewriting Eq. (10) for theunrescaled microscopic disorder correlator at x = x (cid:48) , as ∆(0) − ∆( u − u (cid:48) ) = 12 (cid:68)(cid:2) F ( x, u ) − F ( x, u (cid:48) ) (cid:3) (cid:69) ≥ . (70)Developping the l.h.s. for small u − u (cid:48) with a vanishingfirst derivative implies that ∆ (cid:48)(cid:48) (0) < , valid also for therescaled ˆ∆ (cid:48)(cid:48) (0) .Integrating Eq. (69) with this sign yields ˆ∆ (cid:48)(cid:48) (cid:96) (0) = ˆ∆ (cid:48)(cid:48) (0)1 + 3 ˆ∆ (cid:48)(cid:48) (0) (cid:96) = −
13 1 (cid:96) c − (cid:96) ,(cid:96) c = −
13 ˆ∆ (cid:48)(cid:48) (0) ˆ∆ (cid:48)(cid:48) (0)= − −−−−−−−→ . (71)In the last equality we used the initial condition (68). Withthis, ˆ∆ (cid:48)(cid:48) (cid:96) (0) diverges at (cid:96) = , thus ˆ∆ (cid:96) ( u ) aquires a cups,i.e. ˆ∆ (cid:48) (cid:96) (0 + ) (cid:54) = 0 for all (cid:96) > / . Physically, this is the scale heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles u Δ ( u ) u - - - - - - ( u ) ℓ - - - - - Δ ( + ) u - ( u ) u - - ( u ) ℓ - - - - - Δ ( + ) Figure 5.
Top: Change of ∆( u ) := − R (cid:48)(cid:48) ( u ) under renormalization and formation of the cusp. Left: Explicit numerical integration of Eq. (60), startingfrom ∆( u ) = e − u / (in solid black, top curve for u → ). The function at scale (cid:96) is shown in steps of δ(cid:96) = 1 / . Inset: blow-up. Right: plots of ∆ (cid:48) ( u ) . Inset: ∆ (cid:48) (0 + ) as a function of (cid:96) . The cusp appears for (cid:96) = 1 / (red dot); dashed lines are before appearance of the cusp, and solid lines after.Bottom: ibid for RB disorder, starting from R ( u ) = e − u / ; the cusp appears for (cid:96) = 1 / ; δ(cid:96) = 1 / . where multiple minima appear. In terms of the Larkin-scale L c defined in section 1.7 (cid:96) c = ln( L c /a ) . (72)Our numerical solution shows the appearance of the cusponly approximately, see the inset in the top right plot offigure 5. This discrepancy comes from discretization errors.It is indeed not simple to numerically integrate equation(66) for large times, as ˆ∆ (cid:48)(cid:48) (cid:96) (0) diverges at (cid:96) = (cid:96) c , andall further derivatives at u = 0 + were extracted fromnumerical extrapolations of the obtained functions, in thelimit of u → . Interpreting derivatives in this sense isan assumption , to be justified, without which one cannotcontinue to integrate the flow equations. In this spirit, let usagain look at the flow equation for ˆ∆(0) , now including therescaling terms, ∂ (cid:96) ˆ∆(0) = ( (cid:15) − ζ ) ˆ∆(0) − ˆ∆ (cid:48) (0 + ) . (73)This equation tells us that as long as ∆ (cid:48) (0 + ) = 0 , ζ (cid:96)<(cid:96) c (cid:39) ζ DR = (cid:15) − d , (74)the dimensional-reduction result. Beyond that scale, wehave (as long as we are at least close to a fixed point) ζ (cid:96)>(cid:96) c = (cid:15) − ˆ∆ (cid:48) (0 + ) ˆ∆(0) < (cid:15) , (75) since both ˆ∆ (cid:48) (0 + ) and ˆ∆(0) are positive.Let us repeat our analysis for RB disorder, startingfrom the microscopic disorder ˆ R ( u ) = e − u / ⇐⇒ ˆ∆( u ) = e − u / (1 − u ) . (76)This is shown on the bottom of figure 5. Phenomenolog-ically, the scenario is rather similar, with a critical scale (cid:96) c = 1 / instead of / . We had seen in the last section that integrating theflow-equation explicitly is rather cumbersome; moreover,an estimation of the critical exponent ζ will be ratherimprecise. For this purpose, it is better to directly search fora solution of the fixed-point equation (67), i.e. ∂ (cid:96) ˜∆( u ) = 0 , (cid:15) − ζ ) ˜∆( u )+ ζu ˜∆ (cid:48) ( u ) − ∂ u (cid:104) ˜∆( u ) − ˜∆(0) (cid:105) . (77)We start with situations where u is unbounded, as for theposition of an interface. An important observation is thatthe fixed point is not unique; indeed, if ˜∆( u ) is solution ofEq. (77), so is ˜∆ κ ( u ) := κ − ˜∆( κu ) . (78) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles There is one solution we can find analytically: To thispurpose integrate Eq. (77) from 0 to ∞ , assuming that ˜∆( u ) has a cusp at u = 0 , but no stronger singularity, (cid:90) ∞ ( (cid:15) − ζ ) ˜∆( u ) + ζu ˜∆ (cid:48) ( u ) − ∂ u (cid:104) ˜∆( u ) − ˜∆(0) (cid:105) d u. (79)Integrating the second term by part, and using that the lastterm is a total derivative which vanishes both at and at ∞ yields (cid:15) − ζ ) (cid:90) ∞ ˜∆( u ) d u. (80)This equation has two solutions: either the integralvanishes, which is the case for RB disorder , or ζ RF = (cid:15) . (81)This is the exponent (22) (at N = 1 ) predicted by a Floryargument. Let us remark that Eq. (79) remains valid to allorders in (cid:15) , as long as ∆( u ) is the second derivative of R ( u ) ,s.t. the additional terms at 2- and higher-loop order are alltotal derivatives, as is the last term in Eq. (77).Let us pursue our analysis with the solution (81).Inserting Eq. (81) into Eq. (77), and setting ˜∆( u ) = (cid:15) y ( u ) (82)yields ∂ u (cid:20) uy ( u ) − ∂ u (cid:16) y ( u ) − y (0) (cid:17) (cid:21) = 0 . (83)This implies that the expression in the square bracket is aconstant, fixed to 0 by considering either the limit of u → or u → ∞ . Simplifying yields uy ( u ) + (cid:2) y (0) − y ( u ) (cid:3) y (cid:48) ( u ) = 0 . (84)Dividing by y ( u ) and integrating once again gives u − y ( u ) + y (0) ln( y ( u )) = const. (85)Let us now use Eq. (78) to set y (0) → . This fixes theconstant to − . Dropping the argument of y , we obtain y − ln( y ) = 1 + u . (86)This is plotted on figure 6. For RB disorder (cid:90) ∞ d u ˜∆( u ) = − (cid:90) ∞ d u ˜ R (cid:48)(cid:48) ( u ) = ˜ R (cid:48) (0) − ˜ R (cid:48) ( ∞ ) = 0 . The other option for a fixed point is to have the integral inEq. (80) vanish, (cid:90) ∞ ˜∆ RB ( u ) = 0 . (87)A numerical analysis of the fixed-point equation (77)proceeds as follows: Choose ˜∆(0) = 1 ; choose ζ ; solvethe differential equation (77) for ˜∆ (cid:48)(cid:48) ( u ) . Integrate the latterfrom u = 0 to u = ∞ . In practice, to avoid numericalproblems for u ≈ , one first solves the differential equationin a Taylor-expansion around 0; as the latter does notconverge for large u one then solves, with the informationfrom the Taylor series evaluated at u = 0 . , the differentialequation numerically up to u ∞ ≈ . One then reports,as a function of ζ , the value of ˜∆( u ∞ ) . As in quantummechanics, one finds that there are several discrete valuesof ζ with ˜∆( u ∞ ) = 0 . The largest value of ζ is the onegiven in Eq. (81), where ˜∆( u ) has no zero crossing. Thenext smaller value of ζ is ζ RB = 0 . (cid:15). (88)The corresponding function is plotted on figure 6 (right).It has one zero-crossing. Consistent with Eq. (80), itintegrates to zero. This is the random-bond fixed point. Itis surprisingly close, but distinct, from the Flory estimate(21), ζ = (cid:15)/ .For (cid:15) = 3 we have the directed polymer ( d = 1 ) indimension N = 1 , which has roughness ζ RB d =1 = . Ourresult (88) yields ζ ( d = 1) = 0 . O ( (cid:15) ) . Thisis quite good, knowing that (cid:15) = 3 is rather large. Thisvalue gets improved at 2-loop order, with ζ ( d = 1) =0 . O ( (cid:15) ) . Despite the “strange cusp”, it seemsthe method works!The next solution is at ζ = 0 . (cid:15). (89)It has two zero-crossings, and corresponds to a tricriticalpoint. We do not know of any physical realization. If ζ is not one of these special values, then the solutionof the fixed-point Eq. (77) decays algebraically: Supposethat ∆( u ) ∼ u α . Then the first two terms of Eq. (77) aredominant over the last one, as long as α < . Solving Eq.(77) in this limit one finds ∆ ζ ( u ) ∼ u − (cid:15)ζ for u → ∞ . (90)An important application are the ABBM and BFM modelsdiscussed in sections 4.3 and 4.5, for which ζ ABBM = (cid:15), ∆ ABBM (0) − ∆ ABBM ( u ) = σ | u | , (91)such that the correlations of the random force have thestatistics of a random walk. One easily checks that theflow equation (60) vanishes for all u > . In this case heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3) (cid:5)(cid:2)(cid:1) (cid:5)(cid:2)(cid:3) (cid:6)(cid:2)(cid:1) (cid:1) (cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:8)(cid:1)(cid:2)(cid:9)(cid:4)(cid:2)(cid:1) (cid:2) = ( (cid:1) ) u - ( u ) Figure 6.
Left: The RF fixed point (86) with ζ RF = (cid:15) . Right: The RB fixed point (88), with ζ RB = 0 . (cid:15) . ∆(0) is formally infinite, s.t. the bound (75) does notapply. Generically, however, Eq. (75) applies, implyingthat the exponent in Eq. (90) is negative, and ∆ ζ ( u ) decaysalgebraically. This is what we mostly see in numericalsolutions of the fixed-point Eq. (77). In the above considerations, we had supposed that u cantake any real value. There are important applications wherethe disorder is periodic, or u is a phase between and π .This is the case for the CDWs introduced above. To beconsistent with the standard conventions employed in theliterature [115, 116, 117, 118, 119, 120], we take the periodof the disorder to be . One checks that the following ansatzis a fixed point of the FRG equation (77) ζ RP = 0 , ∆ RP ( u ) = g − g u (1 − u ) , ≤ u ≤ . (92)This ansatz is unique, due to the following three constraints:(i) ζ = 0 , as the period is fixed and cannot change underrenormalization. (ii) ∆( u ) = ∆( − u ) = ∆(1 − u ) due tothe symmetry u → − u , and periodicity. Thus ∆( u ) is apolynomial in u (1 − u ) . (iii) a polynomial of degree in u closes under RG. (iv) the integral (cid:82) d u ∆( u ) = 0 , since ∆( u ) = − R (cid:48)(cid:48) ( u ) , and R ( u ) itself is periodic. The fixedpoint has g = (cid:15) ... (93)Instead of a universal scaling exponent ζ , the lattervanishes, ζ = 0 . As a consequence, the 2-point functionis logarithmic in all dimensions, with a universal amplitudegiven in Eqs. (117)-(118 b ). Apart from geometricprefactors, this amplitude is simply the fixed-point value g . Let us give a simple argument of why a cusp is a physicalnecessity, and not an artifact. The argument is quite old and appeared probably first in the treatment of correlation-functions by shocks in Burgers turbulence. It becamepopular in [121]. Suppose, we want to solve the problem fora single degree of freedom which sees both disorder and aparabolic trap centered at w , which we can view as a springattached to the point w . This is graphically represented onfigure 7 (upper left), with the quenched disorder realizationhaving roughly a sinusoidal shape. For a given disorderrealization V ( u ) , the minimum of the potential as a functionof w is ˆ V ( w ) := min u (cid:20) V ( u ) + m u − w ) (cid:21) . (94)This is reported on figure 7 (upper right). Note that it hasnon-analytic points, which mark the transition from oneminimum to another. The remaining parts are parabolic,and stem almost entirely from the spring, as long as theminima of the disorder are sharp, i.e. have a high curvatureas compared to the spring. This is rather natural, knowingthat the disorder varies on microscopic scales, while theconfining potential changes on macroscopic scales.Taking the derivative of the potential leads to the forcein figure 7 (lower left). It is characterized by almost linearpieces, and shocks (i.e. jumps). Let us now calculate thecorrelator of forces F ( u ) := −∇ ˆ V ( u ) , ∆( w ) := F ( w (cid:48) ) F ( w (cid:48) − w ) c . (95)Here the average is over disorder realizations, or equiva-lently w (cid:48) , on which it should not depend. Let us analyze itsbehavior at small distances, ∆(0) − ∆( w ) = 12 [ F ( w (cid:48) ) − F ( w (cid:48) − w )] = 12 p shock ( w ) (cid:10) δF (cid:11) + O ( w ) (96)As written, the leading contribution is proportional to theprobability to have a shock (jump) inside the window ofsize w , times the expectation of the second moment of theforce jump δF . If shocks are not dense, then the probability heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles formulas u w = wu formulas u w = wu formulas u w = wF ( w ) F ( ) u ˆ | w | Why is a cusp necessary? . . . calculate effective action for single degree of freedom. . . -10 -5 5 10-1.5-1-0.50.511.5
V u
Min ⇥ -10 -5 5 10-1.5-1.25-1-0.75-0.5-0.25
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F u average ⇥ -4 -2 2 40.20.40.60.81 formulas u w = wu formulas u w = wu formulas u w = wu formulas u w = u ( ⇤ t u xt ⇥ x + m ) u xt F ( x , u xt ) = ( ⇤ t u xt ⇥ x + m ) ˙ u xt ⇤ t F ( x , u xt ) = [ F ( w ) F ( )] = ⇥ shock | w | ⇧ ⌅⇤ ⌃ p shock S ⇥ ( w ) = F ( w ) F ( ) ( ⇤ t u xt x + m ) u xt F ( x , u xt ) = ( ⇤ t u xt x + m ) ˙ u xt ⇤ t F ( x , u xt ) = [ F ( w ) F ( )] = ⇥ shock | w | ⇧ ⌅⇤ ⌃ p shock S ⇥ Figure 7.
Generation of the cusp, as explained in the main text. to have a shock is given by the density ρ shock of shockstimes the size w of the window, i.e. p shock ( w ) = ρ shock | w | . (97)Let us now relate δF to the change in u ; as the spring-constant is m , δF = m δu ≡ m S. (98)Here we have introduced the avalanche size S := δu .Putting everything together yields ∆(0) − ∆( w ) = m (cid:10) S (cid:11) ρ shock | w | + O ( w ) . (99)We can still eliminate ρ shock by observing that on averagethe particle follows the spring, i.e. w = u ( w (cid:48) + w ) − u ( w (cid:48) ) = (cid:104) S (cid:105) ρ shock w. (100)This yields ρ shock = 1 (cid:104) S (cid:105) . (101)Expanding Eq. (99) in w , and retaining only the term linearin w finally yields − ∆ (cid:48) (0 + ) = m (cid:10) S (cid:11) (cid:104) S (cid:105) . (102)We just showed that having a cusp non-analyticity in ∆( u ) is a necessity if the system under consideration has shocksor avalanches. The latter are a consequence of metastability,thus metastability implies a cusp in ∆( u ) . The above toy model can be generalized to the field theory[122]. Consider an interface in a random potential, as givenby Eqs. (4)-(6) H w tot [ u ] = (cid:90) x m u ( x ) − w ] + H el [ u ] + H dis [ u ] . (103)Physically, the role of the well is to forbid the interfaceto wander off to infinity. This avoids that observables aredominated by rare events. In each sample (i.e. disorderconfiguration), and given w , one finds the minimumenergy configuration. The ground-state energy, or effectivepotential, is ˆ V ( w ) := min u ( x ) H w tot [ u ] . (104)Let us call u min w ( x ) this configuration. Its center of massposition is u w := 1 L d (cid:90) x u min w ( x ) . (105)Both ˆ V ( w ) and u w vary with w as well as from sample tosample. Let us now look at their second cumulants. Theeffective potential ˆ V ( w ) defines a function R ( w ) , R ( w − w (cid:48) ) := L − d ˆ V ( w ) ˆ V ( w (cid:48) ) c . (106)This is the same function as computed in the field theory,defined there from the zero-momentum action. The factorof volume L d is necessary, since the width u of theinterface in the well cannot grow much more than m − ζ .This means that the interface is made of roughly L/L m pieces of internal size L m ≈ m − pinned independently:Eq. (106) expresses the central limit theorem and R ( w ) measures the second cumulant of the disorder seen by heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles u , u/ for RB ∆( u ) ∆ (cid:48) ( u ) Figure 8.
Filled symbols show numerical results for ∆( u ) , a normalizedform of the interface displacement correlator − R (cid:48)(cid:48) ( u ) [Eq. (109)], for D = 2 + 1 random field (RF) and D = 3 + 1 random bond (RB)disorders. These suggest a linear cusp. The inset plots the numericalderivative ∆ (cid:48) ( u ) , with intercept ∆ (cid:48) (0 + ) ≈ − . from a quadraticfit (dashed line). The points are for confining wells with width given by m = 0 . . Comparisons to 1-loop FRG predictions (curves) are madewith no adjustable parameters. Reprinted from [123]. any one of the independent pieces. The nice thing aboutEq. (106) is that it can be measured. One varies w and computes (numerically) the new ground-state energy,finallying averaging over realizations.In fact, what is even better to measure are thefluctuations of the center-of-mass position u w , related tothe total force acting on the interface. To see this, writethe condition for the interface to be in a minimum-energyconfiguration, − δ H [ u ] δu ( x ) = ∇ u ( x ) − m [ u ( x ) − w ] + F (cid:0) x, u ( x ) (cid:1) ,F ( x, u ) = − ∂ u V ( x, u ) . (107)Integrating over space, and using periodic boundaryconditions, the term ∼ ∇ u ( x ) vanishes. At the minimum-energy configuration u min w ( x ) , this yields m ( u w − w ) = m L d (cid:90) x u min w ( x ) − w = 1 L d (cid:90) x F (cid:0) x, u min w ( x ) (cid:1) =: ˆ F ( w ) . (108)The last equation defines the effective force ˆ F ( w ) . Itssecond cumulant reads ˆ F ( w ) ˆ F ( w (cid:48) ) c ≡ m [ w − u w ][ w (cid:48) − u w (cid:48) ] c = L − d ˆ∆( w − w (cid:48) ) . (109)Taking two derivatives of Eq. (106), one verifies that theeffective correlators for potential and force are related by ∆( u ) = − R (cid:48)(cid:48) ( u ) , as in the microscopic relation (11). Eq.(102) remains valid (without an additional factor of L d ). ∆( u ) and the cusp in simulations A numerical check has been performed in Ref. [123], usinga powerful exact-minimization algorithm, which finds theground state in a time polynomial in the system size. Theresult of these measurements is presented in figure 8. Thefunction ∆( u ) is normalized to at u = 0 , and the u -axis is rescaled (to yield integral 1) to eliminate all non-universal scales. As a result, the plot is parameter free,thus what one compares is purely the shape. It has severalremarkable features. First, it clearly shows that a linearcusp exists in all dimensions. Next it is very close to the1-loop prediction. Even more remarkably the statistics isgood enough to reliably estimate the deviations from the2-loop predictions of [120], see figure 9.While we vary the position w of the center of thewell, it is not a real motion. Rather it means to find thenew ground state given w . Literally moving w is anotherinteresting possibility: It measures the universal propertiesof the so-called depinning transition , see section 3. A technical point.
A field theory is usually defined byits partition function Z [ J ] in presence of an applied field J . To obtain the effective action Γ( u ) , one evaluates thefree energy F [ J ] := − kT ln Z [ T ] , and then performsa Legendre transform from F [ J ] to Γ[ u ] . The effectiveaction, solution of the FRG flow equation, is the 2-replicaterm in Γ[ u ] , and not F [ J ] . When measuring the force-force correlations in Eq. (109), these are technically part of F [ J = m w ] . Passing from F [ J ] to Γ[ u ] is achieved byamputating the correlation function. Due to the statisticaltilt symmetry discussed in section 2.2, the latter doesnot renormalize. For the zero-mode (zero momentum)we consider, this amounts to multiplying twice with m ,resulting in the prefactor of m in Eq. (109). In thedynamics, this remains only true in the limit of a vanishingdriving velocity. These points are further discussed inRefs. [122, 123, 124, 125, 81, 126]. We have successfully applied functional renormalizationat 1-loop order. From a field theory, we demand more.Namely that it(i) be renormalizable,(ii) allows for systematic corrections beyond 1-loop order,(iii) and thus allows us to make universal predictions.This has been a puzzle since 1986, and it was evensuggested that the theory is not renormalizable due to theappearance of terms of order (cid:15) [127]. Why is the nextorder so complicated? The reason is that it involves termsproportional to R (cid:48)(cid:48)(cid:48) (0) . A look at figure 5 or 8 explains thepuzzle. Shall we use the symmetry of R ( u ) to conclude that R (cid:48)(cid:48)(cid:48) (0) is 0? Or shall we take the left-hand or right-hand heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles δ ∆( u ) ∆ simRB ( u ) (1-loop) ∆ simRB ( u ) (2-loop) ∆ simRF ( u ) (1-loop) ∆ simRF ( u ) (2-loop) u δ ∆( u ) u Figure 9.
The measured ∆( u ) with the 1-loop (red) and 2-loop corrections (blue) subtracted. Left: RB-disorder d = 2 . Right: RF-disorder d = 3 . Onesees that the 2-loop corrections improve the precision, and that the second-order correction is stronger in d = 2 than in d = 3 . derivatives, related by R (cid:48)(cid:48)(cid:48) (0 + ) := lim u> u → R (cid:48)(cid:48)(cid:48) ( u ) = − lim u< u → R (cid:48)(cid:48)(cid:48) ( u ) =: − R (cid:48)(cid:48)(cid:48) (0 − ) . (110)Below, we present the solution of this puzzle, obtained at 2-and 3-loop order. This is then extended to finite N (section2.16), compared to large N (section 2.17), and the drivendynamics (section 3).The flow-equation was first calculated at 2-loop orderwithout the anomalous terms ∼ R (cid:48)(cid:48)(cid:48) (0 + ) [128]. Thefull result with the necessary anomalous terms was firstobtained at 2-loop order [118, 120, 129, 130, 131], and laterextended to 3-loop order [40, 41]. ∂ (cid:96) ˜ R ( u ) = ( (cid:15) − ζ ) ˜ R ( u ) + ζu ˜ R (cid:48) ( u )+ 12 ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) ( u ) ˜ R (cid:48)(cid:48) (0)+( + C (cid:15) ) (cid:104) (cid:16) ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) (0) (cid:17) ˜ R (cid:48)(cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48)(cid:48) (0 + ) ˜ R (cid:48)(cid:48) ( u ) (cid:105) + C (cid:110) ˜ R (cid:48)(cid:48) ( u ) (cid:2) ˜ R (cid:48)(cid:48)(cid:48) ( u ) ˜ R (cid:48)(cid:48)(cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48)(cid:48) (0 + ) ˜ R (cid:48)(cid:48)(cid:48)(cid:48) (0 + ) (cid:3) − ˜ R (cid:48)(cid:48) (0 + ) ˜ R (cid:48)(cid:48)(cid:48) ( u ) ˜ R (cid:48)(cid:48)(cid:48)(cid:48) ( u ) (cid:111) + C (cid:2) ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) (0 + ) (cid:3) ˜ R (cid:48)(cid:48)(cid:48)(cid:48) ( u ) + C (cid:2) ˜ R (cid:48)(cid:48)(cid:48) ( u ) − R (cid:48)(cid:48)(cid:48) ( u ) ˜ R (cid:48)(cid:48)(cid:48) (0 + ) (cid:3) (111 a ) C = 136 (cid:2) π − ψ (cid:48) ( ) (cid:3) = − . , (111 b ) C = 34 ζ (3) + π − ψ (cid:48) ( )12 = 0 . , (111 c ) C = ψ (cid:48) ( )6 − π . , (111 d ) C = 2 + π − ψ (cid:48) ( )6 = 1 . . (111 e )The first line contains the rescaling terms, the second linethe result at 1-loop order, already given in Eq. (62). Thethird line is new; setting there (cid:15) = 0 is the 2-loop result.All remaining terms (porportional to C , ..., C ) are 3-loopcontributions, which we put here for completeness. Consider now the last term of the thid line, which in-volves R (cid:48)(cid:48)(cid:48) (0 + ) and which we call anomalous . The hardtask is to fix the prefactor − . There are different prescrip-tions to do this: The sloop-algorithm, recursive construc-tion, reparametrization invariance, renormalizability, poten-tiality and exact RG [120, 118, 41]. For lack of space, letus consider only renormalizability, a necessary property fora field theory. The following 2-loop diagram leads to theanomalous term R (cid:48)(cid:48)(cid:48) R (cid:48)(cid:48)(cid:48) R (cid:48)(cid:48)(cid:48) (112) −→ (cid:104)(cid:0) R (cid:48)(cid:48) ( u ) − R (cid:48)(cid:48) (0) (cid:1) R (cid:48)(cid:48)(cid:48) ( u ) − R (cid:48)(cid:48) ( u ) R (cid:48)(cid:48)(cid:48) (0 + ) (cid:105) . The momemtum integral reads = (cid:90) k (cid:90) p k + m ) p + m k + p ) + m . (113)In units where the 1-loop integral (57) is /(cid:15) , it reads [ (cid:15) ] = 12 (cid:15) + 14 (cid:15) + O ( (cid:15) ) . (114)Note that the integral (112) contains a sub-divergence,which is indicated by the red dashed box, and which yieldsthe /(cid:15) term in Eq. (114). Renormalizability demandsthat its leading divergence (which is of order /(cid:15) ) becanceled by a 1-loop counter-term. The latter is unique;it is obtained by replacing R ( u ) in the 1-loop correction δR ( u ) = R (cid:48)(cid:48) ( u ) − R (cid:48)(cid:48) ( u ) R (cid:48)(cid:48) (0) by δR ( u ) itself; the lastterm then yields δR (cid:48)(cid:48) (0) := lim u → δR (cid:48)(cid:48) ( u ) = lim u → R (cid:48)(cid:48)(cid:48) ( u ) = R (cid:48)(cid:48)(cid:48) (0 + ) . (115)This fixes the prefactor of the last (anomalous) term in thethird line of Eq. (111 a ). heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
19A physical requierment is that the disorder correlationsremain potential, i.e. that forces still derive from a potential.The force-force correlations being − R (cid:48)(cid:48) ( u ) , this means thatthe flow of R (cid:48) (0 + ) has to vanish. (The simplest way to seethis is to study a periodic potential.) From Eq. (111 a ) onecan check that this does not remain true if one changes theprefactor of the last term in the third line of Eq. (111 a ); thusfixing it. RP disorder.
Let us give results for cases of physicalinterest. First of all, for a periodic potential (RP), which isrelevant for charge-density waves, the fixed-point functioncan be calculated analytically. With the notations of Eqs.(92)-(93) this reads (with the choice of period 1, u ∈ [0 , ) R RP ( u ) = − u (1 − u ) g
24 + const. , (116 a ) ∆ RP ( u ) = g − g u (1 − u ) , (116 b ) g = (cid:15) (cid:15) (cid:15) (cid:104) π − ζ (3) − ψ (cid:48) ( ) (cid:105) + O ( (cid:15) ) . (116 c )This leads to a universal amplitude for the 2-point functionat 2-loop [118] and 3-loop order [40], ˜ u ( q )˜ u ( − q ) (cid:12)(cid:12)(cid:12) q =0 = g m d . (117)This leads to a logarithmic growth of the 2-point functionin position space. The amplitude is more complicated toextract, as one needs to extract the asymptotic behavior ofscaling functions involved in this transformation. UsingEqs. (4.13)-(4.18) of [40] , it can be written as
12 [ u ( x ) − u (0)] = gB ( d )6(4 π ) d Γ( d ) ln (cid:18) | x | L (cid:19) , (118 a ) B ( d ) = 1 + 0 . . (cid:15) . (cid:15) + O ( (cid:15) ) . (118 b ) RF disorder.
For random-field disorder, the argumentgiven in Eq. (80) is still valid, and ζ = (cid:15) remains valid,equivalent to the Flory estimate (22). The fixed-pointfunction ∆( u ) changes, and can up to 3-loop order be givenanalytically [40]. RB disorder.
For random-bond disorder (short-rangedpotential-potential correlation function) we have to solveEq. (111 a ) numerically, order by order in (cid:15) . The result is[40] ζ RB = 0 . (cid:15) + 0 . (cid:15) − . (cid:15) + O ( (cid:15) ) . (119)This compares well with numerical simulations, see figure10. It is also surprisingly close, but distinctly different, from Eq. (4.15) of [40] should read F d (0) = 1 . the Flory estimate (21), ζ = (cid:15)/ . For d = 1 ( (cid:15) = 3 ) it getsclose to the exact value [107] ζ d =1RB = 23 . (120)The fixed-point function ∆( u ) can be obtained up to 3-looporder numerically [40]. Having found a fixed point, ∂ (cid:96) ∆( u ) = β [∆]( u ) = 0 , (121)one has to ascertain that it is stable. Linear stability isanalyzed by considering infinitesimal perturbations of thefixed point δβ [∆ , z ]( u ) := dd κ β [∆ + κz ]( u ) (cid:12)(cid:12)(cid:12) κ =0 . (122)Assuming that ∆( u ) a is solution of Eq. (121), theeigenvalue equation reads δβ [∆ , z ]( u ) = − ωz ( u ) . (123)The exponent ω , if it exists, is the standard correction-to-scaling exponent [2]. The solutions to Eq. (123) depend onthe universality class.First, for the periodic fixed point (116 a ), there is adiscrete spectrum of solutions , ω − = − (cid:15), z − ( u ) = 1 . (124 a ) ω = (cid:15) − (cid:15) + 5 + 12 ζ (3)9 (cid:15) + O ( (cid:15) ) , (124 b ) z ( u ) = 1 − u (1 − u ) .ω = 4 (cid:15) − (cid:15) + 56 [13 + 12 ζ (3)] (cid:15) + O ( (cid:15) ) , (124 c ) z ( u ) = 1 − (cid:110) (cid:15) − (cid:15) (cid:104) ζ (3)+5+2 π − ψ (cid:48) ( ) (cid:105)(cid:111) u (1 − u )+ (cid:110) (cid:15) − (cid:15) (cid:104) ζ (3)+5+2 π − ψ (cid:48) ( ) (cid:105)(cid:111) [ u (1 − u )] ω = 25 (cid:15) − (cid:15) (cid:2) ζ (3)+7 (cid:3) (cid:15) + O ( (cid:15) ) (124 d )...The first solution ω − = − (cid:15) is relevant, and comeswith a constant perturbation for ∆( u ) . It is inadmissiblein equilibrium, where (cid:82) u ∆( u ) = 0 , but shows up atdepinning, see section 3. Thus the leading perturbationis z ( u ) , proportional to the fixed-point solution ∆ ∗ ( u ) itself. As the flow in this subspace can be represented bythe flow of a single coupling constant g , the β -functionmust be a polynomial in g , and at leading order it is aparabola. This parabola has two fixed points, with slope − (cid:15) at g = 0 and consequently slope (cid:15) at the non-trivial fixed As in [135] we use the high-energy-physics conventions with ω > foran IR-attractive fixed point. This is opposite to some earlier work, as theleading solution given in [40]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ζ eq d = 3 . ± . [132] d = 2 . ± . [132], . [133] d = 1 Figure 10.
Roughness exponent for random bond disorder obtained by an (cid:15) -expansion in comparison with exact results and numerical simulations. Inthe fourth column is an estimate value using a (2,1)-Pad´e approximant of the 3-loop result. point. This explains why the eigenvalue ω starts with (cid:15) ,making the fixed point stable, exactly as in scalar φ theory[2]. The following solutions z n ( u ) can be classified by theirmaximal order in [ u (1 − u )] n . One sees that the larger n ,the larger ω n . Thus the fixed point is perturbativly stable.The analysis is more difficult for the non-periodic fixedpoints, i.e. those which allow for a non-trivial exponent ζ > . The random-bond and random-field fixed pointsabove belong to this class. While a proof of stability evenfor the 1-loop fixed point is still lacking, there are twoanalytical solutions which can be given ([50], section VII): ω = 0 (125) z ( u ) = u ∆ (cid:48) ( u ) − u ) ω = (cid:15) (126) z ( u ) = ζu ∆ (cid:48) ( u ) + ( (cid:15) − ζ )∆( u ) . The first one is a redundant perturbation: it is a consequenceof the invariance of the β -function under the rescaling ∆( u ) → κ − ∆( κu ) . The dominant solution thus is ω , z ( u ) , which for ζ = 0 reduces (at leading order)to Eq. (124 b ). Subleading solutions can be constructednumerically.To conclude, we believe that all the FRG fixed pointsdiscussed above are perturbatively stable, and that theleading eigenvalue, i.e. correction-to-scaling exponent is ω = (cid:15) + O ( (cid:15) ) . Order- (cid:15) corrections depend on theuniversality class [40]. As we have seen, a cusp non-analyticitynecessary arises at zero temperature, due to the jumpsbetween metastable states. Interestingly, this cusp can berounded by several effects: By non-zero temperature
T > , chaos, or a non-zero driving velocity in the dynamics(discussed below in section 3.11). It is easy to include theeffect of temperature in the FRG equation to one loop [136].The additional 1-loop correction to R ( u ) is δR ( u ) = T (cid:20) (cid:21) = T R (cid:48)(cid:48) ( u ) × , (127) I TP := = (cid:90) k k + m . (128)The combinatorial factor is for the two ends of theinteraction, and / accompanying the second derivative; this an be checked for R ( u a − u b ) = ( u a − u b ) . The RGflow of the tadpole diagram is − m∂ m (cid:20) (cid:21) = 2 m (cid:104) (cid:105) = 2 m I , (129)with I given in Eq. (57). This leads to the β -function ∂ (cid:96) ˜ R ( u ) = ( (cid:15) − ζ ) ˜ R ( u ) + ζu ˜ R (cid:48) ( u )+ 12 ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) ( u ) ˜ R (cid:48)(cid:48) (0) + ˜ T (cid:96) ˜ R (cid:48)(cid:48) ( u ) . (130)The dimensionless temperature ˜ T (cid:96) is ˜ T (cid:96) := 2 T(cid:15) (cid:0) (cid:15)I (cid:12)(cid:12) m =1 (cid:1) m θ = 2 T(cid:15) (cid:0) (cid:15)I (cid:12)(cid:12) m =1 (cid:1) e − θ(cid:96) . (131)The power of m is obtained from Eq. (61) as the scaling of m ˜ R (cid:48)(cid:48) ( u ) , i.e. m − (cid:15) +2 ζ = m d − ζ = m θ . Although ˜ T (cid:96) finally flows to zero since θ > (see Eq. (44)), in Eq. (130)it acts as a “diffusion” term smoothening the cusp. In fact,at non-zero temperature there is no cusp, and R ( u ) remainsanalytic. The convergence to the fixed point is non-uniform.For u fixed, ˜ R ( u ) rapidly converges to the zero-temperaturefixed point, except near u = 0 , or more precisely in aboundary layer of size u ∼ ˜ T (cid:96) , which shrinks to zero inthe large-scale limit (cid:96) → ∞ , i.e. m → . Non-trivialconsequences are: The curvature blows up as R (cid:48)(cid:48)(cid:48)(cid:48) (0) ∼ e θ(cid:96) /T ∼ L θ /T . We show in section 2.20 that this is relatedto the existence of thermal excitations, or droplets in thestatics [137], and of barriers in the dynamics, which growas L θ [138]. An analytic solution for the thermal boundary layer.
Consider the flow equation (130) for RF disorder.Following section 2.4, we solve it analytically. Setting − ˜ R (cid:48)(cid:48) ( u ) ≡ ˜∆( u ) = (cid:15) κ − y t ( κu ) (132) y t (0) = 1 , ˜ T (cid:96) = (cid:15) κ − t, (133)and taking two derivatives of Eq. (130) yields in generalisa-tion of Eq. (83) ∂ u (cid:20) uy t ( u ) − ∂ u (cid:0) y t ( u ) − (cid:1) + ty (cid:48) t ( u ) (cid:21) = 0 . (134)The expression in the square brackets is a constant, fixed to by considering the limit of u → ∞ . Simplifying gives uy t ( u ) + (cid:2) t + 1 − y t ( u ) (cid:3) y (cid:48) t ( u ) = 0 . (135)Dividing by y t ( u ) and integrating once more we arrive at u (cid:0) t + 1 (cid:1) ln (cid:0) y t ( u ) (cid:1) − y t ( u ) = − . (136) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles t = t = t = t = u y t t = t = t = t = u y t T = T = ( exact ) T = ( boundary layer ) u - - Δ T ( u ) Figure 11.
Left: The RF-solution y t given in Eq. (136). Middle: the rescaled solution ˜ y t given in Eq. (139 a ). Right: Solution for the toy model. The blueline is the exact result at t = 0 ; the red dashed line is the numerical integral (147) for t = 0 . ; the green dotted line is the boundary-layer approximation(151). The integration constant was fixed by considering the limitof u → , y t → . This is an explicit analytic solution,plotted on figure 11.It is instructive to relate this to the solution at t = 0 ,which will guide us to a general finite- T approximation. Tothis aim, rewrite Eq. (136) as u t ) = − ln (cid:0) y t ( u ) (cid:1) − − y t ( u )1 + t . (137)It can be reduced to the solution y at t = 0 , by setting y t → (1 + t ) y , u → u (1 + t ) − t )(1 + t ) + 2 t As a consequence, y t ( u ) = (1 + t ) y (cid:32)(cid:114) u − t t + 2 ln(1 + t ) (cid:33) . (138)Finally using the rescaling invariance (132), we find yetanother solution of the flow equation, ˜ y t ( u ) := 11 + t (cid:48) y t (cid:48) (cid:0) u √ t (cid:48) (cid:1) (139 a ) t = t (cid:48) t (cid:48) ⇔ t (cid:48) = t − t . (139 b )Using this and the r.h.s. of Eq. (138) yields ˜ y t ( u ) = y (cid:32)(cid:114) u − tt + 1 + 2 ln( t + 1) (cid:33) ≈ y (cid:0)(cid:112) u + t (cid:1) . (140)This solution, often in the approximate form of the secondline, is commonly used in a boundary-layer analysis. Theidea of the latter is to match a solution in one range, say atsmall u , for which T ˜∆ (cid:48)(cid:48) ( u ) is large but the non-linear termsin ∂ (cid:96) ˜∆( u ) can be neglected, to a solution at large u , wherethe former can be neglected. As a consequence, lim t → lim u → t∂ u ˜ y t ( u ) = lim u → lim t → ∂ u ˜ y t ( u ) . (141)To rewrite this in terms of ˜∆( u ) is not immediate as we donot know the scale κ . However, we can derive a relation directly from the 1-loop flow equation for ˜∆( u ) , obtainedfrom Eq. (130) after taking two derivatives ∂ (cid:96) ˜∆( u ) = ( (cid:15) − ζ ) ˜∆( u ) + ζu ˜∆ (cid:48) ( u ) − ∂ u (cid:104) ˜∆( u ) − ˜∆(0) (cid:105) + ˜ T (cid:96) ˜∆ (cid:48)(cid:48) ( u ) . (142)Suppose that the fixed point is attained and the l.h.s.vanishes. Evaluating Eq. (142) once for ˜ T (cid:96) = 0 , i.e. ˜ T (cid:96) → and then u → , and once for finite ˜ T (cid:96) , where the limit u → is taken first, we obtain ( (cid:15) − ζ ) ˜∆(0) = (cid:26) ˜∆ (cid:48) (0 + ) , ˜ T (cid:96) = 0 − lim ˜ T (cid:96) → ˜ T (cid:96) ˜∆ (cid:48)(cid:48) (0) , ˜ T (cid:96) > . (143)This implies ˜∆ (cid:48) (0 + ) (cid:12)(cid:12)(cid:12) T (cid:96) =0 = − lim ˜ T (cid:96) → T (cid:96) ˜∆ (cid:48)(cid:48) (0) (cid:12)(cid:12)(cid:12) ˜ T (cid:96) > . (144)There is a large mathematics and physics literature on thesubject. Relevant keywords are boundary layer (physicsliterature) or singular perturbation theory (mathematicsliterature); a few references to start with are [139, 140, 141,142]. Check for a toy model.
Consider a particle subject toperiodic disorder (CDW, RP universality class). Supposethat the minimum of the random potential is at u = u + ni , i ∈ Z , and that this minimum is rather sharp. Then for small m , the effective potential ˆ V ( w ) is ˆ V ( w ) = − T ln (cid:32)(cid:88) i ∈ Z exp (cid:16) − ( w − i − u ) m T (cid:17)(cid:33) . (145)The effective force is ˆ F ( w ) = − ∂ w ˆ V ( w ) . (146)We need the force-force correlator, which is obtained as ˆ∆ T ( w ) = (cid:68) ˆ F ( w ) ˆ F (0) (cid:69) c = (cid:90) d u ˆ F ( w ) ˆ F (0) . (147) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ζ N N ζ ζ Figure 12.
The roughness exponent ζ as a function of the number of components N : 1 loop (blue), 2 loops (red), and a 2-loop Pad´e-(1,1) (green). For T = 0 we find at m = 1ˆ∆ ( w ) = 112 − w (1 − w ) . (148)This solution is shown in blue in Fig. 11. There is also thenumerically evaluated integral (147) (red, dashed). Let usfinally consider the FRG-equation for the rescaled disorder ˜∆ T ( w ) := m ˆ∆( w ) , ∂ (cid:96) ˜∆ T ( w ) = 4 ˜∆ T ( w ) − ∂ u (cid:104) ˜∆ T ( w ) − ˜∆ T (0) (cid:105) + 2 T m ˜∆ (cid:48)(cid:48) T ( w ) . (149)The prefactors in their order of appearance are: (cid:15) = 4 , − m∂ m ln I from the 1-loop diagram I , and − m∂ m ln I TP from the tadpole I TP defined in Eq. (128).Thus the FP equation at m = 1 is as above T ( w ) − ∂ u (cid:104) ˜∆ T ( w ) − ˜∆ T (0) (cid:105) + T (cid:48)(cid:48) T ( w ) (150)At m = 1 , ˜∆ T and ˆ∆ coincide, resulting in ˜∆ T ( w ) = ˆ∆ T ( w ) ≈ ˆ∆ (cid:32)(cid:114) w + T (cid:33) + const . (151)The constant is chosen s.t. (cid:82) d w ˆ∆ T ( w ) = 0 . Thisapproximation works quite well, see Fig. 11, right.There are complementary descriptions of the high-temperature regime [143]. When changing the disorder slightly, e.g. by varying themagnetic field in a superconductor, the system may havea different ground state at large scales, a phenomenontermed disorder chaos [144, 123]. Not all types of disorderexhibit chaos. Uusing FRG, one studies a model with twocopies, i = 1 , , each seeing a slightly different disorder V i ( x, u ( x )) in Eq. (6). The latter are mutually correlatedgaussian random potentials with correlation matrix V i ( x, u ) V j ( x (cid:48) , u (cid:48) ) = δ d ( x − x (cid:48) ) R ij ( u − u (cid:48) ) . (152) At zero temperature, the FRG equations for R ( u ) = R ( u ) are the same as in Eq. (62). The one for thecross-correlator R ( u ) satisfies equation (130), with ˜ T (cid:96) replaced by ˆ T := R (cid:48)(cid:48) (0) − R (cid:48)(cid:48) (0) . The flow of thisfictitious temperature must be determined self-consistentlyfrom FRG equations. As for a real temperature the cusp isrounded, leading to a non-trivial cross-correlation function. N Up to now, we have studied the functional RG for onecomponent N = 1 . The general case of finite N is moredifficult to handle, since derivatives of the renormalizeddisorder now depend on the direction in which thisderivative is taken. Define amplitude u := | (cid:126)u | and direction ˆ u := (cid:126)u/ | (cid:126)u | of the field. Then deriving the latter variableleads to terms proportional to /u , which are diverging inthe limit of u → . This poses additional problems in thecalculation, and it is a priori not clear that the theory at N (cid:54) = 1 exists, supposed this is the case for N = 1 . At1-loop order everything is well-defined [127]. A consistentRG-equation at 2-loop order is [145] ∂ (cid:96) ˜ R ( u ) = ( (cid:15) − ζ ) ˜ R ( u ) + ζu ˜ R (cid:48) ( u )+ 12 ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) (0) ˜ R (cid:48)(cid:48) ( u )+ N −
12 ˜ R (cid:48) ( u ) u (cid:34) ˜ R (cid:48) ( u ) u − R (cid:48)(cid:48) (0) (cid:35) + 12 (cid:104) ˜ R (cid:48)(cid:48) ( u ) − ˜ R (cid:48)(cid:48) (0) (cid:105) ˜ R (cid:48)(cid:48)(cid:48) ( u ) + N − (cid:104) ˜ R (cid:48) ( u ) − u ˜ R (cid:48)(cid:48) ( u ) (cid:105) (cid:104) R (cid:48) ( u )+ u ( ˜ R (cid:48)(cid:48) ( u ) − R (cid:48)(cid:48) (0)) (cid:105) u − ˜ R (cid:48)(cid:48)(cid:48) (0 + ) (cid:34) N + 38 ˜ R (cid:48)(cid:48) ( u ) + N −
14 ˜ R (cid:48) ( r ) u (cid:35) . (153)The first three lines are the 1-loop equation, given in [127],with the third line containing additional contributions for N (cid:54) = 1 as compared to Eq. (62). The last three lines heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles R (cid:48)(cid:48)(cid:48) (0 + ) in the last line.The fixed-point equation (153) can be integratednumerically, order by order in (cid:15) . The result, specialized todirected polymers, i.e. (cid:15) = 3 is plotted on figure 12. We seethat the 2-loop corrections are rather big at large N , so somedoubt on the applicability of the latter down to (cid:15) = 3 isadvised. However both 1- and 2-loop results reproduce wellthe two known points on the curve: ζ = 2 / for N = 1 and ζ = 0 for N = ∞ . The latter result will be given in section2.17. As discussed in section 7, the directed polymerin N dimensions treated here, and the KPZ-equation ofnon-linear surface growth in N dimensions are related,identifying z KPZ = 1 /ζ , see Eq. (773). Using the analyticsolution for the latter in dimension N = 1 , ζ N =1KPZ = 1 / (Eq. (807)), and the scaling relation z KPZ + ζ KPZ = 2 (Eq.(771)) leads to ζ N =1 d =1 = 23 . (154)The line ζ = 1 / given on figure 12 plays a specialrole: In the presence of thermal fluctuations, we expect theroughness-exponent of the directed polymer to be boundby ζ ≥ / . In the KPZ-equation, this corresponds to adynamic exponent z KPZ = 1 /ζ ≤ , which due to thescaling relation z KPZ + ζ KPZ = 2 is an upper bound in thestrong-coupling phase. The results above suggest that thereexists an upper critical dimension in the KPZ-problem, with d uc ≈ . . Even though the latter value might be anunderestimation, it is hard to imagine what can go wrong qualitatively with this scenario. The debate in the literatureis far from settled, and we summarize it in section 7.11. N In the last sections we discussed renormalization in a loopexpansion, i.e. an expansion in ε = 4 − d . In orderto check consistency, we now turn to a non-perturbativeapproach which can be solved analytically in the large- N limit. The starting point is a straightforward generalizationof Eq. (30), H [ (cid:126)u,(cid:126)j ] = 12 T n (cid:88) a =1 (cid:90) x [ (cid:126)u a ( x ) − (cid:126)w ]( m − ∇ )[ (cid:126)u a ( x ) − (cid:126)w ] − T n (cid:88) a,b =1 (cid:90) x B (cid:0) ( (cid:126)u a ( x ) − (cid:126)u b ( x )) (cid:1) , (155) B ( u ) = R ( | u | ) . (156)For large N the saddle-point equation reads [146] ˜ B (cid:48) ( w ab ) = B (cid:48) (cid:16) w ab +2 T I TP +4 I [ ˜ B (cid:48) ( w ab ) − ˜ B (cid:48) (0)] (cid:17) . (157)This equation gives the derivative of the effective (renor-malized) disorder ˜ B as a function of the (constant) back-ground field w ab = ( (cid:126)w a − (cid:126)w b ) in terms of: the derivativeof the microscopic (bare) disorder B , the temperature T andthe integrals I and I TP defined in Eqs. (57) and (128). The saddle-point equation can be turned into a closed functionalrenormalization group equation for ˜ B by taking a derivativew.r.t. m . In analogy to Eq. (62), and with the same notationused there, one obtains ∂ (cid:96) ˜ B ( x ) ≡ − m∂∂m ˜ B ( x ) = ( (cid:15) − ζ ) ˜ B ( x ) + 2 ζx ˜ B (cid:48) ( x )+ 12 ˜ B (cid:48) ( x ) − ˜ B (cid:48) ( x ) ˜ B (cid:48) (0) + (cid:15) T ˜ B (cid:48) ( x ) (cid:15) + ˜ B (cid:48)(cid:48) (0) . (158)This is a complicated nonlinear partial differential equation.It is surprising that one can find an analytic solution: Thetrick (reminding the RF-solution (86)) is to examine theflow-equation for the inverse function of y ( x ) ≡ − ˜ B (cid:48) ( x ) ,which is the dominant term at large N for the force-forcecorrelator , m∂ m x ( y ) = ( (cid:15) − ζ ) yx (cid:48) ( y ) + 2 ζx ( y ) + y − y + T m − ( (cid:15)x (cid:48) ) − . (159)Let us only give the results of this analytic solution: First ofall, for long-range correlated disorder of the form ˜ B (cid:48) ( x ) ∼ x − γ , the exponent ζ can be calculated analytically as ζ = (cid:15) γ ) . It agrees with the replica-treatment in [147], the1-loop treatment in [127], and the Flory estimate (23).For short-range correlated disorder, ζ = 0 . Second, itdemonstrates that before reaching the Larkin-length, ˜ B ( x ) is analytic and dimensional reduction holds. Beyond theLarkin length, ˜ B (cid:48)(cid:48) (0) = ∞ , a cusp appears and dimensionalreduction is broken. This shows again that the cusp is notan artifact of the perturbative expansion, but an importantproperty of the exact solution of the problem (here for large N ). /N In a graphical notation, we find [148] δB (1) = ++ + ++ T (cid:34) + ++ + + (cid:35) + T (cid:34) + + + A T (cid:35) (160) The sign of the last term in Eq. (11) of [146] must be reversed. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles = B (cid:48)(cid:48) ( χ ab ) (1 − A d I ( p ) B (cid:48)(cid:48) ( χ ab )) − , = B ( χ ab ) , (161)where χ ab is the argument of the r.h.s. of Eq. (157). Moreexplicit expressions are given in Ref. [148].By varying the IR-regulator, one can derive a β -function at order /N [148]. At T = 0 , it is UV-convergent,and should allow one to find a non-perturbative fixed point.This goal has currently only been achieved to 1-loop order[148]. Another open problem is the behavior at finite T .Since temperature is an irrelevant variable, it makes thetheory non-renormalizable, i.e. in order to define it, oneneeds to keep both an explicit infrared cutoff, and andadditional ultraviolet cutoff, for the tadpole diagram (128). One of the key methods employed in disordered systems is amethod termed replica-symmetry breaking (RSB) [60, 149,150, 151, 152, 153, 61, 154, 62, 57], sometimes referredto as
Gaussian variational ansatz , or simply mean field ,since there is no tractable scheme to go beyond that limit.It is an interesting task to confront this alternative approachto FRG. As we saw above, FRG works very well for theexperimentally most relevant case of N = 1 , whereas theRSB ansatz only holds in the limit of N → ∞ [147].So what is the idea? Ref. [147] starts from Eq. (155) but without a source-term w , i.e. without an applied field, arelevant difference. In the limit of large N , a Gaussianvariational energy of the form H g [ (cid:126)u ] = 12 T n (cid:88) a =1 (cid:90) x (cid:126)u a ( x ) (cid:0) −∇ + m (cid:1) (cid:126)u a ( x ) − T n (cid:88) a,b =1 (cid:90) x σ ab (cid:126)u a ( x ) (cid:126)u b ( x ) (162)becomes exact. The art is to make an appropriate ansatzfor σ ab . The simplest possibility, σ ab = σ for all a (cid:54) = b reproduces the dimensional reduction result, which weknow to break down at the Larkin length. Beyond thatscale, a replica symmetry broken (RSB) ansatz for σ ab issuggestive. To this aim, one breaks σ ab into four blocks ofequal size, and choses two (variationally optimized) valuesfor the diagonal and off-diagonal blocks. This is termed . One then iterates the procedure on the diagonalblocks, proceeding via a to an infinite-step RSB. Thefinal result has the form σ ab = . (163) One finds that the more often one iterates, the more stable(to perturbations) and precise the result becomes. In fact,one has to repeat this procedure infinitly many times. Thisseems like a hopeless endeavor, but Parisi has shown thatthe infinitely often replica symmetry broken matrix can beparameterized by a function [ σ ]( z ) with z ∈ [0 , . In theSK-model, z has the interpretation of an overlap betweenreplicas. While there is no such simple interpretation for themodel (162), we retain that z = 0 describes distant states,whereas z = 1 describes nearby states. The solution of thelarge- N saddle-point equations leads to the curve depictedin figure 6. Knowing it, the 2-point function is given by (cid:104) ˜ u k ˜ u − k (cid:105) = 1 k + m (cid:18) (cid:90) d zz [ σ ] ( z ) k + [ σ ] ( z ) + m (cid:19) . (164)The important question is: What is the relation betweenthe two approaches, which both declare to calculate thesame 2-point function? Comparing the analytical solutions,we find that the 2-point function given by FRG is thesame as that of RSB, if in the latter expression we onlytake into account the contribution from the most distantstates, i.e. those for z between 0 and z m (see figure13). To understand why this is so, we have to rememberthat the two calculations are done under quite differentassumptions: In contrast to the RSB-calculation, the FRG-approach calculates the partition function in presence of anexternal field w , which is then used to give via a Legendretransform the effective action. Even if the field w isfinally tuned to 0, the system remembers its preparation,as does a magnet: Preparing the system in presence of amagnetic field results in a magnetization which aligns withthis field. The magnetization remains, even if finally thefield is turned off. The same phenomenon happens here:By explicitly breaking the replica-symmetry through anapplied field, all replicas settle into distant states, and theclose states from the Parisi-function [ σ ] ( z ) + m (whichrepresent spontaneous RSB) will not contribute. However,the full RSB-result can be reconstructed by remarking thatthe part of the curve between z m and z c is independent of from UV−cutoff 1 FRG [ ]( ) + σ z m m zz c m z IR−cutoff
Figure 13.
The function [ σ ] ( u ) + m as given in [147]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles m . Integrating over m [146] then yields( m c is the mass corresponding to z c ) (cid:104) ˜ u k ˜ u − k (cid:105) (cid:12)(cid:12)(cid:12) RSB k =0 = ˜ B (cid:48) m (0) m + (cid:90) m c m d ˜ B (cid:48) µ (0) µ + 1 m c − m . (165)We also note that a similar effective action has beenproposed in [121]. While it agrees qualitatively, it does notreproduce the correct FRG 2-point function, as it should.To go further, one needs to redo the analysis ofRef. [147] in presence of an applied field, a formidabletask. A first step in this direction was taken in Ref. [121],building on the technique developed in [155]. However,the function R ( u ) defined in that work does not coincidewith the one usually studied in field theory (there isan additional Legendre transform), making a precisecomparison difficult. This goal was finally achieved inRef. [156]. In summary, there are two distinct scalingregimes, ˜ B ( w ) − ˜ B (0) = (cid:26) L − d ˜ b ( w L d ) for w ∼ L − d , (i) N b ( w /N ) for w ∼ N, (ii) . (166)(i) a “single shock” regime, w ∼ L − d where L d isthe system’s volume, and (ii) a “thermodynamic” regime,with w ∼ N , but independent of L . In regime (i)all the equivalent RSB saddle points within the Gaussianvariational approximation contribute, while in regime (ii)the effect of RSB enters only through a single anomaly.When RSB is continuous (e.g., for short-range disorder,in dimension ≤ d ≤ ), regime (ii) yields the large- N FRG function shown above. In that case, the disordercorrelator exhibits a cusp in both regimes, though withdifferent amplitudes and of different physical origin. Whenthe RSB solution is 1-step and non-marginal (e.g. in d < for SR disorder), the correlator ˜ R ( w ) = ˜ B ( w ) in regime(ii) is considerably reduced, and exhibits no cusp. The droplet picture was proposed [157, 56] for Ising spinglasses (more in [158, 159, 160, 161]), as a conceptualalternative to the Parisi solution [60, 149, 150, 151] (more in[152, 153, 61, 154, 62, 57]) of the Sherrington-Kirkpatrick(SK) model [58, 59]. Using the concept of replica-symmetry breaking (RSB), the latter yields infinitely manyextremal Gibbs states at very low temperature, organizedwithin an ultrametric topology, arrangeable as a tree. Astemperature is raised, states at increasing distance coalesceuntil a unique state remains at T = T c . Appropriate forthe infinite-range SK model, its validity for short-rangedspin glasses as the Edwards-Anderson (EA) model [102]was always disputed. The latter assumes an energy H EA = (cid:88) (cid:104) i,j (cid:105) J ij S i S j (167)with uncorrelated random couplings J ij , drawn from aprobability distribution P ( J ) . Existence of the spin-glass phase is detected by the Edwards-Anderson order parameter q EA ( T ) := (cid:104) S i (cid:105) t , (168)where the overline denotes (as above) the disorder average(over J ), and (cid:104) ... (cid:105) t the temporal average over an infinitetime. One expects q EA ( T ) = 0 for T > T c , and q EA ( T ) > for T < T c .In contrast to the RSB scenario with a finite densityof states at q = 0 , the droplet picture proposes that thelow-lying excitations which dominate the long-distance andlong-time correlations are clusters of (nested) droplets ofcoherently flipped spins. Let us denote by F the ground-state free energy, i.e. the infimum of all free energies F i , F := inf i ( F i ) . (169)In a pure system at T = 0 , the energy of a domainwall can be measured by imposing anti-periodic boundaryconditions, which force a domain wall of size L d − andenergy E L (cid:39) Υ L d − . In a disordered system at T > thisgeneralizes to F L (cid:39) Υ L θ , (170)with θ < d − , where F L is the free energy at scale L . Ifone supposes that the free energies F L in the domain wallare uncorrelated, using the central-limit theorem improvesthe bound to θ ≤ d − . The probability of dropletexcitations with free-energy differences F L , given size L ,has the scaling form ρ ( F L | L ) = ρ ( F L / Υ L θ )Υ L θ , ρ (0) > , < θ ≤ d − . (171)Values of θ satisfying the bond (171) have been reported[162].Next consider an ensemble of independent (butpossibly nested) droplets of size between L d and ( L + δL ) d .The probability that a spin is inside such a droplet isindependent of L : while the probability that a spin is insidea droplet scales as L d , the number of droplets scales as L − d .Thus the measure for integration of ρ ( F L | L ) over L is ρ ( L )d L = d LL ρ ( F L | L ) . (172)Stated differently, a given spin has a finite, L -independent,probability to be inside a droplet of size L .Only if both points i and i + r are inside a droplet,the connected spin-spin correlation function (cid:104) S i (cid:105) t (cid:104) S i + r (cid:105) t c ,is non-zero . To give a non-vanishing contribution to thespace-dependent version of the Edwards-Anderson orderparameter (168), the droplet has to be bigger than r , leadingto q EA ( r ) := (cid:104) S i (cid:105) t (cid:104) S i + r (cid:105) t c ∼ r − θ , as r → ∞ . (173) One might argue that if a domain wall is rough, then its size scales as L d f with d − ≤ d f ≤ d , and we get a weaker bound for θ . Thisis incorrect. While the minimum-energy domain wall may be larger, itsenergy is lower; otherwise the minimal-energy interface would be flat. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles L as F b (cid:39) ∆ L ψ , θ ≤ ψ ≤ d − . (174)The upper bound is given by the maximal energy needed toflip a flat wall. The lower bound insures that F L from Eq.(170) satisfies F L < F b .To address dynamical properties, suppose a dropletpersists for a time t = t e F b /T . The line of argumentsused above implies that the autocorrelation function decaysas C ( t ) := (cid:104) S i (0) S i ( t ) (cid:105) t (cid:39) q EA T Υ (cid:20) ∆ T ln( t/t ) (cid:21) θψ . (175)To conclude our excursion into the droplet picture for theEA-spin glass (for further reading see [56, 159, 160, 161,163]), let us mention an interesting result due to Bovier andFr¨ohlich [157], who analyze EA-spin glasses with conceptsfrom gauge theory. They state that for d = 2 the Gibbs stateis unique at all temperatures. In the language used here,this implies θ < . The debate whether EA-spin glasses arebetter described by the droplet picture or RSB is still raging[164, 165, 166, 167, 168, 169, 170, 171]. It is possiblethat depending on the dimension, the distance to T c , andsmall modifications of the EA spin glass energy (167), bothphases are realized in some domains of the phase diagram,while in other domains, none of them is appropriate.Let us finally apply the droplet ideas to the directedpolymer [24], and more generally disordered elasticmanifolds [172, 138, 137, 122]. First of all, the dropletexponent θ as defined in Eq. (170) is the one given in Eq.(44) θ = d − ζ. (176)It is less clear what the exponent ψ is, but there is someevidence [173, 174] that ψ = θ, (177)up to logarithmic corrections, shown to exist at least in onecase [174]. As an immediate generalization to Eq. (173),we expect that at finite temperature [172] (cid:10) [ u ( x ) − u ( y )] n (cid:11) (cid:39) T | x − y | nζ − θ . (178)To build a consistent field theory is a challenge. Tech-nically, one has to connect the thermal boundary layer of ∆( w ) (section 2.14) with the outer region. Physically, oneneeds to make the connection to the droplet picture. Thisproblem is considered in [172, 138, 137, 122]. In Ref. [177] Shigeo Kida solved the problem how a randomshort-ranged correlated velocity field decays under actionof the Burgers equation. As we discuss in sections 7.2–7.7,this is equivalent to minimizing the energy H w ( u ) := m u − w ) + V ( u ) . (179) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8)(cid:9)(cid:5)(cid:1)(cid:9)(cid:8)(cid:1)(cid:9)(cid:5) (cid:2) (cid:1) ) (cid:10) Δ (cid:1) ) Figure 14.
The force-force correlator ∆( u ) for the Kida model (bluesolid), compared to the RB 1-loop FRG result (cyan, dashed) alreadyshowon in Fig. 6 (right), rescaled to have the same value and slopesas Eq. (186). In solid red the potential-potential correlator ˜ R ( w ) , indashed the corresponding 1-loop FRG result. Inset: Numerical simulations[175, 176] for m = 10 − (red solid), m = 10 − (cyan, dashed),and m = 10 − (orange dotted), compared to the theory curve (blue).Convergence is slow. Statistical errors are within the line thickness. The random potential V ( u ) is defined for u ∈ N , andeach V is drawn from a probability distribution P ( V ) . Theeffective potential is defined as in Eq. (94). Kida’s solutionfor the ˆ V ( w ) correlations, rephrased in Refs. [178, 81] forthe minimization problem (179), is constructed in severalsteps: Define P < ( V ) := (cid:90) V −∞ P ( V (cid:48) )d V (cid:48) (cid:39) e − A ( − V ) γ for V → −∞ . (180)The characteristic u -scale is ρ m = (cid:104) m γ ln( m − ) − γ A γ (cid:105) − . (181)For a standard Gaussian distribution, A = 1 / , γ = 2 , thiscan be simplified to ρ Gauss m = 1 m (cid:2) ln( m − ) (cid:3) − . (182)The u -distribution minimizing the energy (179) is P ( u ) ≈ ρ m √ π e − ( u/ρ m ) . (183)In order to proceed, define the auxiliary function Φ( x ) := (cid:90) ∞ d y e − y + xy = (cid:114) π x (cid:104) erf (cid:0) x √ (cid:1) +1 (cid:105) . (184)The effective disorder force-force correlator ∆( w ) and R ( w ) are then given by ∆( u ) = m ρ m ˜∆( w/ρ m ) , R ( u ) = m ρ m ˜ R ( w/ρ m ) , (185) ˜∆( w ) = dd w (cid:90) ∞ w Φ( w + x ) + Φ( w − x ) d x, (186) ˜ R ( w ) = π − (cid:90) ∞ w x Φ( w − x )Φ( w + x ) + Φ( w − x ) d x. (187) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3) (cid:5)(cid:2)(cid:1) (cid:5)(cid:2)(cid:3) (cid:6)(cid:2)(cid:1) (cid:1) (cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7) Δ ( (cid:1) ) (cid:8) (cid:2) ( (cid:1) )- (cid:2) ( (cid:1) ) Figure 15. ˜∆( w ) for the Sinai model (blue), obtained by numericalintegration of Eq. (197). In cyan dashed the solution (82)-(86) of the 1-loop FRG equation, rescaled to the same value and slope at w = 0 . In red,dotted ˜ R (0) − ˜ R ( w ) . Inset: numerical simulation [175, 176] for m =10 − (red, solid), and m = 10 − (cyan, dashed), indistinguishable fromthe theory (blue solid). Statistical errors are within the line thickness. The solutions ˜∆( w ) and ˜ R ( w ) are compared in Fig. 14 tonumerical simulations, and to the fixed point obtained bysolving the 1-loop FRG equation (77), for ζ = ζ RB , Eq.(88). For reference we give for the Kida-solution ˜∆(0) = 1 , ˜∆ (cid:48) (0 + ) = − √ π , ˜∆ (cid:48)(cid:48) (0 + ) = 3 π − . (188)Using Eq. (102), this predicts, among others, the universalavalanche scale, S m := (cid:10) S (cid:11) (cid:104) S (cid:105) = 2 √ π ρ m . (189) In 1983 Y.G. Sinai asked: Consider a random walk X n ,which with probability p n increases by 1 in step n , andwith probability − p n decreases by 1, assuming the p n ∈ [0 , themselves to be (quenched) independent randomvariables . Sinai showed [179] that contrary to a normalrandom walk, which has variance n after n steps, theprocess X n defined above has a variance which grows as ln ( n ) . Interpreting the p n as random field disorder, Sinai’stheorem shows that the walk is localized even at finitetemperature.Let us consider again the model defined in Eq. (179),but with a potential which itself is as a random walk, H w ( u ) := m u − w ) + V ( u ) , (190) V ( u ) = − (cid:90) u F ( u (cid:48) ) d u (cid:48) , (191) F ( u ) F ( u (cid:48) ) = σδ ( u − u (cid:48) ) . (192)Thus
12 [ V ( u ) − V ( u (cid:48) )] = σ | u − u (cid:48) | . (193)Using the methods developed in Ref. [180], therenormalized disorder correlator ∆( w ) for the Sinai model has been obtained in Ref. [81]. Here we give a simplifiedversion : ∆( w ) = m ρ m ˜∆( w/ρ m ) , (194) R ( w ) = m ρ m ˜ R ( w/ρ m ) , (195) ρ m = 2 m − σ . (196)The effective disorder correlator reads ˜∆( w ) = − e − w π √ w ∞ (cid:90) −∞ d λ ∞ (cid:90) −∞ d λ e − ( λ − λ w × e i w ( λ + λ ) Ai (cid:48) ( iλ ) Ai ( iλ ) Ai (cid:48) ( iλ ) Ai ( iλ ) × (cid:20) w (cid:82) ∞ d V e wV Ai ( iλ + V ) Ai ( iλ + V ) Ai ( iλ ) Ai ( iλ ) (cid:21) . (197)For reference we give ˜∆(0) ≈ . , ˜∆ (cid:48) (0 + ) ≈ − . , ˜∆ (cid:48)(cid:48) (0 + ) ≈ . . (198)Using Eq. (102), this predicts, among others, the universalavalanche scale, S m := (cid:10) S (cid:11) (cid:104) S (cid:105) = 0 . ρ m . (199)One can also give an explicit formula for the potential-potential correlator ˜ R ( w )˜ R ( w ) = − √ w e − w π ∞ (cid:90) −∞ d λ ∞ (cid:90) −∞ d λ e − ( λ − λ w × e i w ( λ + λ ) Ai ( iλ ) Ai ( iλ ) (cid:20) − ( λ − λ ) w (cid:21) × (cid:20) w (cid:82) ∞ d V e w V Ai ( iλ + V ) Ai ( iλ + V ) Ai ( iλ ) Ai ( iλ ) (cid:21) + ˜ R (0) . (200)We checked numerically that ˜∆( w ) = ˜ R (cid:48)(cid:48) ( w ) . We find that lim w →∞ ˜ R (0) − ˜ R ( w ) − w . , (201) lim w →∞ − R (cid:48) ( w ) = m ρ m σ. (202)The latter is a consequence of the FRG equation: it cannotrenormalize the tail of R ( w ) , given for the microscopicdisorder in Eq. (193). Finally note that if in the squarebrackets of the second line of Eq. (200) one only retains the“ ”, then the dominant term w/ of Eq. (201) is obtained.A plot, a comparison to the 1-loop FRG fixed point(77), and a numerical verification are presented on Fig. 15. We simplify the result of [81] such that the only appearance of a =2 − / or b = 2 / is in the scale ρ m of Eq. (196). We further correctseveral misprints: First, the formulas given for a and b in [81] can onlybe used for m = σ = 1 , or one would have to rescale the term ∼ w inthe exponential as well. The factor of w in the innermost integral for ∆ ismissing in Eq. (304) of [81], while it is there in Eq. (293) for R . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles - ϵ ln ( ρ ( ϵ )) Figure 16.
The log of the density of states ln ρ ( (cid:15) ) for the REM, n = 23 . The maximal and minimal energies in this sample are . and − . , compared to (cid:15) ∗ = 0 . : The histogram vanishes for | (cid:15) | ≥ (cid:15) ∗ . The random-energy model (REM) was introduced byB. Derrida in 1980 [55, 181] as an exactly soluble, albeitextreme simplification of a spin glass. It was further studiedin [182, 183, 184]. Consider an Ising model with N spins.It can assume N distinct configurations, labeled by i =1 , ..., N . Suppose that the energy E i of each state i is takenfrom a Gaussian distribution P ( E ) = 1 √ πN e − E /N . (203)Thus (cid:104) E (cid:105) = 0 , and (cid:10) E (cid:11) = N/ . The factor of N is chosento obtain a non-trivial limit for N → ∞ below.A sample of the REM consists of N random energies E i drawn from Eq. (203). The partition function attemperature T = 1 /β , and the occupation probabilities are Z = N (cid:88) i =1 e − βE i , p i = e − βE i Z . (204)Consider the number N ( (cid:15), (cid:15) + δ ) of states in the interval [ N (cid:15), N ( (cid:15) + δ )] . Setting I (cid:15) := (cid:90) N ( (cid:15) + δ ) N(cid:15) P ( x ) d x ≡ (cid:114) Nπ (cid:90) (cid:15) + δ(cid:15) e − y d y (205)the expectation and variance of N ( (cid:15), (cid:15) + δ ) are (cid:104)N ( (cid:15), (cid:15) + δ ) (cid:105) = 2 N I (cid:15) , (206) (cid:10) N ( (cid:15), (cid:15) + δ ) (cid:11) c = 2 N I (cid:15) (1 − I (cid:15) ) . (207)This allows us to write the density of states ρ ( (cid:15) ) (cid:39) δ N ( (cid:15), (cid:15) + δ ) as ln ρ ( (cid:15) ) (cid:39) N (cid:2) ln(2) − (cid:15) (cid:3) + 12 ln (cid:18) Nπ (cid:19) . (208)Define (cid:15) ∗ s.t. ln ρ ( (cid:15) ∗ ) = 0 ⇐⇒ (cid:15) ∗ = √ ln 2 + ln( N/π )4 N √ ln 2 + ... (209) It is instructive to run a simulation: as can be seenon Fig. 16, there is not a single state for | (cid:15) | ≥ (cid:15) ∗ .Second, according to Eqs. (206)-(207), relative fluctuationsare suppressed, (cid:10) N ( (cid:15), (cid:15) + δ ) (cid:11) c (cid:104)N ( (cid:15), (cid:15) + δ ) (cid:105) = 2 − N − I (cid:15) I (cid:15) . (210)To good precision, one can therefore approximate at leadingorder in /N ln ρ ( (cid:15) ) = N s ( (cid:15) ) , s ( (cid:15) ) = (cid:26) ln(2) − (cid:15) , | (cid:15) | ≤ (cid:15) ∗ −∞ , | (cid:15) | > (cid:15) ∗ . (211)The quantity s ( (cid:15) ) is interpreted as the entropy of thesystem; s ( (cid:15) ) = −∞ means that the corresponding density ρ ( (cid:15) ) vanishes. Note that the factor of N in Eq. (203) isintroduced in order to render thermodynamic quantities asEq. (211) extensive, i.e. ∼ N .We now proceed to other thermodynamic properties,notably the free energy e − βNf ( (cid:15) ) = Z (cid:39) (cid:90) (cid:15) ∗ − (cid:15) ∗ d (cid:15) e − N [ β(cid:15) − s ( (cid:15) )] , (212)equivalent to f ( (cid:15) ) (cid:39) min (cid:15) ∈ [ − (cid:15) ∗ ,(cid:15) ∗ ] (cid:104) (cid:15) − s ( (cid:15) ) β (cid:105) . (213)Note that this is a Legendre transform, typical ofthermodynamic arguments; the restriction of (cid:15) to [ − (cid:15) ∗ , (cid:15) ∗ ] is implicit in the definition (212), and allows us to use theparabolic form valid in that domain. An explicit calculationyields f ( (cid:15) ) = (cid:40) − β − ln(2) β , β ≤ β c − (cid:112) ln(2) , β > β c , β c = 2 (cid:112) ln(2) . (214)Next define the participation ratio inspired by spin glasses[185, 62] Y ≡ Y ( β ) = (cid:80) i e − βE i (cid:104) (cid:80) i e − βE i (cid:105) . (215)It was shown [182] that all moments can be calculatedanalytically (see Eqs. (10)–(11) of [182]) g ( µ ) = (cid:90) ∞ (1 − e − u − µu ) u − TT c − d u, (216) (cid:104) Y n (cid:105) = ( − n +1 Γ(2 n ) T c T d n ln g ( µ )d µ n (cid:12)(cid:12)(cid:12) µ =0 . (217)The first moments read (cid:104) Y (cid:105) = 1 − TT c , (cid:10) Y (cid:11) c = 13 TT c (cid:18) − TT c (cid:19) . (218)Thus Y is a random variable, with rather large, non-selfaveraging fluctuations.Finally, one can calculate the partition function inpresence of a magnetic field [184], by generalizing itsdefinition to Z ( h ) = N (cid:88) i =1 e − βE i − βM i h , (219) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles w `- D H w ` Lê b m ∆ ZZ ∗ ∆ ZZ w w ` D q H w ` Lê b m ∆ θ w - D H w L ∆ w Figure 17.
Phase portrait of the model, following the conventions ofRef. [186]. The horizontal axis is the strength of V , the vertical axis thestrength of θ . The effective disorder correlators for points A (deep in thediffusive phase), B (deep in the pinned phase), C (infinitesimally smalldisorder) are shown as well. Symbols are from numerical simulations. where M i is the total magnetization of the sample (thenumber of up spins minus the number of down spins). Asthe energies E i are independent of the spin configuration σ i , and its total magnetization M i , the expectation value of Z ( h ) factorizes, (cid:104)Z ( h ) (cid:105) = (cid:104)Z (cid:105) × (cid:10) e − βhM (cid:11) σ . (220)Using this factorization property (which also holds for SK),the partition function for two copies reads (cid:104)Z ( h ) Z ( h ) (cid:105)Z = (cid:88) i = j Z (cid:68) e − β [2 E i +( h + h ) M i ] (cid:69) + 1 Z (cid:88) i (cid:54) = j (cid:68) e − β [ E i + E j + h M i h + M j ] (cid:69) = (cid:104) Y (cid:105) (cid:68) e − β ( h + h ) M (cid:69) σ + (cid:0) − − N (cid:1) (cid:10) e − βh M (cid:11) σ (cid:10) e − βh M (cid:11) σ . (221)The average over spin configurations factorizes, (cid:10) e − βhM (cid:11) = cosh( βh ) N . (222)One sees: in the high-temperature phase where (cid:104) Y (cid:105) vanishes, the partition function factorizes. In the low-temperature phase, the first term dominates for h h > ,whereas the second does for h h < . This behavior leadsto a non-analyticity of the effective action for T < T c . Werefer to [184] for details. FRG was used with success to describe thestatistics of elastic objects (this section 2) and depinning(next section 3), subjected to quenched real disorder. An interesting question is whether it can also be appliedto systems with complex disorder, relevant in quantummechanics. There is one study for quantum creep [187], butwhat we are after is a situation where interference becomesimportant.Let us give a specific example. The hoppingconductivity of electrons in disordered insulators in thestrongly localized regime is described by the Nguyen-Spivak-Shklovskii (NSS) model [188]. The probabilityamplitude J ( a, b ) is the sum over interfering directed paths Γ from a to b [189, 190, 191, 192, 193, 194] J ( a, b ) := (cid:88) Γ (cid:89) j ∈ Γ η j . (223)The conductivity between sites a and b is given by g ( a, b ) = | J ( a, b ) | . Each lattice site j contributes a random sign η j = ± (or, more generally a complex phase η j = e iθ j ).Another example is the Chalker-Coddington model[195] for the quantum Hall (and spin quantum Hall) effect,where the transmission matrix J between two contacts a and b is given by [196, 197] J ( a, b ) = (cid:88) Γ (cid:89) ( i,j ) ∈ Γ S ( i,j ) . (224)The random variables S ( i,j ) on every bond ( i, j ) are U ( N ) matrices, with N = 1 for the charge quantum Hall effectand N = 2 for the spin quantum Hall effect. Γ arepaths subject to some rules imposed at the vertices. Theconductance is given by g ( a, b ) = tr (cid:0) J ( a, b ) † J ( a, b ) (cid:1) .In both models, one would like to understand thedominating contributions to the sum Z over paths withrandom weights J ( a, b ) . In contrast to the thermodynamicsof classical models, where all contributions are positive,contributions between paths with different phases cancancel. One is interested in the expected phase transitions. Definitions.
This is a complicated problem. In Refs. [198,199, 200, 201] simplified models were considered. Here weconsider the toy model of Ref. [201], which allows one todefine the central objects of the FRG. The partition functionat finite T = β − reads in generalization of Eq. (94) Z ( w ) = (cid:114) βm π ∞ (cid:90) −∞ d x e − β (cid:2) V ( x )+ m ( x − w ) (cid:3) − iθ ( x ) =: e − β ˆ V ( w ) − i ˆ θ ( w ) . (225)One wishes to study the correlations ∆ V ( w − w ) := ˆ V (cid:48) ( w ) ˆ V (cid:48) ( w ) , (226) ∆ θ ( w − w ) := ˆ θ (cid:48) ( w )ˆ θ (cid:48) ( w ) . (227)They are related to the correlations of ∂ w Z ( w ) and ∂ w Z ∗ ( w ) by ∆ ZZ ∗ ( w ) = ∆ V ( w ) + β − ∆ θ ( w ) , (228) ∆ ZZ ( w ) = ∆ V ( w ) − β − ∆ θ ( w ) . (229) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Results.
As established by Derrida in [186], there arethree phases: high-temperature phase I, frozen phase II,and strong-interference phase III. This is depicted in thecenter of Fig. 17, accompanied by their specific correlationfunctions [201].
Phase I:
For weak disorder (perturbative regime) one canevaluate the integral (225) by Taylor expanding to leadingorder in both V and θ , to obtain ∆ V ( w ) ∼ ∆ θ ( w ) ∼ − ∂ w e − m w . (230) Phase II:
This phase may be seen as a deformation of thelocalized phase, encountered for short-ranged disorder inthe Kida model (section 2.21), or for long-ranged disorderin the Sinai model (section 2.22). The key change is adeformation of the shocks, which shows up in an additionallogarithmic deformation of the force-force correlator insidea boundary lawyer of sizes w ∼ T , see Fig. 17. Phase III:
Here V ( u ) = 0 . This phase is the one mostclosely related to the NSS or Chalker-Coddington models.Contrary to the Kida or Sinai models which lead to alocalization of the path (the partition function is dominatedby a minimizing path) fluctuations of Z are important,and zero-crossings are observed. They seem to be ratherindependent of the nature of the θ -disorder, which weattribute to θ being a compact variable. The zero crossingslead to a logarithmic divergence of the correlation functions(226) to (229), see Fig. 17. When the magnetic field H is increased in a puresuperconductor, there are two phase transitions: For lowmagnetic fields, H < H c1 ( T ) , vortices are expelled due tothe Meissner effect [203] (red region in Fig. 18). For H >H c1 ( T ) flux-vortices appear. Increasing the magnetic fieldfurther, superconductivity breaks down for H > H c2 ( T ) (white region in Fig. 18).Let us now consider a dirty magnet. There hasbeen a long debate whether the vortex lattice present for H c1 ( T ) < H c1 ( T ) < H c2 ( T ) can be described by aBragg glass, or a vortex glass. In a Bragg glass favoredby [204, 205, 21, 206], the Abrikosov lattice of vortices(see sketch in Fig. 3) is elastically deformed, but thereare no topological defects and each vortex line retains sixnearest neighbors. According to the theory of disorderedelastic manifolds, the correlation function given in Eq.(118 a ) grows logarithmically with distance, preservingenough coherence to show up in a Bragg peak in neutronscattering experiments (hence the name). The alternativetheory, termed vortex glass and favored (for sometimesquite different reasons) in [207, 208, 209, 210, 211, 212,213, 214], assumes that topological defects destroy the Figure 18.
Phase diagram for vortices in a type-II superconductor [202].(Figure reproduced with kind permission from the authors). From bottomto top these are the Meissner, vortex-free region (red), followed by theBragg glass (orange) and a vortex glass (green and blue), above whichsuper-conductivity cedes (white). order, and as a consequence the Bragg peak in neutronscattering experiments.The current experimental situation [202] shown inFig. 18 favors, in agreement with intuition, a Bragg glassfor smaller fields, and a vortex glass for larger ones, witha transition at H cB / V ( T ) , with H c1 ( T ) < H cB / V ( T ) Consider a Majorana (real) fermion,constructed from anti-commuting Grassman fields (section8.2) H Majorana = 12 π (cid:90) d (cid:126)z (cid:2) ¯ ψ (¯ z ) ∂ ¯ ψ (¯ z ) + ψ ( z ) ¯ ∂ψ ( z ) (cid:3) . (236)Its correlation-functions are obtained from Eq. (981) as (cid:104) ψ ( z ) ψ ( w ) (cid:105) = 1 z − w , (237) (cid:10) ¯ ψ (¯ z ) ¯ ψ ( ¯ w ) (cid:11) = 1¯ z − ¯ w . (238)As for the bosons above, the theory can be split into aholomorphic and an antiholomorphic part. Dirac fermion. A Dirac (complex) fermion is made out oftwo Majorana-fermions ψ = ψ ∗ and ψ = ψ ∗ , ψ = ψ + iψ , ψ ∗ = ψ − iψ . (239)Corresponding rules hold for the antiholomorphic fields.Chosing H Dirac = 1 π (cid:90) z ¯ ψ ∗ (¯ z ) ∂ ¯ ψ (¯ z ) + ψ ∗ ( z ) ¯ ∂ψ ( z ) , (240)the correlation functions for the components ψ and ψ have an additional factor of / , resulting in (cid:104) ψ ∗ ( z ) ψ ( w ) (cid:105) = (cid:104) ψ ( z ) ψ ∗ ( w ) (cid:105) = 1 z − w , (241) (cid:104) ψ ( z ) ψ ( w ) (cid:105) = (cid:104) ψ ∗ ( z ) ψ ∗ ( w ) (cid:105) = 0 . (242)Similar relations hold for the anti-holomorphic parts ¯ ψ , andcorrelations vanish between ψ and ¯ ψ . Bosonization. Since a Dirac-fermion and a free bosonhave both central charge c = 1 , we may expect a closerrelation between objects in these theories. Indeed setting The conformal charge c is the amplitude of the leading term in the OPEof the stress-energy tensor with itself. It can be measured from the finite-size corrections of the free energy of a system [219, 220, 221, 3]. (the dots indicate normal ordering, i.e. exclusion of self-contractions at the vertex) ψ ( z ) ˆ= :e iφ ( z ) : ψ ∗ ( z ) ˆ= :e − iφ ( z ) : (243) ¯ ψ (¯ z ) ˆ= :e i ¯ φ (¯ z ) : ¯ ψ ∗ (¯ z ) ˆ= :e − i ¯ φ (¯ z ) : (244)the (diverging part of the) fermion correlation functions arereproduced within the bosonic theory. Products of fermionoperators are obtained from the point-splitted product [ ψ ∗ ψ ] ( z ) := lim z → w ψ ∗ ( w ) ψ ( z ) = :e − iφ ( w ) e iφ ( z ) : 1 w − z = 1 i ∂φ ( z ) . (245)This rule allows one to decouple the 4-fermion interactionas ¯ ψ ∗ (¯ z ) ¯ ψ (¯ z ) ψ ∗ ( z ) ψ ( z ) ˆ= ¯ ∂φ (¯ z ) ∂φ ( z ) . (246)Note that since there are two complex fermions, the onlynon-vanishing combination one can write down is the onegiven in Eq. (246). Introductory texts on bosonisationtechniques can be found in [222, 223]. Thirring and sine-Gordon model. The Thirring model introduced in Ref. [224] is the most general 2-dimensionalmodel with two independent families of fermions, and akinetic term with a single derivative. S Thirring = 1 π (cid:90) z (cid:110) ¯ ψ ∗ (¯ z ) ∂ ¯ ψ (¯ z ) + ψ ∗ ( z ) ¯ ∂ψ ( z )+ λ (cid:104) ¯ ψ (¯ z ) ψ ( z ) + ¯ ψ ∗ (¯ z ) ψ ∗ ( z ) (cid:105) + g ψ ∗ (¯ z ) ¯ ψ (¯ z ) ψ ∗ ( z ) ψ ( z ) (cid:111) . (247)Using the dictionary provided by Eqs. (243), (244) and(246) yields the sine-Gordon model [225] H SG = (cid:90) d (cid:126)z g π (cid:2) ∇ Φ( (cid:126)z ) (cid:3) + λπ cos (cid:0) Φ( (cid:126)z ) (cid:1) . (248) The sine-Gordon model can be treated in a perturbativeexpansion in λ . The leading order correction comes atsecond order, and corrects g . Noting T := 11 + g , (249)it can be written as (cid:18) λ π (cid:19) (cid:90) x,y :e i Φ( x ) : :e − i Φ( y ) := (cid:18) λ π (cid:19) (cid:90) x,y :e i [Φ( x ) − Φ( y )] : | x − y | − T Combinatorial factors are obtained from cos(Φ) = (e i Φ + e − i Φ ) ,leaving two combinations with overall charge neutrality, which cancelagainst the / from the expansion of e −H SG . The dots denote normal-ordering, the change in λ being absorbed therein. Vector notationis suppressed for simplicity of notation. For an introduction into thetechnique see [110]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:1) (cid:1) (cid:1) λ Figure 19. Flow diagram of the Kosterlitz-Thouless transition: Alltrajectories starting from the green line are attracted to ˜ λ = 0 , and ˜ T > ˜ T c (line of fixed points), the remaining ones to the strong-coupling regimewith ˜ λ (cid:29) . (cid:39) (cid:18) λ π (cid:19) (cid:90) x,y (cid:110) − 12 :[Φ( x ) − Φ( y )] : + ... (cid:111) | x − y | − T (cid:39) (cid:18) λ π (cid:19) (cid:90) x,y (cid:110) − 12 : (cid:104) ( x − y ) ∇ Φ( x + y ) (cid:105) + ... (cid:111) × | x − y | − T (cid:39) (cid:18) λ π (cid:19) (cid:90) x,y (cid:110) − ( x − y ) (cid:104) ∇ Φ( x + y ) (cid:105) + ... (cid:111) × | x − y | − T . (250)The first term yields a correction to the free energy; itnecessitates a relevant counter term (UV divergent, IRfinite), but does not enter into IR properties of the theory.The second term is a correction to g , δg = λ (cid:90) L d zz z − T = λ L − T − T . (251)Defining the dimensionless couplings as ˜ g ≡ g eff = g + δg , ˜ λ := λ eff L − T , with λ eff = λ + O ( λ ) , one obtains the β functions β ˜ g (˜ λ, ˜ g ) = ˜ λ + ... (252 a ) β ˜ λ (˜ λ, ˜ g ) = (2 − T )˜ λ + ... = (2 − g )˜ λ + ... (252 b )Subdominant corrections are down by a factor of ˜ λ .Rewritten in terms of ˜ λ and ˜ T = 1 / (1 + ˜ g ) , this yields β ˜ T ( ˜ T , ˜ λ ) = − ˜ T λ + ... = − ˜ T λ + ... (253 a ) β ˜ λ ( ˜ T , ˜ λ ) = ( ˜ T c − ˜ T )˜ λ + ... (253 b )The reader will mostly see these equations expanded around ˜ T c = 2 , as done above. The schematic flow chart is shownon Fig. 19.There is a line of fixed points for ˜ T > ˜ T c (red onFig. 19). All these fixed points have ˜ λ = 0 , thus areGaussian theories. Below ˜ T c , the flow is to strong coupling. Definition of the β functions are as in section 2.1. Eqs. (247) and (248)were tuned to have the simplest coefficients later. Physically, e i Φ and e − i Φ are interpreted as vortices andanti-vortices, topological defects with charge ± , chemicalpotential λ , interacting via Coulomb interactions. For ˜ T > ˜ T c they are bound, and only a finite number is present.For ˜ T < ˜ T c they are unbound, gaining enough entropy toovercome the energetic costs for their core. The transitionat ˜ T = ˜ T c is known as the Kosterlitz-Thouless transition[226].Higher-loop calculations can be performed, both inthe Thirring model (236), as in the sine-Gordon model(248). As our above treatment has shown, they are ratherstraight-forward, and one should be able to go at least to 3-loop order in sine-Gordon, and to even higher order in thefermionic model. It is therefore surprising to read aboutmassive contradictions in the literature already at 2-looporder [227], confirming the original 1980 result of [228],but declaring later calculations in [229, 230, 231] as well as[232] to be incorrect.What the RG approach cannot reach is the strong-disorder fixed point ˜ λ (cid:29) . The latter has been studiedin the Wegner flow-equation approach [233]. The sine-Gordon model (248) with quenched disordercoupling to e i Φ reads (writing (cid:126)z → z ) H rpSG = (cid:90) z (cid:2) ∇ Φ( z ) (cid:3) πT + ξ ( z ) :e i Φ( z ) : + ξ ∗ ( z ) :e − i Φ( z ) : ξ ( z ) ξ ∗ ( z (cid:48) ) = λ π δ ( z − z (cid:48) ) . (254)After replication (see section 1.5), the effective action reads S rpSG = (cid:90) z (cid:88) α [ ∇ Φ α ( z )] πT − λ π (cid:90) z (cid:88) α (cid:54) = β :e i [Φ α ( z ) − Φ β ( z )] : − σ π (cid:90) z (cid:88) α (cid:54) = β ∇ Φ α ( z ) ∇ Φ β ( z ) . (255)We added an additional off-diagonal term since it isgenerated under RG; we will see this shortly.Perturbation theory is performed with (cid:104) Φ α ( x )Φ β ( y ) (cid:105) = − T δ αβ ln (cid:0) | x − y | (cid:1) . (256) First diagram (one loop). We use the graphical notation α β = :e i [Φ α ( x ) − Φ β ( x )] : (257)The contributions to the effective action are δS i ≡ (cid:82) x δs i .The first one is (an ellipse encloses the same replica) − δs ( x ) = xy = 12! (cid:18) λ π (cid:19) (cid:88) α (cid:54) = β (cid:90) y : e i [Φ α ( x ) − Φ β ( x ) − Φ α ( y )+Φ β ( y )] : × e − T ln | x − y | . (258) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1) λ Figure 20. The RG flow for ˜ λ as a function of T . This term contains a strongly UV-divergent contributionto the free energy (which we do not need) and the sub-dominant term − δs ≈ − (cid:18) λ π (cid:19) (cid:88) α (cid:54) = β (cid:90) y | x − y | − T ×× : [( x − y ) · ∇ Φ α ( x ) − ( x − y ) · ∇ Φ β ( x )] := − (cid:18) λ π (cid:19) (cid:88) α (cid:54) = β (cid:90) y | x − y | − T ×× : [ ∇ Φ α ( x ) − ∇ Φ β ( x )] : (259)It corrects σ , δσ = 12 λ × I , (260 a ) I = 12 π (cid:90) d y | y | − T Θ( | y | < L ) = L τ τ , (260 b ) τ := 1 − T. (260 c ) Second diagram (one loop). − δs == 22! (cid:16) λ π (cid:17) (cid:88) α (cid:54) = β (cid:54) = γ (cid:90) y :e i [Φ α ( x ) − Φ γ ( y )] :: e − i [Φ β ( x ) − θ β ( y )] : × e − T ln | x − y | . (261)Projecting onto the interaction yields − δs ≈ (cid:16) λ π (cid:17) × ( n − (cid:88) α (cid:54) = γ :e i [Φ α ( x ) − Φ γ ( x )] : I , (262) I = 12 π (cid:90) d y | y | − T Θ( | y | < L ) = L τ τ . (263)Setting the number of replicas n → , we get δλ = − λ × I . (264)Defining the β -functions as the variation with respect tothe large-scale cutoff L , keeping the bare coupling λ , oneobtains after some algebra [234][235][236][237] T = 0 T = T c Figure 21. The amplitude A ( τ ) , characterizing the super-rough phase(Fig. from [238]). The squares are numerical estimates using the algorithmof [239]. “one loop” indicates the 1-loop result A ( τ ) = 2 τ while “twoloop” refers to Eq. (271), (A Pad´e resummation of it is shown as well). A ff is the result of Ref. [240]. We also show values obtained numerically at T = 0 in the corresponding references. β ˜ λ (˜ λ ) := L ∂∂L ˜ λ = 2 τ ˜ λ − λ + ˜ λ + O (˜ λ ) , (265 a ) β σ (˜ λ ) := L ∂∂L σ = 12 ˜ λ + O (˜ λ ) . (265 b )The additional 2-loop coefficients are obtained in Ref. [241].What are the physical consequences of these β -functions?First of all, there is a non-trivial fixed point for ˜ λ at ˜ λ c = τ + 12 τ + O ( τ ) . (266)Second, integrating the β -function for σ , starting at amicroscopic scale a , yields σ = 12 ˜ λ ln (cid:16) La (cid:17) + ... = (cid:20) τ τ O ( τ ) (cid:21) ln (cid:16) La (cid:17) , (267)equivalent to σ ( k ) (cid:39) − (cid:20) τ τ O ( τ ) (cid:21) ln( ak ) . (268)This allows us to obtain the k dependent 2-point function as (cid:68) ˜Φ ( k ) ˜Φ ( − k ) (cid:69) = (cid:18) πTk (cid:19) σ ( k ) k π . (269)As a consequence , (cid:104) Φ( x ) − Φ(0) (cid:105) = A ln( x/a ) + O (cid:0) ln( x/a ) (cid:1) , (270) A = 2(1 − τ ) (cid:2) τ + τ + O ( τ ) (cid:3) = 2 τ − τ + O ( τ ) . (271) Intermediate steps are (cid:104) Φ ( x )Φ (0) (cid:105) − (cid:104) Φ ( x )Φ (0) (cid:105) = (cid:90) d k (2 π ) πTk e ikx = − T ln | x/a | . (cid:104) Φ ( x )Φ (0) (cid:105) = (cid:90) d k (2 π ) (cid:18) πTk (cid:19) σ ( k ) k π e ikx = − (cid:104) τ + τ + O ( τ ) (cid:105) T ln | x/a | . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles 34A numerical test [238] using a combinatorial algorithmgrowing polynomial in system size (the concept behind thisachievement is discussed in section 2.30) are shown onFig. 21.The result (271) was obtained in a perturbativeexpansion in T − T c . Even if one could calculate thefollowing orders, and resum them properly, one wonderswhether the expansion remains correct down to T = 0 .This is unlikely: we know from the (cid:15) -expansion, see Eq.(92), that the fixed-point at T = 0 has to all orders in (cid:15) theform (with C a constant) ∆(Φ) = C (cid:20) π − Φ(2 π − Φ) (cid:21) ≡ C ∞ (cid:88) q =1 cos( q Φ) q . (272)It contains an infinity of subdominant modes indexed by q , which one can try to incorporate into the perturbativeresult. Naively one expects the mode q to show up at T c ( q ) = T c /q , and it is likely to increase the perturbativeresult. Attempts to do so have ben undertaken in Refs. [242,243, 244, 245]. Consider an observable O ( (cid:96) ) such as the field differencebetween two points a distance (cid:96) apart. Its n -th momentreads (cid:104)O ( (cid:96) ) n (cid:105) ∼ (cid:96) ζ n . (273)Generically there are two possibilities(i) ζ n = nζ : fractal(ii) ζ n (cid:54) = nζ : multifractalIn some cases, e.g. in the critical dimension, one has (cid:104)O ( (cid:96) ) n (cid:105) ∼ (cid:96) ζ n = e ζ n ln( (cid:96) ) → ζ n ln( (cid:96) ) , (274)i.e. the universal anomalous dimension appears as theamplitude of the log, see e.g. Eq. (118 a ) and section 3.8.We still think of these systems as fractal or multifractal,depending on which of the two choices above applies.Most pure critical systems are fractals. Famousmultifractal systems are Navier-Stokes turbulence [246,247, 248], or the more tractable passive advection ofparticles, the passive scalar , [249, 250, 251, 252, 253, 254,255]. This can be generalized to the advection of extendedelastic objects [256].An important question is whether systems with quenched disorder show multifractality. A prominentexample is the wave-function statistics at a delocalizationtransition, such as the Anderson metal-insulator transitionat the mobility edge in three spatial dimensions, or theinteger quantum-Hall plateau transition in two dimensions.Here one considers (see [257] for a concise introduction) P q ( (cid:15) i ) := (cid:90) L d d d r | ψ i ( r ) | q ∼ L − τ ( q ) , (275)where (cid:15) i is the energy of the state i , and ψ i ( r ) its wavefunction. Note that normalization imposes τ (1) = 0 . For extended states inside a band τ ( q ) = d ( q − , while forlocalized states τ ( q ) = 0 . At the band edge τ ( q ) is non-trivial. Define by f ( α ) its Legendre transform,Legendre α ↔ q f ( α ) + τ ( q ) = αq. (276)Then the set of points at which an eigenfunction takes thevalue | ψ ( r ) | = A L − α has weight L f ( α ) . Both f ( α ) and τ ( q ) are convex.The question relevant for this review is whetherdisordered elastic manifolds show multifractality. As longas the roughness exponent ζ > , this does not seem to bethe case. The situation is different for ζ = 0 , i.e. charge-density waves or vortex lattices. Technically, it can beaccessed either via a − (cid:15) expansion [258] or directly intwo dimensions. Multifractality of the random-phase sine-Gordon model indimension d = 2 . The random-phase sine-Gordon modelwas introduced above in section 2.28. The object to beconsidered is C ( q, r ) := (cid:104) e iq [Φ( r ) − Φ(0)] (cid:105) . (277)It was shown in Ref. [259] that with A given in Eq. (271), C ( q, r ) (cid:39) (cid:16) ar (cid:17) η ( q ) exp (cid:18) − A q ln ( r/a ) (cid:19) . (278)The anomalous exponent η defined in Eq. (278) reads η ( q ) = 2 q (1 − τ )[1 + 2(1 − τ ) σ (cid:48) ] + τ η g ( q ) + O ( τ ) . (279)Its nontrivial part η g is [259] η g ( q ) = (cid:26) q [1 − γ E − ψ ( q ) − ψ ( − q )] , q < − , q = 1 (280)Here γ E is Euler’s constant. The result for the correlationfunction (277) enables one to calculate the leading large-distance behavior of all higher powers of the connectedcorrelation functions in the super-rough phase, i.e., for T < T c ( τ > , see Fig. 20). Using η g ( q ) = q + 2 ∞ (cid:88) n =2 ζ (2 n − q n , (281)one sees that odd moments vanish, the second moment isgiven by Eqs. (270)-(271), and higher even moments by ( − n (2 n )! (cid:68) [Φ( r ) − Φ(0)] n (cid:69) c = − τ ζ (2 n − 1) ln (cid:16) ra (cid:17) + ... (282)This system is multifractal. FRG in dimension − (cid:15) . Following [258], define G [ λ ] := (cid:68) e (cid:82) x λ ( x ) u ( x ) (cid:1)(cid:69) = lim n → (cid:68) e (cid:82) x λ ( x ) u ( x ) (cid:69) S , (283) λ ( x ) = iq [ δ ( x ) − δ ( x + r )] . (284) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles T (cid:90) x (cid:88) a,b R (cid:0) u a ( x ) − u b ( x ) (cid:1) → σ T (cid:88) a,b | u a ( x ) − u b ( x ) | , σ = R (cid:48)(cid:48)(cid:48) (0 + ) . (285)The connected part of G [ λ ] reads (the correlation function C ( x ) is defined in Eq. (33 a )) ln( G [ λ ]) = 12 13 + ... = ln (cid:18) det( −∇ + σU ( x ) + m )det( −∇ + m ) (cid:19) (286) U ( x ) = (cid:90) y C ( x − y ) λ ( y ) . (287)Calculating the determinant (286) is a formidable task,usually only possible in perturbation theory in σ . Here wegive analytical results, using three consecutive tricks:(i) Solve the problem for a spherically symmetric source λ ( y ) , assuming a uniformly distributed positive unitcharge on a circle of radius a , and a compensatingnegative charge on a circle of radius L (cid:29) a . Thepotential is U ( x ) = ( r − − L − ) / (2 π ) for a ≤ r ≤ L , and constant beyond.(ii) Write the Laplacian in distance and angle variables, −∇ → H l := − d d r + (cid:0) l + d − (cid:1) (cid:0) l + d − (cid:1) r . (288)(iii) ln( G [ λ ]) is written as a sum of the logarithms of the1-dimensional determinant ratios B l for partial waves,weighted with the degeneracy of angular momenta l , ln( G [ λ ]) = ∞ (cid:88) l =0 (2 l + d − l + d − l !( d − B l ) . (289)(iv) The Gel’fand-Yaglom method [261] explained inappendix 10.8 gives the ratio of the 1-dimensionalfunctional determinants for each partial wave l as B l := det (cid:2) H l + σU ( r ) + m (cid:3) det [ H l + m ] = ψ l ( L )˜ ψ l ( L ) . (290)Here ψ l ( r ) is the solution of (cid:2) H l + σU ( r ) + m (cid:3) ψ l ( r ) = 0 , (291)satisfying ψ l ( r ) ∼ r l +( d − / for r → ; ˜ ψ l ( r ) is thesolution for σ = 0 .(v) After some pages of algebra, one finds (modulo oddterms which vanish) ln( G [ λ ]) = F (cid:16) σq (2 π ) (cid:17) + terms odd in σ (292) F ( s ) = − ∞ (cid:88) l =0 ( l + 1) (cid:16)(cid:112) ( l + 1) + s − ( l + 1) − s l + 1) + s l + 1) (cid:17) . (293) Resummation yields (dropping odd terms) F ( s ) = ∞ (cid:88) n =2 s n Γ(2 n − ) ζ (4 n − √ π (2 n )! . (294)(vi) In a last step one proves perturbatively that all n -pointfunctions remain unchanged if one moves the chargeon the sphere at | r | = L to a single point at distance L from the origin. The combinatorial analyses yields anadditional factor of .Using the FRG fixed point (93), s = (cid:15) q , the n -th cumulantof relative displacements is obtained as (cid:104) [ u ( r ) − u (0)] n (cid:105) c (cid:39) A n ln( r/a ) (295) A n = − (cid:16) (cid:15) (cid:17) n Γ(2 n − ) ζ (4 n − √ π , n ≥ . (296)This correlation function is multifractal. Finding the ground state of a disordered system is in generala very difficult problem, often even NP-hard, meaning noalgorithm exists which is guaranteed to find the groundstate in polynomial time, i.e. within a time which does notgrow faster than N p , with N the system size, and p a finitenumber. This statement should be viewed as “state of theart”: E.g. we know of no algorithm to find the ground stateof the SK model in polynomial time; but this does not implythat no such algorithm can exist. What computer scientistshave proven is that if one day an algorithm is found tosolve one NP-hard problem, all other NP-hard problemscan be solved as well. We refer the reader to the textbooks[262, 263] and collection [264] for a more precise definitionand further information on the subject.While many ground-state calculations are consideredNP-hard, there are some notable exceptions: For the RNA-folding problem [265] a polynomial algorithm exists [266],which evaluates the partition function at all temperatures ina time growing as N , allowing one not only to find theground state but even the phase transition from a frozen toa molten phase [267, 268, 269].Another notable exception are disordered elasticmanifolds, or more specifically the ground state of an Isingferromagnet coupled to either random-bond or random-fielddisorder. It can be solved by the minimum-cut algorithm [270]. This has been used in numerous publications: Tofind the roughness exponent ζ in dimensions 2 and 3 [132],see Fig. 10; to measure the FRG-fixed point function [123],see Fig. 8. or avalanche-size distributions in equilibrium[271], see Fig. 51. Further for flux lines in a disorderedenvironment [272, 273]; or solid-on-solid models withdisordered substrates [234]. There are few experiments which really are in equilibrium.The main reason is that in most cases the exponent θ defined heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles L θ . As a consequence,there is a maximal length L max T up to which the system canequilibrate, i.e. find the minimum-energy configuration. Letus give a list of notable exceptions.(i) domain walls in thin magnetic films with random-bonddisorder have long been interpreted [83, 274, 275] asshowing the roughness exponent of ζ d =1RB = 2 / givenin Eq. (120). This interpretation probably holds onlyon small scales, see the discussion in section 3.21.(ii) hairpin unzipping reported in [276] is consistent witha roughness exponent ζ = 4 / , in agreement withthe value predicted for a single degree of freedom, i.e. d = 0 or (cid:15) = 4 in Eq. (81). We discuss this experimentin detail in section 3.17. There it is confronted withan experiment using the much softer peeling mode,placing it in the different depinning universality classwith ζ = 2 − .(iii) Vortex lattices (section 2.25). 3. Dynamics, and the depinning transition Another important class of phenomena for elastic manifoldsin disorder is the so-called depinning transition : Applyinga constant force to the elastic manifold, e.g. a constantmagnetic field to the ferromagnet mentioned in theintroduction, the latter will move if, and only if, a certaincritical threshold force f c is surpassed, see figure 22. (Thisis fortunate, since otherwise the magnetic domain walls inthe hard-disc drive onto which this article is stored wouldmove, with the effect of deleting all information, deprivingyou from your reading.) At f = f c , the so-called depinningtransition, the manifold has a distinctly different roughnessexponent ζ (see Eq. (7)) from the equilibrium ( f = 0 ). Theequation describing the movement of the interface is ∂ t u ( x, t ) = ( ∇ − m ) u ( x, t ) + F (cid:0) x, u ( x, t ) (cid:1) + f ( x, t ) , (297) F ( x, u ) = − ∂ u V ( x, u ) . (298)There are two main driving protocols, depending whetherone controls the applied force, or the mean driving velocity. Force-controlled depinning. Let us impose a driving force f ( x, t ) = f , and set m → . For f > f c , the manifold thenmoves with velocity v . Close to the transition, new criticalexponents appear: • a velocity-force relation given by (see figure 22) v ∼ | f − f c | β for f > f c , (299) • a dynamic exponent z relating correlation functions inspace and time t ∼ x z . (300) Thus if one has a correlation or response function R ( x, t ) , it will be for short times and distances be afunction of t/x z only, R ( x, t ) (cid:39) R ( x z /t ) . (301) • a correlation length ξ set by the distance to f c ξ = ξ f ∼ | f − f c | − ν . (302)Remarkably, this relation holds on both sides of thetransition: For f < f c , it describes how starting froma flat or equilibrated configuration, the correlationlength ξ , which can be interpreted as the avalancheextension, increases as one approaches f c . Arrivingat f c , each segment of the interface has moved. Above f c , the interface is always moving, and the correlationlength ξ (which now decreases upon an increase in f )gives the size of coherently moving pieces. • The new exponents z , β and ν are not independent,but related [277]. Suppose that one is above f c , andwe witnessed an avalanche of extension ξ . Then itsmean velocity scales as v ∼ ut ∼ ξ ζ ξ z ∼ | f − f c | − ν ( ζ − z ) = ⇒ β = ν ( z − ζ ) . (303)One can make the same argument below f c , byslowing increasing f to f c . • Suppose that below f c the manifold is in a pinnedconfiguration. Increasing f will lead to an avalanche,of extension ξ , and a change of elastic force (per site) ∼ ξ ζ − . This has to be balanced by the driving force,i.e. ξ ζ − ∼ | f − f c | = ⇒ ν = 12 − ζ . (304) Velocity-controlled depinning. If m > , then we canrewrite the equation of motion (297) as ∂ t u ( x, t ) = ( ∇ − m )[ u ( x, t ) − w ] + F (cid:0) x, u ( x, t ) (cid:1) ,w = vt. (305)The phenomenology changes: • Here the driving force acting on the interface isfluctuating as well as the velocity, while the meandriving velocity is fixed ˙ u ( x, t ) = v, f = 1 L d (cid:90) x F (cid:0) x, u ( x, t ) (cid:1) . (306) • The correlation length ξ is set by the confiningpotential, ξ = ξ m = 1 m . (307) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:1) (cid:1) (cid:1) | (cid:1) - (cid:1) (cid:1) β (cid:1) (cid:1) (cid:1) (cid:1) ≫ (cid:1) (cid:1) (cid:1)(cid:2) Figure 22. Left: Snapshot of a contact-line at depinning, courtesy E. Rolley [movie]. Observables derived from this system are shown on Figs. 33 and48. Right: Velocity of a pinned interface as a function of the applied force. f = 0 : equilibrium. f = f c : depinning. For an experimental confirmationof the v ( f ) curve in a thin magnetic film, see Fig. 44. We can enforce the equation of motion (305) with anauxiliary field ˜ u ( x, t ) S [ u, ˜ u, F ] = (cid:90) x,t ˜ u ( x, t ) (cid:104)(cid:0) ∂ t − ∇ + m (cid:1)(cid:0) u ( x, t ) − w (cid:1) − F (cid:0) x, u ( x, t ) (cid:1) − f ( x, t ) (cid:105) . (308)We need to average over disorder, to obtain the disorder-averaged action e −S [ u, ˜ u ] := e − S [ u, ˜ u,F ] , with S [ u, ˜ u ] = (cid:90) x,t ˜ u ( x, t ) (cid:104) ( ∂ t −∇ + m )[ u ( x, t ) − w ] − f ( x, t ) (cid:105) − (cid:90) x,t,t (cid:48) ˜ u ( x, t )∆ (cid:0) u ( x, t ) − u ( x, t (cid:48) ) (cid:1) ˜ u ( x, t (cid:48) ) . (309)We remind the definition of the force-force correlator givenin Eq. (10). Response function and the free theory: The response ofa system is defined as the answer of the system given aperturbation f ( x, t ) . The response can be any observable,as the avalanche-size distribution defined below in Eq.(465), but the simplest one is the response of the field u ( x (cid:48) , t (cid:48) ) itself, R f ( x (cid:48) , t (cid:48) | x, t ) := δδf ( x, t ) u ( x (cid:48) , t (cid:48) ) = (cid:104) u ( x (cid:48) , t (cid:48) )˜ u ( x, t ) (cid:105) . (310)In a translationally invariant system, R f ( x (cid:48) , t (cid:48) | x, t ) doesonly depend on x (cid:48) − x and t (cid:48) − t , and is denoted R ( x (cid:48) − x, t (cid:48) − t ) := R f ( x (cid:48) , t (cid:48) | x, t ) . (311)In the second equality of Eq. (310) we used the averageprovided by the action (309). The formalism is explainedin appendix 10.4, see Eq. (958) and following. This trick is known as the MSR formalism [278, 279, 280]. It is thegeneralization to a field of the relation (cid:82) k e ikx = δ ( x ) : the response field ˜ u ( x, t ) enforces the Langevin equation (305) for each x and t . A shortintroduction is given in appendix 10.4. The most convenient representation is the spatialFourier transform calculated for the free theory in Eq.(972), R ( k, t ) = (cid:104) u ( k, t + t (cid:48) )˜ u ( − k, t (cid:48) ) (cid:105) = e − ( k + m ) t Θ( t ) . (312)We could introduce response functions as the answer todifferent perturbations, e.g. increasing w instead of f , R w ( k =0 , t ) := dd w (cid:104) u ( k =0 , t ) (cid:105) = m R ( k =0 , t ) . (313)This changes the normalization, (cid:90) t R w ( k = 0 , t ) = 1 . (314)While Eq. (313) is the free-theory result, corrected inperturbation theory, Eq. (314) is by construction exact. We now state the famous [281] Middleton Theorem : If F ( x, u ) is continuuos in u , and ˙ u ( x, t ) ≥ , then ˙ u ( x, t (cid:48) ) ≥ for all t (cid:48) ≥ t . Moreover,if two configurations are ordered, u ( x, t ) ≥ u ( x, t ) , thenthey remain ordered for all times, i.e. u ( x, t (cid:48)(cid:48) ) ≥ u ( x, t (cid:48) ) for all t (cid:48)(cid:48) ≥ t (cid:48) > t . Proof : Consider an interface discretized in x . Thetrajectories u ( x, t ) are a function of time. Suppose thatthere exists x and t (cid:48) > t s.t. ˙ u ( x, t (cid:48) ) < . Define t as thefirst time when this happens, t := inf x inf t (cid:48) >t { ˙ u ( x, t (cid:48) ) < } , and x the corresponding position x . By continuity of F in u , the velocity ˙ u is continuous in time, and ˙ u ( x , t ) =0 . This implies that the disorder force acting on x doesnot change in the next (infinitesimal) time step, and theonly changes in force can come from a change in theelastic terms. Since by assumption no other point has anegative velocity, this change in force can not be negative,contradicting the assumption. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles t : xx u u u Here the red configuration is ahead of the blue one, exceptat position x , where they coincide. As in the first partof the proof, we wish to bring to a contradiction thehypothesis that at some time later u ( x ) (blue) is aheadof u ( x ) (red). For this reason, we have chosen t the infimum of times contradicting the theorem, t :=inf t (cid:48) >t { u ( x , t (cid:48) ) > u ( x , t (cid:48) ) } . Consider the equation ofmotion Eq. (305) for the difference between u and u , ∂ t [ u ( x , t ) − u ( x , t )]= ∇ [ u ( x , t ) − u ( x , t )] . (315)The disorder force terms have canceled as well as the termof order m , since by assumption u ( x , t ) = u ( x , t ) .By construction, the r.h.s. is positive, leading to the desiredcontradiction. Remark: Uniqueness. As in the statics, one encountersterms proportional to ∆ (cid:48) (0 + ) ≡ − R (cid:48)(cid:48)(cid:48) (0 + ) . Here the sign-problem can uniquely be solved by observing that due toMiddleton’s theorem the manifold only moves forward, t (cid:48) > t = ⇒ u ( x, t (cid:48) ) − u ( x, t ) ≥ . (316)Thus the argument of ∆ (cid:0) u ( x, t (cid:48) ) − u ( x, t ) (cid:1) has a well-defined sign, allowing us to interpret derivatives atvanishing arguments correctly. Practically this means thatwhen evaluating diagrams containing ∆( u ( x, t ) − u ( x, t (cid:48) )) ,one splits them into two pieces, one with t < t (cid:48) and onewith t > t (cid:48) . Both pieces are well defined, even in the limitof t → t (cid:48) . Consider the field theory defined by the action (309). Toappreciate the problem, let us remind that in equilibriuma model is defined by its Boltzmann weight. As long asthe system is ergodic, it can be sampled with the helpof a Langevin equation, and equilibrium expectations canbe evaluated as expectations in the dynamic field theory.This goes hand in hand with identical renormalizations,as is e.g. known for the effective coupling in φ theory.On the other hand, it does not fix the dynamics. It isindeed well-known that a different dynamics leads to adifferent dynamic universality class , as expemplified bythe Hohenberg-Halperin classification of dynamical criticalphenomena [282], leading to the zoo of models A, B, C,..., F, and J. We might therefore not be surprised if belowwe find the same renormalization for the disorder in the driven dynamics. On the other hand, equilibrium and out-of equilibrium are two distinct phenomena, and may havedistinct critical exponents. As we will see below, at 1-looporder all comparable observables are identical, whereasdifferences are manifest at 2-loop order.Let us start by rederiving the corrections to therenormalized disorder correlator at 1-loop order. Thereplica diagram in Eq. (53) is one of the two contributionsto the effective potential-potential correlator R ( u ) given inEq. (58). In the dynamics, the disorder term in Eq. (309) isthe bare (microscopic) force-force correlator ∆( u ) , whichwe note graphically as (cid:90) x,t ,t ˜ u ( x, t )˜ u ( x, t ) ∆ (cid:0) u ( x, t ) − u ( x, t ) (cid:1) = xt t (317)The arrows are the response fields ˜ u ( x, t )˜ u ( x, t ) ; someauthors represent them by a wiggly line. Since the responsefunction has a direction in time, the static diagram (53) hastwo descendants in the dynamic formulation, x y −→ x y + x y (318)The first descendant with the corresponding times is x yt t t t = − (cid:90) t ,t ,x R ( x − y, t − t ) R ( x − y, t − t ) × ∆ (cid:0) u ( x, t ) − u ( x, t ) (cid:1) ∆ (cid:48)(cid:48) (cid:0) u ( y, t ) − u ( y, t ) (cid:1) × ˜ u ( y, t )˜ u ( y, t ) (cid:39) − (cid:90) k (cid:90) t 12 ∆( u ) I (322)This is the same contribution as given by the diagram inEq. (53), noting that ∆( u ) = − R (cid:48)(cid:48) ( u ) , and using thecombinatorial factor / reported in Eq. (58).To complete our analysis, consider the seconddiagram; it also has two descendants, x y −→ x y + x y (323)After time-integration this yields x yt t t t (cid:39) (cid:90) k k + m ) ∆(0)∆ (cid:48)(cid:48) (cid:0) u ( y, t ) − u ( y, t ) (cid:1) . (324) x yt t t t (cid:39) (cid:90) k k + m ) ∆ (cid:48) (0 + )∆ (cid:48) (cid:0) u ( y, t ) − u ( y, t ) (cid:1) . (325)The last diagram contains a first factor of ∆ (cid:48) (0 + ) ; thedefinite sign results from the causality of the responsefunctions ensuring t < t . It is asymmetric under ex-change of t and t , thus vanishes after integrating overthese times. (B.t.w., inserted into a 2-loop diagram, it isthis diagram which is responsible for the differences seenthere, especially for the 2-loop contribution to ζ .) Together,they give a second contribution to the effective disorder δ ˆ∆( u ) = ∆(0)∆ (cid:48)(cid:48) ( u ) I = ∂ u ∆(0)∆( u ) I . (326)This is the same contribution as given by Eq. (55). The last diagram we drew for the equilibrium wasgiven in Eq. (55). Its descendant reads −→ (327)While the static diagram on the l.h.s. does not contributeto the effective disorder since it is a 3-replica term (threeindependent sums over replicas), the dynamic diagram onthe r.h.s. does not contribute due to the acausal loop, as itdoes not allow for any time integration, thus vanishes.For completeness, we write the effective disorder-force correlator at 1-loop order ˆ∆( u ) = ∆( u ) − ∂ u (cid:20) 12 ∆( u ) − ∆(0)∆( u ) (cid:21) I . (328)This result is the same as when applying − ∂ u to Eq. (58).We thus recover the same flow equation as given in Eq. (58)and first derived in [277, 117, 116, 283] ∂ (cid:96) ˜∆( u ) = ( (cid:15) − ζ ) ˜∆( u ) + ζu ˜∆ (cid:48) ( u ) − ∂ u (cid:2) ˜∆( u ) − ˜∆(0) (cid:3) . (329)While this might not be surprising on a formal level, it is very surprising on a physical level: The effective disorder(58) is for the minimum energy state, while the derivationgiven above is for a state at depinning. We will see inthe next section 3.5 that there are indeed corrections at 2-loop order which account for this difference, and which areimportant to reconcile the physically observed differencesin exponents and other observables with the theoreticalprediction.Before going there, let us complete our analysis withtwo additional contributions not present in the statics, andwhich we will interpret as the critical force at depinning,and a renormalization of friction, leading to a non-trivialdynamical exponent z , as defined in Eq. (300). The diagramin question is xt t = ˜ u ( x, t ) (cid:90) t ,k ∆ (cid:48) (cid:0) u ( x, t ) − u ( x, t ) (cid:1) e − ( t − t )( k + m ) × Θ( t < t ) (cid:39) ˜ u ( x, t ) (cid:90) t ,k (cid:104) ∆ (cid:48) (0 + ) + ∆ (cid:48)(cid:48) (0 + )( t − t ) ˙ u ( x, t ) + ... (cid:105) × e − ( t − t )( k + m ) Θ( t < t )= ˜ u ( x, t ) (cid:90) k ∆ (cid:48) (0 + ) k + m + ∆ (cid:48)(cid:48) (0 + )( k + m ) ˙ u ( x, t ) + ... (330)The first term corresponds to a constant driving force f in Eq. (297), and can be interpreted as the threshold forcebelow which the manifold will not move: f c = − ∆ (cid:48) (0 + ) I TP , (331) I TP = = (cid:90) k k + m . (332) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ∆( u ) um = 1 m = 0 . m = 0 . Figure 23. Doing RG in a simulation: Crossover from RB disorder to RFfor a driven particle [123]. Its value is non-universal, but gives us a pretty good ideahow strong we have to drive. In the driving protocol with aparabola centered at w as given in Eq. (305), it gives us thesize of the hysteresis loop , illustrated on Fig. 26, m (cid:104) u w − w forward − u w − w backward (cid:105) = 2 f c . (333)Let us now turn to the second term in Eq. (330). Restoringthe friction coefficient η in front of ∂ t u ( x, t ) in the equationof motion (305) yields η eff = 1 − ∆ (cid:48)(cid:48) (0 + ) I + ... (334)The dynamical exponent z is then obtained as z = 2 − m∂ m ln η eff = 2 − ˜∆ (cid:48)(cid:48) (0 + ) + ... (335)Taking one derivative of Eq. (77), and evaluating it in thelimit of u → allows us to conclude that ˜∆ (cid:48)(cid:48) (0 + ) = (cid:15) − ζ . (336)Thus z = 2 − (cid:15) − ζ ... = − (cid:15) ... RP disorder − (cid:15) ... RF disorder (337) Renormalization at the depinning transition was first treatedat 1-loop order by Natterman et al. [277], soon followed byNarayan and Fisher [284]. As we have seen, the 1-loopflow-equations are identical to those of the statics. Thisis surprising, since equilibrium and depinning are quitedifferent phenomena. There was even a claim by [284],that the roughness exponent in the random field universalityclass is ζ = (cid:15)/ also at depinning. After a long debateamong numerical physicists, the issue is now resolved:The roughness is significantly larger, and reads e.g. for thedriven polymer ζ = 1 . ± . [53, 285], and possiblyexactly ζ = [286]; this should be contrasted to ζ = 1 at equilibrium, see Eq. (81). Clearly, a 2-loop analysis is necessary to resolve these issues. The latter was performedin Refs. [120, 119].At the depinning transition, the 2-loop FRG flowequation reads [120, 119] ∂ (cid:96) ˜∆( u )= ( (cid:15) − ζ ) ˜∆( u ) + ζu ˜∆ (cid:48) ( u ) − ∂ u (cid:104) ˜∆( u ) − ˜∆(0) (cid:105) + 12 ∂ u (cid:110)(cid:104) ˜∆( u ) − ˜∆(0) (cid:105) ˜∆ (cid:48) ( u ) + ˜∆ (cid:48) (0 + ) ˜∆( u ) (cid:111) . (338)Compared to the FRG-equation (111 a ) for the statics, theonly change is in the last sign on the second line of Eq.(338), given in red (bold). This has important consequencesfor the physics. First of all, the roughness exponent ζ forthe random-field universality class changes: Integrating Eq.(338) the last term yields a boundary term at u = 0 , and dueto the different sign it no longer cancels with the precedingone, resulting in (cid:15) − ζ ) (cid:90) ∞ ˜∆( u ) d u − ˜∆ (cid:48) (0 + ) + O ( (cid:15) ) . (339)Inserting the 1-loop fixed point (82)-(86) leads to ζ depRF = (cid:15) . (cid:15) + . . . ) . (340)Other critical exponents mentioned above can also becalculated. The dynamical exponent z reads [120, 119] ζ depRF = 2 − (cid:15) − . (cid:15) + . . . (341)The remaining exponents are related via the scalingrelations (303) and (304). That the method works wellquantitatively can be inferred from table 25.The random-bond fixed point is unstable and renor-malizes to the random-field universality class. This mightphysically be expected: Since the manifold only moves for-ward, each time it advances it experiences a new disorderconfiguration, and it has no way to “know” whether thisdisorder is derived from a potential or not. This can be seenfrom the integrated FRG equation (339): According to Eq.(87), a RB fixed point is characterized by a vanishing of theintegral in Eq. (339), but this does not solve Eq. (339). Theinstability of the RB fixed point can already be seen for atoy model with a single particle, measuring the renormal-ized disorder correlator at a scale (cid:96) = 1 /m set by the con-fining potential, see figure 23. Generalizing the argumentsof section 2.10 one shows [124] that Eq. (109) remains validin the limit of w = vt , v → . It was confirmed numeri-cally for a string that both RB and RF disorder flow to theRF fixed point [287], and that this fixed point is close to theanalytic solution of Eq. (338), see figure 24.The non-potentiality of the depinning fixed point isalso observed in the random periodic universality class,which is the relevant one for charge density waves. The For details see [119], section IV.A. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ∆( w ) w RF, m = 0 . , L = 512 RB, m = 0 . , L = 512 -0.015-0.01-0.00500.0050.01 0 1 2 3 4 5 z Y ( z ) − Y ( z ) –loop dynamics –loop staticsRF m = 0 . , L = 512 RB m = 0 . , L = 512 FIG. 3: The difference between the normalized correlator Y ( z ) andthe 1-loop prediction Y ( z ) . We have studied the behaviour of the critical force f c ( m ) for the two classes of disorder. Because of (6), one has ! ∆(0) m ∼ m − ζ hence one obtains a parameter free linearscaling shown in Fig.2. For large m the linear scaling doesnot hold, while it holds for smaller m up to the point wherethe correlation length becomes of the order of L ( mL of order10 again ??). Note that c < as discussed below.We now turn to the FP function determination. Since thereare two scales in ∆( u ) hence we write: ∆( u ) = ∆(0) Y ( u/u ξ ) (7)where Y (0) = 1 and one determines u ξ such that " dzY ( z ) =1 hence u ξ = " ∆ / ∆(0) . The function Y ( z ) is then fully uni-versal and depends only on space dimension. We have deter-mined the function Y ( z ) from our numerical data both for RFand RB disorder. For small masses the two function are foundto coincide within statistical errors. We also observe a cusp,i.e. Y ′ (0) = − .. . (show Y ( z ) ??). The predictions from theFRG is that Y ( z ) = Y ( z ) + ϵY ( z ) + O ( ϵ ) with ϵ = 4 − d .The one loop function is the same as for the statics and givenby the solution of Y = Y ( z ) with γz = √ Y − − ln Y and γ = " d y √ y − ln y − ≈ . . Since themeasured Y ( z ) is numerically close to Y ( z ) , as was foundin the statics, we plot in Fig.3 the differential Y − Y . Theoverall shape of the difference function is very similar to theone obtained for the RF statics in d = 3 , , which was foundto exhibit only a weak dependence in d . However the over-all amplitude is larger by a factor of order . . This factorbetween statics and dynamics is consistent with the two loopprediction. We have plotted the function Y ( z ) = ddϵ Y ( z ) | ϵ =0 which, as for the statics turns out to close to the numerical re-sult **Alberto put the zero **Examples of universal amplitudes are ∆ ′′ (0 + ) c d/ , ∆(0) / ( " ∆) c x or ∆ ′ (0 + ) / ∆(0) c y , check exponents of c give one loop predictions ** see what we do about this, Kaycheck the powers of c and one loop predictions ** ∆( w ) / ∆(0) S –loopRFRBFIG. 4: Collapse of the 3 points function for RF and RB disorder.Simlations has been performed for systems of size L = 512 andmass m = 0 . . The line represents the –loop predtiction f ( x ) =(1 − x ) To investigate deeper the validity of FRG we measure thethird cumulant function, de fi ned as: m p ( w ′ − u ( w ′ ) − ( w − u ( w ))) c = L − d S ( w ′ − w ) (8)The lowest order prediction [19] is S ( w ) = m ∆ ′ ( w )(∆( w ) − ∆(0)) . Numerically one fi nds thecorrect sign and to check the scaling in a parameter-free waywe de fi ne S = " w S ( w ) / " ∞ S ( w ) = F (∆( w ) / ∆(0)) . Thefunction F ( x ) hereby de fi ned is expected to be universal.Indeed we fi nd, as can be seen in Fig.4, that RB and RF giveresult identical within statistical errors.The problem of characterizing the universality of the distri-bution of the fi nite size fl uctuations of the critical force bearsome similarity with the problem of the fi nite size fl uctuationsof the ground state energy in the statics. There, for the directedpolymer several ”universal” distributions were found depend-ing on the procedure and the geometry. The Tracy-Widomdistribution (for various β ) was found for fi xed endpoint oruniform KPZ fi eld. On a cylinder the large deviation function ln( e αF ) = L ( κ α − κ G ( κ α )) where G ( z ) is universal.For the critical force problem there are several procedures.Here we study the mass. Another procedure is the cylinder. Athird one is the fi xed center of mass studied with FRG but hardto study numerically. For each procedure there are fully uni-versal quantities (different a priori in each procedure). Fullyuniversal means independent of microscopic details, and ofthe model. It usually requires fi xing two scales one in the u direction the other in the x direction. There are additional uni-versal quantities (usually amplitudes) however which dependof the microscopic details only through renormalized elasticconstant c R and require fi xing only one scale.Here one measures: m p ( w − u ( w )) pc = L (1 − p ) d C ( n ) (0 , .. (9)with C (2) (0 , 0) = ∆(0) . Using the proper scaling δ ∆( w ) w Figure 24. Left: The fixed point ∆( w ) for the force-force correlations in d = 1 , rescaled s.t. ∆(0) = 1 , and (cid:82) w ∆( w ) = 1 , starting both from RB andRF initial condition [287]. Right: Residual error δ ∆( w ) after subtracting the 1-loop correction. The measured difference is consistent with the depinningfixed point, but not the static one. fixed point for a periodic disorder of period one reads(remember ˜∆( u ) = − ˜ R (cid:48)(cid:48) ( u ) ) ˜∆ ∗ ( u ) = (cid:15) 36 + (cid:15) − (cid:18) (cid:15) (cid:15) (cid:19) u (1 − u ) + O ( (cid:15) ) . (342)Integrating over a period, we find (cid:90) d u ˜∆ ∗ ( u ) ≡ (cid:90) d u ˜ F ( u ) ˜ F ( u (cid:48) ) = − (cid:15) . (343)In equilibrium, this correlator vanishes since potentialityrequires (cid:82) d u ˜ F ( u ) ≡ . Here, there are non-trivial contributions at 2-loop order, O ( (cid:15) ) , violating thiscondition and rendering the problem non-potential.If an additional constant term ˜∆ cannot be excludedas is the case in equilibrium, then according to Eq. (338) itflows as ∂ (cid:96) ˜∆ = ( (cid:15) − ζ ) ˜∆ . (344) d (cid:15) (cid:15) estimate simulation ± ± ζ ± ± ± ± ± z ± ± ± ± ± β ± ± ± ± ν ± ± ± Figure 25. Critical exponents at the depinning transition for short-rangedelasticity ( α = 2 ). 1-loop and 2-loop results compared to estimates basedon three Pad´e approximants, scaling relations and common sense. It acts as a Larkin term leading to a roughness exponent[117, 119, 290] ζ CDWobs = ζ Larkin = 4 − d . (345)For the dynamic exponent z , one can go further [291, 292,135, 293], using the equivalence to φ -theory discussed insection 8.9, z = 2 − (cid:15) − (cid:15) (cid:20) ζ (3)9 − (cid:21) (cid:15) − (cid:20) ζ (5)81 − ζ (4)6 − ζ (3)162 + 7324 (cid:21) (cid:15) − (cid:20) ζ (3) 162 + 37 ζ (3)36 + 29 ζ (4)648 + 703 ζ (5)243+ 175 ζ (6)162 − ζ (7)216 + 171944 (cid:21) (cid:15) − . (cid:15) + O ( (cid:15) ) . (346) The RP fixed point at depinning is stableperturbatively, (appendix I of [119]). The leading threemodes are ω − = − (cid:15), z − ( u ) = 1 . (347 a ) ω = (cid:15) + 73 (cid:15) + O ( (cid:15) ) , (347 b ) z ( u ) = 1 − (6 + 4 (cid:15) ) u (1 − u ) .ω = 2 (cid:15) + 4 (cid:15) + O ( (cid:15) ) , (347 c ) z ( u ) = 1 − (15 + 20 (cid:15) ) u (1 − u ) + (45 + 85 (cid:15) )[ u (1 − u )] ,ω = 253 (cid:15) + 1409 (cid:15) + O ( (cid:15) ) . (347 d ) RF fixed point. ω = (cid:15) + 0 . (cid:15) + O ( (cid:15) ) . (348 a ) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles z ( u, (cid:15) ) = (cid:15)z ( u ) + (cid:15) z ( u ) + O ( (cid:15) ) , (348 b ) z ( u ) = ζu ∆ (cid:48) ( u ) + ( (cid:15) − ζ )∆( u ) (cid:12)(cid:12)(cid:12) (cid:15) =1 . (348 c )While the first-order term can rather instructively beexpressed in terms of the fixed point ∆( u ) itself, the higher-order terms are more complicated (section 6.5 of [40]). The fixed points discussed above are also found withinthe non-perturbative functional renormalization group(NP-FRG) [294], leading to slightly varying numericalpredictions in the values of the critical exponents. Theoutput of the NP-FRG approach is z = 1 . ( d = 3 ), z = 1 . ( d = 2 ), and z = 0 . ( d = 1 ). For the roughnessat depinning this yields ζ = 0 . ( d = 3 ), ζ = 0 . ( d = 2 ), and ζ = 1 . ( d = 1 ). [295, 50] consider depinning at the upper critical dimen-sion. To derive this, note that the integral I defined in Eq.(57) has a well-defined limit for (cid:15) → , if one introduces asin Eq. (56) an UV-cutoff Λ ∼ /a , I := = (cid:90) Λ k + m ) = m − (cid:15) − Λ − (cid:15) (cid:15) (cid:15) )(4 π ) d/ → ln( m/ Λ)8 π . (349)This suggests as scale for the RG flow (cid:96) := ln( m/ Λ) . (350)Let us make in generalization of Eq. (61) the ansatz ∆( u ) = 18 π (cid:96) ζ − ˜∆ (cid:96) ( u(cid:96) − ζ ) , (351) ζ = ζ (cid:15) + ζ (cid:15) + ... (352)Then ˜∆ (cid:96) ( u ) satisfies the flow equation ∂ (cid:96) ˜∆ (cid:96) ( u ) = (1 − ζ ) ˜∆ (cid:96) ( u ) + ζ u ˜∆ (cid:48) (cid:96) ( u ) − ∂ u (cid:104) ˜∆ (cid:96) ( u ) − ˜∆ (cid:96) (0) (cid:105) + (cid:88) n> (cid:96) − n β n ( u ) , (353)where β n ( u ) are the n -loop contributions to the β -function.As a consequence, ˜ u q u − q (cid:12)(cid:12)(cid:12) q (cid:28) m (cid:39) ∆(0) m (cid:2) O ( (cid:96) − ) (cid:3) (cid:39) ln( m/ Λ) − ζ π m + ... . (354)This formula is valid both in equilibrium and at depinning.For RF disorder, ζ = 1 / , leading to an additional factorof ln( m/ Λ) / in the 2-point function (354) as comparedto naive expectations, and mean field is invalid at the uppercritical dimension. For a single particle, there is a nice geometrical constructionto obtain the particle trajectories, indicated in figure 26:For given w , draw a line of slope m ( u − w ) . Forforward driving, u ( w ) is the leftmost intersection with thepinning force F ( u ) , while for backward driving it is therightmost such intersection. As indicated by the arrows,this is equivalent to shining light with slope m , eitherfrom the left for forward driving, or from the right forbackward driving. Parts in the shadow are never visited,while illuminated ones are. The jumps are the avalanchesof section 2.9 and are further discussed in section 4.Using this construction, one can obtain both thedistribution of critical forces, as well as the renormalizeddisorder force-force correlator ∆( w ) analytically [296].The distribution of threshold forces corresponds to the threemain classes of extreme-value statistics. Let us according toEq. (109) define ˆ∆( w − w (cid:48) ) := m [ w − u ( w )][ w (cid:48) − u ( w (cid:48) )] c . (355)Each class (discussed below), has its own exponent ζ ,setting a scale ρ m ∼ m − ζ . At small m , force-forcecorrelations are universal, given by ˆ∆( w ) = m ρ m ˜∆( w/ρ m ) . (356)The fixed-point function ˜∆( w ) depends on the universalityclass. The three classes are distinguished by the distributionof the random forces F for the most blocking forces. Gumbel class: P ( F ) (cid:39) e − A ( − F ) γ , as F → −∞ . (357)The threshold forces f c are distributed according to a Gumbel distribution (tested in [126]), P G ( a ) = exp( − a )Θ( a ) , (358) f c = (cid:20) − ln( m a ) A (cid:21) γ = f − ln( a ) m ρ m + ... (359)The constant f , the scale ρ m , and the exponent ζ are f = A − γ (ln m − ) γ ,ρ − m = γA γ m ζ (ln m − ) − γ , ζ = 2 . (360)The effective disorder correlator reads ˜∆ G ( w ) = w Li (1 − e w ) + π . (361)It can be compared with the FRG fixed point, all rescaledto ∆(0) = − ∆ (cid:48) (0) = 1 , see Fig. 27. While astraightforward (cid:15) -expansion is not satisfactory, we can usea Pad´e approximant, ˜∆ Pad´e ( w ) = ˜∆ ( w ) + α(cid:15) ˜∆ ( w )1 + ( α − (cid:15) ˜∆ ( w ) / ˜∆ ( w ) . (362)As one can see on Fig 27, α = 0 . yields a goodapproximation, making it a strong candidate to compare tonumerical simulations or experiments in d = 1 and d = 2 . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles F(u) u wu(w) m w j’F(u) j w’ u Figure 26. Construction of u ( w ) in d = 0 , for the pinning force F ( u ) (bold black line). The two quasi-static motions driven to the right and to theleft are indicated by red and green arrows, and exhibit jumps (“dynamical shocks”). The position of the shocks in the statics is shown for comparison,based on the Maxwell construction (equivalence of light blue and yellow areas, both bright in black and white). The critical force is / (2 m ) times thearea bounded by the hull of the construction. Right: The needles of the discretized particle model (DPM) [296]. u w as a function of w is given by theleft-most intersection of m ( u − w ) with a needle, here u w = j , and u w (cid:48) = j (cid:48) . Fr´echet class: P ( F ) (cid:39) A α ( α + 1)( − F ) − − α Θ( − F ) as F → −∞ . (363)The threshold forces f c are distributed according to a Fr´echet distribution ( α > ), f c = x m ρ m , P F ( x ) = α x − α − e − x − α Θ( x ) . (364)The mean pinning force f c , the scale ρ m , and the exponent ζ are f c = Γ(1 − α ) m ρ m , ρ m = A α m − ζ ,ζ = 2 + 2 α . (365)The effective disorder correlator can be written as anintegral, and is ill-defined for α < , where the secondmoment of the force-force fluctuations vanishes. For α > it has a cusp at small w , and decays algrebraically at large w , ˜∆ F ( w ) (cid:39) Γ (cid:0) α − α (cid:1) − Γ(1 − α ) + α Γ (cid:0) − α (cid:1) w + αw α + 2 + ... for w → , ˜∆ F ( w ) (cid:39) w − α α ( α − α − 1) + ... for w → ∞ . (366) Weibull class: In this class, the random forces arebounded from below, growing as a power law above thethreshold, here chosen to be zero, P ( F ) = A α ( α − F α − θ ( F ) , α > . (367)The threshold forces are distributed according to a Weibull distribution f c = x m ρ m ,P W ( x ) = α ( − x ) α − e − ( − x ) α Θ( − x ) . (368)The mean pinning force f c , the scale ρ m , and the exponent ζ are f c = − A − α m α Γ(1 + α ) ,ρ m = A − α m − ζ , ζ = 2 − α . (369) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8)(cid:9)(cid:2)(cid:8)(cid:9)(cid:4)(cid:8)(cid:9)(cid:6)(cid:8)(cid:9)(cid:10)(cid:1)(cid:9)(cid:8) Δ ( (cid:1) ) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8)(cid:9)(cid:8)(cid:2)(cid:8)(cid:9)(cid:8)(cid:4)(cid:8)(cid:9)(cid:8)(cid:6)(cid:8)(cid:9)(cid:8)(cid:10)(cid:8)(cid:9)(cid:1)(cid:8) δΔ ( (cid:1) ) Figure 27. Main plot: The function (361) rescaled s.t. ∆(0) = − ∆ (cid:48) (0 + ) = 1 (in black). This is compared to the similarly rescaled1-loop prediction (86) (in red), and the straightforward 2-loop predictionobtained from Eq. (338) (blue dashed). To improve convergence, wehave used a Pad´e-(1,1) resummation (green, dashed), defined in Eq. (362).Inset: The same functions with the black curve subtracted. We see that the1-loop result is decent, and that the Pad´e-resummed 2-loop result stronglyimproves on it. The most important class is the box distribution withminimum at 0 ( α = 2 ). Its force-force correlator is ˜∆ α =2W ( w ) = e − w w (cid:104) w − e w √ π (cid:0) w +1 (cid:1) erfc ( w )+ √ π (cid:105) + 12 √ π (cid:104) w e − w − Γ (cid:0) , w (cid:1)(cid:105) . (370)An interesting question is whether one of the casesdiscussed above can be related to the (cid:15) expansion. The mostnatural candidate is the Gumbel class with γ = 2 , as fieldtheory assumes bare Gaussian disorder. In that case ζ = 2 ,close to the 2-loop result (340), i.e. ζ ( (cid:15) = 4) = 2 . .A comparison for the renormalized force-force correlator ∆( u ) , rescaled to a function Y ( u ) with Y (0) = 1 , and (cid:82) ∞ d y Y ( u ) = 1 , is shown on figure 27. The agreement,especially after proper resummation, is quite good. Notehowever, that the functions (361), (366) and (370) are close,so that the (cid:15) expansion might not be enough to discriminatebetween them. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Avalanches and waiting times. For simplicity we restrictourselves to the Gumbel class, where both the waitingdistances w between jumps as well as the avalanches size S have a pure exponential distribution, P ( w ) = ρ − m exp( − w/ρ m ) , (371) P ( S ) = ρ − m exp( − S/ρ m ) . (372) Dynamics. The model defined in Ref. [296] and discussedin this section advances instantaneously. The easiest way toendow it with a dynamics is to consider a Langevin equation[126], η∂ t u ( t ) = m [ w − u ( t )] + F (cid:0) u ( t ) (cid:1) . (373)If the disorder is needle-like as on the right of Fig. 26 (theoriginal construction of [296]), then either the particle isat rest, blocked by a needle, or it moves, and the onlyforce acting on it comes from the spring. Neglecting thatthe spring gets shorter during the movement, the response-function is then given by R ( t ) ∼ P ( S/v ) , where ηv = f c ,resulting for Gaussian disorder (Gumbel class with γ = 2 , A = 1 / ) into R ( t ) = τ − m e − t/τ m , τ m = η m ln( m − ) . (374) The framework of disordered elastic manifolds covers manyexperiments, from contact-line depinning over magneticdomain walls to earthquakes. Many of these experiments,or at least aspects thereof, are successfully described by mean-field theory . For driven disordered systems the firstquestion to pose is: What is meant by mean-field? Letus define mean-field theory as a theory which reduces anextended system to a single degree of freedom , in generalits center of mass u . For depinning, u then follows theequation of motion (297), reduced to a single degree offreedom, ∂ t u ( t ) = m [ w − u ( t )] + F (cid:0) u ( t ) (cid:1) . (375)Specifying the correlations of F ( u ) selects one mean-fieldtheory. However, when the reader encounters the term“mean-field theory” in the literature, it is quite generallyemployed for a model where the forces perform a randomwalk, ∂ u F (cid:0) u (cid:1) = ξ ( u ) , (376) (cid:104) ξ ( u ) ξ ( u (cid:48) ) (cid:105) = 2 σδ ( u − u (cid:48) ) . (377)This model was introduced in 1990 by Alessandro,Beatrice, Bertotti and Montorsi (ABBM) [297, 298] todescribe magnetic domain walls. There F ( u ) are the“coercive magnetic fields” pinning the domain wall, whichwere observed experimentally [299] to change with a seemingly uncorrelated function ξ ( u ) . The decision ofABBM [297] to model ξ ( u ) in Eq. (377) as a whitenoise is a strong assumption, a posteriori justified by the Figure 28. Avalanche-size distribution P ( S ) for a particle evolving due toEq. (373), with forces F ( u ) modeled by the Ornstein-Uhlenbeck process(379). The theoretical curves are the kicked ABBM model as given byEq. (521) (cyan dotted), and the discrete particle model as given by Eq.(372) (blue, dashed). m = 10 − , δw = A = ρ = 1 , δt = 10 − , S m = (cid:10) S (cid:11) / (2 (cid:104) S (cid:105) ) = 2408 . , ρ m = 2329 . , samples. applicability to experiments [298]. It means that F ( u ) hasthe statistics of a random walk.Field theory [118, 119, 120] gives a more differenti-ated view: First of all, mean-field theory should be applica-ble (with additional logarithmic corrections, see section 3.8)in d = d c [295, 50], which contains magnets with strongdipolar interactions [300], earthquakes [79], and micro-pillar shear experiments [301]. As F ( u ) has the statisticsof a random walk, the (microscopic) force-force correlatorof Eqs. (376)-(377) is ∆(0) − ∆( u − u (cid:48) ) = 12 (cid:68) [ F ( u ) − F ( u (cid:48) )] (cid:69) = σ | u − u (cid:48) | . (378)Our argument for RF-disorder in section 1.2, the strongestmicroscopic disorder at our disposal, predicts such a behav-ior for the correlator R (0) − R ( u ) = (cid:10) [ V ( u ) − V (0)] (cid:11) ofthe potential, but not of the force. On the other hand, theeffective (renormalized) force-force correlator ∆( u ) has acusp, so Eq. (378) with σ = | ∆ (cid:48) (0 + ) | is an approximation,valid for small u . The ABBM model (375)-(377) shouldthen be viewed as an effective theory, arriving after renor-malization .If indeed the microscopic disorder has the statisticsof a random walk, then the force-force correlator (378)does not change under renormalization, as is easily checkedby inserting it into the 1-loop (329) or 2-loop (338)flow equation. Counting of derivatives for higher-ordercorrections proves that this statement persists to all ordersin perturbation theory. Thus even an extended (non-MF)system where each degree of freedom sees a random forcewhich performs a random walk, the Brownian-force model(BFM), introduced in [302] and further discussed in section4.5, is stable under renormalization, and has a a roughnessexponent ζ ABBM = (cid:15) . This was indirectly numerically verified in Ref. [303]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ∂ u F (cid:0) u (cid:1) = − F ( u ) + ξ ( u ) . (379)This equation is solved by F ( u ) = (cid:90) u −∞ d u e − ( u − u ) ξ ( u ) . (380)It leads to microscopic correlations ∆( u − u (cid:48) ) = F ( u ) F ( u (cid:48) )= (cid:90) u −∞ d u (cid:90) u (cid:48) −∞ d u e − ( u + u (cid:48) − u − u ) ξ ( u ) ξ ( u )= 2 σ (cid:90) min( u,u (cid:48) ) −∞ d˜ u e − ( u + u (cid:48) − u ) = σ e −| u − u (cid:48) | . (381)Note that the small-distance behavior of ∆( u − u (cid:48) ) is as inEq. (378). The crossover was confirmed numerically [126],and in experiments on magnetic domain walls [304] andknitting [305]. We show a simulation for the crossover inthe avalanche-size distribution (section 4) from τ = 3 / forABBM, given in Eq. (521), to τ = 0 as given by Eq. (372).Eq. (379) also serves as an effective theory for thecrossover observed in systems of linear size L , from aregime with mL (cid:29) described by an extended elasticmanifold (section 3.4), to a single-particle regime asdescribed by the DPM model (section 3.9). This has indeedbe seen in numerical simulations for a line with periodicdisorder [306]. Suppose the system is driven quasi-statically, such thatwhenever we measure, ∂ t u ( x, t ) = 0 . Then the condition(107) derived for equilibrium is valid too. As is illustratedin Fig. 26, the chosen minimum is not the global minimum,but the leftmost stable one (driving from left to right),as obtained by the construction contained in Middleton’stheorem of section 3.2. Thus there are three relevant localminima: From left to right these are the (local) depinningminimum, the equilibrium one, and finally the (local)depinning minimum for driving the system in the oppositedirection. The arguments in the construction entering Eqs.(107), (108) and (109) remain valid, and Eq. (109) is theprescription to measure ∆( w ) at depinning. Defining with w = vt , w (cid:48) = vt (cid:48) ∆ v ( w − w (cid:48) ) := L d m [ w − u w ][ w (cid:48) − u w (cid:48) ] c , (382)the renormalized force-force correlator is ∆( w − w (cid:48) ) = lim v → ∆ v ( w − w (cid:48) ) . (383) In an experiment, the driving velocity v is finite, and it isdifficult to take the limit of v → . However, the observable(382) can be calculated as ∆ v ( w ) = ∞ (cid:90) d t ∞ (cid:90) d t (cid:48) ∆( w − vt + vt (cid:48) ) R w ( t ) R w ( t (cid:48) ) , (384)where R w ( t ) is the response (313) of the center of mass toan increase in w , and (cid:82) t R w ( t ) = 1 . This implies that theintegral of ∆ v ( w ) is independent of v .As an illustration, consider ∆( w ) = ∆(0)e −| w | /ξ , and R w ( t ) = τ − e − t/τ . Then ∆ v ( w ) = ∆(0) e −| w | /ξ − τvξ e −| w | / ( τv ) − (cid:0) τvξ (cid:1) . (385)This is a superposition of two exponentials, with the naturalscales ξ and τ v . Since ∆ (cid:48) v (0 + ) = 0 , (386)the cusp is rounded. This can be proven in general fromEq. (384). Note that ∆ v ( w ) is not analytic, contrary tothe thermal rounding discussed in section 2.14. As long as τ v (cid:28) ξ , the second term decays much faster than the first,allowing us to perform a boundary-layer analysis, alreadyencountered for the thermal rounding of the cusp in Eq.(140). Eq. (384) is approximated by the boundary-layeransatz ∆ v ( w ) (cid:39) A v ∆ (cid:16)(cid:113) w + ( δ BL w ) (cid:17) , (387) δ BL w = vτ, τ := (cid:90) ∞ d t R w ( t ) t , (388) A v = (cid:82) ∞ d w ∆( w ) (cid:82) ∞ d w ∆( (cid:112) w + ( δ BL w ) ) . (389)The amplitude A v ensures normalization. While theresponse function R w ( t ) (and possibly ∆( w ) ) in Eq. (384)may depend on v , our considerations using the zero-velocity expressions in Eq. (384) yield at least the correctsmall-velocity behavior [126].If in an experiment the response function is unavail-able, its characteristic time scale τ can be reconstructed ap-proximatively from ∆ v ( w ) as τ (cid:39) v lim w → ∆ (cid:48) ( w )∆ (cid:48)(cid:48) v (0) . (390)In the numerator we have written lim w → ∆ (cid:48) ( w ) , whichis obtained by extrapolating ∆ (cid:48) v ( w ) from outside theboundary layer, i.e w ≥ δ BL w = vτ , to w = 0 .There are two other, and more precise, strategies toobtain τ , and at the same time reconstruct the zero-velocitycorrelator:(i) use the boundary-layer formula (387) to plot themeasured ∆ v ( w ) against (cid:112) w + ( vτ ) ; find the best τ which removes the curvature of ∆ v ( w ) . This yields τ , and by extrapolation to w = 0 the full ∆( w ) . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:1) (cid:7)(cid:8)(cid:2)(cid:7)(cid:8)(cid:4)(cid:7)(cid:8)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:8)(cid:7) Δ (cid:1) ( (cid:1) ) Figure 29. Rounding of ∆( w ) (green, dashed) at finite v to ∆ v ( w ) (bluesolid) given by Eq. (384) and the boundary-layer approximation (387) (reddotted). (ii) use that ( τ ∂ t +1) R u ( t ) = δ ( t ) to remove the responsefunctions in Eq. (384), ∆ v =0 ( w ) = (cid:34) − (cid:18) τ v dd w (cid:19) (cid:35) ∆ v ( w ) . (391)In both approaches, the fitting parameter τ can ratherprecisely be obtained by plotting − ∆ (cid:48) v =0 ( w ) / ∆ v =0 ( w ) ,and optimizing to render the plot as straight as possible forsmall w , i.e. inside the boundary layer w ≤ δ BL w = vτ .While Eq. (391) is more precise, and reconstructs ∆ v =0 ( w ) down to w = 0 , the boundary analysis is more robust fornoisy data. In order to test the predictions of the field theory, one needsefficient simulation algorithms. There are three categories.(i) Cellular automata are simple to implement, eitherdirectly for the elastic manifold, or for one ofthe related sandpile models (section 5). Directimplementations for the elastic manifold are tricky, asextended moves are necessary [307].(ii) Langevin dynamics is the most realistic approach,and the best approach to access the dynamics [285];even though dynamical simulations are sometimesperformed in cellular automata.(iii) Critical configurations in a continuous setting canbe sampled most efficiently by the Rosso-Krauthalgorithm [308, 309], termed by the authors variantMonte Carle (VMC). The idea is simple: Whenupdating the position of a single site, the lattersite can be moved as far ahead as the equationof motion permits, without updating at the sametime its neighbours. Since Middleton’s theoremguarantees that the such generated configurationcannot surpass the next pinning configuration, thealgorithm converges to the latter. This fictitious zz (cid:48) φ Figure 30. The conformal transformation z → z (cid:48) as given in Eq. (394). dynamics it is much faster than the Langevin one,and gave us precise estimates for the roughness ζ indimension d = 1 [53], and higher [288]. To keep a system translationally invariant, simulationsare usually performed with periodic boundary conditions.Trying to extract critical exponents by plotting simulationresults against distance yields poor results. There are twonatural scaling variables:(i) Polymer Scaling : For a non-interacting polymer ofsize x , or random walk of time x , the probabilitythat monomer x comes close to monomer 0 in d -dimensional space is P ( x ) = A x − d/ . Theprobability that a ring polymer of size L has monomer0 and x close together equals the probability to maketwo rings of size x and size L − x , thus the probabilityis P ( x | L ) = P ( x ) P ( L − x ) = (cid:2) x ( L − x ) (cid:3) − d/ (392)This identifies the natural scaling variables x (1) p := 4 x ( L − x ) L , x (2) p := (cid:0) x (1) p (cid:1) (393)(ii) Conformal Invariance : The conformal mapping fromthe line z = 1 + iy to the circle of diameter 1 as shownin Fig. 30 and known as an inversion at the circle , maps z = 1 + iy −→ z (cid:48) = 1 + iy y = 11 − iy = 12 (cid:0) iφ (cid:1) (394)This implies (for details see [310]) y = tan( φ/ (395)Suppose the 2-point function on the infinite axis is (cid:104)O ( y ) O ( y ) (cid:105) = 1 | y − y | . (396) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:104)O ( φ ) O ( φ ) (cid:105) = (cid:18) ∂y ∂φ (cid:19) ∆ (cid:18) ∂y ∂φ (cid:19) ∆ (cid:104)O ( y ) O ( y ) (cid:105) = t − (397) t = 2 (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) φ − φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . (398)Note that t can be interpreted as the cordal distancebetween two points parameterized by φ and φ , i.e. t = t ( φ − φ ) = | e iφ − e iφ | = 2 (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) φ − φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (399)Similarly, for the 3-point function of three scalaroperators of dimension ∆ , conformal invarianceimplies ( A is an amplitude) (cid:104)O ( φ ) O ( φ ) O ( φ ) (cid:105) = A (cid:16) t t t (cid:17) − ∆ . (400) An anomalously large roughness, ζ = 5 / : Consider thestandard definition of the 2-point function (cid:68) u (2) ( x ) (cid:69) := 12 (cid:10) [ u ( x ) − u (0)] (cid:11) . (401)We expect that (cid:10) u (2) ( x ) (cid:11) ∼ | x | ζ . This is not possible, asis shown by the following simple argument [313]: (cid:10) [ u ( x ) − u (0)] (cid:11) = 12 x (cid:88) i =1 x (cid:88) j =1 (cid:104) [ u ( i ) − u ( i − u ( j ) − u ( j − (cid:105) . (402)The expression inside the expectation values depends on i − j , and is maximal for i = j . Thus (cid:10) [ u ( x ) − u (0)] (cid:11) ≤ x (cid:10) [ u (1) − u (0)] (cid:11) . (403)As can be seen on the middle of Fig. 31, the bound is almostattained. Thus, we expect with some amplitude A (cid:10) [ u ( x ) − u (0)] (cid:11) ≈ A x (2) p L ζ (cid:39) A x L ζ − , x (cid:28) L. (404)The roughness exponent ζ can be observed in the overallscaling, evaluating (cid:10) u (2) ( x ) (cid:11) at its maximum x = L/ ,in Fourier, or by measuring correlations of the discretederivative of u ( x ) . Define (cid:68) ∂u (2) ( x ) (cid:69) := 12 (cid:28)(cid:104)(cid:0) u ( x +1) − u ( x ) (cid:1) − (cid:0) u (1) − u (0) (cid:1)(cid:105) (cid:29) . (405)We expect that (cid:68) ∂u (2) ( x ) (cid:69) = B (cid:0) x (2) (cid:1) ζ − (cid:39) B x ζ − , x (cid:28) L. (406)This is indeed satisfied, see the middle of Fig. 31. Theexponent is consistent with ζ = 5 / , as conjectured in[286]. Skewnews: In equilibrium the connected three-point func-tion (cid:104) u ( x ) u ( y ) u ( z ) (cid:105) c has to vanish, due to the symmetry u → − u . At depinning this symmetry may be broken, butno signs were found yet [314, 315].On figure 32 we show non-vanishing simulation resultsfor the 3-point function [310] (cid:10) u (3) ( x ) (cid:11) := (cid:10) [ u ( x ) − u (0)] [ u ( − x ) − u (0)] (cid:11) = − (cid:104) u ( x ) u (0) u ( − x ) (cid:105) c . (407)Note that the more symmetric-looking variant (cid:104) [ u ( x ) − u ( y )][ u ( y ) − u ( z )][ u ( z ) − u ( x )] (cid:105) c = 0 (408)vanishes as indicated, which can be shown by expansion.The simplest symmetric, non-vanishing combination is (cid:104) ∆ u ( x, y, z )∆ u ( y, z, x )∆ u ( z, x, y ) (cid:105) c = − (cid:104) u ( x ) u ( y ) u ( z ) (cid:105) c (409)where ∆ u ( x, y, z ) := u ( x )+ u ( y ) − u ( z ) . (410)From scaling, we expect that (cid:10) u (3) ( x ) (cid:11) ∼ | x | ζ , x (cid:28) L , (411)as long as we escape the argument that lead to Eq. (404).Figure 31 shows that this is the case. What is tested there inaddition is whether conformal invariance holds. Accordingto Eq. (400), conformal invariance implies that (cid:10) u (3) ( x ) (cid:11) = A ( x (3) ) ζ (412) x (3) = t ( x ) t (2 x ) (413)with t ( x ) introduced in Eq. (399). This does not hold. Itcan, however, not be excluded that conformal symmetry,which is the property of an observable rather than a theory,is present in a different variable. Contact-line depinning can be treated by a modification ofthe theory for disordered elastic manifolds, using the long-range elasticity introduced in section 1.3, Eq. (15). Thetheory was developed to O ( (cid:15) ) in Ref. [317] and to O ( (cid:15) ) in Ref. [120, 119]. Key predictions are [119] (cid:15) = 2 − d, (414) ζ = (cid:15) (cid:0) . (cid:15) (cid:1) + O ( (cid:15) ) , (415) z = 1 − (cid:15) − . (cid:15) + O ( (cid:15) ) . (416)The other exponents are obtained via scaling, ν = 1 / (1 − ζ ) , and β = ν ( z − ζ ) . Simulation results are ζ = 0 . ± . 002 [ ] , (417) z = 0 . ± . 005 [ ] , (418) β = 0 . ± . 005 [ ] . (419)Numerical values both for the (cid:15) -expansion and forsimulations are collected on table 34. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles x - - - u - - - - - - x p ln 〈 u 〉 - - - - - - x p ln 〈∂ u 〉 Figure 31. Left: A critical string at depinning, L = 8000 , mL = 1 . Middle and right: The 2-point function for L = 2000 , mL = 1 . The measuredslopes (in orange) are . , and . , confirming the theoretically expected , and . (in yellow) x / L 〈 u 〉 - x 〈 u 〉 - - - - x 〈 u 〉 Figure 32. The 3-point function (cid:10) u (3) (cid:11) for L = 2000 , mL = 1 . The measured slope is . ± . (orange, dashed), as compared to the expected ζ = 1 . (in yellow). The descending branch of the last curve is consistent with a slope of . . Much bigger systems are needed to confirm this value. w Δ ( w ) Δ ( ) w [ μ m ] Δ ( w ) [ μ m ] Figure 33. Inset: The disorder correlator ∆( w ) for H /Cs, with errorbars estimated from the experiment. Main plot: The rescaled disordercorrelator (green/solid) with error bars (red). The dashed line is the 1-loopresult; figure from [71]. Note that the boundary layer due to the finitedriving velocity (section 3.11) is not unfolded. (cid:15) (cid:15) estimate simulation ζ . ± . . ± . [308] β . ± . . ± . [316] z . ± . . ± . [316] ν ± . . ± . [308] Figure 34. Exponents for the depinning of a line with long-range elasticity,relevant for contact-line depinning and fracture. The last exponent ν wasobtained from ν = 1 / (1 − ζ ) . Contact lines are a nice experimental realization ofdepinning, as one can watch and film the contact line,thus extracting not only its roughness, but also dynamicalproperties. The value of the roughness exponent, givenas ζ = 0 . in [84], but observed smaller ζ ≈ / inearlier work [318], is still debated [319], and many effectiveexponents are found in the literature. Our own theoreticalwork [94, 93] does not allow to exclude an exponent of ζ > ζ LRdep = 0 . , but we do not believe this to be likely.Contact-line depinning is also the first system where heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ∆( w ) was measured,both for liquid hydrogen on a disordered Cesium substrate,and for isobutanol on a randomly silanized silicon wafer[71]. Earlier experiments with water on a glass platewith randomly deposited Chromium islands [84] turnedout to have long-range correlated correlations, both due tothe impurity of water as of the inhomogeneity of glass.Measurements of the renormalized force-force correlator ∆( w ) as defined in Eq. (109) are shown in Fig. 33, using thecleaner of the two systems, liquid hydrogen on a disorderedCesium substrate. The agreement is satisfactory. There are two main types of fracture experiments: Fracturealong a fault plane [321, 308, 322, 323, 320], and fractureof a bulk material [324, 325, 326, 327, 73, 74, 328, 329]. Fracture along a fault plane. Let us start with theconceptionally simpler fracture along a fault plane. It ischaracterized by a roughness exponent ζ ≡ ζ (cid:107) in thepropagation direction. A beautiful example is the Osloexperiment [323, 320], where two transparent plexiglassplates are sandblasted rendering them opaque. Sinteredtogether the sandwich becomes transparent. Breaking thecrack open along the fault plane between the two plates,the damaged parts become again opaque, allowing oneto observe and film the advancing crack, see the insetof Fig. 35. Below a characteristic scale δ ≈ µ m,which also is the correlation length of the disorder, theroughness exponent is ζ (cid:107) ≈ . , which is interpreted[320] as the roughness exponent in directed percolation ζ = 0 . , see Eq. (661). (For reasons discussedin section 5.7, directed percolation is also relevant for Figure 35. Scaling behavior of the height-height correlations with twodifferent roughness exponents ζ (cid:107) ≈ . below the critical scale δ ∗ =100 µ m and ζ (cid:107) ≈ . above [320]. The inset shows the fracture front,moving from bottom to top. anisotropic depinning.) For larger scales, the roughnesscrosses over to a smaller exponent of ζ (cid:107) ≈ . , consistentwith the roughness exponent for depinning of a line withlong-ranged elasticity, ζ = 0 . ± . see Eq. (417)[308]. Long-range elasticity is explained in section 1.3. Forfracture it was introduced in Ref. [92]. We also note thatinterface configurations are non-Gaussian [330]. Fracture of bulk material. Fracture of a bulk material ismore complicated. To get the notations straight, we showthe coordinate system favored in the fracture community(drawing of [328]): Chapter 4: Fracture surfaces for model linear elastic disordered materials z, x )). But the heterogeneities of thematerial induce both in - plane (along x ) and out - of - plane (along y ) perturbationsof the shape of the edge. Schematic views of the in-plane f ( z, t ) and out-of-planeFigure 4.1: Geometry of perturbed cracks subject to mode I loading (large arrowsindicating the direction of macroscopic loading). (a): In-plane perturbations. (b):Out-of-plane perturbations. The shape of the fracture surface is effectively the historyof the out-of-plane perturbations of the crack front. (Taken from Ref. [40]). h ( x = x + f ( z, t ) , z ) displacements are shown in Fig. 4.1. For simplicity, the out-of-plane perturbations are represented for a crack front without in-plane perturbations( f ( z, t ) = 0). The fracture surface is the print of the out-of-plane perturbations h ( x, z )of the crack front. In the following, we will see that, for small enough perturbations,the out-of-plane displacements are independent of the in-plane displacements so thatthe shape of the fracture surface can be predicted independently of f ( z, t ). Thisimplies that the dynamical properties of the crack – the local velocities of the crackfront ∂f∂t ( z, t ) – are decoupled from the crack trajectory h ( x, z ). An experimentalargument based on the analysis of fracture surfaces will also support this statement(see Section 4.2 ). Stress field in the vicinity of a slightly perturbed crack front : We considernow a point M of the crack front characterized by its position ( x = x + f ( z, t ) , y = h ( x, z ) , z ). The local stress field around M determines its trajectory. The stress at adistance r ahead of the point M in the direction θ can be written as the sum of thecontributions of each of the three fracture modes (see Section 1.1), each mode beingdeveloped as a r k/ expansion with k ≥ − σ ij = III ! p = I K p √ πr g ijp ( θ ) + T p k ijp ( θ ) + A p l ijp ( θ ) √ r + ... (4.1)where K p (the so-called stress intensity factors), T p ( T -stress) and A p are constantsdepending on the loading and the geometry of the sample. g ijp , k ijp and l ijp are universal Applying stress in the direction of the fat arrows, the crackadvances on average in the x direction. The crack front as afunction of time is parameterized by x ( z, t ) = w + f ( z, t ) , (420) y ( z, t ) = ˆ h ( z, t ) = h (cid:0) x ( z, t ) , z (cid:1) . (421)The quantity w = vt is the external control parameter usedthroughout this review. Several critical exponents can bedefined. Denote δh ( δx, δz ) := (cid:10) [ h ( x + δx, z + δz ) − h ( x, z )] (cid:11) , (422)where the average is taken over all x and z (and samples,if possible). The critical exponents ˆ β and ζ defined inthe literature are (we changed β → ˆ β in order to avoidconfusion with the exponent β defined in Eq. (299)) δh ( δx, ∼ δx ˆ β , δh (0 , δz ) ∼ δz ζ . (423)Scaling implies that Eq. (422) can be written as δh ( δx, δz ) = δx ˆ β f (cid:18) δzδx /z (cid:19) , (424) f ( u ) ∼ u ζ , for u large, and ζ = ˆ βz. (425)A third exponent ζ (cid:107) can be defined by the fluctuations of f , δf ∼ δz ζ (cid:107) . (426)As measurements are in general post-mortem , ζ (cid:107) isinaccessible, except if the broken material is transparent,and one can observe the crack front advancing. It hasbeen measured in a clever experiment where a crackwas filled with color, and broken open after the colorhad dried, confirming the small-scale regime ζ (cid:107) ≈ . [331]. Numerical simulations suggest [332] that a smallerexponent ζ (cid:107) ≈ . should hold at larger scales; anexperimental confirmation is outstanding [332]. Oneimaging option is to use a synchrotron; this challengingexperiment has to our knowledge not been attempted. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ˆ β ≈ . , and ζ ≈ . to . [331, 328]. The question arises whether there is aconnection to fracture along a fault plane, and depinning.To make contact to the latter, we first observe thatfracture is irreversible, a crucial point for depinning. Toproceed, note the 2-d vector (cid:126)u ( z, t ) = (cid:0) f ( z, t ) , h ( z, t ) (cid:1) . (427)Knowing the elastic kernel (18) for long-range elasticitywith α = 1 [92], the only Langevin equation linear in u ( z, t ) one can write down is (see e.g. [333, 334]) ∂ t (cid:126)u ( z, t ) = γ π (cid:90) z (cid:48) (cid:126)u ( z (cid:48) , t ) − (cid:126)u ( z, t )( z − z (cid:48) ) + η (cid:0) (cid:126)u ( z, t ) , z (cid:1) , (428) η ( x, y, z ) η ( x (cid:48) , y (cid:48) , z (cid:48) ) = σδ ( x − x (cid:48) ) δ ( y − y (cid:48) ) δ ( z − z (cid:48) ) . (429)Assuming that the scenario of Refs. [335, 336] holdsalso for long-range elasticity, then at large scales thelongitudinal exponent ζ (cid:107) should be that of a contact line,with according to Eqs. (417)-(419) an exponent ζ (cid:107) =0 . . The transversal roughness ζ ⊥ should then be thermal , which for α = 1 means logarithmically rough( (cid:82) k e ikz / | k | ∼ ln z ). This agrees with [333], and wasexperimentally verified in [337].The question arises whether this LR universality class,and especially the roughness exponent ζ = 0 . can be seenin an experiment. The first such experiment is in Ref. [73],using a very brittle material. The authors of this studyconjecture that [73] “both critical scaling regimes can beobserved in all heterogeneous materials:” For length scalessmaller than the process zone, the larger exponents ( ζ ≈ . , ˆ β ≈ . , z = ζ/ ˆ β ≈ . ) should be relevant, and thefracture surface was reported to be multi-fractal [338]. Forlarger scales the exponents are those of depinning ( ζ ≈ . , ˆ β = 0 . , z = ζ/β ≈ . ). However, these observationswere made for the transversal roughness, accessible post-mortem, whereas according to the scenario proposed aboveit should hold for the longitudinal roughness which is lessaccessible in experiments. The emerging consensus [332]of the community seems to be that the large-scale roughnessin the longitudinal direction is ζ (cid:107) ≈ . , whereas thetransversal roughness ζ ⊥ = 0 (logarithmic rough), asobserved in [337]; and that whenever a roughness of ζ ⊥ ≈ . has been observed, it has to do with physics related toshort scales (damage zone).What is the process-zone mentioned above? Thestandard theory for fracture is based on work by Griffith[339], with a crucial improvement by Irwin [340]. The ideaof Griffith [339] was to write an energy balance betweenthe stress released by the crack, and the surface energynecessary to create it. Irwin [340] realized that for ductilematerial, part of the released energy goes into a plasticdeformation, i.e. heat, at the crack front. The size of thezone affected is the process zone . It ranges from ξ =50 ± µ m for ceramics, over ξ = 170 ± µ m for aluminumto ξ = 450 ± µ m for mortar [338]. Fracture in thin sheets. In thin sheets, very differentroughness exponents have been reported: ζ = 0 . ± . for polysterene, and ζ = 0 . ± . for paper [341]. Wemay speculate that the largrer one is related to directedpercolation (section 5.8). Random fuse models. Random fuse models, a.k.a. damagepercolation have been proposed [342] as a model forfracture: Consider a regular lattice, where on each bondis placed a fuse of unit resistance, and a random maximumcarrying capacity i c , in most studies drawn from a uniformdistribution, i c ∈ [0 , . The system may be 2 or3-dimensional, with a voltage applied in one direction.To avoid finite-size effects due to the electrodes, it isadvantages to use periodic boundary conditions [343], withan additional voltage gain V in one dimension. The voltageis then ramped up from 0, until one of the fuses exceedsits carrying capacity, at which point it is considered broken,i.e. having an infinite resistance. One then recalculates thecurrent distribution and checks whether another fuse breaks.If not, one increases the applied voltage.This is an interesting model for fracture: (i) by solvingthe Laplace equation to find the current distribution, itincorporates the elasticity of the bulk of the material,providing an effective long-range elasticity; (ii) when a partof the material is broken, it is removed. It incorporatesingredients found in Laplacian walks (section 8.9) and DLA(solving in both cases the Laplace equation to determine themost likely point of action), and cellular automata as TL92(section 5.7).A roughness exponent of the fracture surface in d =2 + 1 was reported to be ζ = 0 . ± . [343], apparentlynot too different from some experiments [343]. Otherauthors focused on the distribution of strength, or brokenfuses upon failure [344, 345, 346]. A variant is the fiber-bundle model [347, 348]. Partial annealFull annealNo anneal RNADNA wu Figure 36. Peeling of a RNA-DNA double strand. The RNA sequence isfrom subunit 23S of the ribosome in E. Coli, prolonged to attach the beads(brown circles, with a much larger radius than drawn here). The DNAsequence is its complement. The beads sit in an optical trap (blue), at adistance w . (Drawing not to scale.) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles w [ µ m] F [pN] Figure 37. Left: A sample force-extension curve. For the data-analysisonly the last plateau part of the curve is used (in red). The effectivestiffness m in Eq. (297) is estimated from the slope of the green dashedlines as m = 55 ± /µ m at the beginning of the plateau, whichremains at least approximately correct at the end of the plateau. Thedriving velocity is about / s . - - w [ m ] ( w ) [ pN ] Figure 38. Measurements of ∆( w ) (in grey), with 1- σ error bars (greenshaded), compared to three theoretical curves: pure exponential decay(dotted red), 1-loop FRG, Eq. (82) (black dot-dashed), and DPM, Eq.(361) (blue dashed), all rescaled to have the same value and slope at u = 0 .Inset: theoretical curves with the data subtracted (same color code). Theblue curve is the closest to the data. The correlation length estimated from ∆( w ) is ξ = 0 . ± . µ m (cid:39) base pairs. There are two ways to open a double helix made out oftwo complementary RNA or DNA strand, or one RNA andits complementary DNA strand: peeling and unzipping.In both cases beads are fixed to the molecules, and thenpulled in an optical or magnetic trap. In the literature, theword peeling is used for the setup of Fig. 36, where forcesact along the helical axis from opposite extremities of aduplex, and one of the two strands peels off. Unzipping denotes an alternative setup where the right bead of Fig. 36 f c T = T > f c f ≫ f c fv Figure 39. Sketch of velocity force curve at vanishing ( T = 0 , depinning)and finite temperature ( T > , creep). For an experimental test see Fig. 44. is attached to the free end of the upper strand. As the readercan easily verify with a twisted thread, unzipping is mucheasier to accomplish than peeling. Let us start with peeling[349], for which a typical force-extension curve is shown inFig. 37. The stationary regime is the plateau part (in red).Averaging over about 400 samples, the effective disorder ∆( w ) defined in Eq. (109) is measured. The resultingcurve, including error bars for the shape [349], is shownin grey in Fig. 37 (right), where it is compared to threetheoretical curves: an exponentially decaying function (red,dotted, top curve), the DPM solution (361) for the Gumbelclass (blue, dashed, middle curve), and the 1-loop FRGsolution given by Eqs. (82) and (86), all rescaled to havethe same value and slope at u = 0 . The experiment clearlyfavors the DPM solution, best seen in the inset of Fig. 37.While this is expected, it is a nice confirmation of the theoryin a delicate experiment.One should be able to extract ∆( w ) also from theunzipping of a hairpin. Interestingly, experiments reportthat the scaling of Eq. (360) is replaced by [276] ρ m ∼ m − / , i.e. ζ = 43 . (430)This is a clear signature of a different universality class,namely “random-field” disorder in equilibrium, for whichthe roughness exponent (81) to all orders in (cid:15) reads ζ = (cid:15)/ ; setting (cid:15) = 4 leads to Eq. (430). An analyticsolution is given in section 2.22. This scenario is possiblethrough the much larger effective stiffness m there, whichmanifests itself in correlation lengths of ξ = 1 to basepairs, as compared to ξ = 186 base pairs for peeling.Equilibrium is observed experimentally [276] through avanishing hysteresis curve. In section 1.7, Eqs. (44)-(46), we had argued that inequilibrium the elastic energy scales as E el ( (cid:96) ) ∼ (cid:96) θ , θ = 2 ζ eq + d − α, (431)and as long as θ > the temperature T is irrelevant atlarge scales. On the other hand, if the driving velocity heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles v = 0 , and leaving the system enough time to equilibrate,it will be in equilibrium. As sketched in Fig. 39, thereare three different fixed points: equilibrium ( v = f = 0 , T → ), depinning ( T = 0 , v → or f → f c ),and large v or f , for which we expect ηv = f . Let usnow consider perturbations of the equilibrium fixed point,i.e. T small, and f (cid:28) f c , commonly referred to as the creep regime . Scaling arguments first proposed by Ioffeand Vinokur [350], and Nattermann [351], were later put onmore solid ground via FRG [352, 136]. Scaling argumentscompare the elastic energy (431) with the energy gainedthrough the advance of the interface, i.e. an avalanche ofsize S , E f ( (cid:96) ) = − f (cid:90) x δu ( x ) ≡ − f S ∼ − f (cid:96) d + ζ eq . (432)As ζ eq < α , the energy E f ( (cid:96) ) dominates over E el ( (cid:96) ) for large (cid:96) , and the optimal fluctuation is obtained for ∂ (cid:96) [ E el ( (cid:96) ) + E f ( (cid:96) )] ! = 0 , resulting in (cid:96) ζ eq − α opt ∼ f, or (cid:96) opt ∼ f − ν eq , ν eq = 1 α − ζ eq , (433) E opt ∼ f − µ eq , µ eq = ν eq θ = 2 ζ eq + d − αα − ζ eq . (434)This identifies the creep law as v ( f, T ) = v e − T ∗ T ( f c f ) µ eq , f (cid:28) f c . (435)We remind that for depinning (see Eqs. (299) and (303)) v ∼ ( f − f c ) β , f ≥ f c , (436)and that for large fv (cid:39) fη , f (cid:29) f c . (437)There are thus three regimes, sketched in Fig. 39: f (cid:28) f c ,the creep regime discussed above, governed by the T = 0 equilibrium fixed point; T = 0 , and f ≈ f c , the depinningfixed point; and the large- f and large- v regime, where thedisorder resembles a thermal white noise, with amplitudeproportional to /v . The latter can be understood from therelation ∆( w ) (cid:39) δ ( w ) = δ ( vt ) = 1 v δ ( t ) . (438)More precisely, it looks like a thermal noise withtemperature T = 1 v (cid:90) ∞ d w ∆( w ) . (439) Creep in simulations. In d = 1 , the creep law (435) wasverified numerically [354, 355, 356, 357, 358, 359] both forrandom-bond and random-field disorder, and short-rangedelasticity ( α = 2 ): ζ RBeq = 23 = ⇒ µ RBeq = 14 , (440) ζ RFeq = 1 = ⇒ µ RFeq = 1 . (441) Figure 40. Experimental confirmation of the creep-law ln( v ) ∼ f − µ inan ultrathin PtCoPt film [83]. Tested is the hypothesis µ RBeq = 1 / ; asthe added orange line shows, a larger value of µ eq ≈ / should improvethe fit, consistent with a value of ζ > / . One might see the beginningof the large-scale regime with ζ = ζ dep = 5 / , see e.g. [353]. A recentexperiment is shown in Fig. 44. The numerical work [354, 356, 357, 358, 359] was possiblethrough the realization that for T → the sequence ofstates, in which the interface rests, becomes deterministic,and can be found by a clever enumeration of all possiblesaddle points. Creep in Experiments. The exponent µ RBeq = 1 / wasfirst found experimentally in Ref. [83], and later confirmedin numerous other magnetic domain-wall experiments [47,360, 361, 362, 363]. These experiments use a Kerrmicroscopic to image the domain wall; a sample image isgiven in Fig. 1. At large scales, the domain-wall roughnessis expected to cross over to ζ qKPZ = 0 . or ζ dep = 1 . ,see the discussion in section 3.21.Creep motion was in less depth also studied invortex lattices [364], fracture experiments [365, 366], andquantum systems [187, 367, 368].For f = f c (critical driving), Ref. [369] claims that v ( T, f = f c ) ∼ T χ , χ = d + 26 − d . (442)In the fixed-velocity ensemble, the scaling (299) suggests lim v → [ (cid:104) f (0 , v ) (cid:105) − (cid:104) f ( T, v ) (cid:105) ] ∼ T χ/β . (443) In most of this review we studied situations where thesystem is equilibrated, either in its ground state, or inthe steady state. One may ask how it reacts to aquench. This question was first considered for model A(Langevin dynamics for φ -theory, classification of [282])in Ref. [370]. There one starts with a system at T (cid:29) T c ,where correlations vanish. At t = 0 one quenches it to heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles w - - - theory ( w )- ( w ) w ( w ) Figure 41. Measured force-force correlator ∆( w ) for a 200nm ribbon ofFeSiB, a bulk magnet with SR elasticity [371] (in grey, with error bars inshaded green, arbitrary units), after correcting for the finite driving velocity(section 3.11). This is compared to several theoretical curves (from top tobottom): an exponential function (red, dotted), the DPM correlator (361)(blue, dashed), FRG resummation (362) for (cid:15) = 2 (orange, dashed), and1-loop (black, dot-dashed). Error bars are at confidence level. Theinset shows theory minus measurement, favoring the FRG fixed point at (cid:15) = 2 (with error bars for this curve only). T = T c . The response function R ( q, t w , t ) then dependson t where one measures the field, and a waiting time t w < t at which a small kick was performed. For disorderedelastic manifolds, the state with vanishing correlations isa flat interface. It can be obtained either by imposing u ( x, t = 0) = 0 , or by moving the interface with a verylarge velocity v (cid:29) up to t = 0 . Or by switching on thedisorder at time t = 0 .Scaling implies that R takes the scaling form R ( q, t w , t )= (cid:16) tt w (cid:17) θ R ( t − t w ) − zz f R ( q z ( t − t w ) , t/t w ) ,f R ( x, y ) → const ∀ x → , or y → ∞ . (444)A similar ansatz holds for the correlation function. Eq.(444) would simplify if θ R ? = z − z . (445)In model A, θ R violates Eq. (445) at 2-loop order [370] .For disordered elastic manifolds at depinning, Eq. (445) issatisfied at 2-loop order [373, 374], but may be violated insimulations in d = 1 [374]; to decide the matter, largersystems need to be simulated [315].Technically, the two calculations are rather different:Imposing φ ( x, t = 0) = 0 in model A amounts tousing Dirichlet boundary conditions. New divergences thenappear between fields φ ( x, t ) , and their mirror images at t < . For disordered elastic manifolds no mirror imagesappear, and one can simply switch on the disorder at t = 0 , Ref. [370] does not state the (cid:15) expansion for θ R , but for a related object η . The missing relation is θ R = − η z , confirmed in [372]. w Δ theory ( w )- Δ ( w ) w Δ ( w ) Figure 42. Measured force-force correlator ∆( w ) as in figure 41, for FeSi7.8 % , a bulk magnet with strong dipolar interactions, making the elasticitylong-ranged [371]. The FRG prediction for ∆( w ) is the 1-loop fixed point(section 3.8). For better visibility, error bars are at confidence level,not accounting for a remaining oscillation from the power grid with periodin w of about . . reducing the possibility for independent divergences; it maythus well be that relation (445) remains valid at higherorders.The situation simplifies in the limit of t w → .Standard power counting then implies that (cid:104) ˙ u ( t ) (cid:105) ∼ t − βνz ≡ t ζz − . (446)Similarly, the squared interface width grows as (cid:68) L − d (cid:82) x,y [ u ( x ) − u ( y )] (cid:69) → t ζz . (447)In Ref. [285] these relations were used in simulations ofsystem sizes up to L = 2 to give the most precise(direct) estimation of the two independent exponents ζ and z in dimension d = 1 , yielding ζ = 1 . ± . , ν =1 . ± . , β = 0 . ± . , and z = 1 . ± . .This should be compared to the values conjectured to beexact reported on Fig. 25.A quench has also been studied in the Manna model[375, 376, 377], and interpreted as a dependence of thedynamical exponent z on the initial condition. As z is abulk property, this is hard to believe. It seems [315] that thesystems used in the simulations are too small to be in theasymptotic regime. d = 2 ) To our knowledge, bulk magnets are the only systemto realize depinning of a 2-dimensional manifold. Twouniversality classes need to be distinguished [300, 378]: • magnets with short-ranged elastic interactions, as theIsing model, for which α = 2 (notations as in section1.3), and (cid:15) = 2 . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Figure 43. Experimentally observed scaling of avalanche sizes (left) andsquared width (right) for magnetic domain walls in ferromagnetic Pt/Co/Ptthin films. The solid lines indicate the exponent expected from depinning( ζ = 1 . ), whereas dashed lines are for qKPZ ( ζ = 0 . ). From [379],with kind permission. • magnets with strong dipolar interactions, which havelong-range elasticity with α = 1 , thus d c = 2 is theupper critical dimension (see section 3.14).For dynamic properties the influence of eddy currents,which varies from sample to sample, needs to be takeninto account. A simple model is discussed in section 4.21.Here we consider the renormalized disorder correlator, forwhich eddy currents are less important. The signal obtainedexperimentally is the current induced in a pickup coil,which we identify as ˙ u ( t ) , the velocity of the center ofmass of the interface. Integrating once yields u ( w = vt ) . ∆ v ( w ) is its auto-correlation function defined in Eq. (382).Using the unfolding procedure of Eq. (391), and discussedin section 3.11, allows one to extract the zero-velocity limit ∆( w ) . The latter is plotted on Fig. 41 for the SR sample,and on Fig. 42 for the LR sample. For the SR sample, weexpect ∆( w ) to be closest to the resummed loop expansionat (cid:15) = 2 , as given in Eq. (362). For the LR sample, weexpect the FRG-fixed point at the upper critical dimension(section 3.8), equivalent to the 1-loop FRG fixed point (82)–(86). In both cases, the agreement is very good, and clearlyallows us to distinguish between the different universalityclasses. Let us stress that while the LR sample has criticalexponents consistent with the ABBM model [300], thedisorder correlator ∆( w ) is clearly distinct from the onein ABBM, given in Eq. (486).The measured ∆( w ) can be compared to the correlatorfor RNA-DNA peeling on Fig. 38 ( d = 0 ), and to contact-line depinning ( d = 1 , α = 1 ) on Fig 33. Note that for thelatter the boundary layer due to the finite driving velocity(section 3.11) was not unfolded. d = 1 ) (i) Guided by a theoretical framework based on creep, thefirst experiments were interpreted as a perturbation ofthe RB equilibrium fixed point with ζ = 2 / [83],even though configurations seem to be frozen. As ∆( w ) for ABBM is not renormalized, we should measure it if themicroscopic disorder were of the ABBM type. Experimental details - The studied sample consists ina Ta(5 nm)/Gd Fe . Co . (10 nm)/Pt(5 nm) trilayerdeposited on a thermally oxidized silicon SiO (100 nm)substrate by RF sputtering. The GdFeCo thin filmis a rare earth-transition metal (RE-TM) ferrimagneticcompound presenting perpendicular magnetic anisotropywith the RE and TM magnetic moments antiferromag-netically coupled. DW motion is measured using PolarMagneto-Optical Kerr E↵ect (PMOKE) microscopy. Weobtain images of DWs before and after the applicationof square-shaped magnetic field pulses of amplitude H and duration t , and measure the resulting mean DWdisplacement x . The mean DW velocity v , characteriz-ing its collective motion, at a given field H is calculatedas x/ t . The maximum investigated field is limitedby the nucleation of multiple magnetic domains, whichimpedes clear DW displacement measurements. Addi-tionally, the microscope is equipped with a cryostat anda temperature controller that give us access to a widetemperature range, from 4 K to 360 K. Zero-temperature-like depinning transition - The con-tribution of thermal e↵ects on the depinning transitionis shown in Fig. 1(a), which compares two velocity-field curves with very similar depinning threshold ( H d ⇡ 15 mT) measured at temperatures that di↵er in morethan one order of magnitude (20 K and 295 K). Asit can be observed, thermal activation blurs the depin-ning transition. Above H d , velocities measured at bothtemperatures are rather similar. However, close to H d the velocity sharply decreases for T = 20 K while it re-mains finite, even well below H d , for T = 295 K. [28].The curve obtained for T = 295 K (Fig. 1(a)) presentsboth the typical thermally activated behavior for lowfields and the thermal rounding around H d , as has beenreported in the literature for most of the field- andcurrent-driven experiments [29, 30]. Below the depin-ning threshold (3 mT < H < H d ), the DWs follow athermally activated creep dynamics. The velocity obeysthe Arrhenius law v ( H, T ) = v ( H d , T ) e E/k B T , with k B the Boltzmann constant. The universal creep bar-rier is given by E = k B T d h ( H/H d ) µ i [29], where T d is the so-called depinning temperature and µ is thecreep exponent ( µ = 1 / 4) [3, 10, 17]. As observed inFig. 1(a) (see the inset), the experimental data measuredat T = 295 K, which covers more than eight orders ofmagnitude, are in good agreement with the creep law.Remarkably, the depinning temperature estimated fromthe fit, T d ⇡ T d reportedin the literature [31]. Moreover, for the lowest temper-ature at which we can observe a clean creep dynamics, T = 100 K, the value of T d is close to ⇡ T d /T ( > µ H [mT]050100150200250300350 v [ m / s ] µ H d (a) T = 20 K T = 295 K . . . ( µ H/ mT) / . . . . . . H H d ) /H d ]4 . . . . . l n [ v / ( m / s ) ] T = 20 K (b) ln[ v / (m / s)] + ln[( H H d ) /H d ] Figure 1. High- and low-temperature velocity curves. (a)Domain wall velocity ( v ) as a function of the external mag-netic field ( µ H ) for a GdFeCo thin film at T = 20 K (circles)and T = 295 K (squares). The depinning field µ H d corre-sponding to T = 20 K (vertical dashed red line), and the fitof Eq. (1) for that temperature (dashed black line) are indi-cated. In the inset, in semi-logarithmic scale, the thermallyactivated creep regime observed for T = 295 K is emphasizedwith the dotted line, a fit to the creep formula. (b) Resultsobtained at T = 20 K and plotted in a logarithmic scale toillustrate an excellent agreement with Eq. (1). The fittedparameters are = 0 . ± . µ H d = 14 . ± . v = 270 ± 40 m / s. The universal velocity depinning exponent - The ra-tio T d /T is strongly enhanced as the temperature de-creases, thus explaining the absence of creep regime andthe quasi-athermal depinning behavior observed for theTa/GdFeCo/Pt film at T = 20 K. Then the velocity-fielddata at T = 20 K can be analyzed as a zero temperature-like depinning transition. Quantitative information isthus obtained by fitting the data using [14, 17, 32–36] v ( H, T = 0 K) = v ✓ H H d H d ◆ . (1)In Eq. (1), the depinning velocity v and H d are ma-terial and temperature dependent parameters [14, 33].As shown in Fig. 1(b), the data presents a good quan-titative agreement with the prediction over the wholerange of measured magnetic fields (see [36] for detailson the fitting procedure). Our direct experimental mea-surement of (= 0 . ± . 08) somehow justifies the valuesassumed in former experimental studies of depinning [14]( = 0 . Figure 44. The velocity as a function of applied force f = µ H for athin GdFeCo film (top). The sharp transition at T = 20 K is round at T = 295 K. The bottom plot shows the quality of the determination of β = 0 . ± . . The inset shows the creep law (435) with µ eq = 1 / .Figure from [380], with kind permission. (ii) Creep exponents are reported [361, 363, 381] withouta measurement of the roughness.(iii) A roughness exponent of ζ ≈ . was reported [382]together with a plateau of the 2-point function forlarge distances due to the confining potential, herea consequence of dipolar interactions. (The plateauwas interpreted as a smaller roughness ζ ≈ . , aconclusion we do not share.)(iv) Bound pairs of domain walls are reported [383], withno clear theoretical interpretation in terms of thephenomena discussed here.(v) Ref. [384] shows clear evidence for the negativeqKPZ class for current induced depinning, and for thepositive qKPZ class for field-induced depinning. Wediscuss this experiment in section 5.12.(vi) Refs. [274, 275] give the most complete analysisof the roughness exponent to date. The beautifulimage shown in Fig. 45 qualitatively confirms thefindings of [384]. Note the strong up-down asymmetrywhich is inconsistent with an equilibrated systems.Nevertheless, the equilibrium RB fixed point with ζ eq = 2 / is still the key theoretical class theexperiments are compared to. In the case of facetingas on Fig. 45, the analysis is applied to fluctuations of heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles R. DÍAZ PARDO et al. PHYSICAL REVIEW B , 184420 (2019) DWs produced by current and magnetic field remains an openquestion. Moreover, the origin of STT-induced DW tilting andits contribution to DW dynamics and universal behaviors arenot well understood.In this paper, we propose an extensive and comparativestudy on DW motion driven by STT and magnetic field.Investigations were conducted with the ferromagnetic semi-conductor (Ga,Mn)As, which presents a sufficiently low de-pinning threshold [24] to cover the creep, depinning, and flowdynamical regimes [25], and to discuss, in particular, DWtilting without pinning. Our stringent test of universality relieson the verification that magnetic-field-driven DW motion inour sample shares the common universal (material and tem-perature independent) behaviors encountered in a variety ofother magnetic materials [4,5,9–11] and on the independentanalysis of DW dynamics and roughness, which are bothconsistent with the conclusion of common universal behaviorsfor STT and field-driven DW motion.The paper is organized as follows. After a descriptionof the experimental techniques (Sec. II), we compare theshape evolution of DWs driven by STT and magnetic fieldand analyze their roughness (Sec. III). Section IV details acomparative study of DW dynamics. In Sec. V, we discuss theorigin of DW faceting produced by STT and its implicationsfor DW dynamics. II. EXPERIMENTAL TECHNIQUES The experiments were performed with rectangles of a4-nm-thick (Ga,Mn)(As,P) / (Ga,Mn)As bilayer film patternedby lithography. The film was grown on a (001) GaAs / AlAsbuffer [26]. It has an effective perpendicular anisotropy and aCurie temperature ( T c ) of 65 K. The sizes of rectangles were133 × × × µ m (see Appendix Afor the details). Two 40- µ m-wide gold electrodes (separatedby 110, 204, and 300 µ m, respectively) were deposited byevaporation parallel to the narrow sides of rectangles. Theywere used to generate a homogeneous current density drivingDW motion by STT. The pulse amplitude varied between0 and 11 GA / m . We verified that the Joule effect had anegligible contribution on DW dynamics (see Appendix B).Perpendicular magnetic field pulses of adjustable amplitude(0–65 mT) were produced by a 75-turn small coil (diameter ∼ ∼ µ m). The DW velocity is defined as the ratio between theaverage displacement ⟨ u ⟩ and the pulse duration ! t , whichvaries between 1 µ s and 120 s. The setup was placed againstthe cold finger of an optical He-flow cryostat, which allowedthe temperature to be stabilized to between 5.7 K and T c . III. DOMAIN-WALL DISPLACEMENT AND ROUGHNESSIN THE CREEP REGIME The time evolution of an initially almost flat DW drivenby magnetic field and STT is compared in Fig. 1. For field-driven DW motion, the average successive displacements arerelatively similar. The initial DW shape is almost conservedduring the motion. The DWs become sometimes stronglypinned and curved [see Fig. 1(a)] but are flattened again when FIG. 1. Time evolution of DW shape. Successive positions of aDW, at T = 28 K, driven by (a) magnetic field ( H = . 16 mT, delaybetween images ! t = . j ≈ . / m , ! t = . depinned due to the combined effects of DW elasticity anddriving force, which acts as a pressure ( f ∝ H ). In contrast,the initial DW shape is significantly altered by the current[see Fig. 1(b)]. DWs are tilted and form faceted structures.The faceting process seems to start close to “strong” pinningcenters [5] and to produce a reduction of DW displacementswith increasing tilting angle θ until DWs stop (on the ex-periment timescale), as already observed in Pt / Co / Pt films[5]. In Sec. V, we will show that the faceting results froman instability of the transverse alignment between current andthe DW, which can be triggered even without any contributionfrom pinning.Let us start with investigations of universal behaviorswith a study on DW self-affinity using the displacement-displacement correlation function [4]: w ( L ) = ! x [ u ( x + L ) − u ( x )] , (1)where u ( x ) is the DW displacement measured parallel to thecurrent and L the length of DW segment along the x axis trans-verse to the current [see Fig. 2(a)]. For a self-affine interface,the function w ( L ) is expected to follow a power-law variation w ( L ) ∼ L ζ , where ζ is the roughness exponent. Typicalvariations of w vs L obtained for field- and current-inducedmotion are compared in Fig. 2(b) in log-log scale. As it can beobserved, both curves present a linear variation with similarslopes ( = ζ ), between the microscope resolution ( ≈ µ m)and L = µ m. In order to get more statistics, the slopes weresystematically determined for successions of DW positionsand a temperature varying over one decade ( T = . ζ j )- andfield ( ζ H )-driven DW motion is reported in Fig. 2 as a functionof temperature. As expected for universal critical exponents,the values of ζ j and ζ H do not vary significantly. Their meanvalues ( ζ j = . ± . 05 and ζ H = . ± . / Co / Pt in Ref.[5]. For (Ga,Mn)(As,P), ζ j presents no significant variation Figure 45. Fig. 1 from [274] (with kind permission), showing the timeevolution of the domain-wall shape. (a) Successive positions at T = 28 K,driven by a magnetic field ( H = 0 . mT, delay between images t = 0 . s,total duration ). (b) ibid. current driven ( j ≈ . / m , t = 0 . ,total duration s .) observed at the same sample location. The DW movesin the direction opposite to the current density, which is indicated by thearrow. The initial DW position is underlined by a thick dashed line. Thetriangles indicate the strongest DW pinning positions. the facets itself.(vii) For a thin antiferromagnetic GdFeCo film, exponentsconsistent with the 1-dimensinal RF depinning classwere found: β = 0 . ± . and ν = 1 . ± . [380], see Fig. 44. In particular the determination of β is remarkable. The sample in question has a vanishingmass, m ≈ . Direct confirmation of a roughness ζ > ζ qKPZ is more tentative [379], see Fig. 43.(viii) Experiments for alternating drive [385].(ix) More experimental results can be found in Ref. [386].To conclude: Evidence for the equilibrium RB universalityclass with ζ = 2 / (sections 2.4 and 2.12) seems toevaporate in favor of the quenched KPZ class with ζ = 0 . (section 5.7). At small scales depinning without KPZ-terms is visible [379], but remains to be confirmed. Ourconclusion is that at short scales KPZ terms are absent,leading to the RF-depinning class with ζ = 1 . . At largerscales, the KPZ term becomes relevant, and one crossesover to one of the qKPZ classes: positive qKPZ for field-induced driving, and negative qKPZ for current-induceddriving, see Fig. 45 and 63 (page 79). As we have seen in sections 3.1, 3.4 and 3.9, hysteresis in adriven disordered system is a sign of a non-vanishing forceat depinning. In a real magnet the overall magnetizationis bounded, thus the critical force depends on where oneis on the hysteresis loop. This allows one to invent aplethora of protocols: One can try to get as close aspossible into equilibrium by ramping up and down themagnetic field, while reducing the amplitude of the fieldin each cycle. One can also study sub-loops , by varyingthe applied field in a much smaller range than necessaryfor a full magnetization reversal. The reader wishing toenter the Science of Hysteresis can find a book with thistitle [378], or one of the many original research articles[387, 388, 389, 390]. There are few analytical result, a notable exception being the hysteresis curve in the ABBMmodel [391]. Inertia plays an important role in everyday-life experienceswith depinning: When we were children, we pulleda block or a cart with the help of an elastic string,observing stick-slip motion, with a certain periodicity. At ahigher frequency stick-slip motion may be observed whenbreaking with our bike or opening a door, often amplified bya resonance excited in the medium. Despite its ubiquity ineveryday life, stick-slip motion is absent in the avalanchephenomenology discussed above. The reason for thisabsence is the modeling of the equation of motion (297)or (305) via an overdamped Langevin equation, neglectinginertia.Here is a good occasion to remind us how theoverdamped Langevin equation (305) is derived fromNewton’s equation of motion for depinning ( w = vt ): M ∂ t u ( x, t ) = − η∂ t u ( x, t ) + ( ∇ − m )[ u ( x, t ) − w ]+ F (cid:0) x, u ( x, t ) (cid:1) . (448)Inertia M times acceleration ∂ t u ( x, t ) are balanced byfriction − η∂ t u ( x, t ) , forces exerted by the elasticity of theinterface ( ∼ ∇ u ( x, t ) ), the confining well m [ u ( x, t ) − w ] and disorder F ( x, u ) . Neglecting inertia, i.e. setting M → yields back the standard equation of motion (305), writtenthere with η = 1 .Assuming an exponential behavior u ( t ) ∼ e − t/τ , thetime scale τ satisfies M τ − − ητ − + m = 0 . (449)The solution for M → starts with τ = η/m ; when M reaches M c = (cid:16) η m (cid:17) (450)this solutions splits into two complex ones, and movementbecomes oscillatory. This can be interpreted as a dynamicalphase transition. One may conjecture that this remains validfor an extended elastic system. This was indeed observedin MF theory [392]. A careful scaling analysis shows thatthis extends to systems below the upper critical dimension,where MF theory is no longer valid. Analytic progress canbe made [393] for various toy models generalizing ABBM(section 4.3), i.e. quenched forces which have the statisticsof a random walk. All these models share a common large-deviation function (a concept discussed below) fora large driving velocity v . They differ in how returns,which are difficult to incorporate in the field theory, aretreated. If instead of returning on the same quencheddisorder, new random forces are generated with the samestatistics of a random walk, then despite dissipation onefinds a new active steady state in the limit of a vanishingdriving velocity v → . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Large-deviation function. An interesting concept, wellstudied in the literature of driven systems, is the large-deviation function, see e.g. [394, 393, 395, 396, 397]. Set F v ( x ) and Z v ( λ ) to be F v ( x ) := − ln (cid:0) P ( xv ) (cid:1) v , Z v ( λ ) := ln (cid:0) e λ ˙ u (cid:1) v . (451)Define F ( x ) and Z ( λ ) as the large- v limits, if they exist, ofthe above functions, F ( x ) := lim v →∞ F v ( x ) , Z ( λ ) := lim v →∞ Z v ( λ ) . (452)The existence of the second limit can be shown in fieldtheory. In simple models, Z v ( λ ) is even independent of λ , see Eq. (516) in a simpler setting. Supposing the latter,we obtain e λ ˙ u = e vZ ( λ ) = (cid:90) ∞ d ˙ u e λ ˙ u P ( ˙ u )= λ (cid:90) ∞ d x e v [ λx −F v ( x )] . (453)If v is large, the latter integral can be approximated by itssaddle point with λ = ∂ x F v ( x ) . We recognize a Legendretransform, Z ( λ )+ F ( x ) = λx, λ = ∂ x F ( x ) , x = ∂ λ Z ( λ ) . (454)As by assumption Z ( λ ) does not depend on v , this showsthat also the first limit in Eq. (452) exists, and F v ( x ) ≡F ( x ) independent of v . The large-deviation function forthe FBM model defined in section 4.5 with an additionalinertia term as in Eq. (448) can then be constructed. Settingfor simplicity m = η = 1 , it reads F ( x, M ) = x − ln( x ) − M (cid:104) x − x − ln( x ) (cid:105) + O ( M ) , = (1 − x ) − x ) M ) + ..., (455)where on the second line terms of order (1 − x ) have beendropped. Units are restored by [393] P v (cid:29) ( ˙ u ) (cid:39) e − v ηm σ F ( ˙ uv , Mm η ) . (456)A similar form holds for the joint distribution of velocitiesand accelerations. Most systems and their deformations discussed so farare elastic , indicating that conformational changes arereversible, and nearest-neighbor relations fixed. There arenumerous systems where this is not the case, as in shearedcolloidal systems, termed plastic for their irreversibledeformations. The question relevant for us is how muchof the phenomenology and methodology developed fordisordered elastic manifolds carries over to plastic systems.The gap has not been bridged yet, but efforts have beenundertaken starting from disordered elastic manifolds, [398,399, 400, 401, 402, 403, 404, 405], and plastic (mostlysheared colloidal) systems [406, 407, 408, 409]. A goodstarting point for the latter is the review [410]. A single vortex line in 3-dimensional space, driven throughquenched disorder is argued [335, 336] to have the statisticsof a depinning line in the driving direction ( ζ (cid:107) = 5 / ,section 3.13) and to have Gaussian fluctuations ( ζ ⊥ = 1 / )in the transversal direction. While the dynamical exponent z (cid:107) = 10 / in driving direction seems to be unchanged, theperpendicular dynamical exponent is argued to be larger, z ⊥ = z (cid:107) + 2 − ζ = 6128 = 2 . ... (457)Note that we updated the values for the exponents of[335, 336] to today’s best estimates (section 3.13).A defect line binds a vortex line more strongly thanpoint disorder. Tilting the sample such that the columnardefect no longer aligns with the magnetic field, oneobserves an unbinding transition of the vortex line, knownas the transverse Meissner effect [25, 411, 412, 215, 413].This is also observed as an effective model for slidingcharge-density waves [414].In the above setting, forces are assumed to bederivatives of a potential, i.e. conservative. If they arenon-conservative, as e.g. in presence of stable advectingcurrents, then a new universality class is reached,accessable perturbatively [415, 416, 417].In section 2.25 we had shown experimental andtheoretical evidence for the existence of an ordered phase invortex lattices (Bragg glass) at weak disorder. Refs. [418,419] argue that the Bragg-glass phase is stable w.r.t. slowdriving, with the lattice responding by flowing throughwell-defined, elastically coupled, static channels. If thelattice is preserved, then after it has moved by a full latticeconstant, it comes back to its original configuration. In thiscase, one expects the velocity to be periodic in time [420].In [421] it was found that translational order in thedriving direction can be destroyed. Exponents are not the only interesting observables: Inexperiments and simulations, often whole distributions canbe measured, as e.g. the width distribution of an interfaceat depinning [422, 50, 423]. Be (cid:104) u (cid:105) its spatial average for a given disorder configuration, then the width w := 1 L d (cid:90) x ( u ( x ) − (cid:104) u (cid:105) ) (458)is a random variable, with distribution P ( w ) . The rescaledfunction Φ( z ) , defined by P ( w ) = 1 /w Φ (cid:16) w /w (cid:17) (459)is expected to be universal, i.e. independent of microscopicdetails and the size of the system.Supposing u ( x ) to be Gaussian, Φ( z ) can becalculated analytically to leading order. It depends on two heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles F ( z ) z analytic resultsimulation, L = 256 Figure 46. Scaling function Φ( z ) for the ( )–dimensional harmonicmodel, compared to the Gaussian approximation for ζ = 1 . . Data from[422]. x - - - u Figure 47. Temporal evolution of an avalanche starting at x = 4 at t = 0 ,evolving to the top until time T . Interface positions at intermediate times t = iT/ are shown for i = 1 , , ... . parameters, the roughness exponent ζ and the dimension d .Numerical simulations [422, 50] displayed on Fig. 46 showagreement between analytical and numerical results. Thedistribution is distinct from a Gaussian.There are more observables of which distributionshave been calculated within FRG, or measured in simu-lations. Let us mention fluctuations of the elastic energy[424], and of the depinning force [260, 48]. 4. Shocks and avalanches When slowly driving a system, long times of inactivity arefollowed by bursts of activity, which on these long timescales look instantaneous. Observables of interest are • center-of-mass position u ( t ) := 1 L d (cid:90) x u ( x, t ) , (460) • center-of-mass velocity ˙ u ( t ) := ∂ t u ( t ) , (461) • duration T , see Fig. 48. This quantity is well-defined,as every avalanche stops at some point (see section4.4), so T := inf t { t, ˙ u ( t ) > } < ∞ . (462) • shape (cid:104) ˙ u ( t ) (cid:105) . • avalanche size S , S := (cid:90) x δu ( x ) = (cid:90) x u ( x ) − u ( x ) , (463)where u ( x ) is the interface position before, and u ( x ) after an avalanche, see Figs. 47 and 48.These are the main observables. Setting and language hereare for depinning. Some observables, as the avalanche-sizedistribution can also be formulated in the statics: These static avalanches , also termed shocks , are the changes inthe ground-state configuration upon a change in the appliedfield, i.e. the position w of the confining potential. We willcomment on differences between these two concepts at theappropriate positions. Key points are • avalanches are the response of the system to anincrease in force. They have a typical size S m := (cid:10) S (cid:11) (cid:104) S (cid:105) ∼ ξ d + ζ ∼ m − ( d + ζ ) . (464)In the literature one sometimes finds the notation D = d + ζ for the fractal dimension of an avalanche. • the avalanche-size distribution per unit force, ρ f ( S ) := δN ( S ) δf (cid:39) S − τ f S ( S/S m ) g S ( S/S ) ,S (cid:28) S m , (465)has a large-scale cutoff S m defined in Eq. (464) due tothe confining potential, and a small-scale cutoff S dueto the size of the kick or discretization effects (as in aspin system). The scaling functions are expected tohave a finite limit when m → , i.e. lim x → f S ( x ) = const, and lim x →∞ g S ( x ) = 1 . • An increase δf in the total integrated force is thenon average given by an increase δu ( x ) in u , whichintegrated over space gives S . On the other hand,we can integrate Eq. (465) over S . Together, theserelations give δf = m (cid:90) x (cid:104) δu ( x ) (cid:105) = m (cid:104) S (cid:105) = δf m (cid:90) ∞ d S Sρ f ( S ) ∼ δf m (cid:2) S − τm − O ( S − τ ) (cid:3) . (466)As we will see below τ < , and the last term can bedropped for m → . This yields m ∼ S τ − m . (467) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ( µ m) ( µ m ) s p a t i a ll y a v e r a g e d h e i g h t plate position avalanches S mean shape at fixed duration avalanche duration T avalanche size S = area under curveand its fluctuationsvelocity Figure 48. Left: The mean spatial height in the contact-line depinning experiment of Fig. 22. Right: Increasing the time resolution to resolve a singleavalanche (jump, marked by the arrow), the velocity inside a single avalanche can be viewed as a random walk with absorbing boundary conditions atvanishing velocity. This allows us to define observables as the mean avalanche shape, the size S (area under the curve), or its duration T . Inserting S m from Eq. (464) then yields τ SR = 2 − d + ζ . (468)In d = 1 and with ζ = 5 / this gives τ d =1SR = 109 = 1 . ... (469) • LR-elasticity : Here one replaces m → m α , leadingto τ α = 2 − αd + ζ . (470) • Alternative scaling argument : In Ref. [425] it wassuggested in the context of sandpile models (seesection 6.6) that a single grain performs a randomwalk which has to reach the boundary, implying that (cid:104) S (cid:105) ∼ L . Using (cid:104) S (cid:105) ∼ S − τm , and L ∼ m − leads to (2 − τ )( d + ζ ) = 2 , equivalent to Eq. (468). • Boundary driving : when a (SR-elastic) system isdriven at the boundary (tip driving [426]) there is adrift (advection) away from this boundary , leadingto a linear scaling, (cid:104) S (cid:105) ∼ L , and as a consequence (2 − τ )( d + ζ ) = 1 [427, 428, 426], τ tipSR = 2 − d + ζ . (471)In d = 1 and for ζ = 5 / this gives τ tip ,d =1SR = 149 = 1 . ... > . (472) • the distribution of avalanche sizes in a submanifold φ of dimension d φ , ρ φf ( S φ ) ∼ S − τ φ , S φm (cid:29) S φ (cid:29) S φ ,τ φ = 2 − αd φ + ζ . (473)The derivation proceeds as for Eq. (468). We do not understand the first-return-to-the-origin argument ofRef. [427]. • avalanche size and duration are related via S m ∼ T γm , γ = d + ζz . (474)This is obtained from the scaling relations S m ∼ m − d − ζ , and T m ∼ m − z . • the (unnormalized) duration distribution per unit forceis ρ f ( T ) ∼ T − ˜ α , T m (cid:29) T (cid:29) T , T m = (cid:10) T (cid:11) (cid:104) T (cid:105) . (475)The integral relation ρ ( S )d S = ρ ( T )d T implies S − τm ∼ T − αm . Setting S m ∼ m − ( d + ζ ) , and T m ∼ m − z yields with the help of Eq. (470) ˜ α = 1 + d + ζ − αz . (476) • the (unnormalized) velocity distribution ρ f ( ˙ u ) ∼ ˙ u − a , ˙ u m (cid:29) ˙ u (cid:29) ˙ u , ˙ u m = S m τ m . (477)The exponent a is obtained from arguments similarto those used in the derivation of Eqs. (468) and(473), with the result that in the denominator thedimension of the obeservable in question appears. Forthe velocity distribution it yields a = 2 − αd + ζ − z . (478) • avalanche extension: In general, avalanches have awell-defined spatial extension (cid:96) , allowing us to definetheir distribution ρ f ( (cid:96) ) . If (cid:96) (cid:28) ξ = 1 /m , then (cid:96) , and not ξ is the relevant scale, and S ∼ (cid:96) d + ζ .Writing ρ f ( S )d S = ρ f ( (cid:96) )d (cid:96) allows us to conclude[429] that for extensions between the lattice cutoff a and ξ = 1 /m , ρ f ( (cid:96) ) ∼ (cid:96) − k , a (cid:28) (cid:96) (cid:28) m , k = d + ζ + 1 − α. (479) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ρ ( S ) ρ ( S φ ) ρ ( T ) ρ ( ˙ u ) ρ ( ˙ u φ ) ρ ( (cid:96) ) S − τ S − τ φ φ T − ˜ α ˙ u − a ˙ u − a φ φ (cid:96) − k SR elasticity τ = 2 − d + ζ τ φ = 2 − d φ + ζ ˜ α = 1 + d − ζz a = 2 − d + ζ − z a φ = 2 − d φ + ζ − z k = d + ζ − LR elasticity τ = 2 − d + ζ τ φ = 2 − d φ + ζ ˜ α = 1 + d − ζz a = 2 − d + ζ − z a φ = 2 − d φ + ζ − z k = d + ζ Table 1. Scaling relations discussed in the main text, specifying to SR elasticity ( α = 2 ), and standard LR elasticity α = 1 . • avalanche volume: in higher dimensions, it is difficultto define the spatial extension of an avalanche, whileits volume is well-defined. Using ρ f ( V )d V = ρ f ( S )d S , and S = m − d − ζ , V ∼ m − ζ , we arrive at ρ f ( V ) ∼ V − k V , a d (cid:28) V (cid:28) m d , k V = 2 − α − ζd . (480) • differences between static avalanches (shocks) andavalanches at depinning: A conceptually and practi-cally important question is whether static avalanchesand avalanches at depinning are in the same universal-ity class. As the roughness exponent ζ differs from oneclass to the other, Eq. (468) implies that they also havea different avalanche-size exponent τ , and thus mustbe different. We will see below that this difference isnot visible at 1-loop order, but shows up at 2-loop or-der. • Phenomenology, and a warning: For magnetic domainwalls, where avalanche phenomena were first observedas Barkhausen noise [68], one distinguishes in generalSR samples ( α = 2 ) from LR samples ( α = 1 ),and samples with noticeable eddy currents from thosewithout. A good review is [300]. The readershould also realize that the ABBM model (section4.3) is often equated with the LR class or mean field ,even though this must be debated [126]. The lineof theory we develop below starts with the ABBMmodel, generalizes it to the Brownian Force Model(BFM) (section 4.5), and then proceeds to short-rangecorrelated disorder (section 4.6). Up to now, our modeling of depinning was based on theequation of motion (305) for the position of the interface.This formulation makes it difficult to extract observablesinvolving the velocity. For this purpose it is better to takea time derivative of Eq. (305), to get an equation of motionfor the velocity ˙ u ( x, t ) , ∂ t ˙ u ( x, t ) = ( ∇ − m ) [ ˙ u ( x, t ) − ˙ w ( t )] + ∂ t F (cid:0) x, u ( x, t ) (cid:1) . (481) The field theory to be constructed below gives a quantitativedescription of avalanches in a force field F ( x, u ) , withshort-ranged correlations in both the x and u -directions. Westart with a toy model for a single degree of freedom, andthen proceed in two steps to short-range correlated forcesfor an interface.The toy model in question is the ABBM model,introduced in 1990 by Alessandro, Beatrice, Bertotti andMontorsi [297, 298], see also [430, 431]. Setting w ( t ) = vt , it reads ∂ t ˙ u ( t ) = m [ v − ˙ u ( t )] + ∂ t F (cid:0) u ( t ) (cid:1) , (482) ∂ t F (cid:0) u ( t ) (cid:1) = (cid:112) ˙ u ( t ) ξ ( t ) , (483) (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 σδ ( t − t (cid:48) ) . (484)The last equation implies that F ( u ) is a random walk, as canbe seen as follows: As ˙ u is non-negative, t is an increasingfunction of u , and we can change variables from t to u , ∂ u F ( u ) = ¯ ξ ( u ) , (cid:10) ¯ ξ ( u ) ¯ ξ ( u (cid:48) ) (cid:11) = 2 σδ ( u − u (cid:48) ) . (485)Thus F ( u ) has the statistics of a random walk. Itscorrelations are ∆(0) − ∆( u − u (cid:48) ) ≡ (cid:68) [ F ( u ) − F ( u (cid:48) )] (cid:69) = σ | u − u (cid:48) | . (486)In the language introduced above, the (bare) disorder has acusp, with amplitude | ∆ (cid:48) (0 + ) | = σ .The ABBM model is traditionally treated [297, 298,430] via the associated Fokker-Planck equation (933), ∂ t P ( ˙ u, t ) = σ ∂ ∂ ˙ u (cid:104) ˙ uP ( ˙ u, t ) (cid:105) + m ∂∂ ˙ u (cid:104) ( ˙ u − v ) P ( ˙ u, t ) (cid:105) . (487)This approach is difficult for time-dependent quantities, butefficient for observables in the steady state. As an example,consider the steady-state distribution of velocities, obtainedby solving ∂ t P ( ˙ u, t ) = 0 , P ( ˙ u ) = ˙ u m vσ − e − ˙ u m σ Γ (cid:0) m vσ (cid:1) (cid:18) m σ (cid:19) m vσ . (488)Setting σ = m = 1 to simplify the expressions yields P ( ˙ u ) = ˙ u v − e − ˙ u Γ( v ) . (489) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles It is important to remark that an avalanche stops at a givenwell-defined time. To see this, we solve Eqs. (482)-(483)for m = 0 , given that at time t = 0 the velocity is ˙ u . Theassociated Fokker-Planck equation is ∂ t P ( ˙ u, t ) = ∂ u [ σ ˙ uP ( ˙ u, t )] . (490)It can be solved analytically, for given initial distribution P ( ˙ u, 0) = δ ( ˙ u − ˙ u ) , as P ( ˙ u, t ) = δ ( ˙ u ) exp (cid:18) − ˙ u σt (cid:19) + exp ( − ˙ u 0+ ˙ uσt ) σt (cid:113) ˙ u ˙ u I (cid:16) √ ˙ u ˙ uσt (cid:17) . (491)( I is the Bessel-function of the first kind.) This canbe checked by inserting the solution into the differentialequation (490). Eq. (491) teaches us that for an initialvelocity ˙ u , with a finite probability exp( − ˙ u σt ) the velocitywill be (strictly) zero after time t . It also means that the endof an avalanche is well defined in time, which is crucial todefine its duration. This would not be the case for a particlein a smooth potential: Linearizing the potential close to theendpoint assumed to be u ( t = ∞ ) = 0 of the avalancheyields ∂ t u ( t ) (cid:39) − αu ( t ) = ⇒ u ( t ) (cid:39) u e − αt . (492)Eq. (491) can serve as an efficient simulation algorithm,replacing t by the time-discretization step δt , andalternately integrating the forcing term δ ˙ u ( t ) = m [ v − ˙ u ( t )] δt and the stochastic process according to Eq. (491).This is not straightforward, due to the appearance ofmultiplicative noise [432]. We can use an additional trick[433]: Observe that the solution (491) can be written as P ( ˙ u, t ) = δ ( ˙ u ) exp (cid:18) − ˙ u σt (cid:19) + ∞ (cid:88) n =1 (cid:0) ˙ u σt (cid:1) n exp (cid:0) − ˙ u σt (cid:1) n ! × σt (cid:0) ˙ uσt (cid:1) n − exp (cid:0) − ˙ uσt (cid:1) ( n − ∞ (cid:88) n =0 p n σt P n (cid:18) ˙ uσt (cid:19) . (493)Here, p n is a normalized probability vector, i.e. (cid:80) ∞ n =0 p n = 1 , and each probability P n ( x ) is normalized, (cid:82) ∞ d x P n ( x ) = 1 . Explicitly, we have p n = (cid:0) ˙ u σt (cid:1) n exp (cid:0) − ˙ u σt (cid:1) n ! , (494) P ( x ) = δ ( x ) , (495) P n ( x ) = x n − exp ( − x )( n − , n ≥ . (496)Given ˙ u , one obtains ˙ u with probability P ( ˙ u, t ) as follows:(i) draw an integer random number n , from the Poissondistribution p n ; the latter has parameter ˙ u / ( σt ) ,(ii) if n = 0 , return ˙ u = 0 , (iii) else draw a positive real random number x , from theGamma distribution with parameter n .(iv) return ˙ u = σtx ,Contrary to a naive integration of the stochastic differentialequation which yields ∂ t F (cid:0) u ( t ) (cid:1) δt = ξ t √ δt , (cid:104) ξ t ξ t (cid:48) (cid:105) = δ t,t (cid:48) , this algorithm is linear in δt .This allows us to define the distribution of durations,given below in Eq. (540), and the mean temporal shape(546), without introducing an (arbitrary) small-velocitycutoff. The model defined in Eqs. (482) and (483) is a model for asingle degree of freedom, not for an interface. A model foran interface can be defined by [302] ∂ t ˙ u ( x, t ) = ∇ u ( x, t ) + m [ v − ˙ u ( x, t )]+ ∂ t F (cid:0) x, u ( x, t ) (cid:1) , (497) ∂ t F (cid:0) u ( x, t ) , x (cid:1) = (cid:112) ˙ u ( x, t ) ξ ( x, t ) , (498) (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = 2 σδ ( t − t (cid:48) ) δ d ( x − x (cid:48) ) . (499)Since each degree of freedom sees a force which is arandom walk, this model is termed the Brownian ForceModel (BFM). Both the ABBM model as its spatial generalization, theBFM model, are pathologic in the sense that the force-force correlator grows for all distances instead of saturatingas expected in short-range correlated systems, and as isreflected in the FRG fixed points discussed in section 2.4.To remedy this, one can add an additional damping term inthe evolution equation of the force, ∂ t ˙ u ( x, t ) = ∇ u ( x, t ) + m [ v − ˙ u ( x, t )]+ ∂ t F (cid:0) x, u ( x, t ) (cid:1) , (500) ∂ t F (cid:0) x, u ( x, t ) (cid:1) = − γ ˙ u ( x, t ) F (cid:0) x, u ( x, t ) (cid:1) + (cid:112) ˙ u ( x, t ) ξ ( x, t ) , (501) (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = 2 σδ ( t − t (cid:48) ) δ d ( x − x (cid:48) ) . (502)As ˙ u ( x, t ) ≥ , the equation of motion for the force isequivalent to ∂ u F ( x, u ) = − γF ( x, u ) + ˜ ξ ( x, u ) , (503) (cid:68) ˜ ξ ( x, u ) ˜ ξ ( x (cid:48) , u (cid:48) ) (cid:69) = 2 σδ ( u − u (cid:48) ) δ d ( x − x (cid:48) ) . (504)This system has the force-force correlator (cid:104) F ( u, x ) F ( u (cid:48) , x (cid:48) ) (cid:105) c = σδ d ( x − x (cid:48) ) e − γ | u − u (cid:48) | γ . (505)For u (cid:28) γ , we recover the correlations (486). heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Consider the equation of motion (481) with generic short-ranged force-force correlators. The dynamical actionis obtained by multiplying the equation of motion with ˜ u ( x, t ) , and averaging over disorder , S = (cid:90) x,t ˜ u ( x, t ) (cid:104) ( ∂ t −∇ ) ˙ u ( x, t ) + m (cid:0) ˙ u ( x, t ) − ˙ w (cid:1)(cid:105) − (cid:90) x,t,t (cid:48) ˜ u ( x, t )˜ u ( x, t (cid:48) ) ∂ t ∂ t (cid:48) ∆ (cid:0) u ( x, t ) − u ( x, t (cid:48) ) (cid:1) . (506)The second line is ∂ t ∂ t (cid:48) ∆ (cid:0) u ( x, t ) − u ( x, t (cid:48) ) (cid:1) = ˙ u ( x, t ) ∂ t (cid:48) ∆ (cid:48) (cid:0) u ( x, t ) − u ( x, t (cid:48) ) (cid:1) = ˙ u ( x, t ) (cid:2) ∆ (cid:48) (0 + ) ∂ t (cid:48) sign ( t − t (cid:48) ) − ∆ (cid:48)(cid:48) (0 + ) ˙ u ( x, t (cid:48) ) + ... (cid:3) = − u ( x, t )∆ (cid:48) (0 + ) δ ( t − t (cid:48) ) − ∆ (cid:48)(cid:48) (0 + ) ˙ u ( x, t ) ˙ u ( x, t (cid:48) ) + ... (507)The terms dropped in this expansion are higher derivativesof ∆( u ) , and they come with higher powers of ˙ u ( x, t ) ,and its time-integral u ( x, t ) − u ( x, t (cid:48) ) = (cid:82) tt (cid:48) d τ ˙ u ( x, τ ) ,reminding that ˙ u ( x, t ) and not u ( x, t ) is the variable forwhich we wrote down the equation of motion.This expression is quite remarkable: The leading termis proportional to δ ( t − t (cid:48) ) , rendering the last term in Eq.(506) local in time. It is therefore appropriate to start ouranalysis of the theory with this term only. The action weobtain is S BFM [ ˙ u, ˜ u ]= (cid:90) x,t ˜ u ( x, t ) (cid:104) ( ∂ t − ∇ ) ˙ u ( x, t ) + m (cid:0) ˙ u ( x, t ) − ˙ w ( x, t ) (cid:1)(cid:105) + ∆ (cid:48) (0 + )˜ u ( x, t ) ˙ u ( x, t ) . (508)This is the action of the BFM introduced in Eqs. (497)-(499). Corrections are obtained by adding the omitted termsperturbatively. The leading order (sufficient at 1-loop order)is S [ ˙ u, ˜ u ] = S BFM [ ˙ u, ˜ u ] (509) + 12 (cid:90) x,t,t (cid:48) ˜ u ( x, t )˜ u ( x, t (cid:48) ) ˙ u ( x, t ) ˙ u ( x, t (cid:48) )∆ (cid:48)(cid:48) (0 + ) + .... We will show below on page 65 several theorems,indicating that at the upper critical dimension the action(506) leads to the results of the BFM model, and that theremaining terms in Eq. (507) lead to loop corrections, oforder (cid:15) = d c − d . The FRG equation (338) for the disorder has the followingstructure ∂ (cid:96) ˜∆( u ) = ( (cid:15) − ζ ) ˜∆( u ) + ζu ˜∆ (cid:48) ( u ) The response field ˜ u ( x, t ) is different from that in Eqs. (308)-(309).One can derive Eq. (506) by substituting in Eq. (309) ˜ u ( x, t ) →− ∂ t ˜ u ( x, t ) , and then integrating by parts in time. + ∞ (cid:88) n =1 ∂ nu (cid:104) ˜∆( u ) − ˜∆(0) (cid:105) n +1 . (510)The n -loop terms are highly symbolic, since the derivativescan be distributed in different ways on the n + 1 disordercorrelators, and we have dropped all prefactors. We nowassume that the microscopic disorder has the form (486),thus ˜∆(0) − ˜∆( u ) has only a linear term in u . This impliesthat the term of order n = 1 may contribute a constantto Eq. (510), while terms with n ≥ vanish. Thus to allorders, the roughness exponent is given by ζ BFM = (cid:15) = 4 − d. (511)As a consequence, the unrescaled disorder is scaleindependent (does not renormalize), and ∆ BFM (0) − ∆ BFM ( u ) ≡ σ | u | . (512)Note that ∆(0) is not well-defined, since random forcesgrow unboundedly.Similarly, the dynamical exponent z has correctionsproportional to ∆ (cid:48)(cid:48) (0) , which vanish. As a consequence, z = 2 , and all exponents can be obtained analytically, z BFM = 2 , β BFM = a BFM = 1 , γ BFM = α BFM = 2 ,τ BFM = 32 , κ BFM = 3 . (513) We want to construct observables for the BFM such as (cid:68) e (cid:82) x,t λ ( x,t ) ˙ u ( x,t ) (cid:69) = (cid:90) D [ ˙ u ] D [˜ u ] e (cid:82) x,t λ ( x,t ) ˙ u ( x,t ) −S BFM [ ˙ u, ˜ u ] . (514)This includes the avalanche-size distribution with λ ( x, t ) = λ , the velocity distribution with λ ( x, t ) = λδ ( t ) , the localavalanche-size distribution with λ ( x, t ) = λδ ( x ) , a.s.o.The key observation is that ˙ u ( x, t ) appears linearly inthe exponent, thus the path integral over ˙ u can be performed exactly , enforcing an instanton equation for ˜ u ( x, t ) , (cid:0) − ∂ t − ∇ + m (cid:1) ˜ u inst ( x, t ) − σ ˜ u inst ( x, t ) = λ ( x, t ) . (515)Here σ ≡ − ∆ (cid:48) (0 + ) > , see e.g. Eqs. (486) and (505). Theexpectation (514) is obtained from Z [ λ, w ] := (cid:68) e (cid:82) x,t λ ( x,t ) ˙ u ( x,t ) (cid:69) = e (cid:82) x,t m ˙ w ( x,t )˜ u ( x,t ) (cid:12)(cid:12)(cid:12) ˜ u =˜ u inst . (516)Let us consider some examples. The simplest example is the avalanche-size distribution.Noting that S = (cid:82) x,t ˙ u ( x, t ) , we have to solve the instantonequation (515) for λ ( x, t ) = λ . The solution for ˜ u = heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ˜ u inst will be constant in space and time, thus the instantonequation (515) reduces to m ˜ u − σ ˜ u = λ. (517)This quadratic equation has two solutions. The relevant onevanishing at λ = 0 reads ˜ u = m − √ m − λσ σ . (518)We now insert this solution into Eq. (516). As ˜ u ( x, t ) isconstant in space and time, the integral on the r.h.s. of Eq.(516) is ˜ u times w := (cid:90) x,t ˙ w ( x, t ) > . (519)This yields with ˜ u given in Eq. (518) (cid:10) e λS (cid:11) = e m w ˜ u . (520)Taking the inverse Laplace transform gives P Sw ( S ) = m w e − m S − w )24 σS √ πσS / . (521)One checks that P Sw ( S ) is normalized, (cid:104) (cid:105) w = (cid:82) ∞ d S P Sw ( S ) = 1 , and that the first avalanche-size mo-ment is (cid:104) S (cid:105) w = (cid:82) ∞ d S SP w ( S ) = w . If w ( x, t ) is con-stant in x , then (cid:104) S (cid:105) w = w is nothing but the displacementof the confining parabola. P w ( S ) is the response of the system to a displacement w , or equivalently to a force kick δf = m w . We now takethe limit of an infinitesimally small displacement w , and tothis purpose define P S ( S ) := lim w → (cid:104) S (cid:105) w w P w ( S )= (cid:104) S (cid:105) m e − m S σ √ πσS / ≡ (cid:104) S (cid:105) e − S Sm √ πS m S τ , (522) τ = τ ABBM = 32 , (523) S m := (cid:10) S (cid:11) (cid:104) S (cid:105) = σm . (524)Since (cid:104) S (cid:105) w = w , by construction all moments which do notnecessitate a small- S cutoff, i.e. (cid:104) S n (cid:105) w with n ≥ have awell-defined small- w limit, given by Eq. (522). What onelooses when taking the limit of w → is normalizability,as formally (cid:104) (cid:105) = lim w → w − = ∞ . The avalanche size-exponent τ = 3 / observed in theABBM model appears in many contexts: It was first studiedin the survival probability of a noble man’s name (maledescendents) [434]. The latter has the equations of motion(482)-(484), where ˙ u ( t ) is the number of descendants ina generation, v = 0 , and m the mean relative decrease inmale descendants in a generation (which could be negative), ∂ t u ( t ) = − m ˙ u ( t ) + √ ˙ uξ ( t ) , (525) (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 σδ ( t − t (cid:48) ) . (526) As ˙ u ( t ) > , the total number of descendants u ( t ) ismonotonically increasing, allowing us to write ˙ u ( t ) ≡ ˙ u ( u ( t )) . The equation of motion (525) then becomes ∂ t ˙ u ( u ( t )) = ˙ u ( u ) ∂ u ˙ u ( u ) = − m ˙ u ( u )+ (cid:112) ˙ u ( u ) ξ ( t ) . (527)As long as ˙ u > , we can divide both sides by ˙ u to arrive at ∂ u ˙ u ( u ) = − m + ˜ ξ ( u ) , (528) (cid:68) ˜ ξ ( u ) ˜ ξ ( u (cid:48) ) (cid:69) = 2 σδ ( u − u (cid:48) ) . (529)Note the change of argument in the noise. Thus ˙ u ( u ) performs a random walk in “time” u with drift − m andabsorbing boundary conditions at ˙ u = 0 . In absence of anabsorbing wall, the probability that ˙ u ( u ) starts close to zeroat u = 0 , and returns to zero behaves for large u as P return ( u ) ∼ e − m u √ u . (530)In presence of an absorbing wall, we need the probabilityto arrive at zero for the first “time” u . The latter is obtainedby taking a “time”, i.e. u -derivative of P return ( u ) , P first ( u ) = − ∂ u P return ( u ) ∼ e − m u u . (531)This is again the ABBM avalanche-size distribution (522)with τ = 3 / , interpreted as first-return probability of arandom walk. To simplify further considerations, we set m → , − ∆ (cid:48) (0 + ) ≡ σ → . (532)To obtain the instantaneous velocity distribution, weevaluate Eqs. (514)-(516) for λ ( x, t ) = λδ ( t ) , setting ˙ w ( x, t ) = v (uniform driving). The instanton equation tobe solved is − ∂ t ˜ u ( t ) + ˜ u ( t ) − ˜ u ( t ) = λδ ( t ) . (533)To impose proper boundary conditions, look at the r.h.s.of Eq. (516): Driving at times t > does not affect thevelocity distribution at t = 0 , thus the instanton solution ˜ u ( t ) must vanish for positive times. Eq. (533) with thisconstraint is solved by ˜ u ( t ) = λ Θ( − t ) λ + (1 − λ )e − t . (534)With the above solution Eq. (516) reduces to (cid:68) e λ (cid:82) x ˙ u ( x, (cid:69) = e vL d (cid:82) t ˜ u ( t ) = e − vL d ln(1 − λ ) = (1 − λ ) − vL d . (535)The inverse Laplace transform is P ˙ uv,L ( ˙ u ) = ˙ u vL d − e − ˙ u Γ( vL d ) . (536)This result is independent of the dimension d , and agreeswith the ABBM-result Eq. (488), there derived for a singledegree of freedom. We can take the limit of v → , anddefine P ˙ u ( ˙ u ) := lim v → P ˙ uv,L ( ˙ u ) vL d = e − ˙ u ˙ u . (537) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles The probability that the avalanche has velocity zero at time after a kick of size w at time t = − T , with T > , canbe obtained from the central result (516) with the instanton(534) as P ( ˙ u ( x, 0) = 0 ∀ x ) = lim λ →−∞ (cid:68) e λ (cid:82) x ˙ u ( x, (cid:69) = lim λ →−∞ e w ˜ u λ ( − T ) = exp (cid:18) − w e T − (cid:19) . (538)This is also the probability that the duration following a kickof size w is smaller than T . The distribution of durations isobtained by taking a derivative w.r.t. T , P duration w ( T ) = ∂ T exp (cid:18) − w e T − (cid:19) = w exp (cid:18) − w e T − (cid:19) e − T ( e − T − = w exp (cid:18) − w e T − (cid:19) T / . (539)This distribution is normalized. As at the end of section4.10, let us define the (unnormalized) probability density inthe limit of w → , P duration ( T ) := lim w → P duration w ( T ) w = 1[2 sinh( T / . (540) In order to obtain the temporal shape of an avalanche, weneed to solve the instanton equation − ∂ t ˜ u ( t ) + ˜ u ( t ) − ˜ u ( t ) = λδ ( t f − t ) + ηδ ( t − t m ) , (541)where t f is the final time (where the avalanche stops) and t m the time at which the velocity is measured. Since we onlyneed its first moment, we can construct ˜ u ( t ) perturbativelyin η . To that purpose write ˜ u ( t ) = ˜ u ( t ) + η ˜ u ( t ) + η ˜ u ( t ) + O ( η ) , ˜ u ( t ) = Θ( t f − t )1 − e t f − t . (542)The solution ˜ u ( t ) is the solution (534), translated to stopat t = t f , in the limit of λ → −∞ . Inserting Eq. (542) intoEq. (541) and collecting terms of order η yields − ∂ t ˜ u ( t ) + ˜ u ( t ) − u ( t )˜ u ( t ) = δ ( t − t m ) . (543)The solution is ˜ u ( t ) = sinh ( t f − t m )sinh ( t f − t ) θ ( t m − t ) . (544)Performing a kick of size w at t = 0 , and constraining theavalanche to stop at time t f = T , the shape can be writtenas (cid:104) ˙ u ( t m ) (cid:105) = ∂ η (cid:12)(cid:12)(cid:12) η =0 ln (cid:16) ∂ t f e w ˜ u ( t ) (cid:17) (cid:12)(cid:12)(cid:12) T = t f = 4 w sinh ( T − t m )sinh ( T ) + 4 sinh( T − t m ) sinh( t m )sinh( T ) . (545) Consider now the limit of w → , for which the first termvanishes: For short durations T , (cid:104) ˙ u ( t m ) (cid:105) converges to aparabola, (cid:104) ˙ u ( t m ) (cid:105) = 2 t m ( T − t m ) T . (546)For long durations, it settles on a plateau at (cid:104) ˙ u ( t ) (cid:105) = 2 , seefigure 49.Pursuing to the next order, one finds for the connectedaverage (cid:10) ˙ u ( t m ) (cid:11) c = 4 w sinh ( T − t m ) sinh( t m )sinh ( T )+ 8 sinh ( T − t m ) sinh ( t m )sinh ( T ) . (547)At w = 0 , quite remarkably the ratio (cid:10) ˙ u ( t m ) (cid:11) (cid:104) ˙ u ( t m ) (cid:105) = 32 (548)is time independent.Further observables, as well as loop corrections areobtained in [435, 391], and compared to experiments in[70]. We now consider avalanches on a codimension-1 hyper-plane, i.e. at a point for a line, or on a line for a 2d interface,a.s.o., by choosing λ ( x, (cid:126)x ⊥ ) = λδ ( x ) . (549)As a consequence, (cid:126)x ⊥ drops from the instanton equation.Setting again σ = m = 1 , one arrives at ˜ u ( x ) − ˜ u (cid:48)(cid:48) ( x ) − ˜ u ( x ) = λδ ( x ) . (550)The only solution which vanishes at infinity and satisfies theinstanton equation at λ = 0 is ˜ u ( x ) = 31 + cosh( x + x ) . (551)It can be promoted to a solution at λ (cid:54) = 0 by setting ˜ u ( − x ) = ˜ u ( x ) . The parameter x = x ( λ ) has tobe chosen to satisfy the instanton equation at x = 0 .Integrating Eq. (550) within a small domain around x = 0 yields λ = − u (cid:48) (0 + ) = 6 sinh( x )[1 + cosh( x )] . (552)On the other hand, the generating function is Z := (cid:90) ∞−∞ d x ˜ u ( x ) = 121 + e x . (553)Solving Eq. (553) for x and inserting into Eq. (552) yields λ = Z ( Z − Z − . (554) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles tT < u > tT < u > Figure 49. Expectation (cid:104) ˙ u ( t ) (cid:105) . Left: T = 10 , and from bottom to top w = 0 , , , and . Right: w = 0 (infinitesimal kick), and from bottom to top T = , , , and . The inverse Laplace transform is a priori difficult toperform, as Z ( λ ) is a complicated function of λ . The trickis to write P w ( S ) := (cid:10) e λS (cid:11) = (cid:90) i ∞− i ∞ d λ πi e − λS e wZ ( λ ) = (cid:90) i ∞− i ∞ d Z πi d λ ( Z )d Z e − λ ( Z ) S + wZ = − S (cid:90) i ∞− i ∞ d Z πi e wZ dd Z e − λ ( Z ) S = wS (cid:90) i ∞− i ∞ d Z πi e wZ e − λ ( Z ) S = 6 e w wπS (cid:90) ∞ d x cos (cid:0) x ( S x + S + 2 w ) (cid:1) = 2 e w w √ S + 2 wπS / K (cid:18) S + 2 w ) / √ S (cid:19) . (555)Note that this can also be written in terms of the Airyfunction (formula (21) of Ref. [429]). In the limit of w → ,this reduces to P ( S ) = 2 πS K (cid:18) S √ (cid:19) . (556)One can also give analytical expressions for the jointdistribution of avalanche size S and local size S , as well asof size S and spatial extension (cid:96) . The interested reader willfind this in [429]. We now turn to the spatial shape of avalanches [303]. Weremind that avalanches have a well-defined extension (cid:96) ,beyond which there is no movement. For a given avalanche,denote its advance by S ( x ) , and its size by S = (cid:82) x S ( x ) .We call avalanche extension (cid:96) the size of the smallest ballinto which we can fit the avalanche. As long as (cid:96) (cid:28) m − , (cid:104) S ( x ) (cid:105) (cid:96) = (cid:96) ζ g ( x/(cid:96) ) , (557) where g ( x ) is non-vanishing in the unit ball. Integratingover space yields S ∼ (cid:96) d + ζ , the canonical scaling relationbetween size and extension of avalanches, confirming theansatz (557).We now want to deduce how g ( x ) behaves close tothe boundary. For simplicity of notations, we write ourargument for the left boundary in d = 1 , which we placeat x = − (cid:96)/ . Imagine the avalanche dynamics for adiscretized representation of the system. The avalanchestarts at some point, which in turn triggers avalanches ofits neighbors, a.s.o. This leads to a shock front propagatingoutwards from the seed to the left and to the right. As longas the elasticity is local, the dynamics of these two shock-fronts is local: If one conditions on the position of the i -thpoint away from the boundary, with i being much smallerthan the total extension (cid:96) of the avalanche (in fact, we onlyneed that the avalanche started right of this point), then weexpect that the joint probability distribution for the advanceof points to i − depends on i , but is independent ofthe size (cid:96) . Thus we expect that in this discretized model the shape (cid:104) S ( x − r ) (cid:105) close to the left boundary r isindependent of (cid:96) . Let us now turn to avalanches of largesize (cid:96) , so that we are in the continuum limit studied inthe field theory. Our argument then implies that the shape (cid:104) S ( x − r ) (cid:105) measured from the left boundary r = − (cid:96)/ ,is independent of (cid:96) . In order to cancel the (cid:96) -dependence inEq. (557) this in turn implies that [303] g ( x ) = B × ( x − / ζ , (558)with some amplitude B . For the Brownian force model in d = 1 , the roughness exponent is ζ BFM = 4 − d = 3 , andone can further show that B = σ/ [303].On the left of figure 50, we show twenty realizationsof avalanches, with mean given by the thick black line. Onthe right we compare numerical averages with the theorysketched below. Note that the latter indeed has a cubicbehavior close to the boundary, as predicted by Eq. (558).We now turn to the theory: In order to get the spatialavalanche shape, one needs to construct a solution of the heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles S ( x ) x (cid:104) S ( x ) (cid:105) x Figure 50. Left: 20 avalanches with extension (cid:96) = 200 , rescaled to (cid:96) = 1 . n = 2871 is the number of samples used for the average. Right: The shape (cid:104) S ( x ) (cid:105) ≡ (cid:104) S ( x/(cid:96) ) (cid:105) (cid:96) /(cid:96) averaged for all avalanches with a given (cid:96) between and . To reduce statistical errors, we have symmetrized this function. instanton equation (515), with a source λ ( x ) = − λ δ ( x − r ) − λ δ ( x − r ) + ηδ ( x − x c ) ,λ , λ → ∞ . (559)Looking at our central result (516), this choice implies thatthe avalanche kicked at x = x does not extend to x = r , .To simplify matters further, one replaces m w ( x ) → f ( x ) ,and considers the response to a kick in the force. Thisallows us to take the limit of m → . Setting further σ = 1 ,the instanton equation to be solved is ˜ u (cid:48)(cid:48) ( x ) + ˜ u ( x ) = − λ ( x ) , (560)The source η generates moments of the avalanche size at x c .While unsolvable for arbitrary η , Eq. (560) can be solvedperturbatively in η , allowing us to construct moments of thespatial avalanche shape. This solution has the form ˜ u ( x ) = ˜ u ( x ) + η ˜ u ( x ) + η ˜ u ( x ) + ..., (561) ˜ u ( x ) = 1( r − r ) f (cid:18) x − r − r r − r ) (cid:19) , (562) f ( x ) = − P (cid:32) x + 1 / g = 0 , g = Γ (cid:0) (cid:1) (2 π ) (cid:33) . (563)Here P is the Weierstrass-P function, diverging at x = 0 and x = 1 . The subdominant terms in η are obtained byrealizing that if f ( x ) is solution of the instanton equation(560), so is κ f ( κx + c ) . The details of this calculation arecumbersome, and can be found in Ref. [303]. On the rightof figure 50 we show (cid:104) S ( X ) (cid:105) predicted by the theory, andits numerical test.Note that the shape of very large avalanches does notscale as (cid:96) but (cid:96) ; it also has a different shape [436]. Inspired by the calculations done so far, one can show thefollowing theorems [302]: Theorem 1: The zero-mode ˙ u ( t ) := L d (cid:82) x ˙ u ( x, t ) of theBFM field theory (508) is the same random process as inthe ABBM model, Eq. (482). Theorem 2: The field theory of this process is the sum ofall tree diagrams, involving ∆ (cid:48) (0 + ) as a vertex. Theorem 3: Tree diagrams are relevant at the uppercritical dimension d c . Corrections involve loops and canbe constructed in a controlled (cid:15) , i.e. loop, expansion aroundthe upper critical dimension d c . Sketch of Proof: One first constructs the generatingfunction for a spatially constant observable, as the velocityor the size in the BFM model. As we saw, these generatingfunctions involve instanton solutions constant in space, thusindependent of the dimension. Graphically this can beunderstood by constructing ˜ u perturbatively, with verticesproportional to σ = − ∆ (cid:48) (0 + ) , and lines which areresponse functions, possibly integrated over time. Sinceby assumption external observable vertices are at zeromomentum, all response functions are at zero momentum.This proves theorems 1 and 2.We now consider models with one of the fixedpoints studied above, be it RB, RF or periodic disorder,at equilibrium or at depinning. Since each vertex isproportional to (cid:15) , the leading order is again given by treesconstructed from ∆ (cid:48) (0 + ) . The only thing which can beadded are loops. Each loop comes with an additional factorof (cid:15) from the additional vertex, of which the leading oneis given in Eq. (509). As long as the ensuing momentumintegrals are finite, thus do not yield a factor of /(cid:15) ,these additional contributions are of order (cid:15) n , where n isthe number of loops. That the momentum integrals arefinite can be checked; it reflects the fact that the theory isrenormalizable, i.e. that all divergences which can possibly heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Loop corrections are cumbersome to obtain, and prone toerrors. To avoid the latter, one should check the obtainedresults by explicitly constructing them perturbatively in ∆( u ) , and λ . This is done in the relevant research literature[302, 437, 438, 391]. Here we sketch the generallyapplicable method of Ref. [302], to which we refer fordetails. Simplified model: Consider the action (509). To leadingorder, we can decouple the term in addition to the BFM via S η [ ˙ u, ˜ u ] = S BFM [ ˙ u, ˜ u ] + (cid:90) x,t η ( x )˜ u ( x, t ) ˙ u ( x, t ) . (564) η ( x ) is an (imaginary) Gaussian disorder to be averagedover, with correlations (cid:104) η ( x ) η ( x (cid:48) ) (cid:105) η = − ∆ (cid:48)(cid:48) (0) δ d ( x − x (cid:48) ) . (565)For each realization η ( x ) , the theory has the same formas in the preceding sections. In particular, the total action(including the sources) is linear in the velocity field, and theonly change is an additional term in the instanton equation(515), (cid:0) − ∂ t − ∇ + m (cid:1) ˜ u ( x, t ) − σ ˜ u ( x, t ) = λ ( x, t ) + η ( x )˜ u ( x, t ) . (566)Our central result (516) remains unchanged. Perturbative solution: To simplify notations, we set m = σ = 1 . We expand the solution of Eq. (566) in powers of η ( x ) , denoting by ˜ u ( n ) ( x, t ) the term of order η n , ˜ u ( x, t ) = ˜ u (0) ( x, t ) + ˜ u (1) ( x, t ) + ˜ u (2) ( x, t ) + .... (567)The hierarchy of equations to be solved is (cid:2) − ∂ t − ∇ x + 1 (cid:3) ˜ u (0) ( x, t ) = λ ( x, t ) + ˜ u (0) ( x, t ) , (568) (cid:104) − ∂ t − ∇ x + 1 − u (0) ( x, t ) (cid:105) ˜ u (1) ( x, t )= η x ˜ u (0) ( x, t ) , (569) (cid:104) − ∂ t − ∇ x + 1 − u (0) ( x, t ) (cid:105) ˜ u (2) ( x, t )= ˜ u (1) ( x, t ) + η ( x )˜ u (1) ( x, t ) . (570)The first line is the usual instanton equation (515). Let usintroduce the dressed response kernel (cid:104) − ∂ t − ∇ x + 1 − u (0) ( x, t ) (cid:105) R x (cid:48) t (cid:48) ,xt = δ d ( x − x (cid:48) ) δ ( t − t (cid:48) ) . (571)It has the usual causal structure of a response function, andobeys a backward evolution equation. It allows us to rewrite the solution of the system of equations (568) to (570) as ˜ u ( x, t ) = (cid:90) x (cid:48) (cid:90) t (cid:48) >t η ( x (cid:48) ) ˜ u ( x (cid:48) , t (cid:48) ) R x (cid:48) t (cid:48) ,xt , (572) ˜ u (2) ( x, t ) = (cid:90) x (cid:48) (cid:90) t (cid:48) >t (cid:104) ˜ u ( x (cid:48) , t (cid:48) ) + η ( x (cid:48) )˜ u (1) ( x (cid:48) , t (cid:48) ) (cid:105) × R x (cid:48) t (cid:48) ,xt . (573)Consider now the average (565) over η ( x ) . Since (cid:104) ˜ u (1) ( x, t ) (cid:105) η = 0 , the lowest-order correction is given bythe average of ˜ u ( x, t ) , Z [ λ ] = Z tree [ λ ] + (cid:90) xt (cid:104) ˜ u (2) ( x, t ) (cid:105) η + .... (574)Inserting Eq. (572) into Eq. (573), and performing theaverage over η , one finds (cid:104) ˜ u (2) ( x, t ) (cid:105) η = − ∆ (cid:48)(cid:48) (0) (cid:90) t The generat-ing function (584) can be inverted analytically [437]. Theresult for avalanches larger than a microscopic cutoff S isto O ( (cid:15) ) P ( S ) = (cid:104) S (cid:105) √ π S τ − m AS − τ exp (cid:32) C (cid:114) SS m − B (cid:20) SS m (cid:21) δ (cid:33) . (585)The coefficients are to O ( (cid:15) ) A = 1 + 18 (2 − γ E ) α, B = 1 − α (cid:16) γ E (cid:17) ,C = − √ π α, α = ζ − (cid:15) , (586)and γ E = 0 . is Euler’s number. The exponent τ is consistent with the scaling relation (468), while the newexponent δ reads δ = 1 + (cid:15) − ζ . (587) The result for the avalanche-size distribution has been verified numerically, both for thestatics [271] as for depinning [439].For the statics (equilibrium) [271], we show plots onfigure 51. The simulations are for a 3-dimensional RFmagnet, with weak disorder s.t. only a single domain wall appears, yielding d = 2 , (cid:15) = 2 , and ζ = ζ RF = 2 / . Thegenerating function Z ( λ ) is verified with high precision.For the avalanche-size distribution, the agreement is good,even though there is appreciable noise due to binning,which is absent from the generating function Z ( λ ) .At depinning, avalanches are simulated for an elasticstring in d = 1 [439]. The results for system sizes up to L = 4000 are shown on figure 52. The statistics is good,allowing to verify Eq. (585) in the tail region, with δ = 7 / . The temporal avalanche shape at fixed duration T : Thetemporal shape at fixed duration T is predicted by the theory[435, 391] as (cid:104) ˙ u ( t = ϑT ) (cid:105) T = 2 N (cid:104) T ϑ (1 − ϑ ) (cid:105) γ − (588) × exp (cid:18) − (cid:15) d c (cid:20) Li (1 − ϑ ) − Li (cid:16) − ϑ (cid:17) + ϑ ln(2 ϑ ) ϑ − 1+ ( ϑ + 1) ln( ϑ + 1)2(1 − ϑ ) (cid:21)(cid:19) . The exponent γ is given in Eq. (474). The temporal shapeis well approximated by (cid:104) ˙ u ( t = ϑT ) (cid:105) T (cid:39) [ T ϑ (1 − ϑ )] γ − exp (cid:0) A [ − ϑ ] (cid:1) . (589)The asymmetry A ≈ − . − d/d c ) is negative for d close to d c , skewing the avalanche towards its end, asobserved in numerical simulations in d = 2 and [440]. For d = 1 the asymmetry is positive in numerical simulations[441]. In experiments on magnetic avalanches (Bark-hausen noise), and in fracture experiments, the asymmetryis difficult to see [441]. The temporal avalanche shape at fixed size S : Thetemporal shape can also be calculated at fixed size S .Scaling suggests that (cid:104) ˙ u ( t ) (cid:105) S = Sτ m (cid:16) SS m (cid:17) − γ f (cid:18) tτ m (cid:16) S m S (cid:17) γ (cid:19) , (590)with (cid:82) ∞ d t f ( t ) = 1 , where f ( t ) may depend on S/S m .In mean field, the scaling function f ( t ) is independent of S/S m [442], and reads f ( t ) = 2 te − t , γ = 2 . (591)To one loop one obtains f ( t ) = f ( t ) − (cid:15) δf ( t ) . Expressionsfor arbitrary S/S m are lengthy. The universal small- S limitreads δf ( t ) = f ( t )4 (cid:20) π (cid:0) t + 1 (cid:1) erfi ( t ) + 2 γ E (cid:0) − t (cid:1) − − t (cid:0) t + 1 (cid:1) F (cid:18) , 1; 32 , t (cid:19) − e t (cid:16) √ πt erfc ( t ) − Ei (cid:0) − t (cid:1) (cid:17)(cid:21) . (592) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles MF MFnumerics H dots L H solid line L l oop - - - - - - Λ- - Z Ž H Λ L H S L I S P H S LM H S L I S P H S LM Figure 51. Results of Ref. [271] for RF disorder, d = 2 . Left: Numerically measured ˜ Z ( λ ) (blue dots). MF result (579) (green dashed), 1-loop result(584) (orange solid). The latter is rather precise, almost up to the singularity at λ = 1 / . Right: Avalanche-size distribution P ( S ) , multiplied by S τ with τ = 1 . from Eq. (468) (dots). The orange solid curve is the prediction from Eq. (585). The dashed line is a constant (guide to the eye). Inset:blow-up of main plot. s τ p ( s ) sL=1000 m=0.01L=2000 m=.005L=4000 m=0.02exponential fitone loop fit s τ p ( s ) sL=1000 m=0.01L=2000 m=.005L=4000 m=0.02exponential fitone loop fit Figure 52. Left: Avalanche-size distribution for Random Field in d = 1 at depinning. The variable s = S/S m . Blow up of the power-law region. Thered solid curve is given by the MF result Eq. (522), the black dashed line by Eq. (585), with A = 0 . , B = 1 . and C = 0 . . Right: the samefor the tail. Data from [439]. It satisfies (cid:82) ∞ d t δf ( t ) = 0 . The asymptotic behaviors are f ( t ) (cid:39) t → At γ − , A = 1 + (cid:15) − γ E ) , (593) f ( t ) (cid:39) t →∞ A (cid:48) t β e − Ct δ , δ = 2 + (cid:15) , β = 1 − (cid:15) ,A (cid:48) = 1 + (cid:15) 36 (5 − γ E − ln 4) , C = 1 + (cid:15) . (594)The amplitude A leads to the same universal short-timebehavior as in Eq. (588). To properly extrapolate to largervalues of (cid:15) , we use f ( t ) ≈ te − Ct δ N exp (cid:18) − (cid:15) (cid:20) δf ( t ) f ( t ) − t ln(2 t ) (cid:21)(cid:19) , (595)with the normalization N chosen s.t. (cid:82) ∞ d tf ( t ) = 1 .Eq. (595) is exact to O ( (cid:15) ) and satisfies the asymptoticexpansions (593) and (594).This result has beautifully been measured in theBarkhausen noise experiment of Ref. [70], see figure 53. The spatial avalanche shape (in d = 1 ): The spatialavalanche shape for the BFM was shown on figure 50.For systems with SR-correlated disorder, it was measuredfor two different driving protocols: tip driven (drivingat a single point), and spatially homogenous drivingby the parabola, the protocol used above. For tip-driven avalanches at the non-driven end, as well as forhomogenously driven avalanches, Eq. (558) predicts thatthe avalanche shape at fixed extension (cid:96) grows close to theboundary point b as (cid:104) S ( x ) (cid:105) (cid:96) ∼ | x − b | ζ . (596)For ζ = 1 . one thus expects this curve to have a slightlypositive curvature at these points, consistent with plots 3and 5 of Ref. [426].Let us also mention the studies of [436] for avalancheswith a large aspect ratio in the BFM which are rare, andwith fixed seed position [443] which are difficult to realizein an experiment. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles t/⌧ m / ( S/S m ) / ⌧ m S / m h ˙ u ( t ) i S S / normalized size S/S m At A t e Ct Figure 53. Scaling collapse of the average shape at fixed avalanche sizes (cid:104) ˙ u ( t ) (cid:105) S , according to Eq. (590), in the FeSiB thin film. The continuousline is the prediction for the universal SR scaling function of Eq. (595).The insets show comparisons of the tails of the data with the predictedasymptotic behaviors of Eqs. (593) and (594), setting (cid:15) = 2 , with A = 1 . , A (cid:48) = 1 . , β = 0 . , C = 1 . , and δ = 2 . . Consistentwith scaling relations, the measured γ = 1 . . +++×××***+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××*********************************************************************+ L × L * L (cid:1) L (cid:2) L - u - - P ( u ) Figure 54. The center-of mass velocity distribution P ( ˙ u ) . The weight ofthe peak at ˙ u = v kick is δt (cid:104) T (cid:105) ∼ L − z ∼ m z , where T is the duration ofan avalanche and δt the time discretization step. The analytic result (blackdashed line) is from Eq. (385) of Ref. [302], the dotted gray line the purepower law P ( ˙ u ) ∼ ˙ u − a , with a = − = − . as given in Eq. (478).There is no adjustable (fitting) parameter, thus convergence to the theoryincluding all scales is read off from the plot. Plot from Ref. [444]. The velocity distribution. The velocity distribution wasanalytically obtained in Refs. [445, 302], and numericallychecked in Ref. [444]. The scaling relation of Eq. (478)actually predicts a negative exponent a = − / ,implying P ( ˙ u ) ∼ ˙ u / . Despite the change in sign, thisis beautifully verified on Fig. 54. In section 2.9, we had asked how avalanche moments areencoded in ∆( w ) , and found the key relation (102). We can further ask how avalanches at w and w are correlated.This is easy to evaluate along the same lines: On one hand, [ u w + δw − u w ][ u w + δw − u w ]= (cid:104) S w S w (cid:105) δw δw ρ + O ( δw ) . (597)On the other hand, [ u w + δw − u w ][ u w + δw − u w ] − δw δw = m − (cid:2) ∆( w + δw − w − δw ) − ∆( w − w − δw ) − ∆( w + δw − w ) + ∆( w − w ) (cid:3) = − δw δw m − ∆ (cid:48)(cid:48) ( w − w ) + O ( δw ) . (598)Using Eq. (97) in Eq. (597), and comparing to Eq. (598) forsmall δw implies (cid:104) S w S w (cid:105) c (cid:104) S (cid:105) ≡ (cid:104) S w S w (cid:105)(cid:104) S (cid:105) − − ∆ (cid:48)(cid:48) ( w − w ) m . (599)These relations, and more, are studied in Refs. [446, 447]. In magnetic systems, a change in the magnetization inducesan eddy current , which in turn can reignite an avalanchewhich otherwise would already have stopped [448]. Thesimplest model exhibiting this phenomena, and whichremains analytically solvable [442] reads ∂ t u ( t ) = F (cid:0) u ( t ) (cid:1) + m (cid:2) w ( t ) − u ( t ) (cid:3) − ah ( t ) , (600) τ ∂ t h ( t ) = ∂ t u ( t ) − h ( t ) . (601)While many observables can be obtained analytically [442]and measured, e.g. the temporal shape given S , otherones are not well-defined, as the duration of an avalanche.Indeed, due to the eddy current h ( t ) , an avalanche canrestart. This complicates the data-analysis in real magnets[300]. Fractional Brownian motion (fBm) is the unique Gaussianprocess X t which is scale and translationally invariant, seee.g. [449, 450, 451, 452]. It is uniquely characterized by its2-point function (cid:104) X t X s (cid:105) = σ (cid:0) t H + s H − | s − t | H (cid:1) . (602)The Hurst exponent H may take values between 0 and 1, < H ≤ . (603)Note that X t is non-Markovian, since the 2-time correla-tions of increments at times t (cid:54) = s (cid:104) ∂ t X t ∂ s X s (cid:105) = 2 H (2 H − σ | s − t | H − (604)do not vanish, except for H = 1 / , for which the fBmreduces to standard Brownian motion.Since X t is a Gaussian process, many observables canbe calculated analytically. This is interesting, since onecan access analytically, in an expansion in H − / , most heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Figure 55. Shocks in a 2-dimensional system with short-ranged correlateddisorder, size L = 500 , and periodic boundary conditions, for twodifferent masses m = 10 − (left) and m = 10 − (right). Decreasing m , shocks merge. Shock fronts are almost straight. variables of interest for extremal statistics [453, 452, 454,455, 456, 457, 458, 459, 460, 461, 462]. An example ofsuch an observable is the maximum relative height of elasticinterfaces in a random medium [463]. Fractional Brownianmotion is also the simplest choice if one only knows thescaling dimension H of a process, without further insightinto higher correlation functions.Returning to depinning, suppose that random forcesare Gaussian and correlated as a fractional Brownianmotion ∆(0) − ∆( u ) = σ | u | H . (605)Solving Eq. (338), and realizing that loop corrections aresubdominant for all H < , we obtain similar to thederivation of Eq. (511) ζ = (cid:15) − H ) . (606)As a consequence of Eq. (468), the avalanche-size exponentis τ = 2 − d + ζ = 2 − − H )4 + d (1 − H ) . (607)Interestingly, in d = 0 , i.e. for a particle, this reduces to τ (cid:12)(cid:12) d =0 = 1 + H. (608)This is consistent with the first-return probability derivedin Refs. [452, 454]. Indeed, the probability to return tothe origin of a fBm X t with Hurst exponent H is P ( t ) = (cid:104) δ ( X t ) (cid:105) ∼ t − H , equivalent to Eq. (40) of [454]. Theprobability to return for the first time is ∂ t P ( t ) ∼ t − (1+ H ) ,equivalent to Eq. (608). These considerations generalizethose leading to Eqs. (530) and (531). Little is known about higher-dimensional shocks oravalanches. As in our understanding of the cusp, the 2-dimensional toy model (94) is helpful here, with V ( u ) drawn as uncorrelated Gaussian random variables withvariance on a unit grid. On Fig. 55 we show the shocks,i.e. the locations where the minimizer u in Eq. (94) changesdiscontinuously. Principle properties are (i) ˆ V ( u ) can be interpreted as a decaying KPZ heightfield, and ˆ F ( u ) := −∇ ˆ V ( u ) as a decaying Burgersvelocity, see section 7.7.(ii) decreasing m , i.e. increasing time t ∼ m − in the KPZ/Burgers formulation, shocks merge andannihilate.(iii) shock fronts are straight lines.(iv) when crossing a shock line, the minimizer u of Eq.(94) jumps perpendicular to the shock.Properties (iii) and (iv) suggest to write (with S = | (cid:126)S | ) [464] P ( (cid:126)S )d S d S = P ( S )d S cos θ d θ. (609)Using d S d S = S d S d θ yields P ( S , S ) = P ( S ) S cos θ. (610)On the other hand, one can again solve the problem inthe mean-field limit [465, 464], valid if the microscopicdisorder R (0) − R ( u ) ∼ | u | . In this limit, shocks are an infinitely divisible process [438]. As a consequence, e (cid:126)λ [ (cid:126)u ( (cid:126)w ) − u (0) − (cid:126)w ] = e wZ ( (cid:126)λ ) = (cid:90) d N (cid:126)S e (cid:126)λ(cid:126)S P ( (cid:126)S, (cid:126)w ) . (611)As in section 3.23, the large-deviation function F ( (cid:126)x ) canbe defined as F ( (cid:126)x ) := − lim w →∞ ln P ( (cid:126)xw, (cid:126)w ) w . (612)Inserting this expression into Eq. (611) yields e wZ ( (cid:126)λ ) = w N (cid:90) d N (cid:126)x e w [ (cid:126)λ(cid:126)x − F ( (cid:126)x )] . (613)This shows that the generating function Z ( (cid:126)λ ) and the large-deviation function F ( (cid:126)w ) are Legendre-transforms of eachother, Z ( (cid:126)λ ) + F ( (cid:126)x ) = (cid:126)λ(cid:126)x, (614) λ i = ∂∂x i F ( (cid:126)x ) , x i = ∂∂λ i Z ( (cid:126)λ ) . (615)It is non-trivial to show [465, 464] that F ( x , x ) = 2 x + (cid:2) x + ( x − x (cid:3) (cid:0) x + x (cid:1) / . (616)Measuring only a single component, equivalent to setting x = λ = 0 , this reduces to F ( x, 0) = (1 − x ) x , Z ( λ, 0) = 12 (cid:16) −√ − λ (cid:17) . (617)This is the same generating function as in Eq. (518), thusthe probability distribution for the longitudinal component S is as given in Eq. (521) (standard Watson-Galton process[434, 296]). The transversal avalanche-size distributionis more involved, but a parametric representation for ˜ Z ( λ ) := Z (0 , λ ) can be given [465], heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles . − λ ∗ − . . λ ∗ ˜ Z ( λ ) λ . . h s p ( s ) i s Figure 56. Left: measured ˜ Z ( λ ) (squares) compared to theprediction (618) (solid line). Right: Plot of s x p ( s x ) (top curve) and s ⊥ [ p ( s ⊥ ) + p ( − s ⊥ )] (bottom curve). Solid lines represents theanalytical predictions. Results from [465]. λ ( θ ) = sin( θ ) (cid:112) − cos(4 θ ) + 2 (cid:2) − cos(2 θ ) + (cid:112) − cos(4 θ ) (cid:3) , ˜ Z ( θ ) = cos( θ )2 (cid:112) − cos(4 θ ) − − cos(2 θ ) + (cid:112) − cos(4 θ ) . (618)This allows one to obtain the graph of ˜ Z ( λ ) , and even toLaplace-invert it. The results and numerical tests are shownon Fig. 56. When elasticity is long-ranged, avalanches can nucleateaway from the part of the avalanche including the firstpoint to have moved. This is an old problem, with manyreferences, see e.g. [466, 467, 468, 469, 470, 471].Suppose that each avalanche of size S is composed of N c ( S ) clusters, distributed as P ( S c | S ) ∼ S − τ c c Θ( S c < S ) , τ c < . (619)Then the typical size of clusters, given avalanche size S , is (cid:104) S c (cid:105) S = (cid:90) ∞ d S c S c P ( S c | S ) ∼ S − τ c . (620)There are N c ( S ) (cid:39) S (cid:104) S c (cid:105) ∼ S τ c − (621)clusters. Suppose that the number of clusters scales as P ( N c ) ∼ N − µc Θ( N c < N c ( S )) . (622)On dimensional grounds, P ( S )d S ∼ P ( N c )d N c . Insertingthe above relations yields τ c − τ − µ − . (623)If one further supposes [469, 470, 471] that the generationof a new cluster is a Galton-Watson process (section 4.11),then µ = 3 / , (624) simplifying Eq. (623) to τ cluster = 2 τ − . (625)Numerically it was checked [469, 470] that this scalingrelation works for all ≤ α < ; it might actually continueto work for α = 2 , if one keeps a finite value for A αd inEq. (17 b ) avoiding to reduce the power-law kernel to short-ranged correlations in that limit (see section 1.3). Gutenberg and Richter [96, 75] first reported that themagnitude of earthquakes in California follows a power-law, equivalent to an avalanche-size exponent of τ = 3 / .Due to its enormous impact on society, much research isdone in the domain, both by geophysicists with the aimof predicting the next big earthquake, and by theoreticalphysicists, trying to put earthquakes into the framework ofdisordered elastic manifolds. The latter is successful to acertain extend: • the elastic object depinning is a 2-dimensional faultplane, to which the relative movement is confined,often with sub-mm precision ( localization ), • driving is through the tectonic plates, equivalent to theparabolic confining potential of Eq. (5), • the elastic interactions on the fault plane are long-ranged since elasticity is mediated by the bulk. Thecalculation is essentially the same as for contact linesin section 1.3, and yields α = 1 in Eq. (16), • the critical dimension d c ( α ) = 2 α = 2 is thedimension of the fault plane. The system is in itscritical dimension. As a consequence ζ = 0 , z = 2 ,and τ = 3 / , which correctly predicts the Gutenberg-Richter law.But there is an additional element: After an earthquake,the fault is damaged , rendering it less resistant to furthermovement before the damage is healed , which happens on amuch longer time scale (see e.g. [473]). As a consequence,immediately after a big earthquake, the likelihood ofanother earthquake is increased. It is indeed found that theprobability for an aftershock to appear decays (roughly) as /t in time t , known today as Omori’s law [474].For further reading, we refer to the original literature[475, 473, 476, 477, 478, 479, 480, 79, 481, 482, 483, 484,485, 486] and to some of the relevant concepts discussed inthis review, long-range correlated elasticity (section 4.24),and inertia (section 3.23). The ABBM model, the BFM, or any other approach basedon a random walk, and commonly summarized as “mean Geophysicist usually consider the cumulative distribution of magnitude.The magnitude was originally defined as “proportional to the log of themaximum amplitude on a standard torsion seismometer” [472]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles τ = 3 / (definedin Eq. (465)), bounding from above all experiments andsimulations on disordered elastic manifolds ζ ABBM = ζ BFM = ζ MF = 32 ≥ ζ d dep ≡ − d + ζ > . (626)On the other hand, since d + ζ > , even in dimension d = 1 , there seems to be a lower bound on τ as well,indicated above. Note that for τ ≤ the avalanche-sizedistribution becomes non-integrable at large S in absenceof an IR cutoff.It is thus quite surprising to learn that in the SK model[59] the exponent τ is smaller [487, 488], τ equilibriumSK = τ dynamicSK = 1 . (627)This result for the equilibrium was obtained [487, 488]within a full-RSB scheme, relevant for SK. Curiously, ex-actly the same exponent is found in numerical simulations[489] out of equilibrium, where one simply increases themagnetic field until one spin becomes unstable, which isthen flipped. While finding the ground-state is an NP-hardproblem, this dynamic algorithm is trivial to implement.Still, the exponent τ is the same. It is also counterintuitiveto learn that avalanches in the SK model involve a finitefraction of its total of N spins, changing the magnetizationon average by √ N for an increase in external field by order / √ N , i.e. S typ := (cid:10) S (cid:11) (cid:104) S (cid:105) = √ N . (628)This means that on the complete hysteresis curve eachspin flips on average an order of √ N times. This is verydifferent to disordered elastic manifolds, where each degreeof freedom moves exactly once. It is compatible with thenon-integrable tail in the size distribution, P ( S ) ∼ /S ,knowing that there is no natural IR cutoff other than thesystem size. 5. Sandpile Models, and Anisotropic Depinning While nowadays sandpile models constitute a domain ofstatistical physics and mathematics on their own, it is worthreminding that they originated in the study of charge-density waves. In the seminal paper [490], the authorsconsidered an array of rotating pendula elastically coupledto their neighbours via weak torsion springs, a mechanicalanalogue of a charge-density wave. In any equilibriumstate the pendulum will almost point down. Consider adecomposition of the positions u i of the pendula, into theirinteger part ¯ u ( i ) and a rest δu ( i ) , u ( i ) = ¯ u ( i ) + δu ( i ) , ¯ u ( i ) ∈ Z . (629) The limit considered in [490] is that of weak springs ascompared to the gravitational forces, implying that δu ( i ) is small. The forces acting on pendulum i are z ( i ) = g cos (cid:0) πu ( i ) (cid:1) + (cid:88) j ∈ nn( i ) u ( j ) − u ( i ) + F ( i )= g cos(2 πδu ( i )) + (cid:88) j ∈ nn( i ) ¯ u ( j ) − ¯ u ( i )+ (cid:88) j ∈ nn( i ) δu ( j ) − δu ( i ) + F ( i ) . (630) F ( i ) are u -independent applied forces, and g is thegravitational constant (with mass and length of the pendulaset to 1). The sum runs over the nearest neighbours j of i , denoted nn( i ) . If a pendulum becomes unstable, ¯ u ( i ) → ¯ u ( i ) + 1 . The model (630) can also be viewedas a charge-density wave at depinning (sections 1.2, 2.8 and3.5). Supposing that the δu ( i ) are small, the update rule for z ( i ) can be written as z ( i ) → z ( i ) − dz ( j ) → z ( j ) + 1 , for j ∈ nn ( i ) . (631)Again neglecting δu ( i ) , the condition for the event (631) is z ( i ) > z c , (632)with z c = g . The Bak-Tang-Wiesenfeld (BTW) model [490] a.k.a. Abeliansandpile model (ASM) uses the update rules (631) com-bined with z c = 2 d. (633)It is interpreted as a sandpile of height z ( i ) . A site topples ,i.e. the rule (631) is performed, when its height exceeds z c .If several sites become unstable at the same time, one has tochoose an order of the topplings. Considering the originalmodel in terms of the u ( i ) , and using Middleton’s theorem,it is clear that the final state is independent of the order ofupdates. Stated differently, the topplings commute. For thisreason the model is also referred to as the Abelian sandpilemodel (ASM). Its algebra was studied in detail, especiallyby D. Dhar [491, 492, 493, 494].In this model, one starts from z ( i ) = 0 for all i ,chosen to belong to a finite lattice with open boundaries,as a chess board. Grains are added at random sites. If a sitebecomes unstable, it topples. If this toppling renders one ofits neighbors unstable, it topples in turn. Grains fall off atthe boundary. When topplings have stopped, a new grain isadded.In the interface formulation, grains falling off at theboundary correspond to an interface where u ( i ) = 0 outsidethe finite lattice (“on the boundary”). As a result, the systemis automatically in a critical state. This phenomenon called heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Figure 57. Stable configuration in a rice pile experiment. (Photo by theauthor). The grains are between two glass plates 5mm apart. The pile wasprepared by slowly increasing the inclination of the plates from horizontalto vertical. Brighter grains sit at the top and are more likely to topple. self-organized criticality (SOC), made the BTW model[490] popular. It is now recognized that if a system canbecome critical, slowly driving it achieves criticality. In thelanguage developed in this review, it is velocity-controlleddepinning , instead of force-controlled depinning (section3.1). Many natural phenomena are self-organized critical,and a large literature exists on the topic [82, 495, 496, 334,497, 498, 494, 499, 500, 501, 502, 503, 504, 505, 489, 492,493, 506, 507, 54, 508, 509, 510, 511, 512, 513, 514, 515,516, 517, 490].Configurations in the ASM can be classified asrecurrent or not. Recurrent configurations can be realizedin the steady state, while non-recurrent ones can not. Anexample for a non-recurrent configuration is the initial state u ( i ) = 0 . Recurrent configurations can be mapped one-to-one onto uniform spanning trees, and the q -states Pottsmodel in the limit of q → . There further is an injectiononto loop-erased random walks. We discuss this in moredepth in section 8.9. We refer the reader to the citedliterature and especially [82, 492] for details. Albeit we used the term “sandpile”, we did not yet motivateits use. To this aim, consider Fig. 57. The system isin a stable configuration, characterised by a mean slope,plus fluctuations. A grain may start to slide, depending onthe local slope, the friction between the neighbors, and itsorientation. The ASM does not contain any randomness,but instead is deterministic. Randomness enters onlythrough the driving, i.e. the order in which grains are added.Any realistic model for a sandpile must contain somerandomness. A simple 1-dimensional model to accomplishthis is the Oslo model.It is defined as follows [518, 510]: Consider the heightfunction h ( i ) of the sand or rice pile as shown in Fig. 58.To each height profile h ( i ) is associated a stress field z ( i ) ih ( i ) iz ( i ) iz c ( i ) Figure 58. A stable configuration of the Oslo model. The latter is a cellularautomaton version of the right half of the rice pile in Fig. 57. The red linesindicate the particle positions of particles used in section 5.4. Note thatthere is one plateau where two particles sit on top of each other, drawnhere slightly apart. defined by z ( i ) := h ( i + 1) − h ( i ) . (634)A toppling is invoked if z ( i ) > z c ( i ) , i > . The topplingrules are equivalent to those of Eq. (631), z ( i ) → z ( i ) − , z ( i ± → z ( i ± 1) + 1 . (635)They can be interpreted as moving a grain from the top ofthe pile at site i to the top of the pile at site i + 1 , h ( i ) → h ( i ) − , h ( i + 1) → h ( i + 1) + 1 . (636)After such a move, the threshold z c ( i ) for site i is updated, z c ( i ) → new random number . (637)In its original version, the random number is or withprobability / . To obtain Fig. 58 we used a randomnumber drawn uniformly from the interval [0 , . This heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles h ( i ) is not one of the usual random-manifold coordinates. As z ( i ) is the discrete Laplacian ofthe interface position u ( i ) , we conclude that h ( i ) = u ( i ) − u ( i − . (638)Note that the random force has been moved into thethreshold z c . The variable u (0) can be identified as thetotal number of grains added to the pile. The Oslo modelcan thus be viewed as an elastic string, pulled at i = 0 . Itsaverage profile is parabolic, (cid:104) u ( i ) (cid:105) ≈ (cid:104) z (cid:105) L − i ) + { grains fallen off at the right } . (639)As the disorder is renewed after each displacement, it fallsinto the random-field universality class.Is this model realistic for the rice pile of Fig. 57?According to [511], this depends on the shape of the grainsand their friction. If the grains are round the system goesinto a self-organized critical state, described by the Oslomodel. On the other hand, if the grains are longish (as onour photo), this does not work. It appears that the directionof the grains is a relevant variable, to be incorporated.For further reading on the Oslo model, we refer toRefs. [501, 519, 286, 520, 521]. ζ dep d =1 = 5 / Let us consider the heights h ( i ) of the plateaus in Fig. 58.They are marked on the left as red lines, which we interpretas particles. If a plateau has length n ≥ , then n particlesare at the same position. (In the figure there is one plateau oflength 2, for which we have drawn the two particle positionsslightly apart.). As h ( i ) is a monotonically decreasingfunction, h ( i ) ≥ h ( i + 1) . This induces a half-order h ( i ) (cid:23) h ( i + 1) on the particle positions. We can extendthis to an order by the convention that if i < i + 1 , and h ( i ) ≥ h ( i + 1) , then h ( i ) (cid:31) h ( i + 1) . Topplings preservethis order. If we identify this process as single-file diffusion[522, 523, 524], then its Hurst exponent is H SFD = 1 / .The additional advection term (grains always topple to theright) converts the temporal correlations into spatial ones,resulting into (cid:10) [ h ( i ) − h ( j )] (cid:11) c ∼ | i − j | H SFD . Using thataccording to Eq. (638) h is the discrete gradient of u , weconclude that [525] (cid:10) [ u ( i ) − u ( j )] (cid:11) c ∼ | i − j | ζ , (640) ζ dep d =1 = 1 + H SFD = 54 . (641)A roughness exponent ζ dep d =1 = 5 / is indeed conjectured inRef. [286]. We introduced the Abelian sandpile model with topplingrules (631), i.e. if z ( i ) ≥ d , then one grain is moved toeach of the d neighbours of site i . In 1991, S.S. Manna[514] proposed a stochastic variant z ( i ) ≥ move 2 grains to randomly chosen neighbours . (642)The chosen neighbors may be identical. Again, we wish tointroduce a random-manifold variable u ( i ) , s.t. a topplingon site i corresponds to u ( i ) → u ( i ) + 1 , while theremaining u ( j ) remain unchanged. To do so, let us definethe discrete Laplacian of u ( i ) as ∇ u ( i ) := (cid:88) j ∈ nn( i ) u ( j ) − u ( i ) . (643)Then write z ( i ) := 1 d (cid:2) ∇ u ( i ) + F ( i ) (cid:3) . (644)Suppose two grains from site i go to sites i and i ,possibly identical. Then choose for the site i and its nearestneighbours ju ( i ) → u ( i ) + 1 , F ( i ) unchanged , (645) F ( j ) → F ( j ) + δF ( j ) , (646) δF ( j ) = d ( δ j,i + δ j,i ) − . (647)The change in the random force vanishes, (cid:80) j ∈ nn( i ) δF ( j ) =0 . We may think of this process as distributing d grainsonto the d neighbours, but instead of doing this uniformlyas in the ASM, twice d grains are moved collectively to arandomly chosen neighbour. Eqs. (642) and (644) implythat the interface position u ( i ) increases by if the r.h.s. ofEq. (644) is larger than . This can be interpreted as a cellu-lar automaton for the equation of motion (297), if F ( i ) hasthe statistics of a random force. One can show that in anydimension dδF ( i ) = (cid:88) j ˙ u ( i + j ) × ∇ (cid:126)η ( j, t ) , (648)where (cid:126)η ( j, t ) is a white noise .As a result, for each i the variable F ( i ) performs arandom walk, which do to Eq. (648) and the equation ofmotion cannot grow unboundedly. In section 6.6 we givea more formal 2-step mapping of the Manna model ontodisordered elastic manifolds. The original version moves all the grains to randomly selectedneighbors. This version is not Abelian, whereas Eq. (642) is. Some authorscall it the Abelian Manna model. In d = 1 it is uncorrelated in space and time, whereas in d = 2 it has anon-trivial spatial structure, but remains short-ranged correlated. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ih ih Figure 59. The cellular automaton model TL92 . Blocking cells, i.e. cellsabove the threshold are drawn in cyan; those below in white. The initialconfiguration is the string at height 1 (dark blue). The interface moves up.An intermediate configuration is shown in red, the final configuration inblack. Open circles represent unstable points, i.e. points which can moveforward; closed circles are stable. Consider a stationary random point process on the line. It issaid to be hyperuniform [286], if the number n L of pointsin an interval of size L has a variance which scales with L asvar ( n L ) ∼ L ζ h , ≤ ζ h ≤ . (649)A Poisson process has ζ = 1 , a periodic function ζ = 0 .For sandpile models, this property was first observedin [526], and later verified in [527, 528, 529]. Recentreferences analysing or using hyperuniformity include [286,530, 531]. Hyperuniformity renders simulations muchbetter convergent, allowing for results from the Mannaor Oslo model to exceed those obtained directly for thedepinning of a disordered elastic manifold. There are intriguing connections between invasion ofporous media, directed percolation (DP), and depinning ofdisordered elastic manifolds when the nearest-neighbourinteractions grow stronger than linearly. Let us start ourconsiderations with the cellular automaton model proposedin Ref. [532]. Variants of this model can be found in [533],where it is applied to experiments on fluid invasion, bothnumerically and experimentally; see also [534].The model TL92 proposed in Ref. [532] uses a squarelattice as shown in Fig. 59. To each cell ( i, j ) is assigneda random variable f ( i, j ) ∈ [0 , . If f ( i, j ) < f c , the cellis considered closed (blocking), drawn in cyan. Open cells(not blocking) are drawn in white. The interface starts asa flat configuration at the bottom (dark blue in Fig. 59). Apoint ( i, h ( i )) on this interface is unstable and can move i = th i = th Figure 60. Simulation of the continuous version of the cellular automatonmodel TL92 . The continuous configurations (in color) converge reliablyagainst the directed-percolation solution (black, with filled circles). d = 1 d = 2 d = 3 ζ z . ± . [536] . ± . [536] Table 2. The exponents of qKPZ. forward by 1, h ( i ) → h ( i ) + 1 , according to the followingrule in meta code: unstable ( i ) if h ( i ) − h ( neighbour ) ≥ return false if f ( i ) > f c return true if h ( neighbour ) − h ( i ) ≥ return true end This cellular automaton models a fluid invading a porousmedia. Invasion takes place if a cell is open (second“if” above), or can be invaded from the side (third “if”).The process stops if all points ( i, h ( i )) are stable. Asis illustrated in Fig. 59, this stopped configuration is adirected path from left to right passing only through blockedsites, commonly referred to as a directed percolation path.One can convince oneself that upon stopping the algorithmyields the lowest-lying directed percolation path. Thiscan be implemented both for open and periodic boundaryconditions. The latter are chosen in Fig. 59. Note thatthe automaton TL92 can straightforwardly be generalizedto higher dimensions [535], but there is a priori no directedpercolation process in the othogonal direction.Two continuous equations of motion may be associ-ated with this surface growth. The first is the (massive)quenched KPZ equation, ∂ t u ( x, t ) = c ∇ u ( x, t ) + λ [ ∇ u ( x, t )] + m [ w − u ( x, t )]+ F ( x, u ( x, t )) . (650)This is almost the equation of motion (297) for a disordered heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles λ is referred to as a KPZ-term, due to its appearancein the famous KPZ equation of non-linear surface growth[537]. The latter accounts for the surface growing in itsnormal direction, and not in the direction of h . For aderivation see section 7.1. For an early reference see [538].The second model one can associate with theautomaton TL92 is depinning of an elastic interface. As TL92 makes no distinction between nearest-neighbourdistances or ± , has strong interactions at distance , andforbids larger distances, the corresponding elastic energy H el [ u ] must be strongly anharmonic. Our choice is H el [ u ] = L (cid:88) i =1 E el (cid:0) u ( i ) − u ( i + 1) (cid:1) , (651) E el ( u ) = (cid:26) , u ≤ ( u − , | u | > . (652)This implies an elastic nearest-neighbour force f el ( u ) := − ∂ u E el ( u ) = (cid:26) , u ≤ − u ( u − , | u | > (653)It evaluates to − at u = 2 , which is sufficient to overcomeany obstacle; and to − at u = 3 , making the latterunattainable. The full equation of motion for site i thenreads ∂ t u ( i, t ) = f el (cid:0) u ( i, t ) − u ( i +1 , t ) (cid:1) + f el (cid:0) u ( i, t ) − u ( i − , t ) (cid:1) + F ( i, u ( i, t )) . (654)The last term is the disorder force, which we choose to be f ( i, j ) if u is within δ close to j . Thus disorder acts as anobstacle close to an integer h . To mimic TL92 , we wish themanifold to advance freely between obstacles, setting there F = f + . Formally F ( i, u ) := (cid:26) f ( i, j ) − f c , ∃ j, | u − j | < δ,f + , else . (655)The parameter δ is a regulator. One checks that δ = 10 − ,and f + = 2 reproduces the time evolution of TL92 ,if movement is restricted to a single degree of freedom i , and one stops when u ( i ) hits the next barrier. Thisis not how Langevin evolution works: the latter beingparallel, we cannot expect trajectories to go through thesame states. However, due to Middleton’s theorem (seesection 3.2), the blocking configurations of both algorithmsare the same. We have verified with numerical simulationsthat the Langevin equation of motion finds exactly the sameblocking configurations as the cellular automaton TL92 .This proves that the critical configurations of the former arestates of directed percolation.While this statement was proven above for a specificnon-linearity, we expect that it is more generic, and appliesto any convexe elastic energy which at large distances growsstronger than a parabola. i = th i = th Figure 61. Directed percolation from left to right. A site ( i, h ) is definedto be connected if it is occupied, and at least one of its left neighbours ( i − , h ) , ( i − , h ± is connected. The index i takes the role of time t . Directed percolation (DP) is a mature domain of statisticalphysics [520, 539, 540]. Consider Fig. 61. Sites are emptyor full with probability p , which in our discussion aboveequals p = f c . A site ( i, h ) is said to be connected tothe left boundary, if it is occupied, and at least one of itsthree left neighbours ( i − , h ) , ( i − , h ± is connectedto the left boundary. The system is said to percolate, if atleast one point on the right boundary is connected to the leftboundary. For small p , this is unlikely, whereas for large p this is likely. There is a transition at p = p c . What iscommonly considered are the three independent exponents β , ν (cid:107) , and ν ⊥ , defined via ρ ( t ) := (cid:42) H (cid:88) h s h ( t ) (cid:43) t →∞ −→ ρ stat , (656) ρ stat ∼ ( p − p c ) β , p > p c , (657) ξ (cid:107) = | p − p c | − ν (cid:107) , (658) ξ ⊥ = | p − p c | − ν ⊥ . (659)The last two relations imply ⇒ ξ ⊥ ∼ ξ ζ (cid:107) , ζ := ν ⊥ ν (cid:107) (660)Hinrichsen [520] gives in d = 1 : ν (cid:107) = 1 . , ν ⊥ = 1 . ,β = 0 . , ⇒ ζ = 0 . . (661)In d = 2 : ν (cid:107) = 1 . , ν ⊥ = 0 . , β = 0 . . (662)In d = 3 : ν (cid:107) = 1 . , ν ⊥ = 0 . , β = 0 . . (663) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ν (cid:107) = 1 + (cid:15) 12 + (cid:15) (cid:2) − 110 ln( ) (cid:3) O ( (cid:15) ) , (664) ν ⊥ = 12 + (cid:15) 16 + (cid:15) (cid:2) − 34 ln( ) (cid:3) O ( (cid:15) ) , (665) β = 1 − (cid:15) (cid:15) (cid:2) − 106 ln( ) (cid:3) O ( (cid:15) ) . (666)This yields ζ := ν ⊥ ν (cid:107) = 12 + (cid:15) 48 + (cid:15) (cid:2) (cid:0) (cid:1)(cid:3) O ( (cid:15) ) . (667)In d = 1 ( (cid:15) = 3) , these values are in decent agreement withthose of Eqs. (661)-(661). To avoid confusion, let us define (cid:10) [ u ( x, t ) − u (0 , t )] (cid:11) ∼ | x − x (cid:48) | ζ for | x − x (cid:48) | (cid:28) m m − ζ m for | x − x (cid:48) | (cid:29) m (668)In d = 1 , the scaling of x and u as a function of p − p c is x ∼ ξ (cid:107) ∼ | p − p c | − ν (cid:107) , (669) u ∼ ξ ⊥ ∼ | p − p c | − ν ⊥ . (670)This implies u ∼ x ζ , ζ = ν ⊥ ν (cid:107) . (671)The exponent ν defined for depinning in Eq. (302) isidentified from Eq. (670) as ν ≡ ν dep = ν (cid:107) . (672)If we drive with a parabolic confining potential, m u (cid:39) | p − p c | . (673)This yields u ∼ m − ζ m , ζ m = 2 ν ⊥ ν ⊥ . (674)Let us define the correlation length ξ m as the x -scale atwhich the crossover between the two regimes of Eq. (668)takes place. This yields ξ m ∼ m − ζmζ , ζ m ζ = 2 ν (cid:107) ν ⊥ . (675)(We remind that in contrast for qEW ζ = ζ m , and ξ m =1 /m , see Eq. (307).) The avalanche-size exponent alsochanges. The avalanche size is S m = ξ d + ζm ≡ ξ dm m − ζ m . (676) Note that the notations in these papers are somehow contradictory. Thedynamical critical exponent z is related to our roughness ζ via z = 1 /ζ .The z defined in [542, 545, 543] is in Reggeon field theory, and equals z Reggeon = 2 ζ . Partial results at 3-loop order are reported in [546]. Since Eq. (467) was derived under the sole assumption thatthe avalanche density has a finite IR-independent limit for f → , it remains valid, implying τ = 2 − d + ζ ζζ m . (677)In dimension d = 1 , the dynamical exponent z = 1 (see below). The depinning scaling relation (303) can berewritten with Eq. (671) and z = 1 as β dep = ν (cid:107) ( z − ζ ) ≡ ν (cid:107) − ν ⊥ . (678)Using the values of Ref. [520] combined with the abovescaling relations, the numerical values in d = 1 are ν dep ≡ ν (cid:107) = 1 . (679) ν ⊥ = 1 . (680) ζ = 0 . , (681) ζ m = 1 . , (682) ζ m ζ = 1 . , (683) τ = 1 . , (684) β dep = 0 . , (685) z = 1 . (686) The dynamic exponent z . In Ref. [547] it was proposedthat the dynamical exponent z is related to the fractaldimension d min of the shortest path connecting two points adistance r apart in a percolation cluster. Denoting its lengthby (cid:96) ∼ r d min , the conjecture is z = d min . (687)This relation was confirmed numerically, and yielded thedynamical exponent z reported in table 2. Curiously, theupper critical dimension of percolation is d = 6 , whereasthe theory for directed percolation used in the precedingsection has an upper critical dimension of d = 4 . Asa consequence, constructing a field theory encompassingboth seems challenging. Let us finally study anharmonic (anisotropic) depinningwithin FRG, and to this purpose consider the standard elas-tic energy (4), supplemented by an additional anharmonic(quartic) term, H el [ u ] = (cid:90) x 12 [ ∇ u ( x )] + c ∇ u ( x )] . (688)The corresponding equation of motion reads ∂ t u ( x, t ) = ∇ u ( x, t ) + c ∇ (cid:110) ∇ u [ ∇ u ( x, t )] (cid:111) + F ( x, u ( x, t )) + f. (689)Since the r.h.s. of Eq. (689) is a total derivative, it issurprising that a KPZ-term can be generated in the limitof a vanishing driving-velocity. This puzzle was solved heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles x (mm) h ( mm ) 20 40 6020406080 −1−0.8−0.6−0.4−0.20 h ( mm ) 20 40 60 80203040 00.51 V f F - F c + −2 −1 −2 −1 −4 −2 −1 −0.5 −0.1 0 0.1 0.5 1 −4 −2 −1 −0.20 0.2 1 KPZ negative QKPZ positiveQKPZ F c F c −4 −2 −1 −0.5 −0.1 0 0.1 0.5 1 2 −4 −2 −1 −0.20 0.2 1 V f V f KPZ a)b) F c F c FF Figure 62. Left: Successive experimental fronts at constant time intervals. Color represents local front velocity. Top left: upward propagating front near F +c . Bottom left: backward propagating front near F − c .Right: Front velocity V f versus the applied force F , in adverse flow. a) experiments (black dots with error bars), b) numerics. Dashed lines are a linearextrapolation of the advancing branch. To put all data on one plot, axes are rescaled according to F → F/ | F | / , V f → V f / | V f | / . Insert: log-logplot of front velocity versus ˆ F − ˆ F c + . The continuous line corresponds to v ( ˆ F ) ∝ ( ˆ F − ˆ F c + ) . ± . . Fig. from [549]. in Ref. [548], where the KPZ term arizes by contractingthe non-linearity with one disorder, following the rules ofsection 3.4: δλ = t’ k p0 t = − c p (cid:90) t> (cid:90) t (cid:48) > (cid:90) k e − ( t + t (cid:48) ) k (cid:0) k p + 2( kp ) (cid:1) × ∆ (cid:48) ( u x,t + t (cid:48) − u x, ) . (690)As u ( x, t + t (cid:48) ) − u ( x, ≥ , Eq. (690) can be written as δλ = − c p (cid:90) t (cid:90) t (cid:48) (cid:90) k e − ( t + t (cid:48) ) k (cid:0) k p + 2( kp ) (cid:1) ∆ (cid:48) (0 + ) . (691)Integrating over t, t (cid:48) and using the radial symmetry in k yields δλ = − c (cid:18) d (cid:19) (cid:90) k ∆ (cid:48) (0 + ) k . (692)This shows that in the FRG a KPZ term is generatedfrom the non-linearity. Field theory does not yet permitto calculate the ensuing roughness exponent, nor explainthe mapping onto directed percolation, even though amechanism for the generation of a branching-like processwas found [548]. Many models nowadays are recognized as being in theuniversality class of Directed Percolation (DP). This startedwith work by Janssen [541] and Grassberger [550], whoconjectured that the findings “suggest another type ofuniversality, comprising all critical points with an absorbingstate and a single order parameter in one universality class”[550]. As a general rule, a model belongs to the universalityclass of directed percolation, as long as it has no additionalsymmetry. A notable exception is the Manna model (seesection 6.6). Note that additional conserved quantities arenot enough, as exemplified by directed percolation withmany absorbing states [551, 552, 553, 554, 495]. Thereader wishing to explore the large literature further canfind a lot of material in the context of Phase Transitionsinto Absorbing States , see [520, 280] for review, as well as[552, 555, 556, 505, 557, 558, 495]. As long as the disorder F ( x, u ) is statistically invariantunder u → − u , the quenched KPZ (qKPZ) equation (650)is invariant under u → − u , λ → − λ , and f → − f . Thisleads to two distinct cases: λf > the positive qKPZ class,and λf < the negative qKPZ class. Consider f > ,and λ > , then the KPZ term facilitates depinning. Inthe opposite case, assuming a tilted configuration allowsthe interface to remain pinned for larger applied forces.It then assumes a sawtooth shape, with the bottom kinks heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles The current-driven roughness growth in these films isdepicted in Figs. 1(a)–1(d). Figure 1(a) shows a linear DWinitially created by the thermomagnetic writing scheme.By injecting current, the linear DW starts to develop theshape of mountains with a constant slope as seen byFigs. 1(b) and 1(c). Figure 1(d) summarizes this evolutionby superimposing the DW lines sequentially with a con-stant time step. We observe this behavior for more than 10samples under examination, over the range of J between0.7 and 2.2 times A = m . This behavior is distinctfrom that of the field-driven roughness as shown byFigs. 1(e)–1(h), where the DW sweeps with a constant speedwithout a significant change in the shape of roughness. These distinct DW shapes belong to different universal-ity classes. Figure 1(i) depicts the log-log plot between theDW segment length L along the x axis and the roughnessamplitude w along the h axis. The linear relation indicatesthe power-law scaling w / L ! , where ! is labeled theroughness exponent [13]. The best fit quantifies that ! J ¼ : " : for the current and ! H ¼ : " : for thefield. The exponent ! H ( ! ) indicates the conventionalself-affinity of the field-driven DW roughness as alreadywell established in a number of studies with ! H ¼ = [4–6]. In contrast, the current-driven exponent ! J ( ’ )corresponds to the self-similarity in the present scalingrange. Thus, very surprisingly, the same DW exhibitsdistinctive roughness scalings between the self-affinityand self-similarity in the same system depending ondriving forces.In the current-driven cases, the shape of the mountains isdeveloped by the pinning of DWs at several strong pinningsites as indicated by purple circles in Fig. 1(d). Once thestructure is fully developed, the strength of pinning is sostrong that the DW is barely depinned over the measure-ment time range of up to a few days. By contrast, suchstrong pinning does not appear in the field-driven case asshown by Fig. 1(h). This suggests that the DW respondsdifferently to pinning sites depending on what the drivingforce is.In quenched disorder systems such as magnetic media,universal behaviors of the interface dynamics are governedby the quenched Kardar-Parisi-Zhang (KPZ) equation[13,14], which describes the temporal change of the inter-face height h ð x; t Þ at the position x and time t by @h=@t ¼ V þ " ð @ h=@x Þ þ ð = Þð @h=@x Þ þ $ q . The first termdenotes the mean speed. The second term describes relaxa-tion caused by the tension " . The third term is called theKPZ nonlinear term and induces growth ( > ) or decay( < ) along the local normal to the interface, dependingon the sign of the coefficient . The fourth term reflectsthe noise induced by the local disorders and thermalfluctuations.The KPZ equation predicts qualitatively different inter-face dynamics depending on the sign of . A scaling ana-lysis predicts the conventional self-affine roughness for apositive [13,15]. However, for a negative , the interfaceforms a typical roughness—called a facet [16,17]—in theshape of mountains with a constant slope, which is trulyaccordant with our observation. It is thus natural to deducethat the DW driven by the current (field) is described by theKPZ equation with a negative (positive) .The opposite sign of readily explains the differentpinning mechanisms [17]. When a driven interface meetsa pinning site, the interface bends at around the pinning siteand thus, the slope j @h=@x j adjacent to the pinning siteincreases accordingly. At this instant, for a positive ,the KPZ nonlinear term adds a force positively to thedriving force. It thus enhances the bending recursively Field Current (i) x h J H µ m (a) (b) (c) (d) (e) (f) (g) (h) x h L w L ( µ m) w ( µ m ) FIG. 1 (color online). Magnetic domain images taken at(a) 0 s, (b) 20 min, and (c) 4 hr after the application of J ( : & A = m ). (d) Time-resolved DW lines superimposed sequen-tially with a constant time step (3 min). Images taken at (e) 0,(f) 10, and (g) 20 min after the application of H (1.0 mT).(h) Time-resolved DW lines superimposed sequentially with aconstant time step (3 min). The purple circles in (d) and (h)designate the strong pinning sites that appeared in the current-driven motion. (i) Log-log scaling plot between the DW segmentlength L along the x axis and the roughness amplitude w alongthe h axis for DWs driven by the current (red) and field (blue). w is defined as the standard deviation of the roughness fluctuationover the length L . Each symbol is obtained by sampling morethan 1000 times. The solid lines show the best linear fit. PRL week ending8 MARCH 2013 and consequently, assists depinning. On the contrary, anegative ! suppresses the bending via the force oppositeto the driving force. Since more bending with a largerdriving force further enhances the negative KPZ force, asignificantly large force is required to overwhelm the KPZforce to trigger depinning.Driving force dependence appears also in the slope-dependent DW displacement. For this, a linear DW isinitially prepared with an angle " ( ¼ tan " ½ @h=@x $ ) andthen subjected to either a magnetic field or current pulse.Figure 2 depicts the results for the current (a)–(d) and field(e)–(h). Each image is obtained by adding two imagesbefore and after the translation and thus, each image showstwo DWs simultaneously. For the field-driven motion, thedisplacement ! h ? normal to the DW is found to be invari-ant irrespective of " . This behavior is due to the rotation symmetry with respect to the axis parallel to the appliedfield H . On the contrary, for the current-driven motion, ! h ? decreases as " increases as we discuss further below.These angular dependences are summarized in Fig. 2(i).We observe the angular dependences for 3 differentsamples with the range of J between 1.0 and 3.4 times A = m . The ordinate is scaled as the displacement ! h along the þ h axis, normalized by ! h for " ¼ . Theplot shows opposite dependences for the field (positive)and current (negative). For the field-driven case, the data fitto a = cos " function, which is attributed to the relation ! h ? ¼ ! h cos " with a constant ! h ? ( ¼ ! h ) as illus-trated in Fig. 2(f).The different slope dependence is a direct indicator ofthe opposite sign of ! . When ! h is converted into the DWspeed V ð¼ ! h= ! t Þ using the pulse duration ! t , the field-driven = cos " dependence indicates a positive ! , since theTaylor expansion of V with respect to @h=@x gives the firstleading term as ð V = Þð @h=@x Þ , which is identical to theKPZ nonlinear term with ! ¼ V ( ¼ ! h = ! t > ). Forthe current-driven case, on the other hand, the oppositeangle dependence implies a negative ! . In both cases, the ð @h=@x Þ term is forbidden due to inversion symmetry ! h ð " Þ ¼ ! h ð" " Þ .We next consider the details of the current-drivenmotion. As depicted in Fig. 2(b), J can be decomposedinto the vector components, parallel ( J k ) and normal ( J ? )to the DW. Since J k makes no macroscopic change alongthe normal direction, it is reasonable to assume that ! h ? ismainly driven by J ? . Then, by defining the new axesparallel ( x ) and normal ( h ) to the DW, it becomes avery typical situation where the DW lies along the hori-zontal x axis and the DW is driven by J ? along the vertical h axis as shown by the green window in Fig. 2(b). Thecreep scaling is then related to the DW roughness in the h axis, of which the scaling exponent J is measured withrespect to the x axis. It is experimentally revealed that J is invariant ( ¼ : ( : ) irrespective of " for the rangeof L smaller than a few tens of micrometers. Note that J ! J , which is a common symptom [18] of an interfacewith facet formation. Interestingly, J is identical to H [4–6]. It is thus natural to infer that the creep scaling in the x - h coordinate system is independent of the driving forceswith the common creep scaling exponent $ ( ¼½ " $ = ½ " $ ) given by = in both cases [3,4]. Therefore, thewell-established creep theory can be applied.For the case that the current is injected normal to theDW in metallic ferromagnetic materials, it has been theo-retically proposed [19] and experimentally verified [12]that the DW speed V follows the creep law V ¼ v exp ½" % f H ) g " $ $ , with respect to the effective field H ) ð H; J Þ ¼ H " & J " ’ J ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H " & J p þ ð = Þ ’ J þ * * * ,where v is the characteristic speed and % is the ratio of ascaling energy constant over the thermal energy. Here, & and ’ are the efficiency constants of the nonadiabatic and θ (degree)Current (i) h h x h J ⊥ J J || x’ h’ µ m (a) (b) (c) (d) (e) (f) (g) (h) J H h / h ∆ ∆∆ ∆ θ FIG. 2 (color online). Displacement driven by a current pulse( : + A = m , 150 s) for DWs with different " , (a) 0,(b) 30 , , (c) 60 , , and (d) 90 , . Displacement driven by a magneticfield pulse (1.0 mT, 150 s) for DWs with different " , (e) 0,(f) 30 , , (g) 60 , , and (h) 90 , . The purple arrows in (b) illustratethe decomposition of J into the parallel J k and normal J ? components. The green window in (b) exemplifies the newcoordinate axes— x and h —parallel and perpendicular to theDW. The equilength green arrows in (e)–(h) guide ! h ? . Thewhite solid and dashed lines in (f) indicate the DWs before andafter the displacement, respectively. The black solid and dashedarrows in (f) depict the displacements ! h and ! h ? , respectively.(i) Plot of ! h= ! h with respect to " for DW motion driven bythe current (red) and field (blue). The error bars correspond to thestandard deviation from data obtained by sampling 10 times. PRL week ending8 MARCH 2013 Figure 63. Magnetic domain walls in a two-dimensional Pt-Co-Pt thin film. (a-d) images for current-driven walls at increasing times. (e)-(h) ibid. field-driven. (i) measurement of the angle-dependent force as extracted from an analysis of the creep laws. From [384], with kind permission. at the strongest pinning centers, and the slope given by f ≈ ( − λ )( ∇ u ) . This was first observed numerically[559, 560], and later confirmed experimentally [549, 384],as beautifully illustrated on Figs. 62 and 63, and discussedin the next section. Experiments for directed percolation seem to be scarce[520]. A notable exception is Refs. [561, 562], wherea transition between two topologically different turbulentstates, called dynamic scattering modes 1 and 2 (DSM1 andDSM2), is observed upon an increase in the applied voltage.This allows them to measure directly the exponent β as β d =2DP = 0 . (693)The remaining exponents are obtained from a quench.Citing only the most precise values, ν (cid:107) = 1 . +(14) − (21) , (694) ν ⊥ = 0 . . (695)The theory values are given in Eq. (662).More experiments have been done for the qKPZ class.A particularly nice example are self-sustained reactionfronts propagating in a disordered environment made bypoly-disperse beads [549], as depicted on figure 62. Themeasured spatial and temporal fluctuations are consistentwith three distinct universality classes in dimension d =1 + 1 , controlled by a single parameter, the mean (imposed)flow velocity. The three classes are(i) the Kardar-Parisi-Zhang (KPZ) class for fast advanc-ing or receding fronts, with a roughness exponent of ζ ≈ . , see Eq. (807). (Purely diffusive motion withthe same roughness exponent is excluded by the tem-poral correlations.) (ii) the quenched Kardar-Parisi-Zhang class (positive-qKPZ) when the mean-flow velocity almost cancelsthe reaction rate. It has a roughness of ζ ≈ . ,in agreement with our discussion in section 5.9. Adepinning transition with a non-linear velocity-forcecharacteristics, v ∼ | F − F c | β is observed, see figure62.(iii) the negative-qKPZ class for receding fronts, close tothe lower depinning threshold ˆ F c − . One observescharacteristic saw-tooth shapes, see Fig. 62, bottomleft.To our knowledge, this system is the first where all threeKPZ universality classes have been observed in a singleexperiments.The qKPZ phenomenology can also be observed indomain walls in thin magnetic films [384] (see section3.21), either driven by an applied field (positive qKPZ)or a current (neagtive qKPZ). The experiment performedin Ref. [384] cleverly extracts the slope-dependent meanforce as a function of the angle, see Fig. 63 (right). Thisfirmly establishes the relevance of the two qKPZ classes fordomain wall experiments. It would be interesting to drivethe system both with a magnetic field and a current, chosens.t. the two effects cancel. 6. Modeling Discrete Stochastic Systems In discrete stochastic processes the elementary degrees offreedom are discrete variables. This can be the numberof colloids in a suspension, the number of bacteria, fishesand their predators in the ocean, or the grains in sandpilemodels. There are two powerful methods to treat thesesystems (for a pedagogical introcution see [563])(i) the coherent-state path integral [564, 565, 566, 567,563], heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles not possible .Real noise appears in a different modeling ofstochastic systems, via effective stochastic equations ofmotion . Here the noise stems from the fact that one triesto approximate a discrete random process by a continuous one, and one has to add back the appropriate shot noise .Another important question we need to deal with isthe notion of the Mean-Field approximation in stochasticequations. We will give a simple and precise definitionof the latter. To our astonishment, we have not found adiscussion of this in the literature. The coherent-state path-integral [564, 565, 566, 567, 563]is constructed by using creation and annihilation operatorsfamiliar from quantum mechanics, (cid:2) ˆ a, ˆ a † (cid:3) = 1 , | n (cid:105) := (ˆ a † ) n | (cid:105) . (696)The state | n (cid:105) is interpreted as n -times occupied. Eigenstatesof of ˆ a are coherent states. They are the building blocks ofthe formalism, giving it its name | φ (cid:105) := e φ ˆ a † | (cid:105) ⇒ ˆ a | φ (cid:105) = φ | φ (cid:105) . (697)Taylor expanding e φ ˆ a † | (cid:105) , one sees that coherent statesare Poisson distributions with n -fold occupation probabilitygiven by p ( n ) = e − φ φ n n ! . (698)Note that (cid:104) n (cid:105) = (cid:10) n (cid:11) c = φ , thus the parameter φ characterizing a coherent state is both its mean andvariance.Consider the reaction-diffusion process with diffusionconstant D and reaction rate A + A ν −→ A . The action, see(Eq. (112) of Ref. [563]) reads S (cid:48) [ φ ∗ , φ ] = (cid:90) x,t φ ∗ ( x, t ) (cid:104) ∂ t φ ( x, t ) − D ∇ φ ( x, t ) (cid:105) + (cid:90) x,t ν (cid:104) φ ∗ ( x, t ) φ ( x, t ) + φ ∗ ( x, t ) φ ( x, t ) (cid:105) . (699) The first two terms are similar to those appearing in theMSR formalism for diffusion, identifying the tilde fieldsthere with star fields here. The next term φ ∗ ( x, t ) φ ( x, t ) is also intuitive: Two particles are destroyed, and one iscreated. The only surprising term is the last one. It appearsin the formalism to ensure that the probability is conserved,and can be interpreted as a first-passage time problem [563].If the last term were not there, then we could interpret theaction as an equation of motion for φ ( x, t ) . To includethe latter, let us decouple the quartic term by introducingan auxiliary field ξ ( x, t ) , to be integrated over in the pathintegral, S (cid:48) [ φ ∗ , φ, ξ ] = (cid:90) x,t φ ∗ ( x, t ) (cid:104) ∂ t φ ( x, t ) − D ∇ φ ( x, t )+ ν φ ( x, t ) − i √ νξ ( x, t ) φ ( x, t ) (cid:105) + 12 ξ ( x, t ) . (700)The corresponding equation of motion and noise correla-tions are ∂ t φ ( x, t ) = − ν φ ( x, t ) + D ∇ φ ( x, t )+ i √ νφ ( x, t ) ξ ( x, t ) , (701) (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) δ ( x − x (cid:48) ) . (702)This noise is imaginary. It has puzzled many researcherswhether this is unavoidable [555, 568, 569, 570], or couldeven be beneficial [571].For the moment, let us restrict our considerations toa single site, starting at time t = t i with the initial state, φ t i = φ i , ∂ ( t ) φ ( t ) = − ν φ ( t ) + i √ νφ ( t ) ξ ( t ) , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) . (703)This equation is integrated from t = t i to t f . On the left offigure 64 we show the result for φ t f for different realizationsof the noise ξ ( t ) . Since φ t f is complex, the question is howto interpret these states. The answer is that the probabilitydistribution is given, in generalization of Eq. (698), by [563] p SEM t ( n ) := (cid:28) e − φ t φ nt n ! (cid:29) ξ . (704)A complex φ ( t ) is necessary, since the final distributionis narrower than a Poissonian . The problem with thestochastic average (704) is that when arg( φ t ) grows intime, it is dominated by those φ t with the smallest realpart, and the estimate (704) breaks down. A stochasticequation of motion for the coherent-state path integral isthus not a valid simulation tool. This is similar to whathappens in the REM model (section 2.23): in both casesrare events give a substantial contribution to the observablewe want to calculate, which is missed in any finite sampleor simulation. It is impossible to construct a probability distribution which is narrowerthan a Poissonian by a superposition of Poissonians with positivecoefficients. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Figure 64. Left: Result of the integration of Eq. (552), with ν = 1 , total time t f − t i = 0 . , and initial state φ i = 15 . The black circle has radius φ f = 3 . , obtained by integrating the drift term ∂ t φ t = − φ t + φ t + t/ . Using an algorithm which splits points which are likely to contribute moreto the final result, the color codes less probable values, from yellow over green, cyan, blue, magenta to red. (Thus a red point has − times the weightof a yellow point.) Right: One trajectory each for process n t , i.e. a direct numerical simulation of A + A → A (red, with jumps), and ˆ n t , Eq. (709)(blue-grey, continuous, rough). The rate is ν = 1 . We have chosen two trajectories which look “similar”. Note that ˆ n t is not monotonically decreasing. In the next sections, we follow a different strategy:We give up on the fact that the number n ( t ) of particlesis discrete, and replace it by a continuous variable ˆ n ( t ) . We now want to derive a stochastic differential equationwith real noise. To this aim let us simulate directly therandom process A + A ν −→ A . Each simulation rungives one possible realization of the process, in the formof an integer-valued monotonically decreasing function n ( t ) . Averaging over these runs, one samples the finaldistribution P f ( n ) , or, equivalently, moments of n f . Wewant to ask the question: Is there a continuous randomprocess ˆ n ( t ) which has the same statistics as n ( t ) ?Let us consider a more general problem: Be n ( t ) thenumber of particles at time t . With rate r + the number ofparticles increases by one, and with rate r − it decreases byone. This implies that after one time step, as long as r ± δt are small, (cid:104) n ( t + δt ) − n ( t ) (cid:105) = ( r + − r − ) δt, (705) (cid:10) [ n ( t + δt ) − n ( t )] (cid:11) = ( r + + r − ) δt. (706)The following continuous random process ˆ n ( t ) has thesame first two moments as n ( t ) , dˆ n ( t ) = ( r + − r − )d t + (cid:112) r + + r − ξ ( t )d t, (707) (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) . (708) Despite our best efforts, we have not been able to locate a source forthis simple argument in the literature. It is applied in [541], but the citedsource [572] dryly states “Our whole work depends on the use of stochasticmaster equations, which, we believe, have a better conceptual and intuitivebasis than the fluctuating force formalism of Langevin equations.” ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ n P ( n ) Figure 65. Result of a numerical simulation, starting with n i = 15 particles, and evolving for t f − t i = 0 . . Blue diamonds: Directnumerical simulation of the process A + A → A with rate ν = 1 .Cyan: Distribution of the continuous random walk (709). Red: The latterdistribution, when rounding n f to the nearest integer. Black boxes: Thesize of the boxes in n -direction to obtain the result of the direct numericalsimulation of the process A + A → A . Both processes have firstmoment . ± . , and second connected moment ± . ; the thirdconnected moments already differ quite substantially, . versus . . This procedure can be modified to include higher cumulantsof n ( t + δt ) − n ( t ) , leading to more complicated noisecorrelations. Results along these lines were obtainedin Ref. [544] by considering cumulants generated in theeffective field theory. For the reaction-annihilation process, the rate r + = 0 , and r − = ν ˆ n ( t )(ˆ n ( t ) − ; the latter, in principle, is only heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles (cid:96) Figure 66. A coarse-grained lattice with box-size (cid:96) = 4 . The yellow boxcontains n = 4 particles. defined on integer ˆ n ( t ) , but we will use it for all ˆ n ( t ) . Thusthe best we can do to replace the discrete stochastic processwith a continuous one is to write dˆ n ( t )d t = − ν n ( t )(ˆ n ( t ) − 1) + (cid:114) ν n ( t )(ˆ n ( t ) − ξ ( t ) , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) . (709)Using n i = 15 , and ν = 1 , we have shown two typicaltrajectories on figure 64 (right), one for the process n ( t ) (red, with jumps), and one for the process ˆ n t (blue-grey,rough). While by construction both processes have (almost)the same first two moments, clearly ˆ n ( t ) looks different: Itis continuous, which n ( t ) is not, and it can increase in time,which n ( t ) can not. One can also compare the distributionfor t f − t i = 0 . , see figure 65. While the distribution of n f is discrete (blue diamonds), the one for ˆ n f is continuous(cyan). Rounding n f to the nearest integer gives a differentdistribution (red). We have also drawn (black lines) the sizeof the boxes which would produce p ( n ) from p (ˆ n ) . Clearly,there are differences. On the other hand, it is also evidentthat these differences diminish when increasing n i . There are several paths to derive a field theory for reaction-diffusion or directed percolation. A beautiful derivation isgiven by Cardy and Sugar [543]. The authors start froman exact microscopic modelization, before introducing anauxiliary field resulting into the action given below in Eq.(716). They then use perturbative results obtained for theequivalent action used in Reggeon field theory. The latter isan effective theory used to handle deep-inelastic scattering[545], quantum gravity (simplicial gravity) [573], vortices in He-II [574], and many more .For pedagogic reasons, we apply the formalismdeveloped in section 6.3 [575]: Denote n ≡ n ( x, t ) thenumber of particles inside a box located around ( x, t ) withsize (cid:96) d , see Fig. 66. There we could draw time as comingout of the plane.In figure 61 a different view is taken: Here n ( x, t ) isthe coarse-grained number of occupied sites connected tothe left border, there drawn in red. Going one step in t tothe right, n grows with rate n r + −→ n + 1 , r + = α + n − βn , (710) α + ≈ p, β ≈ (cid:96) d . (711)Here is the number of left neighbours per site, and p is theprobability that the site itself is not empty, and thus can beconnected. The rate to go down is given by n r − −→ n − , r − = α − n + βn , (712) α − = 1 − p. (713)The first term takes into account that if the site itselfis empty, it cannot be connected. The second termproportional to n ensures that the fraction of connectedsites cannot grow beyond . According to Eq. (707), thisleads to the stochastic equation of motion ∂ t ˆ n ( x, t ) = ∇ ˆ n ( x, t ) + ( α + − α − )ˆ n ( x, t ) − β ˆ n ( x, t ) + √ ˆ n (cid:112) α + + α − + β ˆ nξ ( x, t ) (714)Note that we have added a diffusive term (rescaling x ifnecessary to set its prefactor to 1). In directed percolation(figure 61) it arises since the left neighbor a site is connectedto can be one higher or lower. Multiplying Eq. (714) with aresponse field ˜ n ( x, t ) yields the dynamic action S [˜ n, ˆ n ]= (cid:90) x,t ˜ n ( x, t ) (cid:2) ∂ t ˆ n ( x, t ) − ∇ ˆ n ( x, t ) + ( α − − α + )ˆ n ( x, t ) (cid:3) + (cid:90) x,t β ˜ n ( x, t )ˆ n ( x, t ) + (cid:90) x,t 12 [ α + + α − + β ˆ n ( x, t )]˜ n ( x, t ) ˆ n ( x, t ) (715)Let us rewrite this action. As one can see from theequation of motion, the combination m := α − − α + measures the distance to criticality (without perturbativecorrections). The term proportional to β in the last linegives a quartic term, which is irrelevant. Finally, onecan change normalization of the fields, setting ˆ n → λφ , ˜ n → λ − ˜ φ , which leaves the quadratic terms invariant,but changes the relative magnitude of the two cubic terms.As a result, we obtain the action with coupling const g = Note however, that different theories are associated with the name“Regge”: Sometimes the cubic vertex is an antisymmetrized combinationof a field with two field derivatives, as in Ref. [574]. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles n a i Figure 67. Thick lines: The order parameters of the Manna model, asa function of n , the average number of grains per site, obtained froma numerical simulation of the stochastic Manna model on a grid of size × with periodic boundary conditions. We randomly update asite for iterations, and then update the histogram times every iterations. Plotted are the fraction of sites that are: unoccupied (black),singly occupied (blue), double occupied (green), triple occupied (yellow),quadruple occupied (orange). The activity ρ = (cid:80) i> a i ( i − is plottedin purple. No data were calculated for n < . , where a = e = 1 − n , a = n , and a i> = 0 (inactive phase). Note that before the transition, a = 1 − n and a = n . The transition is at n = n c = 0 . .Thin lines: The MF phase diagram, as given by Eqs. (726) ff. for n ≤ ,and by Eqs. (727) ff. for n ≥ . We checked the latter with a directnumerical simulation. (cid:112) β ( α + + α − ) / , S [˜ n, ˆ n ] = (cid:90) x,t ˜ φ ( x, t ) (cid:2) ∂ t φ ( x, t ) − ∇ φ ( x, t ) + m φ ( x, t ) (cid:3) + (cid:90) x,t g (cid:104) ˜ φ ( x, t ) φ ( x, t ) + ˜ φ ( x, t ) φ ( x, t ) (cid:105) . (716)It has four renormalizations, one for each of the threequadratic terms, plus one for the coupling constant g . Thisleads to three independent exponents given in section 5.8.Results at 2-loop order can be found in Refs. [541, 542, 543,544]. Partial 3-loop results are given in [546]. The action(716), known as Regge field theory [545], is also used asan effective field theory for deep inelastic scattering. There φ and ˜ φ are interpreted as particle annihilation and creationoperators . In this section, we apply our considerations to a non-trivialexample, the stochastic Manna model. We will see that ourformalism permits a systematic derivation of its effectivestochastic equations of motion. While the result is knownin the literature [576, 556, 495, 502], it is there derivedby symmetry principles, which are convincing “up to acertain degree”. Furthermore, they leave undetermined allcoefficients. While many of them can be eliminated byrescaling, our derivation “lands” on a particular line ofparameter space, characterised by the absence of additionalmemory terms, see section 6.9. The Manna model, introduced in 1991 by S.S. Manna[514], is a stochastic version of the Bak-Tang-Wiesenfeld(BTW) sandpile [490]. Let us recall its definition given insection 5.5: Manna Model (MM). Randomly throw grains on a lattice.If the height at one point is greater or equal to two, then withrate 1 move two grains from this site to randomly chosenneighbouring sites. Both grains may end up on the samesite. We start by analysing the phase diagram. We denoteby a i the fraction of sites with i grains. It satisfies the sumrule (cid:88) i a i = 1 . (717)In these variables, the number of grains n per site can bewritten as n := (cid:88) i a i i. (718)The empty sites are e := a . (719)The fraction of active sites is a := (cid:88) i ≥ a i . (720)We also define the (weighted) activity as ρ := (cid:88) i ≥ a i ( i − . (721)Note that ρ satisfies the sum rule n − ρ + e = 1 . (722)In order to take full advantage of this definition, we changethe toppling rules of the Manna model to those of the Weighted Manna Model (wMM). If a site contains i ≥ grains, randomly move these grains to neighbouring siteswith rate ( i − .On figure 67 (thick lines), we show a numericalsimulation of the Manna model in a 2-dimensional systemof size L × L , with L = 150 . There is a phase transitionat n = n c = 0 . . Close to n c , the fraction of doublyoccupied sites a grows linearly with n − n c , and higheroccupancy is small. Indeed, we checked numerically thatfor n > n c the probability p i to find i grains on a sitedecays exponentially with i , i.e. p i ∼ exp( − α n i ) , where α n depends on n , see figure 68. This is to be contrastedwith the initial condition, where we randomly distribute n × L grains on the lattice of size L × L . It yields a Poissondistribution, the coherent state | n (cid:105) , for the number of grainson each site, see inset of figure 68 (left). This result suggeststhat coherent states may not be the best representation forthis system. It further implies that close to the transition, ρ ≈ a , and we expect that the wMM and the original MMhave the same critical behaviour. We come back to thisquestion below. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles i H N H a i LL i H N H a i LL n Α Figure 68. Left: (Unnormalized) histogram after manny topplings for n = 2 ; the probability that a site has i grains decays as e − . i , for all i ≥ .Inset: The initial distribution, a Poissonian. Right: The exponential decay coefficient α as a function of n . The dots are from a numerical simulation. Thedashed red line is the MF result (728). The green dashed line is a fit corresponding to α ≈ ln (cid:18) ( n + n c ) / ( n − n c ) (cid:19) . Inset: blow-up of main plot. In order to make analytical progress, we now study the topple-away or mean field (MF) solution of the stochasticManna sandpile, which we can solve analytically: Mean-Field Manna Model (MF-MM). If a site containstwo or more grains, move these grains to any randomlychosen sites of the system.The rate equations are, setting for convenience a − := 0 : ∂ t a i = − a i Θ( i ≥ a i +2 +2 (cid:104) (cid:88) j ≥ a j (cid:105) ( a i − − a i ) . (723)Using the sum rule (717), they can be rewritten as ∂ t a i = − a i Θ( i ≥ a i +2 +2(1 − a − a )( a i − − a i ) . (724)We are interested in the steady state ∂ t a i = 0 . One cansolve these equations by introducing a generating function.An simpler approach consists in realizing that for i ≥ ,Eq. (724) admits a steady-state solution of the form a i = a κ i − , i > . (725)This reduces the number of independent equations ∂ t a i = 0 in Eq. (724) from infinity to three. Furthermore, there arethe equations (cid:80) ∞ i =0 a i = 1 , and (cid:80) ∞ i =0 i a i = n . Thus thereare 5 equations for the 4 variables a , a , a , and κ . Thereason we apparently have one redundant equation is dueto the fact that we already used the normalisation condition(717) to go from Eq. (723) to Eq. (724).These equations have two solutions: For < n < ,there is always the solution for the inactive or absorbingstate , a = 1 − n, a = n, a i ≥ = 0 . (726)For n > / , there is a second non-trivial solution, a = 11 + 2 n , a i> = 4 n (cid:16) n − n +1 (cid:17) i n − . (727) (Note that a /a has the same geometric progression as a i +1 /a i for i > , which we did note suppose in ouransatz.) Thus the probability to find i > grains on a siteis given by the exponential distribution p ( i ) = 4 n n − − iα n ) , α n = ln (cid:18) n +12 n − (cid:19) . (728)Using these two solutions, we get the MF phase diagramplotted on figure 67 (thin lines). This has to be comparedwith the simulation of the Manna model on the samefigure (thick lines). One sees that for n ≥ , MFsolution and simulation are almost indistinguishable. Wealso checked with simulations that the Manna model hasa similar exponentially decaying distribution of grains persite, with a decay-constant α plotted on the right of figure68. In this section, we give the effective equations of motionfor the Manna model. Let us start from the mean-fieldequations for ρ ( t ) and n ( t ) . For simplicity we use theweighted Manna model. The physics close to the transitionshould not depend on it. Let us start from the hierarchy ofMF equations for the weighted Manna model. These aresimilar to Eq. (724), and can be rewritten as ∂ t a i =(1 − i ) a i Θ( i ≥ 2) + ( i + 1) a i +2 + 2 ρ ( a i − − a i ) . (729)Let us write explicitly the rate equation for the fraction ofempty sites e ≡ a , ∂ t e = a − ρe (730)The first term, the gain r + = a comes from the sites withtwo grains, toppling away, and leaving an empty site. Thesecond term, the loss term, is the rate at which one of thetoppling grains lands on an empty site, r − = 2 ρe . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles ∂ t e = a − ρe + (cid:112) a + 2 ρe ¯ ξ t , (731)where (cid:10) ¯ ξ t ¯ ξ t (cid:48) (cid:11) = δ ( t − t (cid:48) ) /(cid:96) d , and (cid:96) is the size of thebox which we consider. Close to the transition, a ≈ ρ .Inserting this into the above equation, we arrive at ∂ t e ≈ ρ (1 − e ) + √ ρ √ e ¯ ξ t , (732)Next we approximate √ e by the value of e at thetransition, i.e. e → e MFc = , see the mean-field phasediagram in Fig. 67, leading to ∂ t e ≈ ρ (1 − e ) + (cid:112) ρ ¯ ξ t . (733)This equation consistenly gives back e MFc = , used abovein the simplification of the noise term.As the number n of grains is conserved, with the helpof the sum rule n + e = ρ + 1 we can write two moreequations, ∂ t n = 0 , ∂ t ρ = ∂ t e. (734)Finally, we do not have not a single box of size (cid:96) , but alattice of boxes, indexed by a d -dimensional label x . Eachtoppling moves two grains from a site to neighbouring sites,equivalent to a current J ( x, t ) = − D ∇ ρ ( x, t ) + (cid:112) Dρ ( x, t ) ξ ( x, t ) . (735)The diffusion constant is D = 2 × d = d . The first factorof 2 is due to the fact that two grains topple. The factor of d is due to the fact that each grain can topple in any of the d directions, thus the rate D per direction is d , resultinginto D = 1 /d . As discussed above, we drop the noise termas subdominant.This current corrects both the activity ρ ( x, t ) , asthe number of grains n ( x, t ) , resulting into the samecontribution for both ∂ t ρ ( x, t ) , and ∂ t n ( x, t ) . It does notcouple to the density of empty sites. This is consistentwith the sum-rule (722) n − ρ + e = 1 , which implies that ∂ t ρ ( x, t ) ≡ ∂ t n ( x, t ) + ∂ t e ( x, t ) .In conclusion, we have the set of equations ∂ t e ( x, t ) = [1 − e ( x, t )] ρ ( x, t ) + (cid:112) ρ ( x, t ) ξ ( x, t ) , (736) ∂ t ρ ( x, t ) = 1 d ∇ ρ ( x, t ) + ∂ t e ( x, t ) , (737) (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = δ d ( x − x (cid:48) ) δ ( t − t (cid:48) ) . (738)Instead of writing coupled equations for e ( x, t ) and ρ ( x, t ) ,with the help of the sum rule (722) we can also writecoupled equations for ρ ( x, t ) and n ( x, t ) : ∂ t ρ ( x, t ) = 1 d ∇ ρ ( x, t ) + (cid:2) n ( x, t ) − (cid:3) ρ ( x, t ) − ρ ( x, t ) + (cid:112) ρ ( x, t ) ξ ( x, t ) , (739) ∂ t n ( x, t ) = 1 d ∇ ρ ( x, t ) , (740)Eqs. (739)–(740) are known as the equations of motion forthe conserved directed percolation (C-DP) class. They were Random Field, m = 0.005Random Bond, m = 0.005qEW, m = 0.001qEW (SOC)C-DP Langevin, m =0.001C-DP Langevin (SOC)Oslo (SOC)Cons. React.-Diff. (SOC)Eq.(4), DP class (SOC) w ∆( w ) Figure 69. The renormalized disorder correlator ∆( u ) , rescaled to ∆(0) = 1 and (cid:82) u ∆( u ) = 1 , for several situations: RF and RB disorderfor a disordered elastic manifolds, the Oslo and Manna models, as well asconserved directed percolation, all in d = 1 . Reprinted from [495], withkind permission. obtained in the literature [576, 556, 495, 502] by means ofsymmetry principles. This leaves all coefficients undefined,and does not ensure that Eq. (736) is local . This locality willprove essential in the next section. The derivation above isdue to Ref. [563]. It had been conjectured for a long time that the Mannamodel and depinning of disordered elastic manifolds areequivalent, and much work was devoted to clarify thisconnection [497, 577, 495]. The identification of fieldswhich finally led to a simple proof of this equivalence isgiven in [578], followed by [579], ρ ( x, t ) = ∂ t u ( x, t ) (the velocity of the interface), (741) e ( x, t ) = F ( x, t ) (the force acting on it). (742)The second equation (737) is the time derivative of theequation of motion of an interface, subject to a randomforce F ( x, t ) , ∂ t u ( x, t ) = 1 d ∇ u ( x, t ) + F ( x, t ) . (743)Since ρ ( x, t ) is positive, u ( x, t ) is for each x monotonouslyincreasing. Instead of parameterizing F ( x, t ) by space x and time t , it can be written as a function of space x and interface position u ( x, t ) . Setting F ( x, t ) → F (cid:0) x, u ( x, t ) (cid:1) , the first equation (736) becomes ∂ t F ( x, t ) → ∂ t F (cid:0) x, u ( x, t ) (cid:1) = ∂ u F (cid:0) x, u ( u, t ) (cid:1) ∂ t u ( x, t )= (cid:104) − F (cid:0) x, u ( x, t ) (cid:1)(cid:105) ∂ t u ( x, t )+ (cid:112) ∂ t u ( x, t ) ξ ( x, t ) . (744) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles x , this equation is equivalent to the Ornstein-Uhlenbeck [580] process F ( x, u ) , defined by ∂ u F ( x, u ) = 1 − F ( x, u ) + √ ξ ( x, u ) , (745) (cid:104) ξ ( x, u ) ξ ( x (cid:48) , u (cid:48) ) (cid:105) = δ d ( x − x (cid:48) ) δ ( u − u (cid:48) ) . (746)It is a Gaussian Markovian process with mean (cid:104) F ( x, u ) (cid:105) =1 / , and variance in the steady state of (cid:10)(cid:2) F ( x, u ) − (cid:3) (cid:2) F ( x (cid:48) , u (cid:48) ) − (cid:3)(cid:11) = 12 δ d ( x − x (cid:48) )e − | u − u (cid:48) | . (747)Writing the equation of motion (743) as ∂ t u ( x, t ) = 1 d ∇ u ( x, t ) + F (cid:0) x, u ( x, t ) (cid:1) , (748)it is interpreted as the motion of an interface with position u ( x, t ) , subject to a disorder force F (cid:0) x, u ( x, t ) (cid:1) . Thelatter is δ -correlated in the x -direction, and short-rangedcorrelated in the u -direction. In other words, this isa disordered elastic manifold subject to Random-Fielddisorder. As a consequence, the field-theoretic results ofsections 3.2–3.5 are also valid for the Manna model.Eq. (739) has a quite peculiar property, namely thefactor of 2 in front of both n ( x, t ) ρ ( x, t ) and − ρ ( x, t ) .As a consequence, Eq. (736) does not contain a term ∼ ρ ( x, t ) , which would spoil the simple mapping presentedabove. The absence of this term can not be induced onsymmetry arguments only. How this additional term, ifpresent, can be treated is discussed in Ref. [578].The mapping of the Manna model on disorderedelastic manifolds implies that properties of the latter shouldbe measurable in the former. As we discussed in sections2.4 to 2.10, and 3.2 to 3.5, a key feature of the theoryof disordered elastic manifolds is the existence of arenormalized disorder correlator with a cusp. Its existencein the Manna model, and equivalence to the one measured atdepinning was established in the beautiful work [495]. Theresulting (rescaled) disorder correlators ∆( w ) are shownon Fig. 69: it confirms the equivalence of depinning withboth RB and RF disorder, C-DP, Oslo, and several sandpileautomata. It also shows that directed percolation is in adistinct universality class. Remarks on the short-time dynamics of the Manna model The short-time dynamics of the Manna model has beenmeasured in several publications [375, 376, 377], and wasinterpreted as the dynamical exponent z depending on theinitial condition. We cannot follow this logic: the criticalexponent z is a bulk property of the system, and as suchis defined only after memory of the initial state is erased.What is possible is that the initial time critical exponentdiscussed in section 3.19 depends on the initial condition.Simulations for much larger systems are needed to settlethis question. 7. KPZ, Burgers, and the directed polymer In this section we review basic properties for the non-linearsurface growth known as the Kardar-Parisi-Zhang (KPZ)equation [537]. KPZ matters for disordered systems andthe subject of this review for its multiple connections: • mapping of the N -dimensional KPZ equation to the N -component directed polymer (random manifoldwith d = 1 ), • mapping of he N -dimensional decaying Burgers orKPZ equation to a particle (formally a randommanifold with d = 0 ) in N -dimensions, • non-linear surface growth terms `a la KPZ appear fordisordered systems, producing the distinct quenchedKPZ class discussed in section 5.7.For further reading on non-linear surface growth we referto the 1997 review by Krug [581]. A short summary ofmodern developments can be found in Ref. [582]. Notation. We use N for the dimension of the KPZequation instead of d , keeping d for the random-manifolddimension with N components, i.e. living in N dimensions. Consider figure 70. What is seen is an interface betweentwo phases, A and B. Phase A is stable, while phase B isunstable. The interface grows with a velocity λ ( u, t ) inits normal direction, increasing phase A, while diminishingphase B. Using the Monge representation { u, h ( u, t ) } , u ∈ R N the growth in h direction is given by (see figure) δh = (cid:112) ∇ h ( u, t )] λ ( u, t ) δt. (749)Assume that the growth is due to a discrete process.Following the prescription in section 6.3, the growthvelocity λ ( u, t ) has a mean λ plus fluctuations η ( u, t ) , λ ( u, t ) = λ + η ( u, t ) , (cid:104) η ( u, t ) η ( u (cid:48) , t (cid:48) ) (cid:105) = 2 Dδ ( t − t (cid:48) ) δ N ( u − u (cid:48) ) . (750)This leads to ∂ t h ( u, t ) = λ + λ ∇ h ( u, t )] + η ( u, t ) + ..., (751)where the dots indicate higher-order terms in ( ∇ h ) . Thisis (almost) the famous KPZ [537] equation. To derive thelatter, we first subtract the growth for a flat interface, setting h → h − λt , and finally add one more term to the equation ∂ t h ( u, t ) = ν ∇ h ( u, t ) + λ (cid:2) ∇ h ( u, t ) (cid:3) + η ( u, t ) . (752)The additional term proportional to ν describes diffusionalong the interface, rendering it smoother.One typically measures the 2-point function (cid:10) [ h ( x, t ) − h ( x (cid:48) , t (cid:48) )] (cid:11) (cid:39) | x − x (cid:48) | ζ KPZ f ( x z KPZ /t ) , (753) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles h' ( u , t ) h' ( u , t ) δ h AB uh Figure 70. Left: An interface growing in its normal direction, with phase A invading phase B. Right: An experimental realization using two phases of anematic liquid crystal [583]. where f goes to a constant for t → ( x → ∞ ), andtogether with its prefactor becomes independent of x for x → . This defines two exponents, the roughness ζ KPZ and the dynamic exponent z KPZ . The added index allowsus to distinguish it from the exponents of the directedpolymer, especially since we will see later that z KPZ =1 /ζ directed polymer . Taking one spatial derivative of Eq. (752) yields Burgers’ equation [584]. Defining (cid:126)v ( u, t ) := ∇ h ( u, t ) , (754)Burgers’ equation reads ∂ t v ( u, t ) = ν ∇ v ( u, t ) + λ ∇ [ v ( u, t ) ] + ∇ η ( u, t ) , (755) (cid:104) η ( u, t ) η ( u (cid:48) , t (cid:48) ) (cid:105) = 2 Dδ ( t − t (cid:48) ) δ N ( u − u (cid:48) ) . (756)The non-linear term satisfies the identity (cid:88) i ∂ j (cid:2) v i ( u, t ) (cid:3) ≡ (cid:88) i ∂ j (cid:2) ∂ i h ( u, t ) (cid:3) (757) = (cid:88) i (cid:2) ∂ i h ( u, t ) (cid:3)(cid:2) ∂ j ∂ i h ( u, t ) (cid:3) ≡ (cid:88) i (cid:2) v i ( u, t ) ∂ i (cid:3) v j ( u, t ) . Eq. (755) can be written as ∂ t v ( u, t ) = ν ∇ v ( u, t ) + λ (cid:2) v ( u, t ) ·∇ ] v ( u, t ) + ∇ η ( u, t ) . (758)This is identical to Navier-Stokes’ equation for incompress-ible fluids, with the crucial difference that Burgers’ ve-locity is a total derivative, v ( u, u ) = ∇ h ( u, t ) , whereasfor Navier-Stokes it is divergence free, ∇ v ( u, t ) = 0 .For this reason, Burgers equation does not describe turbu-lence encountered e.g. in a fast-flowing river. It has, how-ever, applications to the large-scale structure of galaxies[585, 586, 587]. Consider the N -dimensional KPZ equation (752) in Itˆodiscretization, with noise as given in Eq. (750). We caneliminate the non-linear term by the so-called Cole-Hopftransformation [588, 589] Z ( u, t ) := e λ ν h ( u,t ) − D λ ν t ⇐⇒ h ( u, t ) = 2 νλ ln Z ( u, t ) + Dλ ν t. (759)(The reader might see this transformation without theterm − Dλ t/ (4 ν ) ; this is then done in mid-point, i.e.Stratonovich discretization [2, 279, 280], see section 10.4).Using Itˆo calculus (section 10.2), we obtain d Z ( u, t ) = λ ν Z ( u, t )d h ( u, t ) + λ ν Z ( u, t ) d h ( u, t ) − Dλ ν Z ( u, t )d t = λ ν Z ( u, t ) (cid:110)(cid:16) ν ∇ h ( u, t )+ λ ∇ h ( u, t )] (cid:17) d t + d η ( u, t ) (cid:111) = ν ∇ Z ( u, t )d t + Z ( u, t ) λ ν d η ( u, t ) . (760)Noting λη ( u, t ) ≡ V ( u, t ) this can be written as ∂ t Z ( u, t ) = ν ∇ Z ( u, t ) + 12 ν V ( u, t ) Z ( u, t ) , (761) (cid:104) V ( u, t ) V ( u (cid:48) , t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) R ( u − u (cid:48) ) , (762) R ( u ) = 2 λ D δ N ( u ) . (763) The equation of motion (761) can be solved by Z ( u, t | V ) = (cid:90) u = u ( t ) u ( t i )= u i D [ u ] e − T (cid:82) tt i d τ u (cid:48) ( τ ) − V ( u ( τ ) ,τ ) ,T = 2 ν. (764)This is the path integral of a directed polymer in thequenched random potential V ( u ) , also referred to asthe Feynman-Kac formula [590, 591]. To average over heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles n replicas introducedin section 1.5, Z ( u , ..., u n , t )= n (cid:89) α =1 (cid:90) u α ( t )= u α u α ( t i )= u α, i D [ u α ] (cid:90) D [ V ]e − T (cid:82) tt i d τ (cid:80) nα =1 [ u (cid:48) α ( τ ) + V ( u α ( τ )) ]e − λ D (cid:82) u,τ V ( u,τ ) = n (cid:89) α =1 (cid:90) u α ( t )= u α u α ( t i )= u α, i D [ u α ]e − (cid:82) tt i d τ (cid:80) α T u (cid:48) α ( τ ) − T (cid:80) α,β R ( u α ( τ ) − u β ( τ )) . (765)We had discussed its solution in the T → -limit in section1.5, see Eq. (30).Using Eqs. (764) and (759), the free energy of adirected polymer is related to the KPZ height field h ( u, t ) via F ( u, t ) := − T ln Z ( u, t ) ≡ − λh ( u, t ) + Dλ ν t. (766)Apart from the (last) drift term which is due to thediscretization scheme and which can always be subtracted,this relation is valid in the inviscid limit ν → , equivalentto T → , i.e. for the ground state of the directed polymer.For further reading we refer to [592]. A scaling analysis of the KPZ equation (752) starts at ˜ h ( u, t ) = b − ζ KPZ h ( bu, b z KPZ t ) (767)with a roughness exponent ζ KPZ and a dynamical exponent z KPZ defined in Eq. (753). The rescaled field ˜ h satisfies aKPZ equation (752) with rescaled coefficients ˜ ν = b z KPZ − ν, ˜ D = b z KPZ − d − ζ KPZ D, ˜ λ = b z KPZ + ζ KPZ − λ. (768)If λ = 0 , the scaling of the diffusion equation ζ KPZ =(2 − N ) / , and z KPZ = 2 yields a fixed point of the coarsegraining transformation (767). For λ (cid:54) = 0 , the non-linearity λ grows if the combination ζ KPZ + z KPZ − → − N ispositive. This is always the case in dimension N < .As we will see below in section 7.8 for dimension N > there is a transition between a weak-coupling and a strong-coupling regime.The KPZ equation has an important invariance inany dimension N [593, 537, 581]. Consider the tilttransformation parameterized by a N -dimensional vector c , h (cid:48) ( u, t ) = h ( u + λct, t ) + cu + λ c t. (769)For λ → , this reduces to the statistical tilt symmetry(64). It is reminiscent of the full rotational invariance of thegrowing surface before passing to the Monge representation(749). As a consequence, we expect the KPZ equationto remain invariant under this transformation. Indeed, h (cid:48) satisfies the same KPZ equation (752) as h , with a shiftednoise η (cid:48) ( u, t ) = η ( u + λct, t ) . (770)As long as the temporal correlations of η are sufficientlyshort ranged, the shift does not affect the statisticalproperties of the noise [594], and the statistics of h isinvariant under the transformation (769). In the literaturethis property is referred to as Galilean invariance, as in thecontext of the stirred Burgers equation (755), where it wasfirst discussed in Ref. [593], it appears as a shift in thevelocity, v → v (cid:48) = v + λc .As the tilt transformation explicitly contains the non-linearity λ , the latter should not change under rescaling.From Eq. (768) we conclude that at a fixed point with λ (cid:54) = 0 ζ KPZ + z KPZ = 2 . (771)To make contact with the scaling properties of the directedpolymer, we remind the exponent relation (46) for the freeenergy of an elastic manifold, F ∼ L θ , with θ = d − ζ .The directed polymer has internal dimension d = 1 . Usingthat according to Eq. (759) the free energy identifies with h , and that the length L of the directed polymer is the timein the KPZ equation, we arrive at h (cid:98) = F ∼ L θ (cid:98) = t θ . On theother hand h ∼ u ζ KPZ and u ∼ t /z KPZ , such that θ = 2 ζ − ζ KPZ z KPZ = 2 − z KPZ z KPZ , (772)where for the last identity Eq. (771) was used. This impliesthat z KPZ = 1 ζ . (773)While the notations in the literature are somehow divergent,the prevailing ones seem to be α ≡ χ ≡ ζ KPZ , β = ζ KPZ z KPZ ≡ θ, z = z KPZ ≡ ζ . (774) Another possibility to obtain a field theory for Eqs. (761)–(763) is to write the partition function for the n -timesreplicated field Z α , α = 1 , ..., n , i.e. Z ∈ R n , as Z = (cid:90) D [ Z ] D [ ˜ Z ] D [ V ] e −S CH [ Z, ˜ Z,V ] , (775) S CH [ Z, ˜ Z, V ]= (cid:90) u,t ˜ Z ( u, t ) (cid:20) ∂ t Z ( u, t ) − ν ∇ Z ( u, t ) − ν V ( u, t ) Z ( u, t ) (cid:21) + 14 λ D (cid:90) u,t V ( u, t ) . (776)Performing the integral over V we obtain Z = (cid:90) D [ Z ] D [ ˜ Z ] e −S CH [ Z, ˜ Z ] , (777) S CH [ Z, ˜ Z ] = (cid:90) u,t ˜ Z ( u, t ) (cid:2) ∂ t Z ( u, t ) − ν ∇ Z ( u, t ) (cid:3) − λ D ν (cid:104) ˜ Z ( u, t ) Z ( u, t ) (cid:105) . (778) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles t → t/ν , we arrive at S CH [ Z, ˜ Z ] = (cid:90) u,t ˜ Z ( u, t ) (cid:2) ∂ t Z ( u, t ) − ∇ Z ( u, t ) (cid:3) − g (cid:104) ˜ Z ( u, t ) Z ( u, t ) (cid:105) , (779) g = λ D ν . (780)Note that we do not need to take the limit of n → atthe end. This is allowed as in Itˆo calculus the partitionfunction Z = 1 . To study the flow of the effective couplingconstant g , we need at least two distinct “replicas”, whichcan be thought of as “worldlines” of two particles, startingat different initial positions. To better understand the behavior of the KPZ equation(752), let us consider Eq. (752) for given initial condition h ( u ) := h ( u, t = 0) , in absence of the noise η ( u, t ) , i.e. D = 0 . The Cole-Hopf transformed KPZ equation (761)reduces to a diffusion equation, solved as Z ( u, t ) = (cid:90) u (cid:48) e − ( u − u (cid:48) )24 νt (4 πνt ) d/ Z ( u (cid:48) , . (781)Putting back the definition (759) of Z in terms of h ( u ) := h ( u, t = 0) , we obtain, since D = 0 , e λ ν h ( u,t ) = (cid:90) u (cid:48) e λ ν (cid:20) h ( u (cid:48) ) − ( u − u (cid:48) )22 λt (cid:21) (4 πνt ) d/ . (782)It is interesting to consider the limit of ν → , equivalent to T → for the directed polymer (764). Then the solution toEq. (782) is h ( u, t ) = max u (cid:48) (cid:20) h ( u (cid:48) ) − ( u − u (cid:48) ) λt (cid:21) . (783)This solution is formally equivalent to the solution (94) ofthe toy model introduced in section 2.9, replacing − h ( u ) → V ( u ) (microscopic disorder) (784) λt → m (785) − h ( u, t ) → ˆ V ( u ) (effective disorder at scale m = λt ) . (786)As observed there and shown on Fig. 71 for a randominitial condition, the function h ( u, t ) is composed ofalmost parabolic pieces, continuous everywhere but notdifferentiable at the juncutres. Geometrically, this can beobtained by approaching a parabola of curvature m =1 / ( λt ) from the top, and reporting as a function of its center u the position h ( u, t ) at which it first touches the initialcondition h ( u ) . This construction is shown on Fig. 72.More can be learned via this approach, see Ref. [81]. β -function for KPZ As we had seen after Eq. (768), the Gaussian fixed point( λ = 0 ) is stable for weak disorder as long as the number N of dimensions is larger than 2. We now show thatthere exists a phase transition between the weak-couplingphase (Gaussian fixed point), and a strong-coupling phase.This transition is accessible to a standard perturbative RGtreatment, contrary to the strong-coupling phase which isnot. Perturbative RG treatments of the KPZ equation arenumerous, starting with the original work [537]. They wereextended to 2-loop order in Refs. [595, 596, 597, 598, 599],leading to some controversy finally resolved in Ref. [600].The treatment is much easier for the Cole-Hopf transformedversion [601, 602], allowing us to resum perturbation theoryto all orders. We now calculate the β -function associated tothe model (779)-(780), following Ref. [602]. To this aimwe introduce the graphical notation := (cid:90) u,t (cid:104) ˜ Z ( u, t ) Z ( u, t ) (cid:105) . (787)Then the only diverging diagrams are chains of , ofthe form , and so on. Higher-ordervertices are irrelevant in perturbation theory. As a result,the effective coupling constant is g eff = g + g + g + g + . . . + higher order vertices . (788)In Fourier-representation with incoming momentum p andfrequency ω , each chain in Eq. (788) factorizes, i.e. can bewritten as product of the vertex times a power of theelementary loop diagram (which is a function of p and ω ): p,ω −→ (cid:124) (cid:123)(cid:122) (cid:125) n loops = (cid:0) p,ω (cid:1) n . (789)Eq. (788) is a geometric sum which yields the effective 4-point function g eff = Γ ZZ ˜ Z ˜ Z p,ω = g − g p,ω . (790)The elementary diagram is p,ω −→ ω + ν, p + k ω − ν, p − k p,ω −→ = (cid:90) d N k (2 π ) N (cid:90) d ν π (cid:0) p + k (cid:1) + i (cid:0) ω + ν (cid:1) (cid:0) p − k (cid:1) + i (cid:0) ω − ν (cid:1) = a (cid:18) p + iω (cid:19) N/ − , a = 1(8 π ) N/ Γ (cid:18) − N (cid:19) . (791)This integral is divergent for any p and ω when N → .Renormalisation means to absorb this divergence into a heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles u - - h ( u ) u - - - h ( u ) Figure 71. Left: Evolution of a random initial condition (black) at times λt = 0 (black, bottom), / (orange), / (green), (red), and (blue).Right: ibid. for the Burgers velocity v ( u ) := h (cid:48) ( u ) . h ( u , t ) u u ′ h ( u ′ ) Figure 72. A geometrical solution to Eq. (783) is obtained by movinga parabola of curvature / (2 λt ) and centered at u (in red) down until ithits h (cid:48) ( u (cid:48) ) in blue. Its minimum is then at h ( u, t ) . (This construction isalready discussed in the 1979 paper by Kida [177].) reparametrization of the coupling constant g : We claim thatthe 4-point function (the effective coupling g eff ) is finite(renormalized) as a function of g r instead of g , upon setting g = Z g g r µ − ε , Z g = 11 + ag r , ε = N − . (792) µ is an arbitrary scale, the renormalization scale . As afunction of g r , the 4-point function reads Γ ZZ ˜ Z ˜ Z p,ω = g r µ − ε (cid:0) a − µ − ε p,ω (cid:1) g r . (793)Since ε (cid:0) p + iω (cid:1) ε/ µ − ε is finite for ε > as longas the combination p + iω is finite, it can be read offfrom Eq. (793) that Γ ZZ ˜ Z ˜ Z p,ω is finite even in the limitof ε → . (If useful, either p = 0 or ω = 0 maysafely be taken.) This completes the proof. Note that thisensures that the model is renormalizable to all orders inperturbation-theory, what is normally a formidable task toshow [112, 113, 114, 603, 604, 605, 110].The β -function that we calculate now is exact to allorders in perturbation theory. It is defined as the variation of the renormalized coupling constant, keeping the bare onefixed β ( g r ) = − µ ∂∂µ g g r . (794)From Eq. (793) we see that it gives the dependence of the4-point function on p and ω for fixed bare coupling. Therelation between g and g r is g = g r µ − ε ag r ⇔ g r = gµ − ε − ag , (795)and hence β ( g r ) = − εg r (1 + ag r ) . (796)Using a from Eq. (791), our final result is β ( g r ) = (2 − N ) g r + 2(8 π ) N/ Γ (cid:18) − N (cid:19) g . (797)This equation has a perturbative, IR repulsive, fixed point at g ∗ r = 2(8 π ) N/ ( N − (cid:0) − N (cid:1) . (798)For N > it describes the phase transition between theweak-coupling phase where the KPZ term is irrelevant( g = 0 ), and a strong coupling phase, for which g → ∞ .Note that this is the only fixed point available for N ≤ ;especially, it does not describe KPZ in dimension N = 1 .As a consequence, standard perturbation theory fails toproduce a strong-coupling fixed point, a result which cannotbe overemphasized. This means that any treatment of thestrong coupling regime has to rely on non-perturbativemethods . The FRG approach discussed above qualifies asnon-perturbative in this sense, since FRG follows more thanthe flow of a single coupling constant. It does of course notrule out the possibility to find an exactly solvable model,non-equivalent to KPZ, for which it is possible to expandtowards the strong-coupling regime of KPZ.Let us also note that the β -function is divergent at N = 4 , and therefore our perturbation expansion breaksdown at N = 4 . To cure the problem, a lattice regularized heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles a will enter into the equations and the result isno longer model-independent. This may be interpreted as N = 4 being the upper critical dimension of KPZ, or as asign for a simple technical problem. Compare [606, 391]. A special case is the anisotropic KPZ equation in twodimensions, for which the KPZ-nonlinearity is positive inone direction, and negative in the other. This competitionproduces a perturbative fixed point [607]. When the noise η ( u, t ) in Eq. (752) is long-range corre-lated, (cid:104) η ( u, t ) η ( u (cid:48) , t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) | u − u (cid:48) | ρ − N , ρ > , (799)a different exponent is expected. Using Eq. (23) with γ = N − ρ and d = 1 , we find ζ LRFlory = N − ρ , and as aconsequence of Eqs. (773) and (771) z LRKPZ = 4 + N − ρ , ζ LRKPZ = 2 − N + 2 ρ . (800)This LR fixed point, which is exact as long as the disorderdoes not get renormalized, is in competition with the SR(random bond) fixed point for which disorder renormalizes.As a rule of thumb, the fixed point with the larger ζ or ζ KPZ ,and smaller z KPZ dominates. In dimension N = 1 where ζ KPZ = 1 / , the LR fixed point dominates for ρ > ρ c = .These results can already be found in [608, 609], and werereanalyzed via RG in [610, 611]. A lot of work has been devoted to either proving ordisproving the existence of an upper critical dimension d c ≈ . The arguments in favor are via proof by consistencyor contradiction [612, 613, 614], d c = 4 , or mode-coupling: d c = 4 [615, 616, 617], d c = 3 . or d c = 4 . ,depending on the UV regularization [618]. If these arewrong, presumably one of the underlying assumptions fails.The arguments against are mostly from numericalsimulations, either directly on the KPZ in its RSOSrepresentation [619, 620, 621, 622, 623, 624] or on thedirected polymer [625]. The criticism voiced is that theyare not in the asymptotic regime, or break the rotationalsymmetry of the KPZ equation. Mode-coupling solutionswithout an upper critical dimension have been proposed[626], as well as approximate RG schemes [627], or NPRG[628].The issue is far from settled, and only few distinctarguments can be found: FRG for the directed polymerfavors a critical dimension d c ≈ . [145], approximatelyalso found in [629]. While the work by [601, 602] indicatesthe necessity for an UV-cutoff in d = 4 (see above), an additional scale may appear at all even N , i.e. N = 6 , , etc. [391]. Closure relations in CFT lead to simplefractions for the critical exponents [630, 631], not favoredby numerical simulations. From the newer developments,let us mention the mapping to d -mer diffusion [632], whichseems to give rather precise numerical estimates, see table3. Let us conclude with a quote from the recent review[582]: The “equation proposed nearly three decades ago byKardar, Parisi and Zhang continues to inspire, intrigue andconfound its many admirers.” d ζ KPZ ( d -mer) ζ KPZ z KPZ ( d -mer) z KPZ ( d -mer) ζ KPZ (RSOS)1 / / / / . . . . . . . . . . . . . . . . Table 3. Growth exponent estimates of the d -mer model (d-mer) [632];the results in d = 1 are exact. The last column represents the resultsfor the RSOS model, always taking the most recent and precise values[623, 624, 619, 620, 633, 621]. At least some of the error bars seem overlyoptimistic. Quenched KPZ The quenched KPZ equation is discussed later in section5.7. d = 1 The KPZ equation (752) is formally a Langevin equation.The corresponding Fokker Planck equation for the evolu-tion of its measure P t [ h ] , derived in Eq. (935), reads ∂ t P t [ h ] = D (cid:90) u δ δh ( u ) P t [ h ] (801) − (cid:90) u δδh ( u ) (cid:16) ν ∇ h ( u ) + λ (cid:2) ∇ h ( u ) (cid:3) P t [ u ] (cid:17) . At least for λ = 0 , a solution can be found by asking that D δδh ( u ) P t [ h ] = ν ∇ h ( u ) P t [ h ] . (802)This is solved by P t [ h ] = N exp (cid:18) − ν D (cid:90) u (cid:2) ∇ h ( u ) (cid:3) (cid:19) . (803)Unsurprisingly, this is the measure for a diffusing elasticstring. What are the additional terms for λ (cid:54) = 0 ? Insertingthe measure (803) into Eq. (801) yields ∂ t P t [ h ] = − (cid:90) u δδh ( u ) (cid:16) λ (cid:2) ∇ h ( u, t ) (cid:3) P t [ u ] (cid:17) = λν D (cid:90) u (cid:2) ∇ h ( u ) (cid:3) ∇ h ( u ) . (804) Note that some authors [581] use a different normalization for Eq. (750) D here = D there , reflected in the invariant measure. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles A KPZ Cocktail-Shaken, not Stirred... 797 Fig. 1 stationary-state ( χ ) , transient flat ( χ ) , and radial ( χ ) geometries, stacked up against relevant universal limit distributions Baik–Rains, Tracy–Widom GOE, andGUE, respectively. Inset : skewness minimum, s T M ≈ . 22, exhibited by KPZ equation as experimentalsignature of stationary-state statistics, recently observed in KPZ turbulent liquid-crystal kinetic rougheningwork [84]. Note, in particular, the asymptotic approach from below to TW-GOE skewness s ( τ ≫ ) = s = . runs, suitably averaged, we have extracted the asymptotic growth velocity v ∞ = . A , λ and v ∞ are known exactly, pinning down these quantities represents theprimary challenge, beyond simply aggregating the height fluctuation statistics. In fact, asdone in experiment [82,83] and occasionally in numerics, we have matched the variance ofthe transient regime statistics (i.e., our 1+1 KPZ Euler pdf) to TW-GOE, fitting the secondmoment to ⟨ χ ⟩ = . $ β = . s = ⟨ δ h ⟩ c / ⟨ δ h ⟩ / c = . s = . $ thus determined,can then be employed in the service of a genuine comparison to both the radial TW-GUE andstationary-state BR-F limit distributions. In other words, the KPZ scaling parameters, whilemodel-dependent, are not subclass dependent—in a given dimension, the same $ and v ∞ apply to flat, curved and stationary cases, as noted already in experiment [83]. Thus, here,we take the numerical short-cut, using TW-GOE to determine $ β , privileging the greaterchallenges posed by BR-F & TW-GUE.Hence, we have also included in Fig. 1 the centered, rescaled fluctuations of the h eigh tincrement, ’ h = h ( t o + ’ t ) − h ( t o ) , casting the associated stationary-state KPZ statistics interms of the O ( ) Baik–Rains variable: χ = ( ’ h − v ∞ ’ t )/( $’ t ) / , integrating the KPZequation to a late time t o = , then investigating the temporal correlations at a later time t o + ’ t , with 1 ≤ ’ t ≤ ≪ t o , at a given site, sampling over the entire system of size L = , then averaging over 10 realizations, again yielding a statistical data set with 10 points. Here, for the skewness and kurtosis, we find 0.348 and 0.261, compared to Prähofer–Spohn’s oft-quoted BR-F values: s = . k = . 866 K.A. Takeuchi, M. Sano Fig. 8 Histogram of the rescaledlocal height q ≡ (h − v ∞ t)/( Γ t) / for thecircular ( solid symbols ) and flat( open symbols ) interfaces. The blue circles and red diamonds display the histograms for thecircular interfaces at t = 10 s and30 s, respectively, while the turquoise up-triangles and purpledown-triangles are for the flatinterfaces at t = 20 s and 60 s,respectively. The dashed and dotted curves show the GUE andGOE TW distributions,respectively, defined by therandom variables χ GUE and χ GOE . (Color figure online) For the flat interfaces, one in principle needs to impose a tilt u to measure v ∞ (u) [53], butsupposing the rotational invariance of the system, one again finds v ∞ (u) = √ + u v ∞ andthus Eq. (13). This is justified by practically the same values of our estimates v ∞ in Eq. (10).In passing, the rotational invariance also implies v = χ with the mathematically defined random variables χ GUE and χ GOE . This is achieved by defining the rescaled local height q ≡ h − v ∞ t( Γ t) / ≃ χ (14)and producing its histogram for the circular and flat interfaces (solid and open symbols inFig. 8, respectively). The result clearly shows that the two cases exhibit distinct distribu-tions, which are not centered nor symmetric, and hence clearly non-Gaussian. Moreover,the histograms for the circular and flat interfaces are found, without any ad hoc fitting, veryclose to the GUE (dashed curve) and GOE (dotted curve) TW distributions, respectively, asanticipated from their values of the skewness and the kurtosis as well as from the theoreticalresults for the solvable models. The agreements are confirmed with resolution roughly downto 10 − in the probability density.A closer inspection of the experimental data in Fig. 8 reveals, however, a slight devia-tion from the theoretical curves, which is mostly a small horizontal translation that shrinksas time elapses. To quantify this effect, we plot in Fig. 9 the time series of the differencebetween the n th-order cumulants of the measured rescaled height q and those for the TWdistributions. We then find for both circular and flat interfaces [Fig. 9(a, b)] that, indeed, thesecond- to fourth-order cumulants quickly converge to the values for the corresponding TWdistributions, whereas the first-order cumulant ⟨ q ⟩ , or the mean, shows a pronounced devia-tion as suggested in the histograms. This, however, decreases in time, showing a clear powerlaw proportional to t − / [Fig. 9(c, e)], and thus vanishes in the asymptotic limit t → ∞ .Similar finite-time corrections are in fact visible in other cumulants, though less clearly for χ Figure 73. Left: Numerical verification of the universal distributions for KPZ in one dimension as explained in the text. Figure from [582]. Right:experimental verification in Ref. [583]. The blue circles and red diamonds display the histograms for the circular interfaces at t = 10 s and s,respectively, while the turquoise up-triangles and purple down-triangles are for the flat interfaces at t = 20 s and s, respectively. While written in continuous notation, the calculationshould be made on the discretized version, with propersymmetrization of the [ ∇ h ] term. To go to the second line,we dropped the direct derivative of the latter, as it integratesto 0. In dimension N = 1 , the integrand is a total derivative,thus integrates to zero. In higher dimensions, this is notthe case. The simplest explicit counterexample for periodicboundary conditions ( L = 2 π ) in dimension N = 2 wefound is h ( u , u ) = [ a + cos( u )][ b + cos( u )] , for whichthe last integral in Eq. (804) evaluates to − abL .The measure (803) implies that equal-time correlationfunctions of the nonlinear ( λ > ) theory are given by thoseof the linear theory ( λ = 0 ), (cid:10) ˜ h ( q )˜ h ( q (cid:48) ) (cid:11) = 2 πδ ( q + q (cid:48) ) Dq ν (805) ⇔ (cid:10) [ h ( x ) − h ( x (cid:48) )] (cid:11) = Dν | x − x (cid:48) | ζ KPZ , (806) ζ d =1KPZ = 12 . (807)Moreover, the measure (803) is Gaussian. This can beviewed as due to a fluctuation dissipation theorem (FDT)[634]. Eq. (771) further implies z d =1KPZ = 32 . (808)As a consequence, ζ d =1RB = 1 z d =1KPZ = 23 (809)is the roughness exponent of a directed polymer in dimensions, and the roughness of domain walls in dirty 2dmagnets. Their energy-fluctuation exponent is θ = d − ζ = 13 . (810)This can also be obtained via Bethe ansatz [107, 594]. Since the introduction of the KPZ equation in 1986 [537],much progress has been made in one dimension. Thisstarted in 2000 with the groundbreaking work by Pr¨ahoferand Spohn [635, 636, 637] introducing the polynucleargrowth model (PNG): One starts from a flat configuration.Steps of vanishing size are deposited as a Poisson processupon the already constructed surface. Steps then grow withunit velocity at both ends. When steps meet, they merge.Heuristically it seems clear that this process belongs to theKPZ universality class, similar to its discrete cousin, theRSOS model. Independently, Johansson introduced thesingle-step (SS) model [638], relating surface growth tothe combinatorial problem of finding the longest increasingsubsequence in a random permutation [639] and randommatrix theory [640], relating to older work in this domain[641]. The key observables are constructed from thespatially averaged mean height h ( t ) , which for large timesis assumed to grow with velocity v ∞ , lim t →∞ (cid:104) h ( t ) (cid:105) − v ∞ t = const . As the mean height is proportional to thefree energy of a directed polymer, see Eq. (766), results forthe height fluctuation have an immediate interpretation interms of free-energy fluctuations of a directed polymer.Key observables are χ = h ( t + t ) − h ( t ) − v ∞ t (cid:0) D λ ν t (cid:1) / , (811) χ = h ( t ) − v ∞ t (cid:0) D λ ν t (cid:1) / , (flat initial conditions) , (812) χ = h ( t ) − v ∞ t (cid:0) D λ ν t (cid:1) / , (circular initial conditions) . (813)Each of these observables has a unique universal distribu- heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles χ is distributed according to the Baik-Rains F distribu-tion, χ is distributed according to the Tracy-Widom (TW)Gaussian Orthogonal Ensemble (GOE) distribution, χ is distributed according to the Tracy-Widom GaussianUnitary Ensemble (GUE) distribution.The reader wishing to test these laws himself can find aMathematica implementation online [642].The Bethe-ansatz for the directed polymer wasintroduced in 1987 by Kardar to obtain the roughnessexponent of a directed polymer, ζ = 2 / and θ =1 / . A revival started in 2010, when physicists succeeded[643, 644, 645, 646, 647] to first reproduce the work of[635, 636, 637] via the Bethe ansatz, and then extendit to other situations [648]. At the same time, a newgeneration of mathematicians developed complementarytools [649, 650, 651, 652], joining the work of Spohn andcollaborators [653, 654].Extraordinarily, Takeuchi and Sano [655, 583] suc-ceeded to extract the universal distributions from a turbulentliquid-crystal experiment. A snapshot is shown in Fig. 70,and the two Tracey-Widom distributions for circular and flatinitial conditions in Fig. 73.Let us conclude by mentioning pedagogical presenta-tions [656, 657, 658], as well as attempts to port this at leastnumerically to higher dimensions [659, 660]. In all dimensions: • KPZ [537], • polynuclear growth (PNG) [635, 636, 637], • Rigid-solid-on-solid models (RSOS) [619, 620, 621,622, 623, 624]. • directed polymer in quenched disorder (section 7.4).In one dimension: • longest growing subsequence in a random permutation[639], • the asymmetric simple exclusion process (ASEP)[661, 662, 663].Experimental realizations (one dimension only): • slow combustion of paper [664, 665], • turbulent liquid crystals [655, 583], • particle deposition (with crossover to qKPZ) [666]. In Burgers’ turbulence velocity profiles are locally linear(see e.g. the right of Fig. 71), interrupted by jumps.Phenomenologically it is similar to the force field indisordered systems. This implies that, if non-vanishing, atsmall distances (cid:104) [ v ( u, t ) − v ( u (cid:48) , t )] n (cid:105) (cid:39) A n | u − u (cid:48) | ζ n , (814)and for Burger ζ n = 1 , independent of n . In Navier-Stokes turbulence, the exponent ζ = 1 , as predicted byKolmogorov in 1941 [667]. Other moments obey ζ n = n δζ n , (815)where δζ n is small. Calculating δζ n analytically is theoutstanding problem of turbulence research. One cantry to use FRG for this problem [668], but somethingcrucial is missing: While FRG correctly deals with shocks,the weaker singularities responsible for Eq. (815) are notcaptured by FRG. (It may work in dimension d = 2 ,though [668].) It is possible that the FRG fixed point whichtypically has a cusp, and which usually is implemented forthe second cumulant, instead applies to the third cumulant,as ζ = 1 . How to implement this idea remains an openproblem. 8. Links between loop-erased random walks, CDWs,sandpiles, and scalar field theories Supersymmetry techniques are an alternative to replicasto average over disorder. In these techniques, additionalfermionic or Grassmannian degrees of freedom areintroduced to render the partition function Z = 1 , evenbefore averaging over the disorder. We start by reviewingbasic properties of Grassmann variables, before using themfor disorder averages. Most of the material is standard, andthe reader familiar with it, or wishing to advance may safelydo so.Two points should, however, be retained: While themethod is referred to as supersymmetry technique , super-symmetry is broken at the Larkin scale, i.e. when a cuspappears. The name is due to historical reasons, stemmingfrom a time when people believed that supersymmetry isnot broken.Braking of supersymmetry, and the cusp, can be foundin this framework, as long as one considers at least twophysically distinct copies. The technical reason is that toassess the n -th cumulant of a distribution, one needs at least n distinct copies. Even when supposing the disorder to beGaussian distributed, the variance, i.e. the second cumulantneeds to be assessed, thus two distinct replicas.Apart from these more formal considerations, thetechnique has proven powerful in the mapping of charge-density waves onto a φ -type theory (section 8.6). heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Grassmann variables are anticommuting variables whichallow one to write a path-integral for bosons, in the sameway as one does for bosons. There are only few rules toremember. If χ and ψ are Grassmann variables, then χ ψ = − ψ χ. (816)This immediately implies that χ = 0 . (817)One introduces derivatives, and integrals through the sameformula, known as Berezin integral [669], (cid:90) d χ χ ≡ dd χ χ = 1 . (818)One checks that they satisfy the usual properties associatedto “normal” derivatives, and integrals. An importantproperty is (cid:90) d ¯ χ d χ e − a ¯ χχ = a. (819)This is easily proven upon Taylor expansion. The minussign in the exponential cancels with the minus sign obtainedwhen exchanging ¯ χ with χ , which is necessary since anintegral or derivative is defined to act directly on thevariable following it. This can be generalized to integralsover an n -component pair of vectors (cid:126) ¯ χ , and (cid:126)χ : (cid:90) d (cid:126) ¯ χ d (cid:126)χ e − (cid:126) ¯ χ A (cid:126)χ := n (cid:89) a =1 (cid:90) d ¯ χ a d χ a e − (cid:126) ¯ χ A (cid:126)χ = det( A ) . (820)It is proven by changing coordinates s.t. A becomesdiagonal. For comparison we give the correspondingformula for normal (bosonic) fields, noting φ a := φ ax + iφ ay , ˜ φ a := φ ax − iφ ay , (cid:90) d (cid:126) ˜ φ d (cid:126)φ e − (cid:126) ˜ φ A (cid:126)φ := n (cid:89) a =1 (cid:90) d φ ax d φ ay π e − (cid:126) ˜ φ A (cid:126)φ = 1det( A ) . (821)When combining normal and Grassmanian integrals overthe same number of variables into a product, thecontributions from Eqs. (820) and (821) cancel. This willbe used below. The above formulas permit a different approach to averageover disorder. It is commonly referred to as the supersymmetric method . For concreteness, define H [ u, V ] = (cid:90) x (cid:110) 12 [ ∇ u ( x )] + m u ( x ) + U (cid:0) u ( x ) (cid:1) + V (cid:0) x, u ( x ) (cid:1)(cid:111) . (822) The disorder-average of an observable O is defined as O [ u ] := (cid:90) r (cid:89) a =1 D [ u a ] O [ u ]e − T H [ u a ,j a ,V ] (cid:90) r (cid:89) a =1 D [ u a ]e − T H [ u a , ,V ] . (823)The function U ( u ) is an arbitrary potential, e.g. the non-linearity in φ -theory, U ( u ) = g u . The randompotential V ( x, u ) is the same one used in section 1.2, withcorrelations given by Eq. (9). Its average is indicated bythe overline. We remind that the difficulty in evaluating(823) comes from the denominator. The replica trick usedin section 1.5 allowed us to set r = 0 , effectively discardingthe denominator. Here we follow a different strategy.In the limit of T → only configurations whichminimize the energy survive. These configurations satisfy δ H [ u a ,V ] δu a ( x ) = 0 , and we want to insert a δ -distributionenforcing this into the path-integral. This has to beaccompanied by a factor of det (cid:104) δ H [ u a ,V ] δu a ( x ) δu a ( y ) (cid:105) , such that thepath integral is normalized to 1. The latter can be achievedby an additional integral over Grassmann variables, i.e.fermionic degrees of freedom, using that det (cid:18) δ H [ u, V ] δu ( x ) δu ( y ) (cid:19) (824) = (cid:90) D [ ¯ ψ a ] D [ ψ a ] exp (cid:18) − (cid:90) x ¯ ψ ( x ) δ H [ u, V ] δu ( x ) δu ( y ) ψ ( x ) (cid:19) . This allows us to write the disorder average of anyobservable O [ u ] as O [ u ] = (cid:90) r (cid:89) a =1 D [˜ u a ] D [ u a ] D [ ¯ ψ a ] D [ ψ a ] O [ u ] × exp (cid:20) − (cid:90) x ˜ u a ( x ) δ H [ u a ] δu a ( x ) + ¯ ψ a ( x ) δ H [ u a ] δu a ( x ) δu a ( y ) ψ a ( y ) (cid:21) . (825)This method was first introduced in [32, 670]. Analternative derivation and insight is offered by Cardy [671,672, 673].Averaging over disorder yields with the force-forcecorrelator ∆( u ) := − R (cid:48)(cid:48) ( u ) O [ u ] = (cid:90) (cid:89) a D [ u a ] D [˜ u a ] D [ ¯ ψ a ] D [ ψ a ]exp (cid:0) −S [ u a , ˜ u a , ¯ ψ a , ψ a ] (cid:1) S [˜ u a , u a , ¯ ψ a , ψ a ] , = (cid:90) x r (cid:88) a =1 (cid:110) ˜ u a ( x ) (cid:104) ( −∇ + m ) u a ( x ) + U (cid:48) (cid:0) u a ( x ) (cid:1)(cid:105) + ¯ ψ a ( x ) (cid:104) −∇ + m + U (cid:48)(cid:48) (cid:0) u a ( x ) (cid:1)(cid:105) ψ a ( x ) (cid:111) − r (cid:88) a,b =1 (cid:104) 12 ˜ u a ( x )∆ (cid:0) u a ( x ) − u b ( x ) (cid:1) ˜ u b ( x ) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles − ˜ u a ( x )∆ (cid:48) (cid:0) u a ( x ) − u b ( x ) (cid:1) ¯ ψ b ( x ) ψ b ( x ) − 12 ¯ ψ a ( x ) ψ a ( x )∆ (cid:48)(cid:48) (cid:0) u a ( x ) − u b ( x ) (cid:1) ¯ ψ b ( x ) ψ b ( x ) (cid:105) . (826)We first analyze n = 1 . Suppose that ∆( u ) is even andanalytic to start with, then few terms survive from Eq. (826), S Susy [ u, ˜ u, ¯ ψ, ψ ] = (cid:90) x (cid:110) ˜ u ( x ) (cid:104) ( −∇ + m ) u ( x ) + U (cid:48) (cid:0) u ( x ) (cid:1)(cid:105) + ¯ ψ ( x ) (cid:104) −∇ + m + U (cid:48)(cid:48) (cid:0) u ( x ) (cid:1)(cid:105) ψ ( x ) − 12 ˜ u ( x )∆(0)˜ u ( x ) (cid:111) . (827)(We have used that ¯ ψ a = ψ a = 0 to eliminate the4-fermion-term.) A particularly simple case are randommanifolds, for which U ( u ) = 0 . Then bosons ˜ u and u ,and fermions ¯ ψ and ψ do not interact, all expectation valuesare Gaussian, and dimensional reduction holds. When U ( u ) (cid:54) = 0 , things are more complicated, but as we willsee in the next section, dimensional reduction still holds,at least formally.The reason is that the action (827) possesses anapparent supersymmetry, made manifest by grouping allfields into a (bosonic) superfield, Φ( x, ¯Θ , Θ) := u ( x ) + ¯Θ ψ ( x ) + ¯ ψ ( x )Θ − ¯ΘΘ˜ u ( x ) . (828)Both ¯Θ and Θ are Grassmann numbers. The action (827)can then be written with the super Laplacian ∆ s as S Susy = (cid:90) d ¯ΘdΘ (cid:90) x 12 Φ( x, ¯Θ , Θ)( − ∆ s + m )Φ( x, ¯Θ , Θ)+ U (Φ( x, ¯Θ , Θ)) , (829) ∆ s := ∇ + ∆(0) ∂∂ Θ ∂∂ ¯Θ . (830)As we will see in section 8.5, the action is invariant underthe action of two supergenerators Q := x ∂∂ Θ − ∇ , ¯ Q := x ∂∂ ¯Θ + 2∆(0) Θ ∇ , (cid:8) Q, ¯ Q (cid:9) = 0 . (831)This is sufficient to “prove” dimensional reduction. For more than r = 1 replicas, the theory is richer, and wecan recover the renormalization of ∆( u ) itself. To simplifymatters, set U ( u ) = 0 , and write S [˜ u a , u a , ¯ ψ a , ψ a ] = (cid:88) a (cid:90) x (cid:104) ˜ u a ( x )( −∇ + m ) u a ( x )+ ¯ ψ a ( x )( −∇ + m ) ψ a ( x ) − 12 ˜ u a ( x )∆(0)˜ u a ( x ) (cid:105) − (cid:88) a (cid:54) = b (cid:90) x (cid:104) 12 ˜ u a ( x )∆ (cid:0) u a ( x ) − u b ( x ) (cid:1) ˜ u b ( x ) − ˜ u a ( x )∆ (cid:48) (cid:0) u a ( x ) − u b ( x ) (cid:1) ¯ ψ b ( x ) ψ b ( x ) − 12 ¯ ψ a ( x ) ψ a ( x )∆ (cid:48)(cid:48) (cid:0) u a ( x ) − u b ( x ) (cid:1) ¯ ψ b ( x ) ψ b ( x ) (cid:105) . (832) Corrections to ∆( u ) are constructed by remarking that theinteraction term quadratic in ˜ u is almost identical to thetreatment of the dynamics in the static limit (i.e. afterintegration over times). The diagrams in question are + ++ , (833)where an arrow indicates the correlation-function, x −→−−− y = (cid:104) ˜ u ( x ) u ( y ) (cid:105) = C ( x − y ) . This leads to (in the order givenabove) δ ∆( u ) = (cid:2) − ∆( u )∆ (cid:48)(cid:48) ( u ) − ∆ (cid:48) ( u ) + ∆ (cid:48)(cid:48) ( u )∆(0) (cid:3) × (cid:90) x − y C ( x − y ) . (834)The last term of Eq. (833) is odd, and vanishes. Eq. (834)is equal to the results of Eqs. (318)–(326).A non-trivial ingredient is the cancellation of theacausal loop (327) in the dynamics, equivalent to the 3-replica term in the statics: + = 0 . (835)The first diagram comes from the contraction of two termsproportional to ˜ u a ∆( u a − u b )˜ u b . The second is obtainedfrom contracting all fermions in two terms proportional to ˜ u a ∆ (cid:48) ( u a − u b ) ¯ ψ b ψ b . Since the fermionic loop (orientedwiggly line in the second diagram) contributes a factor of − , both cancel.One can treat the interacting theory completely in asuperspace formulation. The action is S [Φ] = (cid:88) a (cid:90) ¯Θ , Θ (cid:90) x 12 Φ a ( x, ¯Θ , Θ)( − ∆ s + m )Φ a ( x, ¯Θ , Θ) − (cid:88) a (cid:54) = b (cid:90) x (cid:90) ¯Θ , Θ (cid:90) ¯Θ (cid:48) , Θ (cid:48) R (cid:0) Φ a ( x, ¯Θ , Θ) − Φ b ( x, ¯Θ (cid:48) , Θ (cid:48) ) (cid:1) . (836)“Non-locality” in replica-space or in time is replaced by“non-locality” in superspace. Corrections to R ( u ) all stemfrom superdiagrams , which result into bilobal interactionsin superspace, not trilocal, or higher. The latter find theirequivalent in 3-local terms in replica-space in the replica-formulation, and 3-local terms in time, in the dynamicformulation.Supersymmetry is broken, once ∆(0) changes, i.e. atthe Larkin length. A seemingly “effective supersymmetry”,or “scale-dependent supersymmetry” appears, in which theparameter ∆(0) , which appears in the Susy-transformation,changes with scale, according to the FRG flow equations(62) for ∆( u ) , continued to u = 0 . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Let us study invariants of the action. Since total derivativesboth in x and θ or ¯ θ vanish, the crucial term to focus on isthe super-Laplacian. To simplify notations, we set ρ := ∆(0) . (837)By explicit inspection, we find that the two generators ofsuper-translations Q := x ∂∂ Θ + 2 ρ ¯Θ ∇ , ¯ Q := x ∂∂ ¯Θ − ρ Θ ∇ (838)both commute with the Super-Laplacian, and anti-commutewith each other, [∆ s , Q ] = (cid:2) ∆ s , ¯ Q (cid:3) = 0 , (cid:8) Q, ¯ Q (cid:9) = 0 . (839)The following combination is invariant under the action of Q and ¯ Q , ¯ Q (cid:18) x + 4 ρ ¯ΘΘ (cid:19) = Q (cid:18) x + 4 ρ ¯ΘΘ (cid:19) = 0 . (840)Applying the Super-Laplacian (830) gives ∆ s (cid:18) x + 4 ρ ¯ΘΘ (cid:19) = 2( d − . (841)To obtain the super-propagator, inverse of the super-Laplacian plus mass term in Eq. (829), we remark that (cid:18) m − ∇ − ρ ∂∂ ¯Θ ∂∂ Θ (cid:19) (cid:18) m − ∇ + ρ ∂∂ ¯Θ ∂∂ Θ (cid:19) = ( m − ∇ ) . (842)This implies that (cid:0) m − ∆ s (cid:1) − = m − ∇ + ρ ∂∂ ¯Θ ∂∂ Θ( m − ∇ ) . (843)Therefore C ( x − x (cid:48) , Θ − Θ (cid:48) , ¯Θ − ¯Θ (cid:48) ) (844) = m − ∇ + ρ ∂∂ ¯Θ ∂∂ Θ( m − ∇ ) δ ( x − x (cid:48) ) δ (Θ − Θ (cid:48) ) δ ( ¯Θ − ¯Θ (cid:48) ) . The Grassmanian δ -functions are defined as (cid:90) dΘ δ (Θ − Θ (cid:48) ) f (Θ) = f (Θ (cid:48) ) . (845)By direct calculation one finds δ (Θ − Θ (cid:48) ) = Θ (cid:48) − Θ = (cid:90) d ¯ χ e ¯ χ (Θ (cid:48) − Θ) . (846)One can transform (844) into a representation in dual spacesof momentum ( k -space) and super-coordinates ( χ -space) as C ( k, ¯ χ, χ ) = m + k + ρ ¯ χχ ( m + k ) ≡ m + k + ρχ ¯ χ ≡ (cid:90) ∞ d s e − s ( m + k + ρχ ¯ χ ) . (847) This relation comes out incorrectly in [32]. The final proof of dimensional reduction is performed withthis representation of the super-correlator . Any diagramcan be written as (cid:90) k (cid:90) ¯ χ χ ... (cid:90) k n (cid:90) ¯ χ n χ n n (cid:89) i =1 (cid:20)(cid:90) ∞ d s i e − s i ( m + k i + ρχ i ¯ χ i ) (cid:21) , (848)where some δ -distributions have already been used toeliminate integrations over k ’s, i.e. some of the k i ’sappearing in the exponential are not independent variables,but linear combinations of other k j ’s, and the same holdsfor the corresponding χ i and ¯ χ i . The product of exponentialfactors can be written as n (cid:89) i =1 (cid:104) e − s i ( m + k i + ρχ i ¯ χ i ) (cid:105) = exp − k k . . .k n W k k . . .k n × exp − χ χ . . .χ n W ¯ χ ¯ χ . . . ¯ χ n . (849)Integration over the k ’s gives (cid:90) k . . . (cid:90) k n e − (cid:126)k · W · (cid:126)k = (cid:18) π (cid:19) ld/ det( W ) − d/ , (850)where l is the number of loops. Integration over ¯ χ and χ gives (cid:90) ¯ χ χ . . . (cid:90) ¯ χ n χ n e − ρ(cid:126)χ · W · (cid:126) ¯ χ = ( ρ ) l det( W ) . (851)The product of the two factors (850) and (851) is the sameas for a standard bosonic diagram in dimension d − .Remarking that the expansion is in powers of T , andcombining these relations, we obtain after integration overthe s i : l -loop super-diagram in dimension d (852) = (cid:16) ρ πT (cid:17) l × l -loop standard-diagram in dimension d − . This implies that for any observable O ( T ) O d disordered ( ρ ) = O d − (cid:16) T = ρ π (cid:17) . (853)The above proof can be extended to theories with derivativecouplings. The rules are as follows: Consider H s [Φ] given in Eq. (829). To this, we can add an interaction inderivatives for a total of H s [Φ] = (cid:90) ¯Θ , Θ (cid:90) x (cid:20) 12 Φ( x, ¯Θ , Θ)( − ∆ s + m )Φ( x, ¯Θ , Θ)+ U (cid:0) Φ( x, ¯Θ , Θ) (cid:1) + A (cid:0) Φ( x, ¯Θ , Θ) (cid:1) ( − ∆ s + m ) A (cid:0) Φ( x, ¯Θ , Θ) (cid:1)(cid:21) . (854) Note that one can also work in position-space [32]. Then the super-correlator is explicitly d -dependent, and one should check that this d -dependence comes out correctly. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles s a derivative w.r.t. s , taken at s = 0 , (cid:0) m + k + ρχ ¯ χ (cid:1) = − dd s (cid:12)(cid:12)(cid:12)(cid:12) s =0 e − s ( m + k + ρχ ¯ χ ) . (855) φ -theory with twofermions and one boson Consider the fixed point (92) for CDWs. It has the form ∆( u ) = g − g u (1 − u ) = g u + lower-order terms in u. (856)The renormalization, encoded in g , can be gotten byretaining only terms of order u , and dropping lower-orderterms which do not feed back into terms of order u . To thisaim, consider the action (832) with two replicas , replacing ∆( u ) → g u . We further go to center-of-mass coordinates(bosons only), by introducing u ( x ) = u ( x ) + 12 φ ( x ) , u ( x ) = u ( x ) − φ ( x ) , (857) ˜ u ( x ) = 12 ˜ u ( x ) + ˜ φ ( x ) , ˜ u ( x ) = 12 ˜ u ( x ) − ˜ φ ( x ) . (858)The action (826) can then be rewritten as S = (cid:90) x ˜ φ ( x )( −∇ + m ) φ ( x ) + ˜ u ( x )( −∇ + m ) u ( x )+ (cid:88) a =1 ¯ ψ a ( x )( −∇ + m ) ψ a ( x )+ g u ( x ) φ ( x ) (cid:2) ¯ ψ ( x ) ψ ( x ) − ¯ ψ ( x ) ψ ( x ) (cid:3) − g u ( x ) φ ( x ) + g (cid:104) ˜ φ ( x ) φ ( x ) + ¯ ψ ( x ) ψ ( x ) + ¯ ψ ( x ) ψ ( x ) (cid:105) . (859)Note that only ˜ u ( x ) , but not the center-of-mass u ( x ) appears in the interaction. While u ( x ) may have non-trivialexpectations, it does not contribute to the renormalizationof g , and the latter can be obtained by considering solelythe fist and last line of Eq. (859): This is a φ -type theory,with one complex bosonic, and two complex fermionicfields. It can equivalently be viewed as complex φ -theoryat N = − , or real φ -theory with n = − . When supersymmetry was first proposed [32, 670], it wasbelieved to produce an exact result, namely dimensionalreduction. While the latter was found earlier [100, 103, 105]by inspection of diagrams and a combinatorial analysis,the SUSY method proved to be a clever tool to show it Figure 74. Example of a loop-erased random walk on the hexagonal latticewith 3000 steps, starting at the black point to the right and arriving at thegreen point to the left. efficiently. SUSY got discredited when it was realized thatdimensional reduction breaks down at the Larkin scale. Asa remedy, breaking of replica symmetry was invoked, orFRG. As we have seen in section 2.19, RSB and FRG fittogether, and the applied field inherent to FRG explicitlybreaks replica symmetry, and as a consequence super-symmetry. As shown in section 8.4, the SUSY method canbe used to obtain the FRG flow equation for the disorder.Thus dimensional reduction beyond the Larkin length is nota problem of the SUSY method, but of its application.It has been argued that Eq. (825) is inappropriate asit sums over all saddle points, not only the minima, andfor this reason it fails. The objection per se can not bediscarded. But does it invalidate the formalism put in placeabove? We believe not: the formalism makes numerouspredictions, especially for such non-trivial observables asthe FRG fixed-point function ∆( w ) (sections 2.1–2.12).Our intuition is that each RG step merges pairs of closeminima, without invoking any of the higher-lying statespresent in the above argument. And that by this theobjection becomes irrelevant. φ -theorywith two fermions and one bosonIntroduction. A loop-erased random walk (LERW) isdefined as the trajectory of a random walk (RW) in whichany loop is erased as soon as it is formed [674]. Anexample is shown on figure 74, where the underlying RWis drawn in red, and the LERW remaining after erasure inblue. Similar to a self-avoiding walk it has a scaling limitin all dimensions, e.g. the end-to-end distance R scaleswith the intrinsic length (cid:96) as R ∼ (cid:96) /z , where z is thefractal dimension [675]. It is crucial to note that while bothLERWs and SAWs are non-selfintersecting, their fractaldimensions do not agree since they have a different statisticson the same set of allowed trajectories. LERWs appear inmany combinatorial problems, e.g. the shortest path on auniform spanning tree is a LERW. We have collected themany connections on Fig. 74, together with other identities, heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles κ = 2 [680, 681], predicting a fractal dimension z LERW ( d =2) = . Coulomb-gas techniques link this to the 2d O ( n ) -model at n = − [682, 155]. It was recently shown thatLERWs can be mapped in all dimensions to the theory oftwo complex fermions and one complex boson, equivalentto the O ( n ) model at n = − [291, 292, 683, 293]. Perturbative argument. This mapping was first estab-lished perturbatively [291, 292]. Consider x y z −→ xz − g xz − gN xz . (860)The drawing on the l.h.s. of Eq. (860) is a LERW starting at x , ending in z , and passing through the segments numbered1 to 3. Due to the crossing at y , the loop labeled 2 iserased; we draw it in red. The r.h.s of Eq. (860) gives alldiagrams of φ theory up to order g s . The first term isthe free-theory result, proportional to g . The second term ∼ g cancels the first term, if one puts g → . Here it iscrucial to have the same regularization for the interactionas for the conditioning. The third term is proportional to N , due to the loop, indicated in red. Setting N → − compensates for the subtracted second term. Thus setting g → and N → − , the probability to go from x to z remains unchanged as compared to the free theory. This is anecessary condition to be satisfied. Since the first two termscancel, what remains is the last diagram, corresponding tothe drawing for the trajectory of the LERW we started with.Continuing to higher orders, one establishes a one-to-one correspondence between traces of LERWs anddiagrams in perturbation theory. We still need an observablewhich is when inserted into a blue part of the trace, and within a red part. This can be achieved by the operator O ( y ) := Φ ∗ ( y )Φ ( y ) − Φ ∗ ( y )Φ ( y ) . (861)When inserted into a loop, it cancels, whereas inserted intothe backbone (LERW, blue), it yields one for each point.The fractal dimension z of a LERW is extracted from thelength of the walk after erasure (blue part) as (cid:68)(cid:82) y,z O ( x, y, z ) (cid:69)(cid:10)(cid:82) z Φ ∗ ( x )Φ ( z ) (cid:11) ≡ m (cid:28)(cid:90) y,z O ( x, y, z ) (cid:29) ∼ m − z . (862) Proof via Viennot’s theorem. This section is a shortenedversion of [293], itself inspired by [683]. The main toolwe use is a combinatorial theorem due to Viennot [684].It is part of the general theory of heaps of pieces (onlinelectures [685]). Here it reduces to a relation between loop-erased random walks, and collections of loops. To state thetheorem, we need some definitions.Consider a random walk on a directed graph G . Thewalk moves from vertex x to y with rate β xy , and diesout with rate λ x = m x . The coefficients { β xy } x,y ∈G areweights on the graph. In particular, when β xy is positive, G contains an edge from x to y . Denote by r x := λ x + (cid:80) y β xy the total rate at which the walk exits from vertex x .A path is a sequence of vertices, denoted ω =( ω , . . . , ω n ) . The probability P ( ω ) that the random walkselects the path ω and then stops is P ( ω ) = λ ω n r ω n q ( ω ) , (863) q ( ω ) = β ω ω r ω β ω ω r ω . . . β ω n − ω n r ω n − . (864)A loop is a path ω = ( ω , . . . , ω n − , ω n = ω ) wherethe first and last points are identical, and all other verticesdistinct, so it cannot be decomposed into smaller loops.Loops obtained from each other via cyclic permutations areconsidered identical.A collection of disjoint loops is a set L = { C , C , . . . } of mutually non-intersecting loops. Wedenote the set of all such collections by L .To formulate the theorem, fix a self-avoiding path γ . Define the set L γ to consist of all collections ofdisjoint loops in which no loop intersects γ . Then Viennot’stheorem can be written as ( | L | being the number of loops) A ( γ ) := q ( γ ) (cid:88) L ∈L γ ( − | L | (cid:89) C ∈ L q ( C ) = P ( γ ) × Z , (865) P ( γ ) = (cid:88) ω : L ( ω )= γ q ( ω ) , (866) Z = (cid:88) L ∈L ( − | L | (cid:89) C ∈ L q ( C ) . (867)On the l.h.s. one sums over the ensemble of collectionsof loops which do not intersect γ , giving each collectiona weight ( − | L | (cid:81) C ∈ L q ( C ) . The r.h.s. contains twofactors. The first, P ( γ ) , is the weight to find the LERWpath γ , our object of interest. The second is the partitionfunction Z . Conditioning the walk to stop at x , this relationcan be read as P ( γ ) = A ( γ ) / Z .To prove Eq. (865) consider a pair { ω, L } constructedas follows: Take a path ω such that L ( ω ) = γ and an arbitrary collection L of disjoint loops. Our goal is toconstruct another pair { ω (cid:48) , L (cid:48) } by transferring a loop from Warning: a self-avoiding path is a cominatorial object. The loop-erasedrandom walk is one possible distribution on the set of self-avoiding paths.It should not be confused with the self-avoiding walk or self-avoidingpolymer , a distinct distribution on the same set of self-avoiding paths. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles L to ω or vice versa, depending on where the loop originallywas. For example, + ω L ω (cid:48) L (cid:48) = 0 . (868)In the first drawing, the left loop is part of ω , whereas inthe second one it is part of L (cid:48) . These terms cancel, as ( − | L | = − ( − | L (cid:48) | , and all other factors are identical.After each such pair is canceled, we are left with the termsin which it is impossible to transfer a loop from ω to L orvice versa. These are exactly the terms in the l.h.s of Eq.(865).For this procedure to work we need to ensure thatwe cannot obtain the same pair { ω (cid:48) , L (cid:48) } starting form twodifferent pairs { ω, L } . In order to achieve this, we use thefollowing prescription. Start walking along ω , until(i) we reach a vertex ω i that belongs to some C = ( ω i = c , c . . . , c m = ω i ) ∈ L , or(ii) we reach a vertex ω i that does not belong to any C ,but that we have already seen before, i.e., ω j = ω i for j < i .In the first case, we transfer C to ω , i.e., ω (cid:48) = ( ω , . . . , ω i , c , . . . , c m − , ω i , . . . , ω n ) ,L (cid:48) = L \ { C } . (869)In the second case, we apply the one-loop erasure to ω , andtransfer the erased loop to L , ω (cid:48) = ( ω , . . . , ω j , ω i +1 , . . . , ω n ) ,L (cid:48) = L ∪ { ( ω j , ω j +1 , . . . , ω i = ω j ) } . (870)Note that disjointness of the loop collections is preservedunder the transfer, and that the loop erasure of ω (cid:48) remains γ . This completes the proof. For examples see [293]. A lattice action with two complex fermions and onecomplex boson. Our goal is to write a lattice action whichgenerates A ( γ ) , the l.h.s. of Eq. (865). This can be achievedwith an action based on one pair of complex conjugatefermionic fields. While this theory sums over all paths γ ,yielding back the random-walk propagator, it contains noinformation on the erasure. In order to answer whetherthe resulting loop-erased path passes through a given point y it is necessary to use more fields. The simplest suchsetting consists of two pairs of complex conjugate fermionicfields ( φ , φ ∗ ) , and ( φ , φ ∗ ) , as well as a pair of complexconjugate bosonic fields ( φ , φ ∗ ) . When appearing in aloop, the latter cancels one of the fermions.We define the action as φ ∗ ( y ) φ ( x ) := (cid:88) i =1 φ ∗ i ( y ) φ i ( x ) , (871) e −S = (cid:89) x e − r x φ ∗ ( x ) φ ( x ) (cid:104) (cid:88) y β xy φ ∗ ( y ) φ ( x ) (cid:105) . (872) a b cx y z w (1) (1) (2) (2) (3)(3) Figure 75. An example of a diagram that contributes to U . Ourcoloring conventions are blue for ( φ , φ ∗ ) , green for ( φ , φ ∗ ) , and redfor ( φ , φ ∗ ) . The path integral is defined by integrating over the n f = 2 families of fermionic fields, ( φ ∗ i , φ i ) , i = 1 , , and n b = 1 family of bosonic fields, i = 3 . For β xy = 0 , we obtain Z = (cid:89) x r n f − n b x = (cid:89) x r x . (873)Define (normalized) expectation values (cid:104)O ( φ ∗ , φ ) (cid:105) w.r.t.the action (872) and the (normalized) partition function Z as (cid:104)O ( φ ∗ , φ ) (cid:105) := 1 Z (cid:90) D [ φ ] D [ φ ∗ ] e −S O ( φ ∗ , φ ) , (874) Z := (cid:104) (cid:105) . (875)Calculating Z by expansion in β xy is best done graphically:Due to the square bracket in Eq. (872), at each x one canplace exactly one outgoing arrow to one of the neighbors y , with color i , or no arrow. Summing over all possiblecolorings and all graphs, we obtain Z as given in Eq. (867).In order to assess whether a point b belongs to a loop-erased random walk from a to c after erasure, we fix thethree vertices a, b and c , and consider the observable U ( a, b, c ) = λ c r b r c (cid:104) φ ( c ) φ ∗ ( b ) φ ( b ) φ ∗ ( a ) (cid:105) , (876)defined by Eq. (874).The graphs that contribute consist of a self-avoidingpath γ and a collection L of disjoint self-avoiding coloredloops such that (see Fig. 75):(i) γ is a path from a to c passing through b . The edges of γ between a and b have color , and the edges between b and c have color .(ii) Fix C ∈ L . If the color of C is then it cannotintersect γ . If its color is or , it can only intersect γ at the (final) point c .In the latter case, the contribution to U ( a, b, c ) is ( − fermionic loops q ( γ ) (cid:89) C ∈ L q ( C ) . (877)We now sum over all possible colorings of the loops. Sinceloops that intersect c may have either color or , onefermionic and one bosonic, they cancel, leaving only graphsin which loops do not intersect γ . The other loops, asbefore, give a factor of − . We therefore established that heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles | q → CDWFRG at depinning2 fermions + 1 boson φ (cid:12)(cid:12) n = − spin system with2 fermions + 1 boson SUSY[291][293] [ ] KW [686]L [687]L [688] pert. [291]exact [293] MD [491]NM conj. [115]FLW test [689]M [690] Figure 76. Relations between Laplacian walks, loop-erased random walks (LERW), uniform spanning trees (UST), Eulerian Circuits, the AbelianSandpile Model (ASM), the Potts-model in the limit of q → , charge-density waves at depinning (CDW), mapping onto the FRG field theory atdepinning, reducing to φ -theory at n = − , and equivalent to an interacting theory of 2 complex fermions and one complex boson. U ( a, b, c ) = (cid:88) γ ∈ SA( a,b,c ) q ( γ ) (cid:88) L ∈L γ ( − | L | (cid:89) C ∈ L q ( C ) , (878)where the sum is over all self-avoiding paths from a to c passing through b , and denoted SA( a, b, c ) . In view of Eq.(865) this can be written as U ( a, b, c ) = (cid:88) γ ∈ SA( a,b,c ) A ( γ ) . (879)It implies that our object of interest, the probability that aLERW starting at a and ending in c passes through b is (cid:88) γ ∈ SA( a,b,c ) P ( γ ) = U ( a, b, c ) Z . (880) Continuous limit of the lattice action. Let us rewrite theaction S explicitly, S = (cid:88) x (cid:104) r x φ ∗ ( x ) φ ( x ) − ln (cid:16) (cid:88) y β xy φ ∗ ( y ) φ ( x ) (cid:17)(cid:105) . (881)The leading term in S reads (cid:88) x (cid:104) r x φ ∗ ( x ) φ ( x ) − (cid:88) y β xy φ ∗ ( y ) φ ( x ) (cid:105) = (cid:88) x φ ∗ ( x )[ m x − ∇ β ] φ ( x ) , (882) m x = r x − (cid:88) y β yx , ∇ β φ ( x ) = (cid:88) y β yx [ φ ( y ) − φ ( x )] . The subleading term in S is (cid:88) x (cid:104) (cid:88) y β xy φ ∗ ( y ) φ ( x ) (cid:105) = g (cid:88) x (cid:104) φ ∗ ( x ) φ ( x ) (cid:105) + ...g := (cid:104) (cid:88) y β xy (cid:105) , (883)where the dropped terms contain at least one latticeLaplacian ∇ β . Standard arguments [2] show that the latterare irrelevant in a RG analysis, as are higher-order terms inthe expansion of the ln in Eq. (881). Taking the continuum limit, we arrive at the theory defined in Eq. (859), settingthere u, ˜ u → , and identifying ( ¯ ψ i , ψ i ) , i = 1 , there with ( φ ∗ i , φ i ) here, and ( ˜ φ, φ ) there with ( φ ∗ , φ ) here. Perturbative results. Using φ -theory at n = − allowsus to obtain an extremely precise estimate of the fractaldimension z , which can be compared to an even moreprecise Monte Carlo simulation, z = 1 . (6 loops) [135] , (884) z = 1 . (Monte Carlo) [679] . (885)The agreement is quite impressive. There is a plethora of further relations relating CDWs orLERWs to other critical systems, see Fig. 76. Let usdiscuss at least some of them: While LERWs are non-Markovian RWs, their traces are equivalent to those of the Laplacian Random Walk [691, 687], which is Markovian, ifone considers the whole trace as the variable of state. It isconstructed on the lattice by solving the Laplace equation ∇ Φ( x ) = 0 with boundary conditions Φ( x ) = 0 on thealready constructed curve, and Φ( x ) = 1 at the destinationof the walk, either a chosen point, or infinity. The walkthen advances from its tip x to a neighbouring point y ,with probability proportional to Φ( y ) . As Φ( x ) = 0 , Φ( y ) ≡ Φ( y ) − Φ( x ) can be interpreted as the electric field of the potential Φ( y ) .In a variant of this model growth is allowed not onlyfrom the tip, but from any point on the already constructedobject, with a probability ∼ Φ( y ) . This is known as the dielectric breakdown model [692], the simplest model forlightning. The same construction pertains to diffusion-limited aggregation [693].The shortest path on a uniform spanning tree is aLERW [686]. The latter are equivalent to Eulerian circuits heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles q → [491].Many of these exact mappings can be found in the lecture[492]. It was conjectured long ago that Abelian sandpilesmap on charge-density waves [115]. A test on the FRGfield theory was performed in Ref. [689], and validated inRef. [678].It would be interesting to generalize loops to higher-dimensional surfaces, as was done for self-avoidingmanifolds in Refs. [694, 695]. In d = 2 , all critical exponents should be accessiblevia conformal field theory (CFT). The latter is basedon ideas proposed in the 80s by Belavin, Polyakov andZamolodchikov [696]. They constructed a series ofminimal models, indexed by an integer m ≥ , starting withthe Ising model at m = 3 . These models are conformallyinvariant and unitary, equivalent to reflection positive inEuklidean theories. For details, see one of the manyexcellent textbooks on CFT [219, 220, 697, 221]. Theirconformal charge is given by c = 1 − m ( m + 1) . (886)The list of conformal dimensions for allowed operators at agiven m is given by the Kac formula with integers r, sh r,s = [ r ( m + 1) − sm ] − m ( m + 1) , ≤ r < m, ≤ s ≤ m. (887)It was later realized that other values of m also correspondto physical systems, in particular m = 1 (loop-erasedrandom walks), and m = 2 (self-avoiding walks). Thesevalues can further be extended to non-integer n and m ,using the identification n = 2 cos (cid:16) πm (cid:17) . (888)More strikingly, the table of dimensions allowed by Eq.(887) has to be extended to half-integer values, including . This yields: the fractal dimension of the propagator line[698, 699, 700] d f = 2 − h , = 1 + π (cid:0) arccos (cid:0) n (cid:1) + π (cid:1) . (889) ν , i.e. the inverse fractal dimension of all lines, be itpropagator or loops ([700], inline after Eq. (2)) ν = 12 − h , = 14 (cid:18) π arccos( n ) (cid:19) . (890) The conformal charge is the coefficient in the leading term of the OPEof the stress-energy tensor. It also gives the amplitude of finite-sizecorrections [3]. For η , there are two suggestive candidates from the Isingmodel, η = 4 h , = 4 h , , which does not work for othervalues of n ; instead [698, 699, 700] η = 4 h , = 54 − (cid:0) n (cid:1) π − π arccos (cid:0) n (cid:1) + π . (891)It has a square-root singularity both for n = − and n = 2 .There is no clear candidate for the exponent ω [135]. Thecrossover exponent φ c [1, 701, 135] (explained in [135],page 7) becomes φ c = νd f = 1 − h , − h , = 14 + 3 π n ) . (892)To conclude, we remark that ideas identifyingsymplectic fermions with the ASM [702] are overlysimplistic, as they do not catch any of the above exponents. 9. Further developments and ideas Another domain of application of the Functional RG arespin models in a random field (for an introduction see[703]). The model usually studied is H = (cid:90) d d x 12 ( ∇ (cid:126)S ) + (cid:126)h ( x ) (cid:126)S ( x ) , (893)where (cid:126)S ( x ) is a unit vector with N -components, and (cid:126)S ( x ) = 1 . This is the so-called O ( N ) sigma model,to which has been added a random field, which can betaken Gaussian h i ( x ) h j ( x (cid:48) ) = σδ ij δ d ( x − x (cid:48) ) . In theabsence of disorder the model has a ferromagnetic phasefor T < T f and a paramagnetic phase above T f . The lowercritical dimension is d = 2 for any N ≥ , meaning thatbelow d = 2 no ordered phase exists. In d = 2 solelya paramagnetic phase exists for N > ; for N = 2 (XYmodel) quasi long-range order exists at low temperature,with (cid:126)S ( x ) (cid:126)S ( x (cid:48) ) decaying as a power law of x − x (cid:48) . This isthe RP fixed point of sections 2.8 and 2.27. Here we wishto study the model directly at T = 0 . The first step is torewrite the hard-spin constraint (cid:126)S ( x ) = 1 as a field theory.This yields an energy before disorder-averaging H = (cid:90) d d x (cid:2) ∇ (cid:126)φ ( x ) (cid:3) + V (cid:0) (cid:126)φ ( x ) (cid:1) + (cid:126)h ( x ) (cid:126)φ ( x ) . (894)The potential V ( (cid:126)φ ) is the typical double-well potential, ase.g. V ( (cid:126)φ ) (cid:39) ( (cid:126)φ − . The dimensional-reduction theoremin section 1.6, written for this energy indicates that theeffect of a quenched random field in dimension d equals theone for a pure model at a temperature T ∼ σ in dimension d − . Hence one expects a transition from a ferromagneticphase to a disordered phase at σ c as the disorder increasesin any dimension d > , and no order for d < and N ≥ .Not surprisingly, this is again incorrect, as can be seen usingFRG.It was noticed by Fisher [704] that an infinity ofrelevant operators are generated. These operators, which heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles d = 6 [705, 706],the naive upper critical dimension (corresponding to d =4 for the pure O ( N ) model via dimensional reduction).Indeed many early studies concentrating on d around d = 6 missed the anisotropies mentioned above.A controlled (cid:15) -expansion using FRG can be con-structed around dimension d = 4 , the naive lower criti-cal dimension, using the reformulation of the Hamiltonian(893) in terms of a non-linear σ -model, first at 1-loop order[704, 705, 706], and then extended to two loops [33, 38].The FRG includes all operators which are marginal in d =4 . Its action in replicated form reads S = (cid:90) d d x T (cid:88) a (cid:2) ∇ (cid:126)S a ( x ) (cid:3) − T (cid:88) ab ˆ R (cid:0) (cid:126)S a ( x ) (cid:126)S b ( x ) (cid:1) . (895)The function ˆ R ( z ) parameterizes the disorder. The term ˆ R ( z ) ∼ z is a direct result of the disorder average of Eq.(894); higher-order terms are generated within perturbationtheory by contraction with V ( φ ) . The FRG flow equationhas been obtained to order R (one loop) [704, 705, 706]and R (two loops) [33, 38]. It is best parametrized interms of the variable φ , the angle between the two replicas,defining R ( φ ) = ˆ R ( z = cos φ ) . Since the vectors are ofunit norm, z = cos φ lies in the interval [ − , . ∂ (cid:96) R ( φ ) = (cid:15)R ( φ ) + 12 R (cid:48)(cid:48) ( φ ) − R (cid:48)(cid:48) (0) R (cid:48)(cid:48) ( φ )+( N − (cid:20) R (cid:48) ( φ ) sin φ − cot φR (cid:48) ( φ ) R (cid:48)(cid:48) (0) (cid:21) + 12 (cid:2) R (cid:48)(cid:48) ( φ ) − R (cid:48)(cid:48) (0) (cid:3) R (cid:48)(cid:48)(cid:48) ( φ ) +( N − (cid:20) cot φ sin φ R (cid:48) ( φ ) − φ φ R (cid:48) ( φ ) R (cid:48)(cid:48) ( φ )+ 12 sin φ R (cid:48)(cid:48) ( φ ) − 14 sin φ R (cid:48)(cid:48) (0) (cid:16) φ ) R (cid:48) ( φ ) − φR (cid:48) ( φ ) R (cid:48)(cid:48) ( φ ) + (5+ cos 2 φ ) sin φR (cid:48)(cid:48) ( φ ) (cid:17)(cid:21) − N +28 R (cid:48)(cid:48)(cid:48) (0 + ) R (cid:48)(cid:48) ( φ ) − N − 24 cot φR (cid:48)(cid:48)(cid:48) (0 + ) R (cid:48) ( φ ) − N − (cid:104) R (cid:48)(cid:48) (0) − R (cid:48)(cid:48) (0) + γ a R (cid:48)(cid:48)(cid:48) (0 + ) (cid:105) R ( φ ) . (896)The last factor proportional to R ( φ ) takes into accountthe renormalization of temperature, absent in the manifoldproblem . The full analysis of this equation is quiteinvolved. The key observation is that under FRG again acusp develops near z = 1 . Analysis of the FRG fixed pointsshows interesting features already at 1-loop order. For N =2 , the fixed point corresponds to the Bragg-glass phaseof the XY model with quasi-long range order accessiblevia a d = 4 − (cid:15) expansion below d = 4 [707]. Hencefor N = 2 the lower critical dimension is d lc < , and The constant γ a is discussed in Ref. [33]. N = N c N > N c N < N c dd dd d lc F F FD D D D D DQLRO ggg Figure 77. Phase diagram of the RF non-linear sigma model. D = disordered, F = ferromagnetic, QLRO = quasi long-range order.Reprinted from [33]. conjectured to be d lc < [707]. On the other hand Feldman[705, 706] found that for N = 3 , , . . . there is a fixed pointin dimension d = 4+ (cid:15) > . This fixed point has exactly oneunstable direction, corresponding to the ferromagnetic-to-disorder transition. The situation at one loop is thus ratherstrange: For N = 2 , only a stable FP which describes a unique phase exists, while for N = 3 only an unstableFP exists, describing the transition between two phases.The question is: Where does the disordered phase go as N decreases? The complete analysis at 2-loop order [33]shows that there is a critical value of N , N c = 2 . ,below which the lower critical dimension d lc of the quasi-ordered phase plunges below d = 4 , resulting into two newfixed points below d = 4 . This is schematically shown inFig. 77. For N > N c a ferromagnetic phase exists withlower critical dimension d lc = 4 . For N < N c one findsthe expansion d RFlc = 4 − (cid:15) c ≈ − . N − N c ) + O ( N − N c ) . (897)One can then compute the critical exponents at this fixedpoint [705, 706, 33, 38].The expansions discussed above where either in d =6 − (cid:15) , neglecting by construction FRG corrections of thedisorder with the physics of the cusp, or in d = 4 + (cid:15) ,neglecting amplitude flutuations in (cid:126)φ ( x ) := (cid:82) box (cid:126)S ( x ) , asthey were formulated in terms of a non-linear σ -model. Tofind a consistent renormalization-group treatment in the full ( N, d ) -plane is much more complicated, and can to dateonly be achieved within the non-perturbative FRG approach(NP-FRG), i.e. the RG must be both non-perturbative (NP)and functional (FRG). For this formalism to work, and tocorrectly encounter shocks, i.e. the physics of the cusp, onehas to allow for a cusp in the effective disorder correlator. Itis the merit of G. Tarjus and M. Tissier to have transformedthis general idea into a predictive framework [39, 37, 35,36, 34, 708]. We show in Fig. 78 their phase diagram.The behavior in the region around d = 4 and N = N c was obtained above from the non-linear σ model. A novelprediction is that for d = 4 + (cid:15) and N > N ∗ = 18 there is a solution without cusp, and as a consequencedimensiononal reduction and super-symmetry are restored[34, 709, 710, 711]. The critical line starting at d = 4 and N ∗ = 18 can be obtained in an (cid:15) -expansion [712, 38], N ∗ ( d ) = 18 − 495 ( d − 4) + O ( d − . (898) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Figure 78. The phase diagram of Tarjus and Tissier [708], reproduced withkind permission. The added line given by Eq. (898) qualitatively agreeswith the NP-FRG prediction. The FRG in its perturbative and non-perturbative versionscan be applied to a variety of disordered systems in andout of equilibrium, see e.g. [708]. In particular, O ( N ) models with a random anisotropy can be treated. For thisuniversality class, details of the phase diagram, the criticalexponents, and the many subtleties involved, the reader isreferred to Refs. [39, 33, 37, 38, 712, 713, 35, 36, 34, 714,715, 708].For random field, the NP-FRG solution qualitativelyagrees with the perturbative FRG solution, but systemati-cally predicts a smaller N ∗ ( d ) , terminating for N = 1 at d = 5 . , while the analytic solution favors d = 5 . . Weremind that NP-FRG is based on a truncation of the func-tional form of the effective action. By construction it in-cludes loop corrections at all orders, but in an approximateway beyond one loop. Thus for the Ising model, dimen-sional reduction is valid near dimension d = 6 , whereas anon-trivial ordered phase exists down to d = 2 . This hasbeen confirmed numerically in Ref. [27, 716, 717, 718], themost remarkable test being the comparison of diverse cor-relation functions in the 5-dimensional RF model, as com-pared to their 3-dimensional counterparts in a pure systemat T = T c [718].Renewed interest into the RF Ising model comes fromthe conformal bootstrap community [719, 720]. Theyfollow the proposition of Cardy [672] to use n bosonicreplias φ i , i = 1 , ..., n , to introduce fields u := 12 (cid:2) φ + ( n − − ( φ + ... + φ n ) (cid:3) , (899) ˜ u := 12 (cid:20) φ − T ∆(0) ( n − − ( φ + ... + φ n ) (cid:21) , (900)together with ( n − fields ψ j which are linearcombinations of T ∆(0) ( φ , ..., φ n ) chosen to be orthogonalto φ + ... + φ n . As Cardy showed, this choice of fieldsallows one to write an action formally equivalent to Eq.(827). The authors then identify [720] perturbations which destabilize the supersymmetric dimensional-reduction fixedpoint below d c ≈ . . In section 8.4 we showed thatin order to see the renormalization of the disorder, oneneeds more than one physical copy. To be precise, thecusp appears in the renormalized disorder correlations, as afunction of the difference between the two physical copies.In such a calculation, the critical dimension moves up to d c ≈ . [721]. We do not see how this difference betweenreplicas is present in the above choice of coordinates, butwe believe that by doubling the set of Cardy variables thisproblem can be repaired.We would like to conclude by some general remarkson the form of the effective action necessary for a properRG treatment of the RF Ising model. As in all disorderedsystems, it should at least contain a 1-replica and a 2-replicacontribution. Its general form should be as given in Eq.(30) in a replica formulation, in Eq. (309) in a dynamicalformulation, or in Eq. (826) in the Susy formulation. Whilethe 1-replica part may contain an arbitrary function of u and ∇ u , let us concentrate on the 2-replica part parameterizingthe disorder correlations. For disordered elastic manifolds,this is achieved by the function ∆( u − u ) , where weremind that ∆( u − u ) has only one argument due to thestatistical tilt symmetry (64). As the latter is absent for theRF Ising model, one needs to make a more general ansatz,see e.g. [37, 714], ∆( u , u ) = ˆ∆(¯ u, δu ) , (901) ¯ u := 12 ( u + u ) , δu := | u − u | . (902)The absolute value appears since ∆( u , u ) = ∆( u , u ) .Let us apply as in sections 2.9 and 2.10 a field h = m w ,and denote u i ≡ u ( w i ) the expectation of u given w i .Both in the statics and at depinning u ( w i ) is unique. Theconnected correlation function (cid:104) u u (cid:105) c is u ( w ) u ( w ) c = Γ (cid:48)(cid:48) ( u ) − ∆( u , u )Γ (cid:48)(cid:48) ( u ) − . (903)Here Γ (cid:48)(cid:48) ( u ) is the second (functional) derivative of the1-replica contribution to the effective action. Formally,the l.h.s. which depends on w and w is the Legendretransform of the second cumulant ∆( u , u ) in theeffective action depending on u and u . Graphically,the prescription amounts to amputating the 1-particleirreducible contributions to (903). The key point is thatthe observable on the l.h.s. can be measured. For small w − w > it behaves with ¯ w := ( w + w ) / as 12 [ u ( w ) − u ( w )] (cid:39) A ( ¯ w ) | w − w | + O ( w − w ) , (904) A ( ¯ w ) := (cid:10) S (cid:11) (cid:104) S (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ w × u (cid:48) ( ¯ w ) . (905)As indicated, the ratio (cid:10) S (cid:11) / (2 (cid:104) S (cid:105) ) depends on ¯ w . Theserelations are derived similar to Eq. (102), except that whenwriting Eq. (100) as u ( w ) − u ( w ) = (cid:104) S (cid:105) ρ shock | w − w | + O ( w − w ) , (906) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Figure 79. The function V ( φ ) , for φ theory (top, red, dashed), and abounded potential (bottom, blue, solid). the l.h.s. becomes u ( w ) − u ( w ) (cid:39) u (cid:48) ( ¯ w )( w − w ) + O ( w − w ) . (907)Solving Eq. (903) for ∆( u , u ) proves that it has a cusp asa function of u − u , with amplitude given in Eq. (905). Evaluating the partition function of a field theory inpresence of a potential V ( u ) at constant background field u to 1-loop order, and normalizing with its counterpart at V = 0 , one typically gets a flow equation of the form ln (cid:18) Z [ u ] Z [ u ] (cid:19) = − (cid:90) Λ d d k (2 π ) d ln (cid:18) V (cid:48)(cid:48) ( u ) k + m (cid:19) . (908)We have explicitly written an UV cutoff Λ . This equationis at the origin of non-perturbative renormalization group(NPRG) schemes [722, 723, 724, 725], (confusingly) alsoreferred to as exact RG . To leading order, the effectiveaction is Γ( u ) = − ln( Z [ u ] / Z [ u ]) , and denoting its localpart by V ( u ) , we arrive at the following functional flowequation for the renormalized potential V ( u ) − m∂ m V ( u ) = − m∂ m Λ (cid:90) d d k (2 π ) d ln (cid:18) V (cid:48)(cid:48) ( u ) k + m (cid:19) . (909)Keeping only the leading non-linear term [726] leads to thesimple flow equation − m∂ m V ( u ) = − m d − V (cid:48)(cid:48) ( u ) + .... (910)Note that this equation is very similar to the FRGflow equation (59) for disordered elastic manifolds. Itreproduces the standard RG-equation for φ theory; indeed,setting V ( u ) = m − d u g, (911)we arrive with (cid:15) := 4 − d at − m∂ m g = (cid:15)g − g + .... (912) This is the standard flow equation of φ theory, with fixedpoint g ∗ = (cid:15) . One knows that the potential (911) at g = g ∗ is attractive, i.e. perturbing it with a perturbation φ n , n > , the flow brings it back to its fixed-point form.This fixed point, and its treatment with the projectedsimplified flow equation (912) is relevant in manysituations, the most famous being the Ising model. Theform of its microscopic potential, which is plotted in figure79 (red dashed curve), grows unboundedly for large φ . Thisis indeed expected for the Ising model, for which the spin,of which φ is the coarse-grained version, is bounded.There are, however, situations, where this is not thecase. An example is the attraction of a domain wall by adefect. In this situation, one expects that the potential atlarge φ vanishes, as plotted on figure 79 (solid blue line).The question to be asked is then: Where does the RG flowlead?As one sees from figure 79, the bounded potential V is negative. In order to deal with positive quantities, set V ( u ) ≡ −R ( u ) . The flow equation to be studied is − m∂ m R ( u ) = m − (cid:15) R (cid:48)(cid:48) ( u ) + .... (913)As shown in [726], for generic smooth initial conditions asplotted on figure 79:(i) The flow equation (913) develops a cusp at u = 0 , anda cubic singularity at u = u c > .(ii) The rescaled flow equation for the dimensionlessfunction R ( u ) − m∂ m R ( u ) = ( (cid:15) − ζ ) R ( u ) + ζuR (cid:48) ( u )+ 12 R (cid:48)(cid:48) ( u ) + .... (914)has an infinity of fixed points − m∂ m R ( u ) = 0 ,indexed by ζ ∈ [ (cid:15) , ∞ ] .(iii) The solution chosen dynamically when starting fromsmooth initial conditions is ζ = (cid:15) . Its analyticexpression for ≤ u ≤ reads R ζ = (cid:15) ( u ) = (cid:15) (cid:20) 118 (1 − u ) − 172 (1 − u ) (cid:21) . (915)It vanishes for u > , and is continued symmetricallyto u < .This scenario is quite unusual: Normally, the perturbativelyaccessible fixed points of the RG flow have only one fixedpoint. In the few cases where there is more than onefixed point, the spectrum of fixed points is at least discrete .In contrast, here is a spectrum of fixed points. On theother hand, only one of them seems to be chosen. Thusexperiments would only see this one fixed point.It is yet not clear to which physical system it applies.As discussed in the literature [727, 728, 729, 730, 731, 732],experiments describing wetting are usually attributed to aflow equation linear in R ( u ) . Is a nonlinear fixed pointpossible? Let me cite Thierry Giamarchi, one of thepioneers of FRG: “Whenever there is a simple equation, itis somewhere realized in nature.” heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles The aim of this review was to give a thorough overviewover the physics of disordered elastic manifolds, with itsnumerous connections to systems as diverse as sandpilesand loop-erased random walks. We covered all theoreticaltools developed to date, including FRG, replicas, replica-symmetry breaking, MSR dynamics, and super-symmetry.We put emphasis on applications, giving experimentaliststhe necessary tools to verify the theoretical concepts, andgoing beyond critical exponents.Our aim at completeness was seriously challenged bythe shear amount of publications on the subject, and weapologize for any omisions. Please let us know, and wewill try to remedy.While the presented methods are powerful, fundamen-tal questions remain: Can FRG be applied to other sys-tems such as spin glasses, sheared colloids, real glasses, orNavier-Stokes turbulence? Can FRG be applied beyond theelastic limit, i.e. to systems with overhangs or topologicaldefects, or to fractal curves that can not be represented bydirected interfaces?The author is looking forward to exciting newdiscoveries, and the contributions of today’s PhD studentsand postdocs. As for this review, it needs to stop here. Acknowledgements It is a pleasure to thank all my collaborators over thepast years: Pierre Le Doussal for introducing me to thesubject and all his enthusiasm in the many projects weundertook together. A. Rosso, A. Kolton, L. Laurson,and A. Middleton for putting the theoretical concepts totest. M. M¨uller for all his insights in the connection toreplica-symmetry breaking. Many thanks go to my studentsM. Delorme, A. Dobrinevski, C. ter Burg, G. Mukerjee,T. Thierry, Z. Zhu, and postdocs C. Husemann, A.Petkovic, Z. Ristivojevic and A. Shapira, for their helpand pertinent questions. I am grateful to A.A. Fedorenkofor the many enjoyable common projects. S. Atis,F. Bohn, M. Correa, A. Douin, A.K. Dubey, G. Durin,F. Lechenault, S. Moulinet, M.R. Past´o, F. Ritort, P.Rissone, E. Rolley, D. Salin, R. Sommer, and L. Talonwere essential in testing the concepts in experiments.I have benefitted from collaborations with L. Aragon,C. Bachas, P. Chauve, R. Golestanian, E. Jagla, M. Kardar,M. Kompaniets, W. Krauth, A. Ludwig, C. Marchetti,A. Perret, E. Raphael, G. Schehr and J. Vannimenus.Many thanks for discussions go to M. Alava, C. Aron,E. Br´ezin, L. Balents, E. Bouchaud, J.P. Bouchaud,J. Cardy, F. David, H.W. Diehl, T. Giamarchi, P. Goldbart,D. Gross, F. Haake, A. Hartmann, J. Jacobsen, J. Krug,S. Majumdar, M. M´ezard, T. Nattermann, G. Parisi,H. Rieger, L. Ponson, S. Rychkov, S. Santucci, L. Sch¨afer,S. Stepanov, L. S¨utterlin, G. Tarjus, M. Tissier, F. Wegner,J. Zinn-Justin and A. Zippelius. This review is based on lecture notes for the ICTP master program at ENS, and Ithank all students for their feedback. 10. Appendix: Basic Tools In Markov chains the state at time t N := N τ is given bythe product of transition probabilities P ( x N , x N − , ..., x , x ) = N (cid:89) i =1 P τ ( x i | x i − ) . (916)Transition probabilities are drawn from a Gaussian distri-bution. The probability to be at x (the variable) given x (cid:48) (prime as previous), reads P τ ( x | x (cid:48) ) = 1 (cid:112) πτ D ( x (cid:48) ) e − [ η ( x − x (cid:48) ) − τF ( x (cid:48) )]24 ητD ( x (cid:48) ) . (917)Both F and D depend on the previous position (Itˆodiscretization). As a stochastic process, this reads η ( x i +1 − x i ) = τ F ( x i ) + √ τ ξ i , (918) (cid:104) ξ i (cid:105) = 0 , (cid:104) ξ i ξ j (cid:105) = 2 δ ij ηD ( x i ) (919)The formal limit of τ → is the Itˆo-Langevin equation , η ˙ x ( t ) = F (cid:0) x ( t ) (cid:1) + ξ ( t ) , (920) (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 ηδ ( t − t (cid:48) ) D (cid:0) x ( t ) (cid:1) . (921)The factor of η is the friction coefficient in Newton’sequation of motion. Indeed, for the problem at hand thelatter reads M ∂ t ˙ x ( t ) = F (cid:0) x ( t ) (cid:1) + ξ ( t ) − η ˙ x ( t ) . (922)On the l.h.s. is the mass M (or inertia) of the particle (notto be confounded with the mass m in field theory), times itsacceleration. This defines a characteristic time scale τ M = Mη . (923)For times t (cid:29) τ M , inertia plays no role, M can be set to 0,and we arrive at Eq. (920).The situation is different, when the noise is correlatedon a time scale τ (cid:29) τ M . Then in the equation of motion F ( x ( t )) changes, since x ( t ) changes, and it is better todiscretize this limit as η ( x i +1 − x i ) = τ F (cid:18) x i + x i +1 (cid:19) + √ τ ξ i . (924)This prescription is known as mid-point or Stratonovichdiscretization.Let us finally rescale time, t → ηt , which effectivelysets η → . The friction coefficient η can always be restoredby multiplying each time derivative with η . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Consider (with η = 1 ) g ( x i +1 ) − g ( x i ) = g (cid:0) x i + τ F ( x i ) + √ τ ξ i (cid:1) − g ( x i )= g (cid:48) ( x i ) (cid:2) τ F ( x i ) + √ τ ξ i (cid:3) + 12 g (cid:48)(cid:48) ( x i ) τ ξ i + O ( τ / )= g (cid:48) ( x i ) (cid:2) τ F ( x i ) + √ τ ξ i (cid:3) + g (cid:48)(cid:48) ( x i ) τ D ( x i ) + O ( τ / ) . (925)The last relation is justified since in any time slicemaximally two powers of ξ i can appear. (If there could be4 then one would have to use Wick’s theorem to decouplethem pairwise.) It is implicitly understood that the noiseis independent of x , thus (cid:104) g ( x i ) ξ i (cid:105) = 0 , and (cid:10) g ( x i ) ξ i (cid:11) =2 g ( x i ) D ( x i )d t .Mathematicians prefer setting x i → x , τ → d t , ξ i √ τ → d ξ , and write the Langevin equation as d x = F ( x )d t + d ξ, (926) (cid:104) d ξ (cid:105) = 0 , d ξ = (cid:10) d ξ (cid:11) = 2 D ( x )d t. (927)The stochastic evolution of a function g ( x ) can then bewritten with these “differentials” as d g ( x ) = g (cid:48) ( x )d x + 12 g (cid:48)(cid:48) ( x )d x + ... = g (cid:48) ( x )[ F ( x )d t + d ξ ] + 12 g (cid:48)(cid:48) ( x )[ F ( x )d t + d ξ ] + ... = [ g (cid:48) ( x ) F ( x ) + g (cid:48)(cid:48) ( x ) D ( x )] d t + g (cid:48) ( x )d ξ (928)This is known as Itˆo calculus . The rule of thumb toremember is that when expanding to first order in the timedifferential d t , as d ξ ∼ √ d t , one has to keep all terms upto second order in d ξ . The forward Fokker-Planck equation can bederived from Itˆo’s formalism. Consider the expectation of atest function g ( x ) at time t : (cid:104) g ( x t ) (cid:105) ≡ (cid:90) x g ( x ) P t ( x ) . (929)Taking the expectation of the first line of Eq. (928) yields (cid:104) d g ( x t ) (cid:105) = (cid:104) g (cid:48) ( x t )d x (cid:105) + 12 (cid:10) g (cid:48)(cid:48) ( x t )d x (cid:11) + ... (930)Averaging over the noise gives dd t (cid:104) g ( x t ) (cid:105) = (cid:104) g (cid:48) ( x t ) F ( x t ) (cid:105) + (cid:104) g (cid:48)(cid:48) ( x t ) D ( x t ) (cid:105) . (931)Expressing the expectation values with the help of Eq.(929), we obtain (cid:90) x g ( x ) ∂ t P t ( x )= (cid:90) x g (cid:48) ( x ) F ( x ) P t ( x ) + g (cid:48)(cid:48) ( x ) D ( x ) P t ( x ) . (932) Integrating by part, and using that g ( x ) is an arbitrary testfunction, we obtain the forward Fokker-Planck equation ∂ t P t ( x ) = ∂ ∂x (cid:16) D ( x ) P t ( x ) (cid:17) − ∂∂x (cid:16) F ( x ) P t ( x ) (cid:17) . (933)Our derivation is valid for any initial condition, thus thepropagator P ( x f , t f | x i , t i ) also satisfies the forward FokkerPlanck-equation as a function of x = x f , t = t f .If there are several degrees of freedom x u , u =1 , ..., L , then Eq. (933) generalizes to an equation for thejoint probability P t [ x ] ≡ P t ( x , x , ..., x L ) ∂ t P t [ x ] = L (cid:88) u =1 ∂ ∂x u (cid:16) D u [ x ] P t [ x ] (cid:17) − ∂∂x u (cid:16) F u [ x ] P t [ x ] (cid:17) . (934)Passing to the continuum limit, this yields the functionalFokker-Planck equation ∂ t P t [ x ] (935) = (cid:90) d u δ δx ( u ) (cid:16) D u [ x ] P t [ x ] (cid:17) − δδx ( u ) (cid:16) F u [ x ] P t [ x ] (cid:17) . The backward Fokker-Planck equation: Let us study P ( x f , t f | x i , t i ) as a function of its initial time and position.To this purpose, write down the exact equation, using thenotations of Eq. (926), P ( x f , t f | x, t ) = (cid:104) P ( x f , t f | x + d x, t + d t ) (cid:105) . (936)The average is over all realizations of the noise η duringa time step d t . Expanding inside the expectation value tofirst order in d t and second order in d x , and taking theexpectation, we find (cid:104) P ( x f , t f | x + d x, t + d t ) (cid:105) = (cid:68) P ( x f , t f | x, t ) + d t ∂ t P ( x f , t f | x, t )+ d x ∂ x P ( x f , t f | x, t ) + d x ∂ x P ( x f , t f | x, t ) (cid:69) = P ( x f , t f | x, t ) + d t (cid:104) ∂ t P ( x f , t f | x, t )+ F ( x ) ∂ x P ( x f , t f | x, t ) + D ( x ) ∂ x P ( x f , t f | x, t ) (cid:105) . (937)Comparing to Eq. (936) implies that the term of order d t vanishes, thus − ∂ t P ( x f , t f | x, t )= F ( x ) ∂∂x P ( x f , t f | x, t ) + D ( x ) ∂ ∂x P ( x f , t f | x, t ) . (938)This is the backward Fokker-Planck equation . Note thatcontrary to the forward equation, all derivatives act on P ( x f , t f | y, t ) , not on F or D . heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Remark on Consistency: The form of the backward andforward equations is constraint by an important consistencyrelation: Using that the process is Markovian, we can writethe Chapman-Kolmogorov equation P ( x f , t f | x i , t i ) = (cid:90) x P ( x f , t f | x, t ) P ( x, t | x i , t i ) . (939)This relation must hold for all t between t i and t f . Taking a t derivative and using the backward Fokker-Planck equationfor the first propagator P ( x f , t f | x, t ) , and the forwardequation for the second P ( x, t | x i , t i ) , we find cancelationof all tems upon partial integration in x , due to the specificarrangement of the derivatives in Eqs. (933) and (938). Remark on Steady State: Let us find a steady-statesolution of Eq. (933), i.e. a solution which does not dependon time. Integrating once and dropping the time argumentyields ∂∂x (cid:2) D ( x ) P ( x ) (cid:3) = F ( x ) P ( x ) + const. (940)Let us suppose that the probability P ( x ) vanishes when x → ∞ . This implies that the constant vanishes. Thesolution is obtained as ( x is arbitrary) P ( x ) = N D ( x ) exp (cid:18)(cid:90) xx F ( y ) D ( y ) d y (cid:19) , (941) N − = (cid:90) ∞−∞ d x D ( x ) exp (cid:18)(cid:90) xx F ( y ) D ( y ) d y (cid:19) . (942)The simplest case is obtained for thermal noise, i.e. D ( x ) = T , and when the force F ( x ) is the derivative of a potential, F ( x ) = − V (cid:48) ( x ) . Eq. (941) can then be written as P ( x ) = N e − V ( x ) /T , N − = (cid:90) ∞−∞ d x e − V ( x ) /T . (943)This is Boltzmann’s law [733]. The transition probability (917) from x (cid:48) to x was P τ ( x | x (cid:48) )d x = d x (cid:112) πτ D ( x (cid:48) ) e − [ x − x (cid:48)− τF ( x (cid:48) )]24 τD ( x (cid:48) ) . (944)This is ugly: our standard field-theory calculations workwith polynomials in the exponential. We therefore rewritethis measure as P τ ( x | x (cid:48) )d x = d x (cid:90) i ∞− i ∞ d˜ x πi e −S τ [ x, ˜ x ] , (945) S τ [ x, ˜ x ] = ˜ x (cid:0) x − x (cid:48) − τ F ( x (cid:48) ) (cid:1) − τ ˜ x D ( x (cid:48) ) . (946)The term S τ [ x, ˜ x ] is termed action . Reassembling all timeslices, it is normally written in the limit of τ → as P ( x | x )d x = (cid:90) x ( N )= xx (0)= x D [ x ] D [˜ x ]e −S [ x, ˜ x ] , (947) S [ x, ˜ x ] = (cid:90) t ˜ x ( t ) (cid:2) ˙ x ( t ) − F (cid:0) x ( t ) (cid:1)(cid:3) − ˜ x ( t ) D (cid:0) x ( t ) (cid:1) , (948) D [ x ] D [˜ x ] = N (cid:89) i =1 (cid:90) ∞−∞ d x i (cid:90) i ∞− i ∞ d˜ x i πi . (949)This is known as the MSR formalism (Martin-Siggia-Rose)[278], the action also as Martin-Siggia-Rose-Janssen-DeDominicis action, in honor of their respective work[279, 734, 735] . Changing the discretization: Let us turn back to a singletime slice, as given in Eq. (945). The variables ˜ x and x areconjugate, i.e. (cid:90) d x d˜ x πi e − ˜ x ( x − x (cid:48) ) ˜ x n f ( x, ˜ x )= (cid:90) d x d˜ x πi e − ˜ x ( x − x (cid:48) ) ∂ nx f ( x, ˜ x ) , (950) (cid:90) d x d˜ x πi e − ˜ x ( x − x (cid:48) ) ( x − x (cid:48) ) n f ( x, ˜ x )= (cid:90) d x d˜ x πi e − ˜ x ( x − x (cid:48) ) ∂ n ˜ x f ( x, ˜ x ) . (951)We can thus change our discretization scheme, i.e. replace F ( x (cid:48) ) → F (¯ x ) , D ( x (cid:48) ) → D (¯ x ) , where ¯ x = αx + (1 − α ) x (cid:48) , ≤ α ≤ . (952)There are cases where this change is advantageous. On theother hand, the microscopic dynamics may be such that F and D depend on ¯ x instead of x (cid:48) (see end of section 10.1).The consequences of the reparametrization (952) isunderstood from the following example: Expand e τ ˜ xF (¯ x ) − to linear order in τ , τ (cid:90) d x d˜ x πi e − ˜ x ( x − x (cid:48) ) ˜ xF (¯ x ) f ( x, ˜ x )= τ (cid:90) d x d˜ x πi e − ˜ x ( x − x (cid:48) ) ∂ x [ F (¯ x ) f ( x, ˜ x )] . (953)As the derivative acts on F (¯ x ) , this depends on α , as ∂ x ¯ x = α . Luckily, we can compensate this by an explicit x derivative: Wherever we change x → ¯ x , we also replace ˜ x by ˜ x − ∂ x . This yields for the action of a single time slice S τ [ x, ˜ x ]= ˜ x (cid:0) x − x (cid:48) ) − τ (˜ x − ∂ x ) F (¯ x ) − τ (˜ x − ∂ x ) D (¯ x )= ατ F (cid:48) (¯ x ) − α τ D (cid:48)(cid:48) (¯ x )+ ˜ x (cid:2) x − x (cid:48) − τ F (¯ x ) + 2 ατ D (cid:48) (¯ x ) (cid:3) − τ ˜ x D (¯ x ) . (954)The noise-correlator D ( x ) did not change, but there is anadditional contribution to the force F ( x (cid:48) ) → F (¯ x ) − αD (cid:48) (¯ x ) . (955)The first two terms, αF (cid:48) (¯ x ) − α D (cid:48)(cid:48) (¯ x ) can be interpretedas a change in the integration measure. Let us stress thatthe change in the action leaves the physics of the probleminvariant . One may arrive at α = also when the bath isevolving more slowly than the time scale set by viscosity(see end of section 10.1). Then the choice α = 1 / , knownas Stratanovich discretization , is natural; one can use theabove procedure to get back to Itˆo’s discretization. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles Interpretation of the field ˜ x ( t ) : Let us now turn to aninterpretation of the two fields x ( t ) and ˜ x ( t ) , and modifyequation (920) to ˙ x ( t ) = F (cid:0) x ( t ) (cid:1) + ξ ( t ) + f δ ( t − t ) . (956)Thus at time t = t , we kick the system with an infinitlysmall force f . Then, the probability changes by ∂ f (cid:12)(cid:12)(cid:12) f =0 P ( x | x )d x = ∂ f (cid:12)(cid:12)(cid:12) f =0 (cid:90) x ( t )= xx (0)= x D [ x ] D [˜ x ] e −S [ x, ˜ x ] = (cid:90) x ( t )= xx = x (0) D [ x ] D [˜ x ] ˜ x ( t )e −S [ x, ˜ x ] (957)Multiplying with x ( t ) and integrating over all finalconfigurations, we obtain R ( t, t ) = ∂ f (cid:12)(cid:12)(cid:12) f =0 (cid:104) x ( t ) (cid:105) = (cid:104) x ( t )˜ x ( t ) (cid:105) . (958)The expectation is w.r.t the measure D [ x ] D [˜ x ] e −S [ x, ˜ x ] . AsEq. (958) is the response of the system to a change in force, ˜ x is called response field , and R ( t, t ) response function . Correlation functions are similarly obtained as C ( t, t (cid:48) ) = (cid:104) x ( t ) x ( t (cid:48) ) (cid:105) . (959)Since the probability is normalized, (cid:90) P ( x | x ) d x = 1 , (960)for all forces, one shows by taking derivatives w.r.t. forcesat different times that expectations of the sole response fieldvanish, (cid:104) ˜ x ( t ) (cid:105) = (cid:104) ˜ x ( t )˜ x ( t (cid:48) ) (cid:105) = (cid:104) ˜ x ( t )˜ x ( t (cid:48) )˜ x ( t (cid:48)(cid:48) ) (cid:105) = ... = 0 . (961) Consider theories with spatial dependence, and let ussuppose that the energy is given by H [ u ] = (cid:90) x 12 [ ∇ u ( x )] + m u ( x ) . (962)This corresponds to an elastic manifold inside a confiningpotential of curvature m . The elastic forces acting on apiece of the manifold at position x are given by F ( x ) = − δ H [ u ] δu ( x ) = (cid:0) ∇ − m (cid:1) u ( x ) . (963)Its Langevin dynamics reads ∂ t u ( x, t ) = (cid:0) ∇ − m (cid:1) u ( x, t ) + ξ ( x, t ) , (964) (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 T δ ( t − t (cid:48) ) δ ( x − x (cid:48) ) . (965)The action in Itˆo discretization is S [ u, ˜ u ] = (cid:90) x,t ˜ u ( x, t ) (cid:104) ∂ t −∇ + m (cid:105) u ( x, t ) − T ˜ u ( x, t ) . (966) It can be diagonalized in momentum and frequency space, S [ u, ˜ u ] = (cid:90) k,ω ˜ u ( − k, − ω ) (cid:2) iω + k + m (cid:3) u ( k, ω ) − T ˜ u ( − k, − ω )˜ u ( k, ω )= 12 (cid:90) k,ω (cid:18) u ( − k, − ω )˜ u ( − k, − ω ) (cid:19) M (cid:18) u ( k, ω )˜ u ( k, ω ) (cid:19) , (967) M = (cid:18) − iω + k + m iω + k + m − T (cid:19) . (968)This implies M − = (cid:32) T ( iω + k + m )( − iω + k + m ) 1 iω + k + m − iω + k + m (cid:33) . (969)As a consequence, R ( k, ω ) := (cid:104) u ( − k, − ω )˜ u ( k, ω ) (cid:105) = 1 iω + k + m , (970) C ( k, ω ) := (cid:104) u ( − k, − ω ) u ( k, ω ) (cid:105) = 2 T | iω + k + m | . (971)Inverse Fourier transforming R leads to R ( k, t ) = (cid:104) u ( − k, t )˜ u ( k, (cid:105) = (cid:90) ∞−∞ d ω π e iωt iω + k + m = e − ( k + m ) t Θ( t ) . (972)We used the residue theorem to evaluate the integral: Thereis a pole at ω = i ( k + m ) , i.e. in the upper complex half-plane. If t > , then the integral converges in the upper halfplane, and closing the contour there yields the residuum aswritten. For t < , one has to close the path in the lowerhalf-plane, and there is no contribution, thus the Θ( t ) .The response function (972) satisfies the diffusionequation, ( ∂ t + k + m ) R ( k, t ) = δ ( t ) . (973)Performing one more inverse Fourier transform yields theresponse function in real space, a.k.a. the diffusion kernal(we completed the square) R ( x, t ) = (cid:90) ∞−∞ d d k (2 π ) d e ikx − ( k + m ) t Θ( t )= e − m t − x t (4 πt ) d/ Θ( t ) . (974)Setting d = 1 this is identical to Eq. (944) (setting there η = D = 1 , F = x (cid:48) = 0 , and τ = t ). Correlation functionscan be obtained as C ( k, t − t (cid:48) ) = (cid:104) u ( k, t ) u ( − k, t (cid:48) ) (cid:105) = 2 T (cid:90) ∞−∞ d τ R ( k, t − τ ) R ( k, t (cid:48) − τ )= 2 T (cid:90) min( t,t (cid:48) ) −∞ d τ e − ( k + m )( t + t (cid:48) − τ ) = Tk + m e − ( k + m ) | t − t (cid:48) | . (975) heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles C ( k, 0) = (cid:104) u ( k, t ) u ( − k, t ) (cid:105) = Tk + m . (976)The correlation function (975) satisfies the differentialequation ∂ t C ( k, t − t (cid:48) ) = T [ R ( k, t (cid:48) − t ) − R ( k, t − t (cid:48) )] . (977)This relation is known as fluctuation-dissipation theorem. Itis more generally valid, see e.g. [279, 734]. ∇ (cid:20) − d ) S d | (cid:126)z | − d (cid:21) = δ d ( (cid:126)z ) , (978)where the volume of the unitsphere is defined as S d = 2 π d/ Γ( d/ . (979)Proof: ∇ | (cid:126)z | − d + η = (2 − d + η ) η | (cid:126)z | − d + η . Integratingthe last term against a test function f ( (cid:126)z ) yields (2 − d + η ) S d f (0) . Taking the limit of η → completes the proof. The inverse Laplacian in d = 2 . In d = 2 , we set (cid:126)z := ( x, y ) , and z = x + iy , ¯ z = x − iy . Eq. (978) reducesto ∇ ln( (cid:126)z )4 π = δ ( (cid:126)z ) . (980)Our notations imply ln( (cid:126)z ) = ln( z ¯ z ) = ln z + ln ¯ z . Since ¯ ∂∂ = ∇ (check of norm: ∂ ¯ ∂ ( z ¯ z ) = 1 , ∇ ( x + y ) = 4 ), ∂ = 4 ¯ ∂ ∇ = ¯ ∂ ln( z ¯ z ) π = 1 π ¯ z . (981)As a consequence ∂ π ¯ z = ¯ ∂ πz = δ ( (cid:126)z ) ≡ δ ( x ) δ ( y ) . (982) Consider a random variable x with probabil-ity distribution P ( x ) , and cumulative distributions P > ( x ) := (cid:90) ∞ x P ( y ) d y, (983) P < ( x ) := (cid:90) x −∞ P ( y ) d y = 1 − P > ( x ) . (984)Suppose x i , i = 1 , ..., N are drawn from the measure P ( x ) .We are interested in the law of their maximum m , m := max( x , ..., x N ) . (985)The probability that the maximum is smaller than m isequivalent to the probability that x i < m for all i , P max < ( m ) = P < ( m ) N = [1 − P > ( m )] N . (986)For large N , this can be approximated by P max < ( m ) (cid:39) e − NP > ( m ) , (987)with density P max ( m ) = ∂ m P max < ( m ) (cid:39) N P ( m )e − NP > ( m ) . (988) - (cid:1) - (cid:2) (cid:2) (cid:1) (cid:3) (cid:4) (cid:5) (cid:1) (cid:6)(cid:7)(cid:1)(cid:6)(cid:7)(cid:4)(cid:6)(cid:7)(cid:8)(cid:6)(cid:7)(cid:9)(cid:2)(cid:7)(cid:6) (cid:2) < (cid:1) ( (cid:1) ) (cid:10) (cid:1) (cid:2) (cid:1) ( (cid:1) ) Figure 80. The cumulative Gumbel distribution P G < ( y ) (blue, solid)and its derivative P G ( y ) (red, solid, rescaled by a factor of 2 for betterreadability). This is compared to the law exact law (986) for N = 4 (dashed, note the bounded support) and N = 10 (dotted). For N = 100 no difference would be visible on this plot. - (cid:1) - (cid:2) (cid:2) (cid:1) (cid:3) (cid:4) (cid:5) (cid:1) (cid:6)(cid:7)(cid:1)(cid:6)(cid:7)(cid:4)(cid:6)(cid:7)(cid:8)(cid:6)(cid:7)(cid:9)(cid:2)(cid:7)(cid:6) (cid:2) < (cid:1) ( (cid:1) ) (cid:10) (cid:1) (cid:2) (cid:1) ( (cid:1) ) Figure 81. The cumulative Gumbel distribution P G < ( y ) (blue, solid) andits derivative P G ( y ) (red, solid). This is compared to the exact law (986)for a Gauss-distribution, using the equality in Eq. (993), for N = 100 (dashed) and N = 10 (dotted). Gumbel distribution: Suppose that P ( x ) = e − x Θ( x ) ⇔ P > ( x ) = e − x Θ( x ) . (989)This implies that for large NP max < ( m ) (cid:39) e − N e − m Θ( m ) = e − e − m +ln( N ) Θ( m ) . (990)The variable y = m − ln( N ) (991)is distributed according to a Gumbel distribution [736] P G < ( y ) = e − e − y , P G ( y ) = ∂ y P G < ( y ) = e − y − e − y . (992)A plot elucidating the convergence is shown in Fig. 80. TheGumbel class has a large basin of attraction, encompassingall distributions which decay as P > ( m ) ∼ e − x α , α > ,including in particular the Gauss distribution. The ideais that a particular point x c in the distribution of P ( x ) will dominate P max ( m ) ; it then suffices to approximate heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles - (cid:1)(cid:2)(cid:3) - (cid:4)(cid:2)(cid:5) - (cid:4)(cid:2)(cid:3) - (cid:6)(cid:2)(cid:5) - (cid:6)(cid:2)(cid:3) - (cid:3)(cid:2)(cid:5) (cid:1) (cid:3)(cid:2)(cid:4)(cid:3)(cid:2)(cid:7)(cid:3)(cid:2)(cid:8)(cid:3)(cid:2)(cid:9)(cid:6)(cid:2)(cid:3) (cid:2) < (cid:1) ( (cid:1) ) (cid:10) (cid:2) (cid:1) ( (cid:1) ) Figure 82. The cumulative Weibull distribution P W < ( y ) (blue, solid) andits derivative P W ( y ) (red, solid) for α = 2 . This is compared to the lawexact law (986) for N = 4 (dashed) and N = 10 (dotted). For N = 100 virtually no difference would be visible on this plot. ln P > ( x ) by a linear fit at m = m c . For the standard Gauss-distribution P ( x ) = e − x / √ π , (993) P > ( x ) = 12 erfc (cid:16) x √ (cid:17) (cid:39) e − x / √ πx . (994)A strategy is to replace m − e − m / → m − e − m / − mm c ,and then to find the best m c to get rid of the N -dependence.This yields after some algebra y (cid:39) m (cid:112) ln( N ) − ln( N ) + 12 ln (cid:0) π ln( N ) (cid:1) . (995)A numerical check reveals a very slow convergence tothe asymptotic form: while the right tail and the centerof the density are correct even for small N , the lefttail converges very slowly (from above), while the peakamplitude converges slowly from below. Note that thiscould not be repaired by changing the parameters in Eq.(995), which work for the peak-position and the right tail. Weibull distribution: Suppose that P ( x ) is distributedaccording to a power-law, bounded from above by x = 0 , P ( x ) = α ( − x ) α − Θ( − ≤ x ≤ , (996) P > ( x ) = ( − x ) α Θ( − ≤ x ≤ . (997)This implies that for large N (we suppress the lower boundat x = − for compactness of notation, and since it doesnot matter in the final result) P max < ( m ) (cid:39) e − N ( − m ) α Θ( m ) , = exp (cid:16) − (cid:104) − mN α (cid:105) α (cid:17) Θ( − m ) . (998)The variable y = mN α (999) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:1) (cid:6)(cid:7)(cid:2)(cid:6)(cid:7)(cid:4)(cid:6)(cid:7)(cid:8)(cid:6)(cid:7)(cid:9)(cid:1)(cid:7)(cid:6) (cid:2) < (cid:1) ( (cid:1) ) (cid:10) (cid:2) (cid:1) ( (cid:1) ) Figure 83. The cumulative Fr´echet distribution P W < ( y ) (blue, solid) andits derivative P W ( y ) (red, solid) for α = 2 . This is compared to the lawexact law (986) for N = 4 (dashed) and N = 10 (dotted). For N = 100 virtually no difference would be visible on this plot. is distributed according to a Weibull distribution [737] withindex αP W α,< ( y ) = e − ( − y ) α Θ( − y ) , (1000) P W α ( y ) = ∂ y P W < ( y ) = α ( − y ) α − e − ( − y ) α Θ( − y ) . (1001) Fr´echet distribution: Suppose that P ( x ) is distributedaccording to an unbounded power law, α > , P ( x ) = αx α − Θ( x > , (1002) P > ( x ) = x − α Θ( x > . (1003)This implies that for large NP max < ( m ) (cid:39) e − Nm − α Θ( m > (cid:18) − (cid:104) mN − α (cid:105) − α (cid:19) Θ( m > . (1004)The variable y = mN − α (1005)is distributed according to a Fr´echet distribution [738] withindex α , P F α,< ( y ) = e − y − α Θ( y ) , (1006) P F α ( y ) = ∂ y P W < ( y ) = αy − α − e − y − α Θ( y ) . (1007)Note that P F α,< ( y ) has an algebraic tail ∼ y − α , thus decaysmuch more slowly than the Gumbel distribution for large y . We want to compute functional determinants of the form f ( α, m ) := det[ −∇ + αV ( x ) + m ]det[ −∇ + m ] , (1008)with Dirichlet boundary conditions at x = 0 and x = L ,at α = 1 . In order for the problem to be well-defined, −∇ + αV ( x ) + m must have a discrete spectrum. heory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles d = 1 , this can efficiently be calculatedusing the Gel’fand Yaglom method [261]. Considersolutions of the ODE [ −∇ + αV ( x ) + m ] ψ α ( x ) = 0 , (1009)with boundary conditions ψ α (0) = 0 , ψ (cid:48) α (0) = 1 . (1010)Define g ( α, m ) := ψ α ( L ) ψ ( L ) . (1011)Then the ratio of determinants is given by f ( α, m ) = g ( α, m ) . (1012) Proof: Set Λ α := −∇ + αV ( x )+ m . Call its eigenvalues λ i ( α ) , ordered, and non-degenerate. Consider the analyticstructure of f ( α, m − λ ) and g ( α, m − λ ) , as a functionof λ . By definition f is a product over eigenvalues, f ( α, m − λ ) = (cid:89) i λ i ( α ) − λλ i (0) − λ . (1013)Note that for large i the ratio λ i ( α ) /λ i (0) goes to 1, thusthe product should converge; that was the reason why theratio of determinants was introduced in the first place. Ifwe want to make the proof rigorous, we can put the systemon a lattice, replacing the Laplacian by its lattice version.Then the spectrum is finite, and the product converges. Asa consequence of Eq. (1013), f ( α, m − λ ) is an analyticfunction of λ , which vanishes at λ = λ i ( α ) . Now consider g ( α, m − λ ) . If λ is an eigenvalue, λ = λ i ( α ) , then thesolution of Eq. (1009) vanishes at x = L . Playing aroundwith solutions of differential equations, we can convinceourselves that for λ − λ i ( α ) → , ψ α ( L ) ∼ λ − λ i ( α ) . (1014)We further expect g to be analytic in λ . Thus, as a functionof λ , f and g have the same analytic structure, i.e. the samezeros and poles. The latter cancel in the ratio r ( λ ) := f ( α, m − λ ) g ( α, m − λ ) . (1015)The only possibility for a zero or a pole we have to check isfor | λ | → ∞ .Consider f in the limit of λ → −∞ : Each factor in(1013) will go to 1, s.t. also f goes to 1. (For the disrcetizedversion, this is evident, and does not depend on the phaseof λ ; for the continuous version one has to work a little bit,and take the limit away from the positive real axes, wherethe spectrum lies.)Now consider the differential equation (1009) with m → m − λ , in the same limit λ → −∞ . In thiscase, one can convince oneself that both solutions growexponentially, and that V ( x ) is a small perturbation, s.t.again g ( α, m − λ ) → . Thus r ( λ ) is a function in thecomplex plane which has no poles. As a consequence, r ( λ ) is bounded. According to Liouville’s theorem it is a constant. This constant can be extracted from the limit of λ → ∞ , which shows that r ( λ ) = 1 . 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