Complexity of waves in nonlinear disordered media
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Complexity of waves in nonlinear disordered media
C. Conti , , L. Leuzzi , ∗ ISC-CNR, UOS Roma, Piazzale A. Moro 2, I-00185, Roma, Italy IPCF-CNR, UOS Roma, Piazzale A. Moro 2, I-00185, Roma, Italy Dipartimento di Fisica, Universit`a di Roma “Sapienza,” Piazzale A. Moro 2, I-00185, Roma, Italy
The statistical properties of the phases of several modes nonlinearly coupled in a random systemare investigated by means of a Hamiltonian model with disordered couplings. The regime in whichthe modes have a stationary distribution of their energies and the phases are coupled is studied forarbitrary degrees of randomness and energy. The complexity versus temperature and strength ofnonlinearity is calculated. A phase diagram is derived in terms of the stored energy and amountof disorder. Implications in random lasing, nonlinear wave propagation and finite temperatureBose-Einstein condensation are discussed.
The interplay between disorder and nonlinearity inwave-propagation is a technically challenging process.Such a problem arises in several different frameworksin modern physics, as nonlinear optical propagation andlaser emission in random systems, Bose-Einstein conden-sation (BEC) and Anderson localization (as, e.g., in Refs.[1–9,9–19]). Related topics are the super-continuum gen-eration and condensation processes.
When disorder has a leading role, nonlinear processescan be largely hampered as due to the fact that wavesrapidly diffuse in the system. Conversely, if the struc-tural disorder is perturbative, its effect on nonlinear evo-lution is typically marginal, leading to some additionallinear or nonlinear scattering-losses, but not radically af-fecting the qualitative nonlinear regime expected in theabsence of disorder. When disorder and nonlinearity playon the same ground, one can envisage novel and fascinat-ing physical phenomena; however, the technical analysisis rather difficult, as the problem cannot be attacked byperturbational expansions.Physically, disorder and nonlinearity compete in thoseregimes when wave scattering affects the degree of local-ization, eventually inducing it (as in the Anderson lo-calization), and nonlinearity couples the modes in thesystem. These may in general exhibit a distribution oflocalization lengths (determined by the amount of disor-der) and a strength of the interaction depending on theamount of energy coupled in the system.Our interest here is to provide a general theoreticalframework, whose result is the prediction of specific tran-sitions from incoherent to coherent regimes, which arespecifically due to the disorder and display a glassy char-acter, associated with a large number of degenerate statespresent in the system.We adopt a statistical mechanics perspective to theproblem, which allows to derive very general conclusions,not depending on the specific problem, and our focus ison the case in which many modes are excited. This im-plies that energy is distributed among many excitationsin an initial stage of the dynamics. The overall coherence(i.e., the statistical properties of the overall wave) will bedetermined by the phase-relations between the involvedmodes. Here we show that there exist collective disor-dered regimes, where coherence is dictated by the fact that the system is trapped in one of many energeticallyequivalent states, as described below.Representing mode phases by means of continuous pla-nar XY-like spins and applying a statistical mechanic ap-proach we can identify different thermodynamic phases.For negligible nonlinearity, all the modes will oscil-late independently in a continuous wave noisy regime(“paramagnetic”-like phase). For a strong interactionand a suitable sign of the nonlinear coefficients, allthe modes will oscillate coherently (“ferromagnetic”-likeregime). This corresponds, for example, to standard pas-sively mode-locked laser systems , which we found totake place even in the presence of a certain amount of dis-order. In intermediate regimes, the tendency to oscillatesynchronously will be frustrated by disorder, resulting ina glassy regime.These three regimes are identified by a set of order pa-rameters (up to 10 for the most complicated phase, asdetailed below), which can be cast into two classes: the“magnetizations” m , and the “overlaps” q . As the sys-tem is in the paramagnetic-like phase all m and q vanish;in the ferromagnetic regime they are both different fromzero; while in the glassy phase m = 0 and (at least someof) the overlap parameters are different from zero.The paramagnetic and ferromagnetic phases may bepresent even in the absence of disorder; conversely, anecessary condition to find a glassy phase is frustration(disorder induced in our case, see Sec. III 1). The glassyphase is characterized by the occurrence of a rugged -complex - landscape for the Gibbs free energy functionalin the mode phases space: a huge number of minimaare present, corresponding to a multitude of stable andmetastable states in the system, separated by barriers ofvarious heights and clustered in basins. This is a resultof the competition between disorder and nonlinearity.The existence of a not-vanishing complexity (whichmeasures the number of energetically equivalent states)for the possible distributions of mode-phases is the basicingredient for explaining a variety of novel phenomenalike speckle pattern fluctuations and spectral statisticsfor disordered, or weakly disordered nonlinear, systems,ergodicity breaking, glassy transitions of light or BEC,and ultimately the onset of a coherent regime in a ran-dom nonlinear system.Our work extends previously reported results, cf. Ref.[27] and it includes an arbitrary degree of disorder andthe discussion of its application to nonlinear Schroedingermodels, relevant, e.g., for BEC, spatial nonlinear opticsand supercontinuum generation.The paper is organized as follows: in Sec. I we intro-duce the model and we discuss some of its possible fieldsof application, namely random lasers, Bose-Einstein con-densates, optical propagation; in Sec. II we discuss theeffect of disorder in the coupling of light modes and thenew expected phenomena; we dedicate Sec. III to an ex-tremely basic introduction to the statistical mechanics ofsystems with quenched disorder, to the replica method,and to the definition of complexity; in Sec. IV we studythe model within the replica approach, details of the com-putation are reported in App. A; in Sec. V we discussthe presence of excited metastable states and we com-pute the complexity functional; in Sec. VI we show thephase diagrams of our model and discuss the propertiesof its thermodynamic phases; eventually, in Sec. VII wedraw our conclusions. I. THE LEADING MODEL
Here we review some of the disordered systems wherea relevant non linear interaction may arise and our modelapplies. The basic Hamiltonian of N adimensional angu-lar variables φ ∈ [0 : 2 π ] is given by H J [ φ ] = − ,N X i
We start from the electromagnetic energy inside a di-electric cavity (due to the generality of the consideredmodel similar examples can be found in a variety of sys-tems): E EM = Z E ( r ) · D ( r ) dV (2)The displacement vector is written in terms of a positiondependent relative dielectric constant ǫ r ( r ): D ( r ) = ǫ ǫ r ( r ) E ( r ) + ǫ P NL ( r ) (3)with P NL the nonlinear polarization. In absence of thelatter, for a closed cavity, the field can be expanded interms of the modes of the system. In the presence of dis-order these modes may display a different degree of local-ization as, e.g., in a disordered photonic crystal (PhC). For a closed cavity these modes form a complete set andthe field can be expanded in terms of the modes E = ℜ " N X n =1 a n ( t ) exp ( − ıω n t ) E n ( r ) (4)with E = { E x , E y , E z } . As far as a nonlinear po-larization is not present, the coefficients a n are time-independent. Conversely, in the general case, taking for P NL a standard third order expansion, one has for thenon-linear interaction Hamiltonian H = −h Z ǫ E · P NL dV i = − X ω j + ω k = ω l + ω m ℜ [ G jklm a j a k a ∗ l a ∗ m ] (5)where h . . . i is the time average over an optical cycle andthe sum ranges over all distinct 4-ples for which the con-dition ω j + ω k = ω l + ω m (6)holds, with j, k, l, m = 1 , . . . , N . The effective interactionoccurring among mode-amplitudes reads: G jklm = ı √ ω j ω k ω l ω m (7) × Z V d r χ (3) αβγδ ( ω j , ω k , ω l , ω m ; r ) × E αj ( r ) E βk ( r ) E γl ( r ) E δm ( r )with α, β, γ, δ = x, y, z . This coefficient represents thespatial overlap of the electromagnetic fields of the modesmodulated by the non-linear susceptibility χ (3) . The dis-order is induced, e.g., by the random spatial distributionof the scatterers (as in random lasers) that leads to ran-domly distributed modes and, hence, to random suscep-tibilities and couplings among quadruple of modes.If the cavity is open, the mode set is no more complete,the modes whose profile decays exponentially out of thecavity are taken for the expansion (4), all the others formthe radiation modes. Under standard approach thecoefficients in the expansion that weight the radiationmodes can be expressed in terms of the disordered cavityone, and this results into linear terms in the Hamiltonian(open cavity regime). Thus, for an open cavity, Eq. (5)becomes H = −ℜ " X j A similar situation is found in the finite temperatureBose Einstein condensation with random potential. Thezero temperature Gross-Pitaevskii equations reads as ı ¯ h ∂ Φ ∂t = − ¯ h m ∇ Φ + V ext ( r )Φ + g | Φ | Φ (20)where V ext ( r ) is an externally set disordered potentialand g = 4 πℓ ¯ h /m , with ℓ being the s -wave scatter-ing length. An analogous model holds for reduced-dimensionality cases.The modes satisfy the time-independent linearSchroedinger equation − ¯ h m ∇ Φ n + V ext ( r )Φ n = E n Φ n (21)Their interaction can be treated variationally by lettingΦ( r , t ) = X n a n ( t )Φ n ( r ) exp (cid:18) − ı E n ¯ h t (cid:19) . (22)A finite temperature model for BEC is the Stoofequation, which is here written as ı ¯ h ∂ Φ ∂t = (cid:20) h β K K ( r , t ) (cid:21) (23) × (cid:20) − ¯ h m ∇ Φ + V ext ( r )Φ + g | Φ | Φ (cid:21) + η ( r , t )with β K = 1 /k B T ( k B is the Boltzmann constant) andwhere the finite temperature noise is such that h η ∗ ( r ′ , t ′ ) η ( r , t ) i = ı ¯ h K ( r , t ) δ ( t − t ′ ) δ (3) ( r − r ′ ). (24)Σ K ( r , t ) being the Keldish self-energy, which is imagi-nary valued (for its expression see Ref. [48]) and ¯ h Σ K ∝− ıβ − K (see Ref. [49]). Expanding over the complete setof the zero temperature equations, one obtains ı ¯ h ˙ a n ( t ) = − ı X j α jn a j E j e − ıt ¯ h ( E j − E n ) (25)+ X jkl ( G jkln − ıK jkln ) a ∗ l a j a k e − ıt ¯ h ( E j + E k − E l − E n ) + η n ( t )where η n ( t ) = R d r η ( r , t ) φ n ( r , t ), and the mode-overlapcoefficients are defined as: G jklm = g Z Φ j Φ k Φ l Φ m d r (26)and K jklm = ıβ K ¯ hg Z Σ K ( r )Φ j Φ k Φ l Φ m d r . (27)Finally, the linear coupling coefficients come out to be α jk = ıβ K ¯ h Z Σ K ( r )Φ j Φ k d r (28)While retaining the synchronous terms (such that E j + E k − E l − E n = 0), the resulting equations are,hence, of the same form of those reported in Sec. I Afor the disordered electromagnetic cavity, being the en-ergy of the eigenstates in place of the angular frequency.Indeed, a strong coupling regime is attained when thereis an enhanced region for the density of states. Con-versely, in other spectral regions, both the linear and thenonlinear coupling terms are averaged out by the rapidlyoscillating exponential tails.Let us consider, for example, a periodic external po-tential with some degree of disorder. In this case, a Lif-shitz tail is present, that is, a region with energies in-side the forbidden gap corresponding to localized modes.This modes will all have approximately the same energy E ∼ = E B where E B is the band-edge energy, and willcouple both among each other and with the delocalizedBloch modes at the band-edge. Correspondingly, the rel-evant equations for the strongly coupled modes are ı ¯ h ˙ a n ( t ) = − ı X j α jn a j E B (29)+ X jkl ( G jkln − ıK jkln ) a ∗ l a j a k + η n ( t )The other modes (those far from the spectral gap) willbe those mediating the thermal bath. The quenchedamplitude approximation eventually leads to the phase-dependent Hamiltonian, Eq. (14).As discussed in the following section of the manuscript,even in the zero temperature limit a transition is ex-pected. This corresponds to the existence of a replicasymmetry breaking transition in Bose Einstein conden-sates for finite and vanishing temperature, mediated bythe degree of disorder and heuristically following thephase diagram reported in Fig. 1 below. C. Nonlinear optical propagation in disorderedmedia and the zero temperature limit The nonlinear optical propagation of a light beam isdescribed by the paraxial equation ı ∂A∂z + 12 k ∇ x,y A + ∆ n kn A = 0 (30) where A is the optical amplitude, k the wavenumber, n is the bulk refractive index and ∆ n is its perturbationdue to disorder and optical nonlinearity (Kerr effect):∆ n kn = U ( x, y ) + n | A | . (31)The nonlinear coefficient n can be either positive (fo-cusing) or negative (defocusing), while U ( x, y ) can be aperturbed (by disorder) periodical potential or a com-pletely disordered (speckle pattern) external potential.The resulting equation reads as ı ∂A∂z + 12 k ∇ A + U ( x, y ) A + n kn | A | A = 0. (32)This formally corresponds (with different meanings forthe variable) to the zero-temperature two-dimensionallimit of the Gross-Pitaevskii equations detailed above,cf. Eq. (20).In this case, as well, the field can be expanded in termsof transversely localized (in two dimensions they are al-ways localized) electromagnetic modes, the energies be-ing replaced by their propagation wave-vectors. Whenthere are bunch of modes such that their wave-vectorsare similar, these will be strongly coupled and result intodynamical equations like Eqs. (9), (29). This approachcan be extended to three-dimensional propagation, en-compassing the dynamics of ultra-short pulses in randommedia as will be reported elsewhere.The replica symmetry breaking transitions investi-gated in the following will in general correspond to vary-ing coherence properties of the beam, eventually result-ing in unstable speckle patterns. The β → ∞ limit of thestatistical mechanical formulation of the problem has tobe taken in this case (see, e.g., Ref. [51] for a simplecase example in the framework of constraint satisfactionproblems). II. RANDOMNESS IN MODE-COUPLINGCOEFFICIENTS Let us consider our model Hamiltonian, Eq. (13), inthe mean-field fully connected approximation in whichthe non-vanishing components of the four index tensor J i ,i ,i ,i = J i are distributed as J i = J /N (33)( J i − J i ) = σ J /N (34)The coefficient J was already introduced in the case ofrandom lasers, cf. Eq. (18), and N is the number ofdynamic variables (mode phases) of the system, propor-tional to the volume V . The overbar denotes the averageover the disorder.To quantify the amount of disorder, we introduce the“degree of disorder” parameter, i.e., a size independentratio between the standard deviation of the distributionof the coupling coefficients J i and their mean: R J ≡ σ J J (35)The limits R J → R J → ∞ correspond, respec-tively, to the completely ordered and disordered case.The other relevant parameter for our investigation is theinverse temperature β . For random lasers it is related tothe normalized pumping threshold for ML, defined in ourmodel as, cf. Eq. (19), P = p βJ = s ¯ βR J (36)where ¯ β ≡ βσ J . In general, β increases as the strengthof nonlinearity increases or the amount of noise is re-duced. A. The ordered limit, saturable absorbers inrandom lasers, defocusing versus focusing With specific reference to the laser systems, as J grows the effect of disorder is moderated and for smallenough R J the model corresponds to the ordered case,previously detailed in Ref. [52]. As also previously re-ported in Ref. [53], a passive mode-locking (PML) tran-sition is predicted as a paramagnetic/ferromagnetic tran-sition occurs in β .Indeed, in our units, when R J → P = P PML ∼ = 3 . 37 91 Asexplained below, the deviation from this value quantifiesan increase of the standard ML threshold P PML due todisorder. The specific value for P PML will depend on theclass of lasers under consideration (e.g., a fiber loop laseror a random laser with paint pigments), but the trend ofthe passive ML threshold with the strength of disorder R J in Fig. 1 has a universal character. The pumpingrate P contains J : for a fixed disorder the threshold willdepend on the nonlinear mode-coupling.A key point here is that the transition from continuouswave to passive mode-locking (PM → FM) only occursfor a specific sign of the mean value of the coupling coef-ficient J , as shown in Fig. (2). Comparing Eqs. (9) and(11) one observes that this formally corresponds to thepresence of a saturable absorber in the cavity (see alsoRef. [26] and Sec. I A). In typical random lasers sucha device is not present, and, hence, this ferromagnetictransition is not expected.On the other hand, the reported phase diagram, Fig.(2) predicts that starting from a standard laser support-ing passive/mode-locking and increasing the disorder thesecond order transition acquires the character of a glasstransition. A notable issue is that this phase-lockingtransition (normally ruled out for ordered lasers withouta saturable transition), spontaneously occurs increasing β , as an effect of the disorder and the resulting frustra-tion.With reference to nonlinear waves, the spontaneousphase-locking process is expected for a specific sign of thenonlinear susceptibility (corresponding to repulsive inter-actions for BEC and defocusing nonlinearities for opticalspatial beams), for T = 0, amounting to J /σ J > J /σ J ∼ = 4). For example,for a nonlinear optical beam propagating in a disorderedmedium, it is expected that above a certain degree of dis-order, there is a transition from a coherent regime to a“glassy coherent phase”, characterized by a strong vari-ation from shot to shot of the speckle pattern and, morein general, of the degree of spatial coherence. III. FUNDAMENTALS OF STATISTICALMECHANICS OF DISORDERED SYSTEMS Hereby we report an extremely concise summary ofideas and techniques developed to deal with disorderedsystems. The aim is to let the non-expert reader findhis/her way through the computation of the propertiesof our model that we present in Sec. IV and App. A. 1. Disorder and frustration:quenched disorder as technical tool. The main issue determining complex features, notpresent in ordered systems and involving collective pro-cesses that cannot be understood just looking at localproperties, is frustration . This is usually a the conse-quence of disorder, not necessarily quenched disorder,though. Indeed, also in materials whose effective statis-tical mechanic representation is carried out through de-terministic potentials (as, e.g., for colloidal particles), ageometry-induced disorder can set up, determining frus-tration and a consequent multitude of degenerate stableand metastable states typical of glasses and spin-glasses. Quenched disorder, i.e., the explicit appear-ance of random coefficients in the Hamiltonian, allows ananalytic computation, but the results are general and donot depend on the specific source of frustration. 2. Statistical mechanics of a disordered system:the replica trick. In the presence of quenched disorder, one can computethe statistical mechanics of the system, averaging overthe probability distribution of the disorder. In order todo this the so-called replica trick can be adopted, or,else, the equivalent cavity method. The free energy of a single disordered system sample,denoted by J , is Φ J = − /β log Z J . Correspondingly, thephysically relevant average free energy can be written asΦ = − β log Z J (37)where the overbar denotes the average over the distribu-tion of the J ’s. The latter coincides with the thermo-dynamic limit of any Φ J according to the self-averagingproperty required in order to have macroscopic repro-ducibility of experiments (the thermodynamics of a hugesystem does not depend on the local distribution of in-teraction couplings).To perform the average in Eq. (37) is highly non triv-ial and one can proceed by considering n copies of thesystem, Eq. (13), H [ { φ } ] → n X a =1 H [ { φ ( a ) } ] (38)The average free energy per spin can, then, be computedin the replicated system, as β Φ = − lim N →∞ N log Z J = − lim N →∞ lim n → Z nJ − N n (39)where the average of the generic power of the partitionfunction Z nJ is somehow computed for a finite integer n and, eventually, the analytic continuation to real n andthe limit n → 3. Oddities of the replica formulation. Actually, to evaluate Z nJ , one makes use of the saddlepoint approximation holding for large N (see AppendixA for the specific case considered in this work). Thatis, one practically inverts the limits N → ∞ and n → 4. A probability distribution as an order parameter. The main novelty of the characterization of the spin-glass phase, historically first obtained by the replicamethod and subsequently confirmed by other methods,is that the order parameter is a whole probability distri-bution function describing how different thermodynamicstates are correlated. The degree of the correlation be-tween two states is called overlap . In mean-field the-ory different states exist that can be more or less corre-lated according to their distance on a tree-like hierarchi-cal space called ultrametric . 5. Complexity as a well-defined thermodynamic potential. Besides numerous and hierarchically organized glob-ally stable states, glasses also display a large number ofmetastable states, that is, excited states of relatively longlifetime. In the mean-field theory such lifetime is, actu-ally, infinite in the thermodynamic limit because of thedivergence of the free energy barriers with the size ofthe system, see, e.g., Ref. [73]. This means that, con-trarily to what happens in real glasses, the number ofmetastable states at a given observation timescale doesnot change with time (after a given transient period). Be-low a certain temperature (called dynamic or mode cou-pling temperature), the number N of metastable statesgrows exponentially with the size N of the system ( N being the number of modes in our cases). One can thendefine an entropy-like function counting the metastablestates as Σ ≡ N log N . (40)This is called configurational entropy in the framework ofstructural glasses, else complexity in spin-glass theory andits applications to constraint satisfaction and optimiza-tion problems. One can further look at the metastablestates of equal free energy density f : N ( f ) = exp N Σ( f )and at the free energy interval, above the equilibriumfree energy f eq , in which the complexity is non zero: f ∈ [ f eq : f ⋆ ]. IV. STATISTICAL MECHANICALPROPERTIES Starting from the Hamiltonian, Eq. (13), replicated ac-cording to the prescription Eq. (38), and averaging overthe disorder with the Gaussian probability expressed byEqs. (33)-(34), one obtains the following expression forthe average of the n -th power of the partition function,cf. Appendix A: Z nJ = Z D Q D Λ e − N n G [ Q , Λ ] (41) n G [ Q , Λ ] = n A [ Q , Λ ] + log Z φ [ Λ ] n A [ Q , Λ ] ≡ − β σ J n X a =1 (cid:16) | ˜ r a | (cid:17) − βJ n X a =1 | ˜ m a | − β σ J ,n X a
Under this Ansatz, taking the n → β Φ reads, cf. Appendix A, β Φ( m ; Q (1)sp , Λ (1)sp ) = G ( m ; Q (1)sp , Λ (1)sp ) = (45)= − ¯ βR J | ˜ m | − ¯ β h − (1 − m ) (cid:0) q + | r | (cid:1) − m (cid:0) q + | r | (cid:1) + | r d | i − ℜ h − m λ q + ¯ µ r )+ m λ q + ¯ µ r ) − ¯ µ d r d − ¯ ν ˜ m i + λ − m Z D [ ] log Z D [ ] (cid:20)Z π dφ exp L ( φ ; , ) (cid:21) m where = { x , ζ R , ζ I } , = { x , ζ R , ζ I } , D [ a ] is theproduct of three Normal distributions and L ( φ ; , ) ≡ ℜ n e ıφ h ¯ ζ p ∆ λ − | ∆ µ | + ¯ ζ p λ − | µ | + x p µ + x p µ + ¯ ν i + e ıφ (cid:16) ¯ µ d − ¯ µ (cid:17)o (46)with ∆ λ = λ − λ , ∆ µ = µ − µ . For later conveniencewe define the following averages over the action e L , cf. Eq. (46): c L ≡ h cos φ i L ≡ R π dφ cos φ e L R π dφ e L (47) s L ≡ h sin φ i L ≡ R π dφ sin φ e L R π dφ e L (48)The values of the order parameters λ , , µ , , µ d and ν are yielded by λ , = ¯ β q , ) ; µ , = ¯ β | r , | r , (49)˜ µ = ¯ β | ˜ r | ˜ r ; ν = ¯ βR J | ˜ m | ˜ m (50)The parameter m (without tilde!), whose meaning will bediscussed below, takes values in the interval [0 , q = hh c L i m i + hh s L i m i (51) q = hh c L i m i + hh s L i m i (52) r = hh c L i m i − hh s L i m i + 2 ı hh c L s L i m i (53) r = hh c L i m i − hh s L i m i + 2 ı hh c L i m i hh s L i m i (54)˜ r = hhh e ıφ i L i m i ; ˜ m = hhh e ıφ i L i m i (55)where the averages are defined as h ( . . . ) i m ≡ R D [ ]( . . . ) hR π dφ e L ( φ ; , ) i m R D [ ] hR π dφ e L ( φ ; , ) i m (56) h ( . . . ) i ≡ Z D [ ]( . . . ) (57)These equation are solved numerically by an iterativemethod. The overlap parameters q , are real-valued,whereas r , , ˜ r and ˜ m are complex. “One step” parame-ters X , ( X = q, r ) enter with a probability distributionthat can be parametrized by the so-called replica sym-metry breaking parameter m , such that P ( X ) = m δ ( X − X ) + (1 − m ) δ ( X − X ) . (58)The resulting independent parameters (there are ten ofthem) that can be evaluated by solving Eqs. (51)-(55)must be combined with a further equation for the pa-rameter m . This is strictly linked to the expression forthe complexity function of the system. V. COMPLEXITY In the order parameter Eqs. (49)-(55) m is left unde-termined. An additional condition is needed to fix thevalue for this parameter. The first possibility is treating m as a standard order parameter: in this case the thermo-dynamic state corresponds to extremizing the replicatedfree energy (thus, maximizing it ), i.e., implementingthe self-consistency equation ∂ Φ( m ; Q sp , Λ sp ) ∂m = 0 (59)The highest temperature at which a solution exists with m ≤ static tem-perature ( T s ). This is an equilibrium thermodynamic phase transition.This approach, however, does not reflect the knownphysical circumstance that a glassy system exhibits ex-cited metastable states also at temperature T above T s , where the equilibrium phase is paramagnetic. Vitrifica-tion, indeed, is due to the presence of a not vanishingcomplexity at a temperature above T s (and below some T d > T s ), i.e., to the presence of a number of energeticallyequivalent states with free energy f > f eq ( T ). Since,however, energy barriers tend to infinity in the thermo-dynamic limit in the mean-field approximation, the sys-tem dynamics is forever trapped in one of these statesfor T < T d . The temperature T d is, thus, called dynamic transition temperature.Across this transition the complexity Σ, defined inEq. (40), starts being different from zero. Exactly at T = T d the complexity as a function of free energy, Σ( f ),has a delta-shaped non-zero peak at the free energy f which corresponds to a maximum of Σ( m ) for a value of m = m ( f ) = 1. In our 1RSB formalism: ∂ Σ( m ; Q sp , Λ sp ) ∂m = 0 (60)As T decreases ( T s < T < T d ), the complexity is not van-ishing for an increasing range of free energies f ∗ > f > f that corresponds to a range for m : m ∗ < m < 1. Thecomplexity shows a maximum for m ≤ m ∗ ( f = f ∗ )solution of d Σ /dm = 0, while it is at its minimum valuefor m = 1 and f = f . We stress that this is not a solu-tion to Eq. (59).Lowering the temperature, at T = T s the minimumvalue of complexity - corresponding to m = 1 - vanishes,i.e., it is a solution to Eq. (59), and f = f eq correspondsto the free energy density of the global glassy minima ofthe free energy landscape: as mentioned above, we arein presence of a thermodynamic phase transition and thethermodynamic stable phase is a glass.The physically significant value for m is m ∗ , corre-sponding to the maximum of Σ. It denotes the value offree energy f ∗ where the number of states is maximumand exponentially higher than the number of states atany f < f ∗ , and, hence, the most probable (among thoseof the metastable states). At the thermodynamic tran-sition point from the paramagnetic state to the glassy( T = T s ) it holds f P M = f = f eq = Φ. Below T s f < f eq (hence, Σ( f ∗ ) < Σ( f eq ) = 0) and the physicallyrelevant Σ( f ) has a support [ f eq , f ∗ ].In the following we will analyze the whole complexity vs. free energy curve Σ( f ) at given β, J and the behav-ior of the minimal positive complexity Σ( T ) (and Σ( P ))between T s and T d . A. Computing the complexity functional In Eq. (40) one needs to know the number ofmetastable states, that are the local minima of thefree energy landscape. Would we know the landscape,though, we would have solved the problem already. Ifself-consistency equations for local order parameters areknown, a possible analytic approach to get informationon the complex landscape is to guess a trial free energyfunctional whose stationary equations lead back to theself-consistency equations. This is what Thouless, An-derson and Palmer (TAP) proposed in the framework ofspin-glasses starting from the self-consistency equationsfor local magnetizations. Starting from TAP functionaland TAP equations and considering solutions to the TAPeqs. as states (with some assumptions to be a posteriori satisfied) one can build the functional Σ from Eq. (40),cf., e.g., Refs. [78–85].A comparative study to the TAP-derived complexityfunctional and the replicated free energy, computed in ageneral scheme that includes the Parisi Ansatz, allowsto show that the Legendre Transform of Φ with respectto the single state free energy coincides with Eq. (40).According to this approach, in our model the complexitycan, thus, be explicitly computed as the Legendre trans-form of Eq. (45):Σ( m ; Q sp , Λ sp ) (61)= min m [ − βm Φ( m ) + βmf ]= βm ∂ Φ ∂m = 34 β m (cid:0) | q | + | r | − | q | − | r | (cid:1) + Z D [ ] log Z D [ ] (cid:20)Z π dφ exp L ( φ ; , ) (cid:21) m − m Z D [ ] h log Z π dφ exp L ( φ ; , ) i m where the single state free energy f = ∂ ( m Φ) ∂m (62)is conjugated to m . Since the above expression is propor-tional to ∂ Φ /∂m , equating Σ = 0 provides the missingequation to determine the order parameters values. VI. PHASE DIAGRAM AND COMPLEXITY By varying the normalized pumping rate P and thedegree of disorder R J , we find three different phases, as0 pu m p i ng r a t e P degree of disorder R J PM continuous wave SG glassymode-locking FM passive mode-locking FIG. 1: Phase diagram in the P , R J plane. Three phases arepresent: PM (low P ), FM (high P /weak disorder) and SG(high P /strong disorder). The full lines are thermodynamictransitions, the dashed line represents the dynamic PM/SGtransition. T / σ J J / σ J PM continuous wave SG glassymode-locking FM passivemode-locking dynamic ☛ random first order ☛ f i r s t o r de r ☛ FIG. 2: Phase diagram in the plane J , T in σ J units. Alsonegative J are considered. Three phases are found: PM(high T , low J ), FM (low T /large J ) and SG (low T /low ornegative J ). The full lines are thermodynamic transitions: random first order between PM and SG and standard firstorder between PM and FM and between SG and FM. Thedashed line represents the dynamic PM/SG transition. shown in Fig. 1 in the ( P , R J ) plane and in Figs. 2, 3 inthe ( T, J ) plane. Paramagnetic phase — For low P the only phasepresent is completely disordered: all order parameters arezero and we have a “paramagnet” (PM); for the randomlaser case this phase is expected to correspond to a noisycontinuous wave emission, and all the mode-phases areuncorrelated. Actually, this phase exists for any degreeof disorder and pumping, yet it becomes thermodynami-cally sub-dominant as P (or β ) increases and, dependingon the degree of disorder, the spin-glass or the ferromag-netic phases take over. Glassy phase — For large disorder, as P / β grows, a dis- T / σ J J / σ J PM continuous wave SG glassymode-locking FM passivemode-locking F M s p i noda l F M s p i n o d a l F M s p i noda l ( R S ) f i r s t o r de r S G / F M FIG. 3: (Color online) Detail of the J , T phase diagramaround the tricritical point. Full lines are thermodynamictransitions. Also the transition between the SG (1RSB)and the approximated RS solution for the FM phase is dis-played (double-dotted line) showing no appreciable differencewith the exact one. The dashed line represents the dynamicPM/SG transition. The dotted bold line represents the FMspinodal lines both inside the PM and the SG phases. Thespinodal of the RS FM phase is shown as well (smaller dots). continuous transition occurs from the PM to a spin-glass(SG) phase in which the phases φ are frozen but do notdisplay any ordered pattern in space. First, along the line P d = p ¯ β d /R J , in Fig. 1, or at T /σ J = 1 / ¯ β d = 0 . P )is displayed for three different values of R J ; the thresh-old pumping for non-zero minimal complexity grows asthe degree of disorder R J decreases, as well as the cor-responding P range. In the right panel Σ( T ) is plottedand it is independent of R J . In Figs. 6 and 7 we displaytwo instances of the whole complexity curve both vs. f and m at T = T s and at a higher temperature T < T d .Across the full line P s ( R J ) = p ¯ β s /R J , in Fig. 1 or,alternatively, across T /σ J = 1 / ¯ β s = 0 . q (the Edwards-Anderson parameter q EA87 ), discontin-uously jumps at the transition from zero q > q = 0,while ˜ m = r = r = r d = 0 (see Fig. 8, bottom panel).The SG phase exists for any value of R J and ¯ β > ¯ β s .1 Σ ( P ) P P d = . P s = . R J =0.5 FIG. 4: Complexity Σ( P ) of the lowest lying glassy statesin free energy between the values of the pumping rate corre-sponding to the dynamic and static transition from the PMto the SG phase at R J = 0 . σ J T d = . σ J T s = . σ J Σ P R J = . R J = . R J = . FIG. 5: Left: Complexity of the lowest lying glassy states infree energy Σ( P ) between dynamic and static transition fromthe PM to the SG phase along R J = 0 . , . . R J > ∼ . 3. Right:Σ vs. the effective temperature T in σ J units. Σ fT=T s =0.141 0.8 0.85 0.9 0.95 1mR J >0.3 FIG. 6: Σ( f ) (left) and Σ( m ) (right) in the glassy phase atthe static transition effective temperature, T = 0 . R J > ∼ . Σ fT=0.1500 P M f r ee ene r g y f f * J >0.3m * FIG. 7: Σ( f ) (left) and Σ( m ) (right) in the glassy phase at thestatic transition effective temperature, T = 0 . < T d . Thelowest state free energy of metastable glassy states is denotedby f (i.e., corresponding to m = 1 in the right panel, seetext). The free energy of maximum complexity is denoted by f ∗ , correspondingly m ∗ in the right hand side plot. In the stable SG phase, metastable states (with infinitelifetime) continue to exist so that the thermodynamicstate is actually unreachable along a standard dynamicsstarting from random initial condition. In Fig. 9 we plotthe typical behavior of the complexity versus the singlestate free energy at T /σ J = 0 . 1, qualitatively identicalto the left panel of Fig. 6 displaying Σ( f ) at T = T s . Ferromagnetic phase — For weak disorder a ran-dom ferromagnetic (FM) phase turns out to dominateover both the SG and the PM phases. The transitionPM/FM line is the standard passive ML threshold (seee.g. [53,88]) and it turns out to be first order in theEhrenfest (i.e., thermodynamic) sense . From Fig.1 we see that it takes place at growing pumping rates P for increasing R J until it reaches the tricritical pointwith the SG phase. In the ( T, J ) plane it occurs at large- positive - J , cf. Fig.2To precisely describe the FM phase in the 1RSB Ansatzwe have to solve eleven coupled integral equations [Eqs.(51)-(55) and Eq. (59) (Σ( m ; Q sp ) , Λ sp ) = 0), cf. Eq.(61)]. In evaluating their solutions we have to considerthat, in the region where the FM phase is thermodynam-ically dominant, both the PM and the SG solutions alsosatisfy the same set of equations. Besides, unfortunately,the basin of attraction of the latter two phases - in termsof initial conditions - is much broader than the FM one.Starting the iterative resolution from random initial con-ditions, determining the FM transition and spinodal linesbecomes, thus, numerically demanding.An approximation can be obtained by consideringthe Replica Symmetric (RS) solution for the FM phase(FM rs ). This reduces the number of independent param-eters to seven ( q = q , r R,I = r R,I , r R,Id and ˜ m R,I ). Thecorresponding transition line is shown as a2 P R J =0.4 P M / S G q q 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 P R J =0.4 ( P M / S G ) r r~m~ 0 0.2 0.4 0.6 0.8 R J =0.26189 P M / F M F M / S G q q R J =0.26189 P M / F M F M / S G r r r~m~ 0 0.2 0.4 0.6 0.8 1 3 4 5 6 7 8 R J =0.101 P M / F M q =q R J =0.101 P M / F M r =r r~m~ FIG. 8: Discontinuity of the order parameters at the transition points for three values of R J . Top Left panel: jump in q , ,at the PM/FM transition in P for small disorder, R J ≃ . 1; top right: discontinuities in r , , ˜ r and ˜ m at the same transition.For such small R J the replica symmetry breaking is practically invisible: q ≃ q , r ≃ r [to the precision of our computation, O (10 − )]. Mid left panel (across tricritical region in Fig. 1: q , vs. P at R J ≃ . 26 where, increasing the pumping rate, firsta PM/FM transition occurs followed by a FM/SG one. Mid right panel: r , , ˜ r and ˜ m vs. P for the same interval. First ordertransition point are signaled by vertical lines. Left bottom panel: q , vs. P for large disorder, R J = 0 . r , , ˜ r and ˜ m are always zero in the SG and in the PM phase. dashed-dotted line in Fig. 3, where, around the tran-sition , we observe no practical difference with the exactSG/FM, even though the replica symmetry is clearly bro-ken.In Fig. 8 we show the discontinuous behavior of theorder parameters across various transitions. As disorderis small (top panel) one can observe that the RSB ofthe solution representing the passive mode-locking phasevanishes, at least for what concerns the limit of precisionof our computation. As the degree of disorder takes val-ues around the tricritical point the RSB is clearly visible(mid panel), both in the FM and in the SG phases. Forincreasing disorder the FM is absent ( R J > ∼ . T and R J . This, as anticipated, also im-plies the occurrence of a dynamic transition besides thethermodynamic one. In the phase diagrams, Figs. 1, 2,3, this takes place between PM and SG, where the statestructure always displays a non-trivial Σ, for any ¯ β > ¯ β d .Whether an exclusively dynamic transition can occur asa precursor to the FM phase, as well, could not be di- β -=9.9 SG Σ (f) . 10 f-f eq β -=9.9R J =3.54FM Σ (f) . 10 f-f eq FIG. 9: Complexity vs. free energy curve is plotted in the SGphase (left) at T /σ J = 0 . rectly established in the present work. Indeed, the regionof expected dynamic transition lies beyond the spinodalFM line, already very difficult to obtain numerically be-cause of the competition with the SG and PM solutions.However, the existence of a metastable FM phase (cf.spinodal line in Fig. 3) with an extensive complexity, cf.e.g., Figs. 9 and 10, might well correspond to an arrestof the dynamic relaxation towards equilibrium of the sys-tem.In the right inset of Fig. 9 we show, e.g., Σ( f ) in the3 Σ ( f ) fComplexity in Passive Mode-LockingR J =3.54T=0.082T=0.114T=0.124T=0.134T=0.139 FIG. 10: Complexity curves of the FM phase at R J = 3 . T = 0 . σ J (right most) and T = 0 . σ J (left most). Both the magnitude of the maximalcomplexity and the free energy interval in which Σ( f ) > -1.39-1.37-1.35-1.33-1.31-1.29 0.25 0.26 0.27 0.28 Φ T=0.0785 S G t r an s i t i on F M f i r s t o r de r FMSG r r~m~ q R J q (SG)q (FM)q (FM) FIG. 11: Top panel: free energy of the FM and SG phasesvs. R J at T = 0 . r , r , ˜ r and the magnetization ˜ m are shown vs. R J .Beyond the transition point their values drop to zero in theSG phase. Bottom panel: q order parameters for the FM andthe SG phase. FM phase at ( R J , P ) = (0 . , . R J : the maximum complexity dropsof about two orders of magnitude at the SG/FM transi-tion, thus unveiling a corresponding high to low complex-ity transition .In Fig. 11, at a relatively low temperature T = 0 . R J . VII. CONCLUSION We have reported on an extensive theoretical treat-ment of the thermodynamic and dynamic phases of non-linear waves in a random systems. The approach allowsto treat nonlinearity and an arbitrary degree of disorderon the same ground, and predict the existence of complexcoherent phases detailed in a specific phase-diagram. Thewhole theoretical treatment is limited to the quenched-amplitude approximation, which allows to catch the ba-sic phenomenology and to demonstrate the existence ofphases with a not-vanishing complexity in a variety ofphysical systems, and specifically random lasers, finitetemperature BEC and nonlinear optics. This approxi-mation will be removed in future works, and novel exoticphases of light in nonlinear random system will be de-tailed.Our theoretical work shows that the interplay of non-linearity and disorder leads to the prediction of substan-tially innovative physical effects, which bridge the gapbetween fundamental mathematical models of statisticalmechanics and nonlinear waves. This allows to identifyfrustration and complexity as the leading mechanisms fora coherent wave regime in nonlinear disordered systems.Natural extension of this work will be considering thequantum counterpart of the predicted transitions, andthe analysis of out of equilibrium nonlinear waves dy-namics. Acknowledgments The research leading to these results has receivedfunding from the European Research Council underthe European Community’s Seventh Framework Pro-gram (FP7/2007-2013)/ERC grant agreement n. 201766and from the Italian Ministry of Education, Univer-sity and Research under the Basic Research Investiga-tion Fund (FIRB/2008) program/CINECA grant codeRBFR08M3P4.4 Appendix A: Replica computation of thethermodynamic properties The replicated partition function of the system de-scribed by the Hamiltonian H [ { φ j } ], cf. Eq. (13), reads Z nJ = Z n Y a =1 N Y j =1 dφ aj exp " − β n X a =1 H [ { φ aj } ] (A1)In order to compute the free energy of the system usingthe replica trick, cf. Eq. (39), Eq. (A1) has to be av-eraged over the probability distribution of i.i.d. randombonds: P ( J ) ≡ s N πσ J exp (cid:20) − N ( J − J /N ) σ J (cid:21) (A2)Eq. (41), then reads Z nJ = Z D φ exp ( N " β σ J n X a =1 (cid:18) (cid:12)(cid:12)(cid:12) ˜ R a ( { φ } ) (cid:12)(cid:12)(cid:12) (cid:19) + β σ J X a
1. Replica Symmetric Ansatz In this Ansatz Eqs. (A16,A17) become Z RS φ = Z n Y a =1 dφ a e − β H eff [ { φ a } ] (A27) β H eff = λ R n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X a =1 e ıφ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (A28) −ℜ ¯ µ n X a =1 e ıφ a ! + (cid:16) ¯˜ µ − ¯ µ (cid:17) n X a =1 e ıφ + ¯ ν n X a =1 e ıφ The second term in the rhs can be rewritten as ℜ ¯ µ n X a =1 e ıφ a ! = ℜ µ n X a =1 e − ıφ a ! (A29)= ℜ √ ¯ µ n X a =1 e ıφ a + √ µ n X a =1 e − ıφ a ! − | µ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X a =1 e ıφ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) The squared terms in the exponent of the integrand canbe linearized by using e | w | / = Z dζ R dζ I π e −| ζ | / e ℜ (¯ ζw ) (A30) e w R / = Z dx √ π e − x / e xw R (A31)6thus yielding Z RS φ = Z D p ( x ) D p ( ζ R ) D p ( ζ I ) (cid:20)Z π dφe L ( φ ; x,ζ ) (cid:21) n (A32) L ( φ ; x, ζ ) ≡ ℜ (cid:20) e ıφ (cid:16) ¯ ζ p λ − | µ | + x p µ + ¯ ν (cid:17) (A33)+ e ıφ (cid:16) ¯˜ µ − ¯ µ (cid:17)(cid:21) D p ( w ) = dw √ π e − w / (A34)The replicated free energy eventually reads: β Φ = − β σ J (cid:2) − | q | − | r | + | ˜ r | (cid:3) (A35) − βJ | ˜ m | + λ R − q R ) − ℜ [¯ µr − µ ˜ r − ν ˜ m ] − Z D p ( x ) D p ( ζ R ) D p ( ζ I ) log Z π dφ e L ( φ ; x,ζ ) Deriving w.r.t. to Q ’s parameter we obtain the specifi-cation of Eqs. (A18-A21) for the replica overlap param-eters q , r and for ˜ r and ˜ mλ = β σ J q (A36) µ = β σ J | r | r (A37)˜ µ = β σ J | ˜ r | ˜ rν = βJ | ˜ m | ˜ m Taking the derivative of β Φ in Eq. (A35) w.r.t. ˜ µ and ν we obtain ˜ r d = hh e ıφ i L i x,ζ (A38)˜ m = hh e ıφ i L i x,ζ (A39)where we define h . . . i L ≡ R π dφ . . . e L ( φ ; x,ζ ) R π dφ e L ( φ ; x,ζ ) (A40)Deriving G w.r.t. λ and µ and equating to zero weobtain q R = (cid:10) c L + s L (cid:11) x,ζ (A41) r = (cid:10) c L − s L + 2 ı c L s L (cid:11) x,ζ (A42) c L ≡ h cos φ i L s L ≡ h sin φ i L after having integrated by part in the Gaussian measures.To help the non-expert reader to easily derive the self-consistency equations we exemplify the calculation of Eq.(A18). 2 ∂G∂λ R = 0 = 1 − q R (A43) − (cid:10)(cid:0) ζ R c L + ζ I s L (cid:1)(cid:11) x,ζ / q λ R − | µ | The latter term can be simplified by integrating by part Z ∞−∞ D p ( y ) y F ( y ) = Z ∞−∞ D p ( y ) ∂F ( y ) ∂y (A44)with y = ζ R , ζ I in Eq. (A44), yielding (cid:10) ζ R c L + ζ I s L (cid:11) x,ζ = q λ R − | µ | × (A45) (cid:10) cos φ − c L + sin φ − s L (cid:11) x,ζ The self-consistency equation can thus be rewritten as,cf. Eq. (51), 1 − q R = 1 − (cid:10) c L + s L (cid:11) x,ζ q R = (cid:10) c L (cid:11) x,ζ + (cid:10) s L (cid:11) x,ζ We recall that since q ab is real, and so is λ ab , cf. Eq.(A18), in the RS Ansatz the equations q I = λ I = 0.Before deriving Eq. (A42), we rewrite the part of Eq.(A33) involving the integrating variable x as: ℜ (cid:20) e ıφ x p µ (cid:21) = (A46) x p | µ | cos φ s µ R | µ | + sin φ s − µ R | µ | ! In determining the above expression one can use, e.g.,the trigonometric law of tangents to yield12 arctan µ I µ R = arctan s − µ R / | µ | µ R / | µ | (A47)and the relationships between trigonometric and inversetrigonometric functions:sin[arctan( θ )] = θ √ θ cos[arctan( θ )] = 1 √ θ Using Eq. (A47), together with Eqs. (A38) and (A41),we have:2 ∂G∂µ R = 0 = r Rd − r R + µ R | µ | (1 − q ) (A48) − * x c L s µ R | µ | + s L s − µ R | µ | !+ x,ζ / p | µ | Integrating by part with Eq. (A44), y = x , we find * x c L s µ R | µ | + s L s − µ R | µ | !+ x,ζ (A49)= p | µ | *(cid:20) h cos 2 φ i L − c L + s L + µ R | µ | (cid:0) − c L − s L (cid:1)(cid:21)+ x,ζ r is analogously determined from ∂G∂µ I = 0.Above a given critical temperature (depending on J )the solution to Eqs. (A38,A39,A41,A42) is paramagnetic,i.e., q = r = ˜ m = ˜ r = 0 and the free energy is β Φ P M = − ¯ β − log 2 π (A50)Below T c ( J ), depending on the value of J the solutioncan either be ferromagnetic ˜ m = 0 or spin-glass ˜ m =0. The latter solutions are, however, not stable againstfluctuations in the space of replica overlaps and, thus,we have to try an Ansatz different from Eq. (A26) toprovide a self-consistent thermodynamics. 2. One step of Replica Symmetry Breaking In order to obtain a thermodynamically consistent re-sult the symmetry cannot be conserved. We are in pres-ence of a spontaneous Replica Symmetry Breaking. Theway to break the symmetry must be a-priori hypothe-sized, since there has been found, so far, no way to deduceit. The correct way to express the elements of the over-lap matrices is called Parisi Ansatz and, dependingon the kind of system, can consist of one or more RSB’s.According to what happens in other spin models with p -body quenched random interactions ( p being larger than2), the right Ansatz for the matrices of our model is theone-step RSB, that is, we have a n × n matrix divided insquare blocks of m × m elements q ab = q ; r ab = r if I (cid:16) am (cid:17) = I (cid:18) bm (cid:19) (A51) q ab = q ; r ab = r if I (cid:16) am (cid:17) = I (cid:18) bm (cid:19) (A52)For instance, for n = 6 and m = 3. q ( αβ ) = q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q The one replica index observables are instead still RS,as exemplified in Eqs. (A23-A25). Now, let us write the”vectorial” replica index a → a = ( a , a ) φ a → φ a = φ a ,a n X a =1 O a = n/m X a =1 m X a =1 O a a Take a 1RSB matrix K ab and two replicated observables g a and h a . The following expressions hold for the sum of a generic product2 X a
G/n with respect to theparameters we obtain the twelve self-consistency equa-tions determining the order parameter values at givenexternal pumping intensity and amount of disorder. • Deriving w.r.t. to Q ’s parameter we obtain thespecification of Eqs. (A18-A21) for each 1RSBreplica matrix sector and for ˜ r and ˜ m , cf. Eqs.(49,50), λ , = β σ J q , | q , | (A70) µ , = β σ J | r , | r , (A71)˜ µ = β σ J | ˜ r | ˜ r (A72) ν = βJ | ˜ m | ˜ m (A73) • Deriving w.r.t ˜ µ and ν we obtain Eqs. (55), wherewe define h . . . i L ≡ R π dφ . . . e L ( φ ; , ) R π dφ e L ( φ ; , ) (A74) c L ≡ h cos φ i L ; s L ≡ h sin φ i L (A75) • Deriving G w.r.t. λ , and µ , and equating to zerowe obtain Eqs. (51-54), after having integrated bypart in the Gaussian measures. To help the non-expert reader to easily derive the self-consistencyequations we exemplify the calculation of Eq. (51).2 ∂G∂λ R = 0 = 1 − (1 − m ) q R (A76) − Z D [ ] (cid:10) ζ R c L + ζ I s L (cid:11) m / q ∆ λ R − | ∆ µ | The latter term can be simplified by integrating bypart Z ∞−∞ D p ( y ) yF ( y ) = Z ∞−∞ D p ( y ) ∂F ( y ) ∂y (A77)with y = ζ R , ζ I in Eq. (A77), yielding (cid:10) ζ R c L + ζ I s L (cid:11) m = q ∆ λ R − | ∆ µ | × (A78) (cid:10) cos φ − (1 − m ) c L + sin φ − (1 − m ) s L (cid:11) m − (1 − m ) q R = 1 − (1 − m ) Z D [ ] (cid:10) c L + s L (cid:11) m q R = (cid:10)(cid:10) c L (cid:11) m (cid:11) + (cid:10)(cid:10) s L (cid:11) m (cid:11) (A79) Eqs. (52-54)) are analogously derived. We noticethat since from the equations ∂G/∂λ I , = 0 oneobtains q I , = 0 the values of the q overlap arereal-valued and so are the values of λ . ∗ Electronic address: [email protected], [email protected] T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature , 52 (2007). G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. 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Since in our model the dynamic variables are continuousphases, the whole low T phase is consistently described bythe 1RSB solution, unlike models with discrete variablessuch as the Ising p -spin model where a further transitionoccurs at the so-called Gardner temperature. Technically speaking, this is due to the fact that all theterms of the free energy functional depending on two repli-cas observables have n − n − m factors in front, cf.App. A, and in the n → The very existence of a Kauzmann temperature, also calledthe ideal glass transition temperature , in structural glassesis, actually, a matter of debate. The paramagnetic phase exists as metastable also at T