Universality Classes of Critical Points in Constrained Glasses
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Universality Classes of Critical Points in Constrained Glasses
Silvio Franz (1) and Giorgio Parisi (2) (1) Laboratoire de Physique Th´eorique et Mod`eles Statistiques,CNRS et Universit´e Paris-Sud 11, UMR8626,Bˆat. 100, 91405 Orsay Cedex, France(2) Dipartimento di Fisica, Universit`a di Roma La Sapienza,INFN, Sezione di Roma I, IPFC - CNR,P.le Aldo Moro 2, I-00185 Roma, Italy
Abstract
We analyze critical points that can be induced in glassy systems by the presence of constraints.These critical points are predicted by the Mean Field Thermodynamic approach and they areprecursors of the standard glass transition in absence of constraints. Through a deep analysis ofthe soft modes appearing in the replica field theory we can establish the universality class of thesepoints. In the case of the “annealed potential” of a symmetric coupling between two copies of thesystem, the critical point is in the Ising universality class. More interestingly, is the case of the“quenched potential” where the a single copy is coupled with an equilibrium reference configuration,or the “pinned particle” case where a fraction of particles is frozen in fixed positions. In these caseswe find the Random Field Ising Model (RFIM) universality class. The effective random field is a“self-generated” disorder that reflects the random choice of the reference configuration. The RFIMrepresentation of the critical theory predicts non-trivial relations governing the leading singularbehavior of relevant correlation functions, that can be tested in numerical simulations. . INTRODUCTION Recent times have seen a renewed interest for glassy systems in presence of constraints.Glassy relaxation in liquids is dominated by the presence of metastable states. Accordingto the Mean Field picture of the glass transition [1, 2], also known as Random First OrderTransition (RFOT) [3], these states have a well defined thermodynamic meaning and canbe probed and stabilized by imposing suitable constraints that modify the Hamiltonian.The simplest procedure consists in considering two copies of the system and introduce anattraction between the particles of the first and the second copy [4]. The free-energy as afunction of the overlap, which is the conjugate parameter to the strength of the attraction,is often referred to as the “annealed potential” function. A second, more refined procedureconsists in fixing a reference configuration and biasing the Boltzmann probability of thesystem in the direction of this configuration [5]. One can consider an external potential thatprovides an attraction for the particles of the system to the position they take in the referenceconfiguration. In this case, the free-energy as a function of the overlap is called “quenchedpotential”. Finally, the bias towards the reference configuration can be imposed by fixingsome of the degrees of freedom -in practice the position of a fraction of the particles- to thevalues they take in the reference configuration [6–10]. This is called the “pinned particles”method.The three ways of constraining the system have different advantages and reveal differentaspects of metastability. In both the annealed and quenched potentials metastability isrevealed by the shape of the potential function. It is well known that many aspects ofdynamical Mode Coupling Theory-like transitions [11], including dynamical heterogeneitiesand growth of correlations, can be seen studied from the quenched potential construction[12–16]. Using the overlap as an order parameter, as it is done in the annealed and quenchedpotential cases, allows to discriminate the thermodynamic view of the glass transition, wherethe overlap among configurations is the relevant order parameter, from a purely kinetic one,where the overlap does not allow to discriminate different metastable states [17, 18].The pinned particles method on the other hand, does not uses directly the overlap as anorder parameter, but is attractive because it can add stability to metastable states in a waythat the equilibrium state of the system is not perturbed.A marking feature of glassyness as we know from Mean-Field theory is the fact that in2ll three cases, the imposed constraint induces new phase transitions in the system [19, 20]-[9]. The nature of these transitions differs in the different procedures [21]. In the casesof attractive interactions one finds a first order transition line in the plane of temperatureand interaction strength [19, 20]. In the case of pinned particles, the constraint inducesa line of phase transition in the plane of temperature and fraction of blocked particles.Here the nature of the transitions depends in the detailed procedure of pinning, one caneither find a first order transition line as in the case of the coupled systems, or instead aline of ideal glass transition with Kauzmann entropy crisis that crosses over to a line ofsecond order glass transition [22]. In all cases, but this last one, the line of phase transitionterminates in a critical point. The existence of these critical points is a crucial predictionof the thermodynamic mean field approach. On the contrary, in purely dynamic theoriesof glassyness [23] and in exactly solvable kinetically constrained models [24] (such as e.g.the Fredickson-Andersen on random graph or similar models) the lines of thermodynamicphase transition and the critical point are not present [25]. Their existence is therefore oneof the few discriminating predictions that are different between the two approaches. Thethermodynamic scenario has started to receive confirmations in numerical simulations ofliquid systems. In [26] it was provided evidence for a coupling induced first order transitionsin the quenched potential setting. In [27] this result was confirmed, both for the annealedand the quenched and it was convincingly shown that the line of phase transition terminatesin a critical point. Other numerical results in this sense will be presented soon [28]. In thispaper we address the problem of the characterization of the universal properties of thesecritical points. Simple arguments can be put forward to understand these properties. Inthe annealed potential case, the only source of overlap fluctuations is the thermal noise.The critical point is described by a quartic field theory and is in the Ising universalityclass. In the quenched and pinned cases, however, a second source of fluctuations can beidentified in the choice of the reference configuration [14]. This acts as a random field inthe system and the resulting universality class is the one of the Random Field Ising Model(RFIM) [29, 30]. In order to turn these qualitative arguments into an accomplished theorya deep analysis of the soft modes emerging at the critical point and the properties of theperturbation theory should be performed. Replica Field Theory (RFT), in terms of whichthe constrained free-energy can be in principle computed, provides the natural formal settingto frame the problem. The three different procedure are found to correspond to different3nderlying symmetries and/or analytic continuations in the number of replicas that oneshould consider.The analysis of perturbation theory of replica field theories describing glassy criticality hasbeen initiated in [14] where it was shown how the description of dynamical heterogeneitiesin the beta regime, close to a mode coupling (MCT) dynamical transition could be mappedin a spinodal point of a RFIM with cubic interaction. In a subsequent paper [31] it wasanalyzed the case of a replica symmetric theory where the leading cubic interaction termvanishes. This theory describes higher order glass singularities as well as the critical point ofthe symmetric pinned particle construction where the the pinned particles are blocked froma configuration equilibrated at a temperature equal to the one at which the free particlesevolve. In that case the universality class of the ( φ ) RFIM was found within a perturbativeone loop calculation. Here we extend our analysis to the annealed and the quenched potentialand asymmetric pinning where the pinned particles are blocked from a temperature smallerthan the one of the free particles, that we are able to treat at all orders of perturbationtheory. In all cases we find that the expectations from the qualitative argument are met.While the annealed case is attractive for the simplicity of the result and the possibility toverify it in numerical simulations, most interesting from the theoretical point of view, forits implication on the nature of fluctuations and heterogeinities in glassy systems are thequenched potential case [14] and the asymmetric pinning case, where our analysis showshow the Parisi-Sourlas supersymmetry [32] of the RFIM naturally emerges at criticality.The plan of the paper is the following: In the next section we shortly review the theoryof glassy systems under constraints. In section 3 we briefly discuss the annealed case. Thenin section 4 we state the problem of the critical point for the quenched potential and pinnedparticle case. We analyze the zero modes of the mass matrix in section 5. In section 6 wederive the RFIM action by dimensional analysis. In section 7 we discuss physical correlationfunctions and their relations. We finally summarize and conclude the paper. An appendixpresents some technical details. II. GLASSY SYSTEMS UNDER CONSTRAINTS
In this section we briefly review the use of constraints to unveil glassyness and metasta-bility. Let us consider a system described by the Hamiltonian H ( X ) where X specifies4he configuration of all the particles in the system. We suppose a-priori that there is noquenched disorder in H , even though this could be included. As we will see, in all cases thecomputation of the constrained free-energy can be tackled through the use of the replicamethod. The order parameter of the theory is an overlap matrix, and fluctuations will bedescribed by a Landau expansion of the free-energy around a saddle point. In the specific theproblems differ in the number of replicas, which is 2 in the annealed potential problem and n → S n − in the quenched potential and asymmetric pinning, S n in the symmetricpinning. A. Annealed Potential Construction
The simplest setting consists in considering two copies in the system interacting throughan attraction [4, 20] H ( X, Y ) = H ( X ) + H ( Y ) + N ǫq ( X, Y ) . (1)where for a system with N particles, the overlap q ( X, Y ) among two configurations X = { x , ..., x N } and Y = { y , ..., y N } can be defined in terms of a short range attractive inter-action potential w ( x ) as q ( X, Y ) = 1 N X i,j w ( x i − y j ) . (2)Space dependent overlap fields q ( x ; X, Y ) can be defined restricting the sum in (2) to theparticles in some neighborhood of x .The free-energy of the system F ( ǫ, T ) involves a sum over the configurations of the twocopies -or replicas- of the system. One can see this sum as a particular case of a replicatedsystem where the number of replicas n here is just equal to 2, and, just as in the case ofuncoupled systems, the study of liquid phases can be addressed without need for analyticcontinuations. Conversely, the study of glassy phases requires analytic continuations in thenumber of replicas, but since, as we will see, the critical point we are interested lies in aliquid region we will not need to consider these continuations.The Legendre transform of F ( ǫ, T ), W ( q, T ) = F ( ǫ, T ) + ǫq is called annealed potentialfunction. We refer broadly to this procedure of symmetric coupling as annealed potentialconstruction. 5n Mean Field models with a glass transition, like e.g. p-spin or Potts spin glasses[5,33] or liquids in the HNC approximation [34] the coupling induces temperature dependentphase transitions in the system. A typical phase diagram is presented in figure II A Inthe temperature-coupling plane, one finds both a line of ordinary liquid-glass transitionwhere the overlap between the two copies is non singular [33], and a line of first order phasetransition that separates a low overlap or deconfined, phase, where the two copies are weaklycorrelated, from a high overlap or confined phase where the two replicas stay close to eachother. The first order transition line, which departs from the ideal Kauzmann transitiontemperature T k for ǫ = 0, terminates in a critical point ( T Cr , ǫ Cr ). Interestingly, this lineand the glass transition line meet in a point, and while the deconfined phase is always aliquid, depending on the temperature, the confined phase can be either a liquid or a glass.What is important for us is that a whole part of the line, which includes the critical point,marks the border of a liquid-liquid transition. The critical point lies at a finite distance fromthe line of glass transition and a description with just two replica is appropriate.The inset of figure II A shows the typical isothermal lines and coexistence curve in theoverlap-coupling plane for temperatures close to the critical point. Notice the similarity tothe isothermal of the gas-liquid phase transition in the V − p plane. As in this case, thecritical fluctuations can be described expanding the free-energy with respect to the the localfluctuation of the order parameter around its (space homogeneous) average value q ∗ . Wecan notice the similarity of figure II A with the coexistence diagram recently obtained innumerical simulations of a realistic liquid model by Berthier [27]. B. Quenched Potential Construction
The symmetric coupling between replicas in the annealed procedure introduces strongbiases to the equilibrium. In order to faithfully explore the vicinity of typical equilibriumstates at temperature, one considers instead a quenched procedure where one fixes a referenceconfiguration X extracted with Boltzmann probability at a temperature T ref , and uses theparticle positions in the reference configuration to define an external potential in which theparticles of the constrained system X evolve [5, 19, 20]. The Hamiltonian of the system forfixed X is H ǫ ( X ) = H ( X ) + N ǫq ( X, X ) . (3)6 C oup li ng ε Temperature T 0.12 0.16 0.2 0.2 0.3 0.4 0.5 0.6 C oup li ng ε Overlap q ‘ C oup li ng ε Temperature T
FIG. 1. Typical Mean Field phase diagram in the T − ǫ plane for the annealed and the quenchedconstructions. Left panel : The annealed construction phase diagram. The central (red) line marksthe first order phase transition between a confined high overlap phase above and an unconfinedphase below. The green line is the line of dynamical (MCT-like) glass transition. The confinedphase is a glass to the left of the brown point where the red line and the green line meet andit is a liquid to its right. The blue lines are the spinodal lines of the confined (lower curve) andunconfined (upper curve) phases. In the inset we show the typical isothermal and coexistencelines in the q − ǫ plane. From top to bottom we have an isothermal in the single phase region T > T Cr , the critical isothermal T = T Cr and a isothermal in the two phase region T < T Cr , thehorizontal line corresponds to Maxwell construction. The cyan line is the Widom line: the locusof points where the potential has an inflection and g = 0 and the susceptibility χ = d h p i /dǫ hasa maximum. Right panel : Phase diagram of the quenched construction The transition line andits corresponding spinodal are similar to the annealed case, however here the coupling does notinduces new glass transitions. Glassyness appears at the dynamical glass transition temperature T d of the unconstrained model (vertical green line). Notice the different scales in the two panels.The quenched critical point lies at lower temperature and coupling than the annealed one. Wehave used here the spherical p -spin model [35] for p = 3 for which the dynamical (MCT) transitiontemperature is T d = 0 .
612 and the Kauzmann transition temperature is T k = 0 . It should be noted the fundamental asymmetry between X , which is just a random equilib-rium configuration and the system X which feels an attraction towards X . The referenceconfiguration can be considered as a sort of quenched disorder in which the system evolves.7he constrained free-energy F Q ( ǫ, T, T ref ) is defined as F Q ( ǫ, T, T ref ) = − TN Z ( T ref ) X X e − β ref H ( X ) log X X e − β ( H ( X )+ Nǫq ( X,X )) ! . (4)One usually chooses T ref = T , but the case in which the temperature of the system isdifferent from the temperature has also been considered [20, 36] to study the evolution ofmetastable states with temperature. The quenched average over the distribution of thereference configuration is usually dealt with the replica method. One needs to replicatethe system X a number n ′ of times and perform a continuation n ′ → n = 1 + n ′ which should be sent to 1. Due to the asymmetryin the interaction between reference configuration and the system, replica number “0” turnsout to be privileged. In fact the effective Hamiltonian reads: H eff ( X , X , ..., X n − ) = n − X a =0 H ( X a ) + ǫ n − X a =1 q ( X , X a ) (5)(notice that the index a runs over different ranges in the two sums). Instead of possessingthe familiar symmetry S n under permutations of all the n replicas, the problem is symmetriconly under the permutations S n − of replicas with index a >
0. Analogously to the annealedcase, mean field theory predicts the existence of a line of phase transition in the ǫ − T plane that terminates in a critical point [20, 34, 37] and separate a confined phase with highoverlap with the reference configuration from a deconfined phase with low overlap. C. Particle pinning
Just as in the quenched potential case, in the case of particle pinning one fixes a referenceconfiguration in X from the equilibrium distribution at a temperature T ref , but then oneconsiders configurations X in which a fraction θ of the variables are fixed to the values theytake in X [6–10]. Also in this case Mean Field Theory predicts that the reduction of degreesof freedom induces new phase transitions in the system. Interestingly, the nature of the phasetransitions depends on the details of the pinning procedure. As discussed in much detail in[22], if T ref = T /α with α > T ref < T ) one finds a pattern of phase transition similarto the one of the annealed and quenched potential, there is line of confinement first orderphase transition in the θ − T plane that terminates in a critical point. Conversely, if α < T ref > T ) one finds a line of ideal RFOT Kauzmann-like transition of the discontinuous1RSB kind that crosses-over into a line of second order glass transition of the continuous1RSB kind [9, 22]. The nature of the terminating point of the first order transition in thefirst case and the RFOT transition in the second case is rather different as we will discussin the next section.Within the replica method this procedure still requires an analytic continuation in thenumber of replicas n which tends to 1. If the temperature of the reference configuration T ref is different from the temperature of the non-pinned particles, still the reference configurationis singled out and the symmetry is S n − . In the important case T ref = T however, one canshow that the unpinned particles remain at equilibrium [38, 39][40]. As a consequence,within the replica formalism there is full S n replica symmetry. The problem of criticalpoint in n → S n symmetric replica field theories has been addressed in [31]. In that casethrough the analysis of the soft modes of the replica field theory the critical point was shownto belong to the RFIM universality class. In this paper we extend our analysis to the caseof n → S n − symmetric theories. III. THE ANNEALED CRITICAL POINT
As we stated in the previous section, in the annealed potential case if we describe liquidphases, the complexity of the replica method is reduced to minimal terms. There are justtwo replicas and the n × n overlap order parameter matrix q a,b which appears in the replicamethod has here a single independent entry q with a = b . This represents the overlap betweenthe two copies, it is the only order parameter of the problem. [41] The resulting Landaufree-energy is a functional of a single field φ ( x ) = q ( x ) − q ∗ representing the fluctuation ofthe overlap around its average, F [ φ ] = Z dx k ( ∇ φ ( x )) + V ( φ ( x )) V ( φ ) = 12 m φ + g φ + g φ . (6)The coefficients m , g , g as well as q ∗ smoothly depend on the control parameters T and ǫ . Away from the critical point, where g = 0 the quartic term is irrelevant in perturbationtheory, however, as in the gas-liquid transition case, at the critical point, both m and g vanish. We find therefore that the critical point is described by an ordinary scalar field9heory with φ interaction and is in the universality class of the ordinary Ising model. Thisis coherent with the recent analysis of [42, 43]. IV. THE QUENCHED AND PINNED CRITICAL POINTS
In order to be defined, we consider the context of quenched potential however, the mainingredients within Replica Field Theory being the number replicas and the relative sym-metry, with little modifications, that we will specify on the way, one can treat the pinnedparticle construction with T ref < T . Replicas can be used to average over the choice of thereference configuration and we would like to describe the class of universality of the criticalpoint within replica field theory. The starting point will be a replica field theory with n → n × n space dependent matrix Q ab ( x ) of the kind: F [ Q ab ( x ) , ǫ ] = F [ Q ab ( x )] − ǫ n − X a =1 Z dx Q a ( x ) . (7)The term F [ Q ab ( x )] is symmetric under permutation of all replicas. The last term in theaction breaks this symmetry, in fact replica number 0, which corresponds to the referencestate is privileged with respect to the others. The symmetry S n − under permutations ofreplicas 1 , ..., n − ǫ -couplingterm is absent, however, if the temperature of the reference configuration T ref is differentfrom the temperature T at which the free particles evolve, the reference configuration issingled out and again F [ Q ab ( x )] contains terms that break S n into S n − .As usual we will start from a Landau expansion of the free-energy close to the criticalpoint, supposing that the non-diagonal elements of the matrix Q ab ( x ) can be written as Q ab ( x ) = Q ∗ ab + φ ab ( x ) (8)where Q ∗ ab is the saddle point value of the matrix order parameter. The diagonal elements,which are related to the structure factor, in general are non critical, regular across thetransition point and will not be discussed here.The saddle point matrix Q ∗ ab is homogeneous in space and for the critical points weconsider here has the replica symmetric form Q ∗ a,b = q + ( δ a + δ b )( p − q ). The parameters p = q represent respectively the overlap between the system and the reference configurationand the self-overlap of the system with itself. Instead of trying to write the most general10 n − -invariant polynomial expansion of the free-energy in terms of φ ab ( x ), our strategy willconsist first to analyze the properties of a S n − invariant mass matrix close to criticality andthen after identified the soft modes, in writing directly the generic field theory describingtheir interaction disregarding completely the massive modes. As a preliminary let us studylongitudinal fluctuations, i.e. just fluctuations of p and q . At the saddle point level, thefree-energy as a function of ǫ readsΓ[ ǫ ] = ∂∂n | n =1 F [ Q ∗ ab , ǫ ] (9)Γ[ ǫ ] = N ( W [ p, q ] − ǫp ) ∂W∂q = 0 ∂W∂p = ǫ. As usual one can interpret the effective potential V ( p, ǫ ) = W [ p, q ( p )] − ǫp at the point q ( p )defined by ∂W∂q = 0 as the value of the free-energy when the system to reference configurationoverlap takes the value p . The physical value of p is fixed by the stationary condition V ′ [ p ] = 0. At a critical point terminating a first order line one should have in additionthat the second and the third derivatives of V vanish, V ′′ [ p ] = V ′′′ [ p ] = 0, conditions thatgenerically fix the values of T and ǫ . Let us remark that close to a generic saddle pointvalues, away from the critical point, the function W [ p, q ] must admit an expansion of thekind: W [ p + δp, q + δq ] − W [ p, q ] = 12 h ˆ M pp δp + 2 ˆ M pq δpδq + ˆ M qq δq i + X r =0 C r δp r δq − r + O ( p ) (10)This function describes longitudinal fluctuations which are constant in space and the form of φ is the same as the one of the saddle point. By definition, at the critical point, longitudinalfluctuations for which δq = dq ( p ) dp δp = − ˆ M pq / ˆ M qq δp + O ( δp ) are long ranged, the quadraticand the cubic forms vanish and the quartic terms become important. Of course, the pointswhere the quadratic form vanishes but the cubic form remains finite are also critical. Thesecorrespond rather to spinodal points than to thermodynamic critical points. A well knownexample is the one of dynamical glass transitions points that correspond to S n symmetriccubic theories [14–16]. In the present case, as it can be seen in fig. II A there are just twospinodal lines, for the confined and the deconfined phases that converge into the criticalpoint for ( T, ǫ ) → ( T Cr , ǫ Cr ). Our analysis shows that generically, despite their differentnature [21], both these spinodals belong to the φ -RFIM universality class.11f one considers longitudinal fluctuations that are not constant in space, an additional“kinetic term” of the kind K [ δp ( x ) , δq ( x )] = 12 Z dx [ k p ( ∇ δp ) + k q ( ∇ δq ) ] (11)is present in the longitudinal Landau expansion.Close to the critical point, the mass of the soft mode can be simply related to the coeffi-cient to the quadratic form, to the lowest order,ˆ m = ˆ M pp ˆ M qq − ˆ M pq ˆ M pp + ˆ M qq + O ( ˆ m ) (12) V. THE MASS MATRIX AND ITS EIGENSPACES
In this section we would like to go beyond longitudinal fluctuations, and identify all thezero modes of the problem in order to build up the suitable critical theory that describes theirinteraction. The physical meaning of the relevant modes will result from their contributionto the various kinds of correlation functions that we discuss in section VIII.Let us study the most general mass matrices of small fluctuations, actually a 4-index“matroid”, M [ a, b ; c, d ] that is symmetric under the operations ( ab ) → ( ba ), ( ab ; cd ) → ( cd ; ab ), vanishes if a = b or c = d and respects the S n − replica symmetry. Such a matrixhas at most 7 distinct elements that can be parametrized in the following way: (all indexesare assumed to be different among themselves and different from 0 in the next formulae) M [0 , a ; 0 , a ] = m + µ + µ ; M [0 , a ; 0 , b ] = µ µ M [0 , a ; a, b ] = ν + ν ; M [0 , a ; b, c ] = ν M [ a, b ; a, b ] = m + m + m ; M [ a, b ; a, c ] = m m M [ a, b ; c, d ] = m The parameters m , m , m , µ , µ , ν , ν can be supposed to be distinct. Notice thatthe usual S n symmetric matrix is recovered if one poses µ = ν = m and µ = ν = m .We now look at the eigenspaces of M proceeding analogously to the classical De Almeida-Thouless analysis of fully S n invariant matrices [44]. In full generality the eigenspaces of M can be related to the representation of S n − over symmetric (two index) matrices withvanishing diagonal elements. These representations are well known and consist in replica12ymmetric matrices, matrices that break the symmetry privileging one replica and matri-ces that privilege two replicas. In the following we use the terminology usually employedin spin glass theory, calling respectively Longitudinal the Anomalous and Replicon theseeigenspaces. A. The longitudinal space
The simplest eigenvector are the longitudinal ones that have the same structure of thesaddle point Q ab , for a = b : L ab = ( u − v )( δ a + δ b ) + v (13)to which there correspond the two eigenvalues λ ± LO that are given in the appendix. One ofthese, that we call λ LO vanishes at the critical point while the other remains finite. Noticethat u and v can be identified respectively with the variations δp and δq of the previoussection. Comparing the quadratic form h L | M | L i = P ab,cd L ab M [ a, b ; c, d ] L cd in the limit n → M pp , ˆ M qq and ˆ M pq asˆ M pp = 2 m + µ , ˆ M qq = m − m , ˆ M pq = − ν (14)Strictly speaking the longitudinal eigenvalues of the matrix M do not coincide with theeigenvalues of the quadratic form in (10). This due to the fact that if we write L ab = uw ab + vw ab the vectors w and w are not normalized to 1. However, it is easy to see thatif one of the eigenvalue of the quadratic form is zero, so it is the corresponding eigenvalue of M . Simple linear algebra shows that for n → λ LO reads λ LO | n =1 ≡ m = ˆ M pp + ˆ M qq M qq − ˆ M pp ˆ m + O ( m ) . (15)The corresponding eigenvector should be such that for n → v = dq ( p ) dp u = − ˆ M pq / ˆ M qq u .For future reference we introduce the notation γ = dq ( p ) dp . In general it can be expected γ >
0, implying strong correlations between the fluctuations of p and these of q . B. The anomalous space
The second family of eigenvectors are the so-called anomalous ones a µab , where in additionto replica 0 a replica µ > a µ µ = u + u ; a µ a = u a µµa = v + v ; a µab = v . (16)If we impose orthogonality between the anomalous and longitudinal spaces we find: u = − ( n − u (17) v = −
12 ( n − v (18)This fixes two parameters out of four and also in this case there are two independent eigen-values. Notice that u and v that are responsible for the difference between a µ and L , are oforder n − u and v , this implies, on a very general basis that the anomalous andlongitudinal eigenvalues form degenerate doublets for n →
1. Their difference which shouldbe linear in u and v is of the order n −
1. The values of v and u become degenerate withthe values of the parameters v and u in the corresponding longitudinal eigevectors. This isconfirmed by the explicit computation of eigenvalues and eigenvectors as a function of themass matrix parameters with Mathematica. As we will see this eigenvalue degeneracy is atthe origin of typical random field terms in the action.The total dimension of the anomalous space is 2( n −
2) as it can be realized taking intoaccount the orthogonality with the longitudinal space.The meaning of the anomalous vectors can be understood within the replica formalismlooking at the projection of the fluctuating field: h φ | a µ i = u [ − ( n − φ µ + n − X b =1 φ b ]+ v [ − ( n − n − X a =1 φ µa + n − X a,b =1 φ ab ] . (19)These are replica symmetry breaking fluctuations where of φ µ and P n − a =1 φ µa differ fromtheir averages over the index µ . Soft modes in these directions have as physical consequencedeep relations among different correlation functions that can be defined, as we discuss insection VIII. 14 . The replicon space The last family of eigenvectors is the one of so-called replicons. These can be characterizedin two equivalent ways: either as matrices that besides replica number “0” privilege tworeplicas µ, ν > R ab such that R a = 0; X b R ab = 0 ∀ a. (20)The replicon space concerns fluctuations of the φ ab which are independent from the onesof φ b and induce replica symmetry breaking of the type familiar from spin glass theory[45]. The dimensionality of the replicon sub-space is ( n − n − /
2. Together with thelongitudinal and anomalous spaces it exhausts the n ( n − / m . The behavior of thereplicon eigenvalue marks the difference between critical points terminating the first ordertransition lines of the quenched construction and the pinned particles one for T ref < T andthe critical points marking the passage from a RFOT Kauzmann (or discontinuous 1RSB)transition to a continuous 1RSB glass transition for the pinned particles construction with T ref > T . Generically, in the former case, m does not have reasons to vanish at the criticalpoint. For example it can be checked that m indeed remains finite at the critical point ofthe spherical p-spin model in the quenched potential setting. Conversely, in the latter case m vanishes at the transition and is zero on the whole second order glass transition line.In this paper we concentrate on the case that m remains positive at criticality, and treattherefore the terminating critical points of the first order lines of the quenched constructionand the pinned particle problem with T ref < T . The case T ref = T where full S n replicasymmetry is recovered marks the boundary between the two behaviors. This case has beenanalyzed within a one loop approximation in [31]. The analysis performed there shows thatthe additional degeneracy of the small eigenvalue does not change the universality classof the problem. In the replica formalism, we think this could be related the peculiaritiesof the n → T ref > T . Further work will be needed to extend the analysis of [31] to all orders inperturbation theory. In conclusion, generically, at critical points terminating first order15ines, the singularities come from the fact the longitudinal and anomalous fluctuations gosoft at the transition, while the replicon ones remain massive.We notice that an alternative approach to analyze the eigenspaces of M (that leadsto same results) consists in treating separately the first line and column of the matrix φ a,b a = b = 0 , ..., n − n − z a = φ a in replica space with a = 1 , ..., n − φ ab matrix with a = b = 1 , ..., n − n − × n − VI. A VECTORIAL REPRESENTATION
To build up the relevant critical theory we can disregard massive directions and consideran interacting field theory for fluctuating fields φ ab ( x ) which are linear combinations of thecritical modes. We therefore concentrate on the zero mode subsector of the longitudinal andanomalous spaces and ignore the sectors corresponding to hard modes. In order to havetheory that keeps explicitly the S n − symmetry it is convenient to combine longitudinal andanomalous vectors L and a µ , that we suppose to be defined up to a normalization to befixed a-posteriori, into vectors A µ A µ = a µ + L (21)that form an orthonormal basis h A µ | A ν i = P ab A µab A νab = δ µν . Let us state a few propertiesof these vectors and fix the normalizations of a µ and L . Notice that while replica symmetryimplies that P µ a µ should be proportional to L , the orthogonality condition h a µ | L i = 0 saysto us that P µ a µ = 0. Using again the replica symmetry we can write h a µ | a ν i = δ µν ( α − α ) + α . (22)Summing over µ we obtain that α = − α n − . Imposing orthonormality we have h A µ | A ν i = α + h L | L i = 0 µ = ν h A µ | A µ i = α + h L | L i = 1 (23)which implies α = 1 − h L | L i = ( n − h L | L i and h L | L i = n − .16e can now evaluate the matrix element h A µ | M | A ν i . If we write the longitudinal eigen-value as λ LO = m + ( n − η LO and the anomalous as λ AN = m + ( n − η AN and define η = η LO − η AN , we can easily see that h A µ | M | A ν i = m δ µν + η + O ( n − . (24)Let us now expand the critical field on the basis of the A µ φ ab ( x ) = n X µ =2 ψ µ ( x ) A µab . (25)We can now formulate the critical theory in terms of the single index fields ψ µ . This theoryshould of course be invariant under all permutation of the indexes µ = 1 , ..., n −
1. Genericallythe action of the theory could be written as a sum of a local term which is a polynomialin the fields, and a kinetic term sensitive to space fluctuations of the ψ µ . We are led thento the study of the replica symmetric low order local polynomial invariants of the n − ψ µ . These can be build up explicitly, starting from the monomialsof lower orders: • The only linear invariant is I = X µ ψ µ . (26) • The quadratic invariants are: I , = I , I , ≡ J = X µ ψ µ • The cubic ones are: I , = I , I , = I I , , I , ≡ J = X µ ψ µ . The higher order invariants can be obviously generated in a recursive way. In general, theonly invariant of order k which can not be expressed as a product of lower order ones is: J k = P µ ψ kµ . In addition to purely local invariants, we should consider the lowest orderinvariant in ∇ ψ µ , namely, the kinetic term K [ ψ µ ] = X µ ( ∇ ψ µ ) . (27)17 II. DIMENSIONAL ANALYSIS
The quadratic form h φ | M | φ i in the bases of the ψ µ is readily computed: h φ | M | φ i = m X µ ψ µ + η X µ ψ µ ! = m I , + ηI . (28)We can remark at this point that (28) has the typical form that appears in the replicatreatment of the RFIM with random field δ -correlated in space. The coefficient η , that hereoriginates corresponds in that case to the variance of the random field. Its appearance herestems from the degeneracy of the longitudinal and replicon eigenvalues for n →
1. It can bechecked in specific problems that while m → η remains positive.It is well known from the theory of the RFIM that the inversion of the form (28) hassingle pole and double pole propagators. The field ψ µ cannot have a well defined scalingdimension. The same conclusion can be reached observing that this is incompatible withthe fact that the two terms in eq. (28) are of the same order of magnitude for m → η finite.In order to use fields with well defined scaling dimension we can make a further changeof basis as originally suggested by Cardy [46] and write: ψ µ = ψ + δ µ ˆ ψ + χ µ χ = 0 X µ χ µ = 0 . (29)This is a legitimate change of basis since for all x it contains n − φ can be written as φ = ( n − ψ L + ˆ ψ A + X µ χ µ A µ = [( n − ψ + ˆ ψ ] L + ˆ ψ a + X µ χ µ a µ (30)Purely longitudinal fluctuations correspond to ˆ ψ = χ µ = 0. Notice that in this case, theinvariants J k are of order n −
1, (in fact J k = ( n − ψ k ) while all composite invariants areof higher order. Since the effective potential is equal to the derivative of the free-energywith respect to n in n = 1, this implies that the only invariants that enter in the effective18otential are the J k . We would like to argue that the same invariants are also the ones thatgovern fluctuations.In the basis (29) the linear and quadratic invariants read: I = ( n − ψ + ˆ ψJ = ( n − ψ + 2 ψ ˆ ψ + X µ χ µ + ˆ ψ (31)so that the quadratic form writes: h φ | M | φ i = m " ( n − ψ + 2 ψ ˆ ψ + X µ χ µ + ˆ ψ + η (cid:16) ( n − ψ + ˆ ψ (cid:17) . (32)We now proceed with dimensional analysis, which as it is well known in general, is equivalentto the analysis of the leading singularities in perturbation theory.Imposing that for n → m ψ ˆ ψ , m P µ χ µ and η ˆ ψ share the same superficialscaling dimension, as m → ψ ] = 2 + [ ψ ][ χ µ ] = 1 + [ ψ ] . (33)where we have set [ m ] = 2. Among the invariants of order k the ones of lower scalingdimension are these which contain the lower power of ˆ ψ and χ µ for n →
1. These are theterms J k = P µ ψ kµ , which are the only ones that contain ψ k − ˆ ψ and ψ k − P µ χ µ . Thisis enough to say that the spinodal lines, where the coefficient of J is non-null, belong tothe universality class of the φ -RFIM theory, (the spinodal of the RFIM). At the criticalpoint, by definition the coefficient of J in the effective action vanishes, however in general,the coefficients of the other cubic invariants I , and I , are non-zero. In order derive theRFIM, we should argue that close to the upper critical dimension of the RFIM, D c = 6,these invariants have superficial scaling dimension higher than the one of J .The scaling dimension of J is [ ˆ ψψ ], the ones of I , and I , are respectively [ ˆ ψ ψ ] and[ ˆ ψ ]. Since the dimension of x is −
1, in order to make the action adimensional we need[ ψ ] = D/ −
2. We see that [ J ] = 2 D − I , ] = 3 / D − I , = 3 / D . We findtherefore that close to the dimension 6, if the coefficient of J vanishes, the leading singularterm becomes J and the critical point is in the RFIM class. Calling g the coefficient of J in the Landau expansion of the free-energy, we can write explicitly, close to the critical19oint and for n → F [ ψ ] = Z dx ˆ ψ ( x ) (cid:16) − k ∆ ψ ( x ) + m ψ ( x ) + gψ ( x ) + η ˆ ψ ( x ) (cid:17) + 12 Z dx X µ (cid:0) k ( ∇ χ µ ) + [ m + 3 gψ ( x ) ] χ µ (cid:1) (34)It is well known that since there are n − → − χ µ integrationover them is equivalent to a Fermionic determinant, and (34) is equivalent to the Parisi-Sourlas action of the RFIM. Analogously to the case of the dynamical transition [14], thefluctuations of the potential with respect to the reference configurations can be effectivelyparametrized by a random field term, with Gaussian statistics, uncorrelated from site tosite. VIII. CORRELATION FUNCTIONS
So far we have proceeded to a formal analysis of the replica soft modes, we found thatthere are different components of the fluctuations that have different scaling dimensions andwe derived the RFIM on the basis of a dimensional analysis. We anticipated in the previoussections that the emergence of the RFIM comes from fluctuations with respect to the choiceof the reference configuration. To substantiate this statement we study correlation functionsin our system relating them to corresponding function in the RFIM. We show that functionsthat are sensitive to thermal fluctuations relate to thermal fluctuations of the RFIM, whilefunctions that are sensitive to the choice of the reference configuration relate to functionssensitive to the choice of the random field in the RFIM.We define therefore two averages: we denote by angular brackets h·i the thermal averageconditioned by the choice of the reference configuration and can involve several replicas, andby square brackets [ · ] the average over the choice of the reference configuration. Moreover,inside the averages, index 0 is assigned to the reference configuration, while indexes 1 , , , g ( x ) = [ h φ ( x ) φ (0) i ]; g ( x ) = [ h φ ( x ) φ (0) i ] = [ h φ ( x ) ih φ (0) i ] g ( x ) = [ h φ ( x ) φ (0) i ]; g ( x ) = [ h φ ( x ) φ (0) i ] g ( x ) = [ h φ ( x ) φ (0) i ]; g ( x ) = [ h φ ( x ) φ (0) i ] g ( x ) = [ h φ ( x ) φ (0) i ] (35)To the order of the leading singularity, in which we can use the RFIM, however, startingfrom the fields ψ , ˆ ψ and χ µ we can construct at most four independent correlations, namely h ψ ( x ) ψ (0) i , h ψ ( x ) ˆ ψ (0) i , h ˆ ψ ( x ) ˆ ψ (0) i , and P µ h χ µ ( x ) χ µ (0) i , where we have taken into accountthe condition P µ χ µ = 0. This number is reduced to two by replica symmetry, that impliesthat ,n − X a,b h φ a ( x ) φ b (0) i = ( n − g − g ] + O (( n − ) ,n − X a,b,c,d h φ ab ( x ) φ cd (0) i = ( n − − g + 8 g − g ] + O (( n − ) (36)The explicit expression in terms of ψ , ˆ ψ and χ µ , that we have computed with our Mathe-matica script, shows that (36) are of order n − h ˆ ψ ( x ) ˆ ψ (0) i = 0 X µ h χ µ ( x ) χ µ (0) i = −h ψ ( x ) ˆ ψ (0) + ψ (0) ˆ ψ ( x ) i . (37)These are known identities in the supersymmetric formalism for the RFIM and related prob-lems [47–49] if we identify the χ µ with Fermion fields. It follows that the correlations (35)are linear combinations of h ψ ( x ) ˆ ψ (0) i and h ψ ( x ) ψ (0) i , which in the RFIM represent respec-tively the thermal fluctuations and the sample to sample fluctuations correlation functions.The coefficients of the combination depends on γ = dq ( p ) dp which is the only parameter of thesystem appearing in the eigenvectors.The correlations have connected and disconnected components with respect to the angu-lar brackets. Correlations which are connected measure thermal fluctuations. Disconnectedcorrelations measure instead fluctuations with respect to changes of the reference configura-tion. Omitting the position indexes, we can write the connected combinations g − g ,21 − g , g − g and g − g , and the disconnected ones g , g , g , and g . Exact expressions can be readily obtained with Mathematica and read: g ( x ) − g ( x ) = 12 − γ h ψ ( x ) ˆ ψ (0) + ψ (0) ˆ ψ ( x ) i g ( x ) − g ( x ) = γ g ( x ) − g ( x )) g ( x ) − g ( x ) = γ g ( x ) − g ( x )) g ( x ) − g ( x ) = γ g ( x ) − g ( x )) (38)The non connected components are g ( x ) = 12 − γ h ψ ( x ) ψ (0) i − − γ − γ ) h ψ ( x ) ˆ ψ (0) + ψ (0) ˆ ψ ( x ) i g ( x ) = γ g ( x ) g ( x ) = γ − γ h ψ ( x ) ψ (0) i − γ (6 − γ )4(2 − γ ) h ψ ( x ) ˆ ψ (0) + ψ (0) ˆ ψ ( x ) i g ( x ) = γ g ( x ) (39)As announced, we find that the connected correlations only contain h ψ ( x ) ˆ ψ (0) + ψ (0) ˆ ψ ( x ) i that in the RFIM represents the thermal correlation function for fixed random field whilethe disconnected correlations also contain h ψ ( x ) ψ (0) i which is the correlation sensitive torandom field changes in the RFIM.The relations (38,39) between the different overlap correlation functions are an importantconsequence of our analysis, they are valid at all orders of perturbation theory and can betested in numerical simulations. The computation of all correlations (35) requires to simulatea maximum of four independent replicas besides the reference one. The parameter γ can alsoin principle be measured in simulations by looking on the dependence on p of the overlap q between two distinct replicas that have overlap p with the reference.The relations (38,39) express the fact that to the leading order the overlap of the sys-tem with the reference configuration and the self-overlaps are strongly correlated and thefluctuations verify φ ( x ) = δq ( x ) ∼ γφ ( x ) = δq ( x ). The second of (38) for examplecan be derived observing that both the LHS and the RHS combinations can be expressedas derivative of local overlap averages with respect to space dependent coupling ǫ ( x ), g ( x ) − g ( x ) = 1 T δ [ h q ( x ) i ] δǫ (0) ; g ( x ) − g ( x ) = 12 T δ [ h q ( x ) i ] δǫ (0) (40)22nd using the chain rule for the derivative δ [ h q ( x ) i ] δǫ (0) = dq ( p ) dp δ [ h q ( x ) i ] δǫ (0) . (41)It can be noted however that when integrated over space the second of (38) holds exactlyeven beyond the level of leading singularity described by the RFIM and can be used tomeasure γ .Finally, we remark that as it is well known in the theory of the RFIM, to the one-looporder of Gaussian fluctuations, the non-connected components are more singular than theconnected ones. In momentum space, connected correlations behave as ( k + m ) − , whiledisconnected ones behave as ( k + m ) − . IX. SUMMARY AND CONCLUSIONS
In this paper we have analyzed the universality class of critical points terminating firstorder transition lines in glassy systems in presence of constraints. After the analysis ofthe case of a symmetric coupling between two replicas, for which the Ising universalityclass is found, we have considered the case in which a coupling with a quenched referenceconfiguration is present. We extend in this way the analysis of the equal temperature pinnedparticle critical point presented in [31]. This includes the quenched potential constructionand the pinned particle construction for T ref < T . A full analysis of the soft modes withinReplica Field Theory at all order of perturbation theory leads to the universality class ofthe Random Field Ising Model. The effective random field appearing in the final descriptionparametrizes the randomness in the reference configuration. We have analyzed the variousfour-body correlation functions that appear in the theory and found that there are only twoindependent combinations that become dominant close to the critical point. The existenceof the critical point, its universality class and the relation between correlation functionsconstitute important predictions of the Thermodynamic theory of glasses based on MeanField Theory, and now, on its loop expansion. We hope that in a next future they can betested in numerical simulations of realistic glass forming liquid models. The critical pointand line of continuous glass transition present in the particle pinning problem for T ref > T are excluded by the present analysis. In that case the replicon modes are critical and theirinteraction with the longitudinal and anomalous modes that we have seen to give rise to theRFIM should be included. This is a fascinating research project that we leave for the future.23After completion of our work we came to know that G. Biroli, C. Cammarota, G. Tarjusand M. Tarzia have also considered the problem of quenched critical points with a similarapproach (arXiv:1309.3194). Acknowledgments
We thank L. Berthier, F. Ricci-Tersenghi, T. Rizzo, S. Sastry and P. Urbani, for usefuldiscussions and exchanges. We also thank our JSTAT anonymous referee for insightfulremarks and constructive comments. SF thanks the Dipartimento di Fisica Universit`a diRoma “La Sapienza” for hospitality. The European Research Council has provided financialsupport through ERC Grant 247328.
X. APPENDIX
The algebra of multiplication of the four replica index mass matrix M [ a, b ; c, d ] by a tworeplica index vectors v ab , which is necessary to perform the explicit computation of theeigenvalues, can be implemented in Mathematica. In fact, given the structure of M and itseigenvalues, one just needs to be able to perform sums over replica indexes of constants anddelta functions. A detailed script performing this task was published in [31]. The one we usehere is an adaptation of that one. The eigenvalues of the mass matrix can be then explicitlyfound. At the end, this amounts to solving second order algebraic equations. We presentthe result of the computation to the order ( n − λ ± LO = 14 (cid:18) m − m + µ ± q ( µ + 2 m ) − ν (cid:19) + (42)+ 14 ( n − m − m + µ + 4 µ ± ν ( ν − ν ) − ( µ + 2 m ) ( − µ − µ + 2 m − m ) p ( µ + 2 m ) − ν ! The zero mode corresponds to the positive determination of the square root λ LO = λ + LO .This can be argued be the condition that in the S n symmetric limit µ = ν = m and µ = ν = m the leading order m become degenerate with the replicon eigenvalue λ RE = m .In this way we find: m = 14 (cid:18) m − m + µ + q ( µ + 2 m ) − ν (cid:19) (43) η LO = 14 m − m + µ + 4 µ + 4 ν ( ν − ν ) − ( µ + 2 m ) ( − µ − µ + 2 m − m ) p ( µ + 2 m ) − ν ! . (44)24e notice that for n → ν = ± p | m µ | . This condition is met at the critical point of junction of the discontinuousand continuous glass transition of the pinned particle problem for T ref > T , and actually onthe whole line of continuous glass transition.Analogously one can compute the anomalous eigenvalues that turn out to be: λ ± AN = 14 (cid:18) m − m + µ ± q ( µ + 2 m ) − ν (cid:19) + (45) ± ( n − ν (cid:16) µ (cid:16) µ ± p ( µ + 2 m ) − ν + 2 m (cid:17) − ν (cid:17)p ( µ + 2 m ) − ν (cid:16) µ ± p ( µ + 2 m ) − ν + 2 m (cid:17) with λ AN = λ + AN and η AN = ν (cid:16) µ (cid:16) µ + p ( µ + 2 m ) − ν + 2 m (cid:17) − ν (cid:17)p ( µ + 2 m ) − ν (cid:16) µ + p ( µ + 2 m ) − ν + 2 m (cid:17) . (46)The expression of η = η LO − η AN is not particularly illuminating. The consistency of theapproach requires that it should remain positive at the critical point. [1] A. Cavagna, Physics Reports , 51 (2009)[2] P. G. Wolynes and V. Lubchenko, Structural Glasses and Supercooled Liquids: Theory, Ex-periment, and Applications (Wiley. com, 2012)[3] T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes, Phys. Rev. A , 1045 (Jul 1989)[4] S. Franz, G. Parisi, and M. Virasoro, Journal de Physique I , 1869 (1992)[5] S. Franz and G. Parisi, Journal de Physique I , 1401 (1995)[6] K. Kim, EPL (Europhysics Letters) , 790 (2003)[7] L. Berthier and W. Kob, Phys. Rev. E , 011102 (Jan 2012), http://link.aps.org/doi/10.1103/PhysRevE.85.011102 [8] W. Kob and L. Berthier, Physical Review Letters , 245702 (2013)[9] C. Cammarota and G. Biroli, Proceedings of the National Academy of Sciences , 8850(2012)[10] S. Karmakar and G. Parisi, Proceedings of the National Academy of Sciences , 2752 (2013)[11] W. G¨otze, Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory: A Mode-Coupling Theory , Vol. 143 (Oxford University Press, 2008)
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