Long time limit of equilibrium glassy dynamics and replica calculation

Long time limit of equilibrium glassy dynamics and replica calculation

A. Crisanti andrea.crisanti@roma1.infn.it Dipartimento di Fisica, Università di Roma “La Sapienza” and SMC,
P.le Aldo Moro 2, I-00185 Roma, Italy
2007/11/27
Abstract

It is shown that the limit of the equilibrium dynamic self-energy can be computed from the limit of the static self-energy of a -times replicated system with one step replica symmetry breaking structure. It is also shown that the Dyson equation of the replicated system leads in the limit to the bifurcation equation for the glass ergodicity breaking parameter computed from dynamics. The equivalence of the replica formalism to the long time limit of the equilibrium relaxation dynamics is proved to all orders in perturbation for a scalar theory.

pacs:
05.50.Gg, 03.50.-z, 65.60.+a

I Introduction

Spin glass and structural glasses are characterized by the presence of a complex structure of stable and metastable states metastable (). The basic idea KirThi87 (); CriSom92 (); CriHorSom93 (); BouMez94 (); MarParRit94 (); CriSom95 (); BouCugKurMez96 (), supported by the behavior of some mean-field spin glass models, is that the glassy behavior arises because of the emergence of an exponentially large number of metastable states that breaks the ergodicity and prevents the system from reaching the true thermodynamic equilibrium state. The logarithm of the number of metastable state is commonly called “complexity” or “configurational” entropy. The glass transition is then associated with the emergence of a non-zero configurational entropy. By extending the replica method, originally developed for disordered systems, the occurrence of a non-zero configurational entropy can be conveniently studied within the replica formalism Monasson95 ().

The basic idea of the replica approach Monasson95 (); MezPar99 (); ColMezParVer99 (); MezPar00 (); ParZam05 (); ParZam06 () is to consider the equilibrium a thermodynamics of copies of the original system interacting among them via an infinitesimal attractive coupling. If the free-energy landscape breaks down into (exponentially) many metastable states, the copies will then condensate into the same metastable state since their relative distances in such states are smaller than in the liquid (paramagnetic) phase. The replica method allow for an equilibrium analysis of the properties of the metastable states, in spite of the fact that these are originally defined in a dynamic framework. In particular the properties of the original system is recovered continuing the replica number to and taking in the limit .

If one applies the present method to mean-field -spin spin glass models, whose low temperature phase is described by a one replica symmetry breaking step (1RSB), the result agrees with that from a dynamical study. We observe that in this case the number of replicas corresponds to the break point of the 1RSB solution of the conventional replica approach, and hence the limit clearly corresponds to the onset of the glass phase.

All checks of the equivalence between the replica approach and the dynamical approach we are aware of GroKraTarVio02 (); WesSchWol03 (); MiyRei05 (); AndBirLef06 (), use some approximations to compute the relevant correlation functions of the many-body problem. These leave open the possibility that the equivalence is just a consequence of the limited number of dynamical diagrams considered in the various approximations.

To our knowledge the equivalence of two approaches for a general system is not yet proved. The proof of this statement is the main result reported in this paper. To be more specific, in this paper we show under rather general assumptions, that the long time limit of the equilibrium two-times correlation function of a glassy system can be computed from the limit of the static theory of an -times replicated system with a 1RSB structure. The proof is based, following similar works Naketal83 (); Gozzi83 () on the connection between static and equilibrium dynamics, on a suitable summation of classes of dynamical diagrams of the dynamic perturbation theory generated by the Langevin dynamics, and showing that in the limit these approaches those of the limit of the equilibrium theory of an -times replicated system with a 1RSB structure.

The reader will not find in this paper the physical consequences that can be obtained for specific systems or models by using the results reported here. For these the interested reader is referred to, e.g., Ref. [GroKraTarVio02, ; WesSchWol03, ; MiyRei05, ; AndBirLef06, ] or to the forthcoming paper BirCriprep () where these will be applied to a simple toy system.

To keep the notation as simple as possible we shall consider only scalar fields. The generalization to more complex fields, e.g., vector field, is straightforward. Our starting point is hence the Langevin equation of the form

(1)

which describes the purely relaxation dynamics towards equilibrium of the scalar stochastic field in presence of the stochastic force . In general both and can be functions of both time and space coordinates however, since we are interested into the time behavior of correlations, space coordinates can be safely neglected. As a consequence we shall drop any explicit space dependence of fields. The Hamiltonian governs the behavior of the system. Here, when needed to illustrate the calculations, we shall use

(2)

which describes a zero-dimensional scalar theory.

The stochastic force has a Gaussian distribution of zero mean and second moment fixed by the Einstein’s relation

(3)

where is the temperature. Indeed, as can be verified by means of the associated Fokker-Planck equation risken (), eq. (3) guarantees that the time dependent probability distribution function eventually converge for to the equilibrium distribution

(4)

where .

Given an initial condition the expectation value of a generic observable over the stochastic process generated by the Langevin equation (1) can be written as the path integral MarSigRos73 (); DeDom75 (); DeDom76 (); Janssen76 (); BauJanWag76 (); DeDomPel78 ()

(5)

where

(6)

The dynamical functional takes the form of a statistical field theory with two independent sets of fields, the original field and the response field , so that all well established machinery of statistical and quantum field theory can be applied. The field is called “response field” since it is conjugated to an external field . As a consequence arbitrary response functions can be generated by taking correlators involving -fields. In particular the dynamic susceptibility or response function reads

(7)

From now on we absorb the factor into the definition of the response function.

We note that in deriving the dynamical functional we have assumed the Ito prescription for the stochastic calculus. This implies that . Accordingly, in a perturbative expansion of averages (5) all diagrams with at least one loop of the response function can be neglected.

In equilibrium the response function is related to the two point correlation function by the fluctuation-dissipation theorem (FDT):

(8)

We note that time translation invariance of equilibrium implies that all two-points correlators are function of the time difference only. This functional dependence is always assumed, even if not explicitly shown, every times we consider equilibrium correlators.

In equilibrium the calculation of the two point correlation function can be reduced to that of the dynamic self-energy . Indeed by using standard methods of statistical field theory zinn () and the FDT relation (8) one obtains for the following equation, valid for :

(9)

where

(10)

is the quadratic part of the Hamiltonian . For simplicity we assume that in equilibrium .

In glassy systems the correlation function does not vanish in long-time limit :

(11)

signaling the breaking of the ergodicity gotze (). The value of the “ergodicity breaking parameter” is obtained by taking limit of eq. (9):

(12)

where is the limit of , while and are the equal-time value of the correlation and self-energy. The latter can be eliminated using the relation

(13)

that follows form the limit of equation (9) and . By combining together eq. (12) and eq. (13) we finally obtain the bifurcation equation

(14)

In this approach the glass transition is signaled by the appearance of a non-trivial solution of the bifurcation equation.

The calculation of the dynamic self-energy is in general a non trivial task and approximations are usually required to deal with the diagrams of the dynamical perturbation theory. In this paper we shall show that in the limit of the calculation of the dynamic diagrams simplifies and the bifurcation equation (14) can be obtained from a purely static calculation of a -times replicated system with a 1RSB structure.

The paper is organized as follows. In Section II we shall study the structure of the equilibrium diagrammatic expansion of the dynamic self-energy . We shall show that for can be decomposed into the sum of classes of diagrams obtained by a suitable rearrangement and partial summation of dynamical diagrams. In Section III we shall consider the limit of the self-energy . Here, using both a sum rule approach and a diagrammatic approach, we shall prove that in the limit all time integrals of the diagrammatic expansion can be evaluated and that the self-energy is a function of and only. In Section IV we develop the replica calculation. We shall show that evaluated in Section III is equal to the static self-energy of an -times replicated system with a 1RSB structure when the limit is taken. This result allows us to derive the bifurcation equation (14) on a purely static calculation.

Ii Equilibrium Dynamic Self-Energy: finite

The self-energy gives the correction to the free theory correlators when interactions are taken into account. Its calculation is in general rather difficult and one is obliged to use some approximation schemes. Following standard field theory procedures zinn (); lebellac () a systematic perturbative calculation of the self-energy can be established in terms of the so called proper vertex functions. Diagrammatically these quantities are represented by the sets of one-particle irreducible (1PI) diagrams, i.e., by diagrams that do not split into two subdiagrams by cutting any single propagator line, with all the external incoming and outgoing legs amputated.

ii.1 Equilibrium Dynamic Diagrams

The self-energy is related to the proper vertex with the two external outgoing legs removed. The equilibrium diagrammatic expansion of the dynamic self-energy takes then the following form

(15)

where each line connecting the two vertices represents the full correlation function DeDom63 (); CorJacTom74 (); Haymaker91 ()

(16)

The full (left) vertex with external legs is the sum of all connected 1PI dynamic diagrams with the outgoing leg at , the wiggly line in (15), and incoming legs at times () removed. To each internal -line connecting the internal vertices at times and is associated the full correlation function while to each internal -line connecting the internal vertices at times and is associated the full response function DeDom63 (); CorJacTom74 (); Haymaker91 ()

(17)

with . All internal times are integrated from to .

The empty (right) vertex with external legs is built from the same 1PI diagrams of the full vertex but with all external times times equal to . As a consequence, since in equilibrium the correlation and response functions are related by FDT, it is given by the topological equivalent vertex obtained from the associated static equilibrium theory described by the canonical distribution (4) Naketal83 ().

The structure just described can be understood as follows. First of all we note that since closed loops of response function vanish, each internal vertex of the 1PI diagrams contributing to the self-energy is connected via -lines to one or the other of the external wiggly lines, but not to both. This means that the diagram can be divided into two subdiagrams made of all the vertices connected to the same external wiggly line. The two subdiagrams are clearly 1PI and joined together by or more -lines. It is easy to recognize the two subdiagrams as the full/empty vertices.

Alternatively one can invoke the general diagrammatic expansion of the correlation function as the matching of two tree expansions ma (), one for and one for , and note that the lines connecting the full/empty vertices are the lines joining the two tree expansions. We note that the assumption that the system is at equilibrium implies that all two-times quantities depend only on time difference. Then the contribution from the tree expansion of averaged over noise and equilibrium initial conditions, the empty vertices, cannot depend on time and must be equal to that obtained from the equilibrium static theory described by the canonical distribution (4) with all lines equal to the static equilibrium correlation function .

The internal structure of the empty vertices can be inferred by using the following dynamical functional, see Appendix C,

(18)

to impose statistical equilibrium with the canonical distribution (4) at the initial time . The analysis of the dynamic diagrams generated by shows that the effect of the last term is that of canceling out from all dynamic diagrams for the contribution from times yielding for the equilibrium dynamic diagrammatic expansion discussed above. This can be understood on a general ground as follows. The canonical distribution (4) is a stationary solution of the associated Fokker-Planck equation, and hence the probability distribution of remains canonical for any time past . Consequence of this is that all quantities evaluated from cannot depend on . This guarantees, for example, that all two-times quantities depend only on time differences. In evaluating for we can then choose for in (18) any value . The invariance property ensures that we always get the same diagrams. Clearly the simplest choice is which, in turn, implies that in the diagrammatic dynamical perturbative expansion all free times must be integrated from . We have seen that the dynamical diagrams for generated by can be divided into two subdiagrams, joined by correlation lines, by grouping together all vertices connected by response -lines to the same external (amputated) wiggly line at or , respectively. The dynamical diagrams generated by can be divided in a similar way into two subdiagrams connected by correlation lines just grouping together all vertices connected by response -lines to the external (amputated) wiggly line at . It is easy to realize that this procedure leads to the same full vertex obtained from , while the (putative) empty vertex, the one connected to external (amputated) leg at , contains now only contributions from the last term in (18) since the equal time response function vanishes. Stated in a different way, the empty vertex contains only the diagrams generated by the equilibrium initial condition, i.e., it is equal to topological equivalent 1PI diagram with external (amputated) legs of the associated static theory described by the canonical distribution (4) with all -lines equal to the static equilibrium correlation function . The same conclusion can be obtained by first dividing the dynamical diagrams generated by as described above, and then taking the limit directly on diagrams.

The definition of empty vertex given above uses and follows from the observation that the equilibrium FDT relation between response and correlation function guarantees that setting all external times of a dynamic diagram equal to each other reduces the dynamic diagram to the topological equivalent diagram of the associated static equilibrium theory Naketal83 ().

We can then summarize the rules for writing down the equilibrium dynamic diagrams for :

  1. Write down the dynamic diagrams generated by the dynamical functional using the standard dynamical rules, neglecting all numerical symmetry factors.

  2. In each diagram remove the minimal number of -lines needed to divide the diagram into two disjoint sub-diagrams so that in the first (left) sub-diagram all vertices are connected to time through -lines, while in the second (right) sub-diagram are connected through -lines to time .

  3. In the second (right) sub-diagram, the one connected to , replace all response -lines by correlation -lines, and set all time variables to .

  4. If after replacement two or more different dynamic diagrams lead to the same diagram count the latter only once.

  5. Multiply each diagram so obtained by the appropriate numerical symmetry factor and evaluate it with the usual rules integrating all left internal times from to .

To illustrate the above rules consider the third order dynamic diagrams shown in Fig. 1 generated by the dynamical functional (18) for the scalar zero-dimensional theory (2).

Figure 1: Third order dynamic diagrams with equilibrium condition at .

By using the standard dynamic rules the contribution of these diagrams is

(19)

With the help of the FDT relation (8) the contribution can be rewritten as

(20)

The relevant third order diagrams for the equilibrium dynamic rules, point 1), are the first two diagram in Fig. 1. The decomposition of points 2)-4) leads to the equilibrium diagrams shown in Fig. 2. The two diagrams are easily evaluated and one recovers the result (20).

Figure 2: Third order equilibrium dynamic diagrams.

ii.2 Self-Energy Base Diagrams

By reversing the rules to write down the equilibrium dynamic diagrams it follows that each equilibrium dynamic diagram can be obtained by a suitable decoration of a base diagram, i.e., of the diagram with the same topology of the equilibrium dynamic diagram. For example the base diagram for the dynamic diagrams of the previous example is the one shown in Fig. 3.

Figure 3: Third order base diagram.

The equilibrium dynamic diagram is generated from the base diagram by:

  1. dividing the base diagram into two sub-diagrams by cutting some lines to reproduce the topology of point 2) of the equilibrium dynamic rules.

  2. In the (left) sub-diagram associated to the full vertex attach a -line to each vertex to generate the desired -line connection structure.

It is clear that the base diagram is by construction the dynamic diagram in which all -lines are replaced by -lines. As a consequence the self-energy base diagrams are the static self-energy diagrams of the associated equilibrium static theory described by the canonical distribution (4).

ii.3 Partial summation of full vertex diagrams

From the structure (15) of the diagrammatic expansion of the self-energy it is easy to realize that also each dynamic diagram contributing to the full vertex with external legs can be obtained from a suitable (static) base diagram by attaching to each internal vertices one -line to generate the desired -line structure. Figure 4 shows a base diagram for the full vertex and the three different dynamic diagrams that can be generated.

Figure 4: A base diagram for the full vertex and the three different dynamic diagrams that can be generated. The numerical symmetry factors are not displayed.

A convenient way of imposing that to each vertex is attached one and only one -line is by means of anti-commuting Grassman variables. Following Ref. [Naketal83, ] we then introduce for each vertex the pair of conjugated Grassman variables , where is the time variable label of the vertex. By using the FDT relation (8) and the properties of Grassman variables it is easy to see that

(21)

The last equality follows from even parity of the correlation function . Equation (21) has the following simple diagrammatic representation:

(22)

Consider now a 1PI base diagram for the full vertex made of internal bare vertices, i.e., vertices without external legs, and external bare vertices, i.e., vertices with at least one external leg:

(23)

Each bare external vertex has external legs, with . Then from (22) and the properties

(24)

of Grassman variables it follows that the sum of all dynamic diagrams of the full vertex that can be generated from the base diagram can be obtained as:

  1. Order the bare vertices of so that: the label corresponds to the external vertex attached to the out-going -leg at time , the labels to the remaining external vertices and labels to the internal vertices.

  2. Assign to vertex the time variable , while to each vertex labeled by assign the time variable and the pair of conjugate Grassman variables . From the time-oriented nature of the dynamic diagram it follows that for .

  3. Assign to each line connecting the vertices and , with , the function

    (25)

    and to each line connected to the vertex the function

    (26)

    with .

  4. Integrate over all Grassman variables,

    (27)

    to ensure that all vertices have one, and only one, -line attached to them.

  5. Integrate all internal time variables from to ,

    (28)

When these steps are translated into formulae we end up with:

(29)

where is symmetry factor of the base diagram and

give the connection topology of . The last equality in eq. (29) follows from the observation that for all .

It is easy to verify that the integration over Grassman variables produces all possible dynamic diagrams, with the correct weighting factor, that can be generated from the base diagram .

Iii Equilibrium Dynamic Self-Energy: The limit

In the limit some simplifications occur in the calculation of the dynamic self-energy diagrams. Each bare vertex making up the full vertex is connected to the bare vertex attached to the external -leg at time through -lines then, since

(30)

it follows that the time variables of the remaining external legs are for . As a consequence in the diagrammatic expansion of the self-energy we can replace for all correlation lines connecting the full and empty vertex pair with and the generic diagram factorizes as shown in Fig. 5.

Figure 5: Factorization of generic dynamic self-energy diagram for .

The overbar on the full vertex means that all external time variables are integrated from to .

The integration in the full vertex can be done considering first the contribution of all dynamic diagrams that can be generated from a full vertex base diagram and then summing up the contributions from all possible base diagrams. Neglecting the full-empty vertex connection symmetry factor, the contribution of all dynamic diagram generated by the full vertex base diagram with external and internal vertices, can be written as:

(31)

where is the number of external legs of the -th external bare vertex of , and is the contribution of the empty vertex.111For the empty vertex a decomposition into base diagrams similar to that of the full vertex can be done.

In the limit the first term of eq. (31) is not zero only if all and we can replace all in the second term with . Equation (31) then factorizes, cfr. Fig. 5, as

(32)

where

(33)

Finally by adding the appropriate full-empty vertex connection symmetry factor and summing eq. (32) over all possible base diagrams one recovers the complete dynamic diagram contribution in the limit from the vertex to .

iii.1 full vertex base diagram

From the definition of (33) and the expression (29) it follows that

(34)

where is the total number of bare vertices making up , excluding the final one connected to the external -leg at time . This expression can be simplified by integrating over in a fixed order because from FDT it follows that:

(35)

Thus ordering so that:

(36)

the integrand of eq. (34) can be rewritten as

(37)

Inserting this expression into (34) and using the identity:

(38)

where are the permutations of , we end up with

(39)

The integration over Grassman variables is now diagonal and can be performed. A straightforward algebra leads to

(40)

As simple example of eq. (40) consider the base diagram shown in Fig. 4. The bare vertices of are numbered as shown in Fig. 6.

Figure 6: Base diagram vertex numbering.

There are two possible orderings, namely: and . We use the short-hand notation for , for and so on. Then from eq. (40) the contribution of this base diagram is

(41)

The factor is the symmetry factor of the base diagram .

In the direct calculation we have to evaluate the three dynamic diagram generated by shown in Fig. 4 using the dynamic rules and FDT, and then add the results. The first diagram from left, with the vertex numbered as in Fig. 6, leads to:

(42)

while the second to:

(43)

and finally the third to:

(44)

The factor in the third diagram follows from the symmetry of the diagram. By adding the three contributions, and using the identity

(45)

one easily recovers the result (41).

We note that all integrations in eq. (41) can be done. Indeed the two integrals in eq. (41) can be combined together to give

(46)

and performing the integral over and then over one ends up with

(47)

where and .

The possibility of carrying out all integrals over is not a properties of this special example, but it is a general result valid for any base diagram , as shown in the next subsections.

iii.2 Recursive integration: sum rules approach

Equation (40) can be written as