Static replica approach to critical correlations in glassy systems
We discuss the slow relaxation phenomenon in glassy systems by means of replicas by constructing a static field theory approach to the problem. At the mean field level we study how criticality in the four point correlation functions arises because of the presence of soft modes and we derive an effective replica field theory for these critical fluctuations. By using this at the Gaussian level we obtain many physical quantities: the correlation length, the exponent parameter that controls the Mode-Coupling dynamical exponents for the two-point correlation functions, and the prefactor of the critical part of the four point correlation functions. Moreover we perform a one-loop computation in order to identify the region in which the mean field Gaussian approximation is valid. The result is a Ginzburg criterion for the glass transition. We define and compute in this way a proper Ginzburg number. Finally, we present numerical values of all these quantities obtained from the Hypernetted Chain approximation for the replicated liquid theory.
- I Introduction
- II An illustration of the main results of this paper in the simpler case of a standard ferromagnetic transition
- III Dynamical heterogeneities and replicas: definitions
- IV Landau expansion of the free energy around the glassy solution
- V Gradient expansion
- VI Ginzburg criterion
- VII Explicit calculations in the Hypernetted Chain (HNC) approximation
- VIII Conclusions
- A Double Counting Problem on the theory
Much progress in the recent understanding of glassy relaxation of supercooled liquids has come from the study of dynamical heterogeneities For a review, see the contributions (2011). Space-time fluctuations of the density field result in a distribution of regions of different mobility with typical size that grows upon decreasing the temperature, and that persist for time scales of the order of the relaxation time. Remarkably, the present theories of glassy relaxation are able to predict the qualitative features of this effect. Many progresses have been achieved neglecting activated processes. Both theories based on dynamics (Mode Coupling Theory Götze (2009)) and on constrained equilibrium (e.g. the molecular liquid theory Mezard and Parisi (2012)) agree in a critical instability of the liquid phase at a finite transition temperature. As far as universal aspects are concerned, a first theoretical insight originally came from spin glass theory, that suggested to look at the critical behavior of four point density correlation functions Kirkpatrick and Thirumalai (1988); Franz and Parisi (2000). In the context of p-spin models where both the equilibrium and the dynamic approach can be pursued exactly, it was possible to predict the qualitative features of dynamical correlations growth and associated dynamic criticality observed in numerical simulations Kob et al. (1997) to well defined features of static correlations in constrained equilibrium. Beyond these schematic models a direct connection between equilibrium and dynamics is more difficult to make. However the static hallmarks of dynamical criticality are generically present whenever one has a one step replica symmetry breaking transition (or mean-field random first order transition Kirkpatrick and Thirumalai (1987a); Wolynes and Lubchenko (2012)), as it is found e.g. for liquids in the Hypernetted Chain (HNC) approximation Mézard and Parisi (1996). At the level of dynamical liquid theory, the growth of correlations has been found with generality within Mode Coupling Theory Biroli et al. (2006) through a one-loop diagrammatic expansion. Despite these important results, in order to progress further it is necessary to build a field theoretical description of dynamical fluctuations capable in principle to go systematically beyond the zero-loop Gaussian level. Unfortunately, the dynamic approach is rather problematic in this respect for several reasons. The MCT is not a self-consistent theory as it requires the input of the static structure factor. Moreover, the equation for the dynamic correlator does not follow from a variational principle. Clearly this is not the ideal starting point to build up a perturbative computation. In addition an expansion around mean field in dynamics is prohibitively difficult even in the simplest cases.
A static formulation would be therefore very useful to make additional progresses. The connection between statics and dynamics is based on the idea that the emergence of slow dynamics is due to the appearance of long lived metastable states Kirkpatrick and Wolynes (1987); Kirkpatrick and Thirumalai (1987b). Within Mode-Coupling Theory and similar dynamical theories of glass formation, one can distinguish two separate dynamical regimes: the so-called regime, that corresponds to the long time dynamics inside a metastable state, and the regime, that corresponds to transitions between different metastable states. A general theory of fluctuations in the regime based on the replica method was proposed in Franz et al. (2011) on the basis of general symmetry considerations. It was found that the dynamical transition is in the universality class of a cubic random field Ising model. Moreover in Caltagirone et al. (2012) it has been shown at the level of schematic models that the same theory can be used to evaluate the Mode-Coupling dynamical exponents and that they are related to the amplitude ratio between correlation functions that are in principle measurable. At the Gaussian level, we can consider this phenomenological theory as the static Landau theory of the glass transition (see Andreanov et al. (2009) for a discussion of a dynamical Landau theory of the glass transition). In summary, the advantage of the approach discussed in Franz et al. (2011) is that one can compute the long time behavior of the dynamical correlations in the regime starting from a completely static replicated (and constrained) Boltzmann measure Kirkpatrick and Thirumalai (1989); Monasson (1995); Franz and Parisi (1995). The first consequence of this is that all the calculations simplify drastically. Moreover one can look at the dynamical transition through a static measure and one can see systematically where the mean field prediction is valid and where it fails.
The aim of this paper is to construct a framework to study the glass transition in the regime by following as closely as possible the standard treatment of critical phenomena using field theoretical tools. Furthermore, we want to obtain the replica field theory of critical fluctuations starting directly from the microscopic grand-canonical expression for the replicated partition function for the liquid, in such a way that the couplings appearing in the field theory can be computed starting from the microscopic potential. This program can be achieved through an analysis of the soft modes that appear at the dynamical transition and that are responsible for the criticality. By computing these soft modes we can focus on fluctuations that are along them: in this way we construct a gradient expansion for the field theory of the critical fluctuations and we compute in full details the Gaussian correlation functions, thus obtaining the critical part of the long time limit of the dynamical four point functions in the regime. Then we look at the corrections to the Gaussian theory and we introduce a Ginzburg criterion in order to see where the Gaussian theory is valid. The Ginzburg criterion can be used in two ways. On the one hand it gives the upper critical dimension for the model, on the other hand it provides a measure of how much one has to be close to the transition line in order to see the non Gaussian fluctuations that cannot be treated by the mean field approach.
A short account of our findings appeared in Franz et al. (2012). This paper is organized as follows. In Section II we review all the line of reasoning we just discussed for the standard Ising model. Then, in Section III we discuss how to obtain a replica description for the dynamical correlation functions in the regime. In this way we rephrase the problem from a dynamical one to a standard static computation. In Section IV we discuss the expansion of the free energy around the glassy solution. By studying the Hessian matrix we identify the soft modes that appear at the dynamical transition and we compute the expression for the exponent parameter that can be related to the dynamical Mode-Coupling exponents and that describe the critical slowing down of the two point correlation functions at the dynamical transition. Then, in Section V we show how we can perform a gradient expansion of the replica field theory in order to study the long distance physics and in Section VI we use this effective theory to compute where the Gaussian level computation fails by introducing a Ginzburg criterion for the dynamical transition. Up to that point we will remain completely general. The only assumption that we will make is that the replicated system has a glassy phenomenology, namely that the replica structure of the two point density function is non trivial below a certain dynamical transition point. In Section VII we report concrete calculations in the framework of the replicated HNC approximation by giving explicit expressions for all the couplings and masses of the effective replica field theory starting from the microscopic potential and we show the numerical results for several physical systems.
Ii An illustration of the main results of this paper in the simpler case of a standard ferromagnetic transition
The aim of this paper is to build a theory of the glass transition by following closely the first steps of the standard field theory formulation of critical phenomena. Namely, we want to start from a Landau theory (including microscopic parameters) and deduce from it a set of mean field critical exponents. This is done first by studying the behavior of the uniform order parameter in the mean field theory, and then considering a gradient expansion for slowly varying order parameter to compute the correlation length. Finally, taking into account the interaction terms lead to a perturbative loop expansion that allow to establish the region of validity of mean field theory (hence the upper critical dimension) through a Ginzburg criterion. The aim of this paper is to repeat all of these steps in the more complex case of a system undergoing a glass transition. The final step would be of course to set up an epsilon expansion around the upper critical dimension using renormalization group methods. We will not make any attempt in this direction in this paper. For pedagogical reasons, we find useful to briefly describe how these steps are carried out in a simple ferromagnetic system before turning to the glass case. The reader should keep in mind that this section is just a short reminder of the main steps, for a reader already accustomed with the modern theory of critical phenomena.
ii.1 Landau theory
Suppose we consider a microscopic system undergoing a ferromagnetic transition. In the following we will consider the ferromagnetic Ising model with nearest neighbor interactions on a -dimensional cubic lattice and a properly scaled coupling constant:
Starting from the microscopic Hamiltonian, we can construct the free energy as a functional of the order parameter, the magnetization , as follows. The free energy (here we will ignore some factors of temperature by rescaling some variables) in presence of an external magnetic field is
Taking a Legendre transform Cornwall et al. (1974), we define
or in other words
where is the solution of . The function is the free energy of the system as a function of the magnetization field.
In order to detect the phase transition, we want to investigate the small behavior of . Let us make a crucial assumption, that is an analytic function of around . This assumption is plain wrong in finite dimensional systems at the critical point and below. However, let us for the moment forget about this problem and proceed with our discussion. We can consider a uniform magnetization profile and expand at small . From the symmetries of the problem, we know that
which is the celebrated Landau free energy (here is the volume of the system). A very practical way to compute the coefficients and is to perform a systematic high temperature expansion Georges and Yedidia (1991). For example, at the leading order, for the -dimensional Ising model with coupling constant we obtain . Adding more terms leads to an expansion of in powers of . For the Ising model (1), the true expansion parameter is actually , i.e. the temperature in units of the coupling constant. Because the latter has to be chosen equal to to obtain a good limit , the expansion parameter is . In other words, the high temperature expansion is also a large dimension expansion around the mean field limit.
The equilibrium value of , called , is obtained by minimizing . From the high temperature expansion we obtain that vanishes linearly at a given temperature , in such a way that . Note that for instance in it is enough to consider the cubic term in the small expansion to obtain a fairly accurate estimate of . When becomes negative, the magnetization becomes non-zero with which gives one of the critical exponents. The other critical exponents are obtained in a similar way.
ii.2 Gradient expansion
The subsequent step is to compute the correlation length. This is done by considering a gradient expansion for a slowly varying magnetization profile, again under the assumption that the expansion is regular at small . One can perform a continuum limit to simplify the notations: we denote by the continuum limit of the spin field , while is the local average magnetization. The Landau free energy becomes at the quadratic order:
The correlation function of the magnetization is given by
Hence at this order the correlation function is
which is often called the bare propagator. The calculation is performed by using and changing variable to . Then
When , a saddle point calculation shows that , from which Eq. (8) follows. This expression shows that the correlation length is and the magnetic susceptibility is .
ii.3 From the microscopic Hamiltonian to a field theoretical formulation
The above analysis relies on the assumption that can be expanded as an analytic function around . Although this is certainly true at the mean field level (where the Landau theory provides the exact result), this is not the case in finite dimensional systems, because critical fluctuations induce a singular behavior of at small . Hence we now want to assess the limits of validity of the Landau theory by studying the effect of critical fluctuations.
The problem is that the definition of given in Eq. (4) is not very convenient to perform a systematic expansion in the fluctuations around the mean field theory, although the computation could be done in principle. It would be much more convenient to write the effective action as a functional integral over a continuous spin field :
with the following requirements:
The mean field approximation should correspond to a saddle point evaluation of the above integral, in such a way that at the mean field level . For consistency, must therefore have a Landau form:
The bare coefficients and entering in must be reasonable approximations to the microscopic coefficients as deduced from the Hamiltonian, in such a way that the mean field approximation is already a good approximation, and that loop corrections improve systematically over it. In this way we can guarantee that the criterion of validity of mean field theory has a quantiative meaning for the original microscopic Hamiltonian .
So we want to give an appropriate definition of the continuum spin field and the corresponding action in such a way that the requirements above are satisfied.
There are many recipes for such a construction. Probably the best one is given by the non-perturbative renormalization group construction Delamotte (2007), in which one defines a functional by integrating the small-scale spin fluctuations on length scales smaller than , see e.g. (Delamotte, 2007, Eq. (28)). If one chooses a “coarse-graining” length that is quite bigger than the lattice spacing, but still quite small with respect to the correlation length (which diverges at the critical point), the function is an analytic function of at small , because the singularity is only developed at the critical point for Delamotte (2007). Then, Eq. (10) is basically exact with replaced by . One can then expand at small and use this as the bare action in Eq. (10). Although this strategy can be generalized to the physics of liquids Parola and Reatto (1985); Caillol (2006), calculations are quite involved so we need to consider something simpler.
An alternative and very convenient prescription is the following. Let us call the truncation at a finite order of the high temperature expansion of , as given in Georges and Yedidia (1991). We know that is an analytic function of for any finite , hence cannot be a good approximation of at the critical point, because we know that is not analytic: in fact the high temperature expansion is divergent at the critical point. However, we can assume that gives a good approximation for . Note that our two requirements are satisfied by the prescription that . In fact, for the first requirement, at the saddle point level we obtain , and we already know that for this is the correct mean field result, while for we will obtain an “improved” mean field result. For the second requirement, we have already mentioned that the coefficients of give, for large enough , a good estimate of the microscopic properties of the model (e.g. the critical temperature). Furthermore, we can argue that the high temperature expansion, at a given order , is only sensitive to local physics up to a scale that grows with . Hence, truncating the high temperature expansion at a finite order in should be morally equivalent to perform an integration over the microscopic fluctuations on a scale smaller than . We will see that this procedure is easily generalized to the case of liquids where the high temperature expansion is replaced by the low-density virial expansion.
We will then use the prescription , expand in the form of Eq. (11), and use it in the functional integral Eq. (10) to compute in a loop expansion around mean field. Loop corrections give some non-singular contributions to , which were already in part taken into account in the bare action because it was obtained from the high temperature expansion: hence we might have some “double counting” of non-singular contributions related to the short range physics. This double counting problem is discussed in more details in Appendix A. Still, our aim here is to find a Ginzburg criterion that identifies the region where these singular loop corrections are small, and the mean field approximations remains correct: we find that if the Ginzburg criterion is formulated in terms of physical quantities, then double countings are irrelevant. This is shown in next section II.4 and in Appendix A.
ii.4 Ginzburg criterion
We can use the above construction to perform a loop expansion in the coupling and check whether fluctuations are small such that they do not spoil the main assumptions we made above on the behavior of at small , hence they do not change the critical behavior of the system. We will follow closely the derivation of Amit (1974). Our bare action is
Here we will need to consider explicitly the presence of an ultraviolet cutoff (which will be of the order of the scale mentioned above). The bare propagator is
and the one loop correction to the propagator is Parisi (1988)
We can consider the inverse propagator and write
which can be seen either as a Dyson resummation of the “tadpole” diagrams, or as an inversion of the perturbation expansion to obtain directly the second derivative of the Legendre transform of the generating functional. Physically, is the “renormalized mass” or inverse magnetic susceptibility:
where the last equality of course holds at first order in . The replacement of by is needed, because the perturbation theory must be done at fixed , i.e. at fixed distance from the true critical point, otherwise corrections cannot be small Parisi (1988). The critical point is defined by the condition that , or in other words the susceptibility is divergent, hence at the critical point
We see that the shift of the critical temperature is divergent in the ultraviolet (UV divergent) for : indeed, this is a non-universal quantity and depends on the details of the UV regularization. Now if we define the distance from the critical point as
we can write Eq. (16) as
This is the crucial relation that relates the inverse susceptibility to the distance from the critical point. The Ginzburg criterion is obtained by imposing that the one loop corrections do not change the mean field behavior .
We can now distinguish two cases:
For , the correction is UV convergent and infrared (IR) divergent. In this case, we can safely send the cutoff to infinity because the renormalized theory exists. We obtain
and the integral over is finite. We clearly see that because , the second term will be dominant over the first close enough to the critical point. By imposing that the first term dominates, we obtain the criterion in the form
where we used that in the mean field region the correlation length . Hence the Ginzburg number is a universal constant in this case. This shows that loop corrections will always be relevant close enough to the critical point and gives a precise value of the correlation length at which they will become relevant, .
Instead, for , the correction is UV divergent and IR convergent. In this case the Ginzburg criterion is non-universal and strongly dependent on the details of the regularization. For a fixed cutoff , the integral is finite at and the mean field behavior is always correct:
However one loop corrections provide a strong renormalization of the coefficient relating to . Imposing that these conditions are small we obtain
This provides a condition on for a given UV cutoff . When the condition is satisfied the mean field calculation is not only qualitatively, but also quantitatively correct. Note that the integral is upper-bounded by its value in . Then, if , the Ginzburg criterion is always satisfied and one loop corrections are small even at the critical point. Instead, if is of order 1 of bigger, then we obtain a non-trivial condition on and one loop corrections are large close enough to the critical point.
Iii Dynamical heterogeneities and replicas: definitions
The aim of this paper is to repeat the steps outlined in Sec. II in the case of a glass transition. As we will see, the calculation is in this case complicated by several problems:
We will need to introduce replicas to define a proper static order parameter.
The order parameter is in general not a real number (e.g. the magnetization) but a function that encodes the replica-replica correlations. Hence we will have to introduce a smoothing function to define a scalar order parameter . We will then have to show that the choice of the function is irrelevant.
The glass transition is discontinuous in the order parameter, which jumps to a finite value at the transition. Hence the transition is not an instability of the high temperature solution, but rather a spinodal point where the low temperature solution first appear. Because of that, we need to control the effective free energy at values of the order parameter that are very far away from the high temperature solution. Keeping only a few terms in the high temperature expansion is not enough, and we will have to resum an infinite number of terms to obtain a good starting point for the mean field theory.
Because the glass transition is akin to a spinodal point, the resulting effective action is a cubic theory. Hence the theory is not really defined (spinodals do not exist in finite dimension). This will not be a problem for the mean field and loop calculations, but we expect it to be a serious problem if one wants to go beyond mean field and construct a systematic epsilon expansion (which we do not attempt here).
In this section we will better explain the first two points: we will give some important basic definitions on criticality at the glass transition (as encoded by the so-called dynamical heterogeneities) and we will show how the problem can be tackled using replicas. In Sec. III.1 we introduce the basic dynamical order parameter of the glass transition, and in Sec. III.2 its correlation function. In Sec. III.3 we show how both quantities can be written as static correlations in a replicated theory. In Sec. III.4 we set up the general form of this replicated theory and give some useful definitions.
Throughout this paper we consider a system of particles in a volume interacting through a pairwise potential in a dimensional space. The basic field is the local density at point and time :
We will consider a generic dynamics that can be either Newtonian or stochastic, e.g. of Langevin type. In both cases, we will consider equilibrium dynamics, that starts from an equilibrated configuration of the system. It will be convenient to separate the dynamical average in two contributions Franz et al. (2011). A dynamical history of the system will be specified by the initial configuration of the particles , and by a dynamical noise. For Newtonian dynamics, this noise comes from the initial values of the velocities, extracted by a Maxwell distribution; for stochastic dynamics, it comes from the random forces that appear in the dynamical equations. We will denote by the average over the dynamical noise for a fixed initial condition; and by the average over the initial condition. Hence, for instance, the equilibrium average of the density will be .
iii.1 Two-point functions: the dynamical order parameter
The dynamical glass transition is characterized by an (apparent) divergence of the relaxation time of density fluctuations, that become frozen in the glass phase. The transition can be conveniently characterized using correlation functions. Consider the density profiles at time zero and at time , respectively given by and . We can define a local similarity measure of these configurations as
Here is an arbitrary “smoothing” function of the density field with some short range , which is normalized in such a way that . As an illustration, let us choose , where is the volume of a sphere of radius , and suppose that is much smaller than the inter-particle distance and that is short enough. In this situation, vanishes unless , and we get
Therefore, counts how many particles that are around point have moved less then in time and is often called “mobility” field. Alternatively, Eq. (26) can be taken as the definition of a self two-point correlation function. Different choices of lead to other correlations that have been used in different studies. We will show later that the choice of the function is irrelevant as far as the critical properties are concerned.
Let us call
the spatially and thermally averaged connected correlation function. Typically, on approaching the dynamical glass transition , displays a two-steps relaxation, with a fast “-relaxation” occurring on shorter times down to a “plateau”, and a much slower “-relaxation” from the plateau to zero Götze (2009). Close to the plateau at , one has in the -regime. The departure from the plateau (beginning of -relaxation) is described by . One can define the -relaxation time by . It displays an apparent power-law divergence at the transition, . All these behaviors are predicted by MCT Götze (2009), which in particular relates all these exponents to a single parameter through the formulae
and gives a microscopic expression of in terms of liquid correlation functions Götze (2009). In low dimensions, a rapid crossover to a different regime dominated by activation is observed and the divergence at is avoided; however, the power-law regime is the more robust the higher the dimension Charbonneau et al. (2010); Charbonneau et al. (2012a) or the longer the range of the interaction Ikeda and Miyazaki (2011).
iii.2 Four point functions: the correlations of the order parameter
It is now well established, both theoretically and experimentally, that the dynamical slowing is accompanied by growing heterogeneity of the local relaxation, in the sense that the local correlations display increasingly correlated fluctuations when is approached Franz and Parisi (2000); Donati et al. (2002); Berthier et al. (2005); For a review, see the contributions (2011). This can be quantified by introducing the correlation function of , i.e. a four-point dynamical correlation
This function describes the total fluctuations of the two-point correlations, and it decays as with a “dynamical correlation length” that grows at the end of the -regime and has a maximum that also (apparently) diverges as a power-law when is approached. MCT Götze (2009) and its extensions Biroli et al. (2006); Biroli and Bouchaud (2007); Berthier et al. (2007a, b); Szamel (2008); Szamel and Flenner (2010) give precise predictions for the critical exponents.
For later convenience, we can also consider a modified four-point correlation:
This function describes the isoconfigurational fluctuations of the two-point correlations, i.e. the fluctuations due to the noise of the dynamical process at fixed initial condition. It describes the in-state susceptibility: indeed, the initial condition selects a typical glass state, which is then explored by the dynamics. A final average over initial conditions is taken to ensure that the initial condition is a typical one.
For each of these correlations, we can define the corresponding susceptibility
iii.3 Connection between replicas and dynamics
The dynamical glass transition can be also described, at the mean field level, in a static framework. This has the advantage that calculations are simplified so that the theory can be pushed much forward, in particular by constructing a reduced field theory and setting up a systematic loop expansion that allows to obtain detailed predictions for the upper critical dimension and the critical exponents Franz et al. (2011). Moreover, very accurate approximations for the static free energy of liquids have been constructed Hansen and McDonald (1986), and one can make use of them to obtain quantitative predictions for the physical observables.
In the mean field scenario, the dynamical transition of MCT is related to the emergence of a large number of metastable states in which the system remains trapped for an infinite time. At long times in the glass phase, the system is able to decorrelate within one metastable state. The regime is identified with the dynamics “inside a metastable state”, while the regime is identified with “transitions between different states”. Hence we can write (introducing two new averages):
In fact, if one performs a dynamical average at fixed initial condition, the system is trapped in a single metastable that can be explored, and at long times the dynamical average can be replaced by the average in the metastable state selected by the initial condition. The average over the initial condition then induces an average over the metastable states with equilibrium weights, that we denoted by an overline.
For the four-point functions we obtain
The reason for the particular structure of the second term of is that the densities at time 0 come from the same initial condition and are therefore correlated, but they then evolve separately and therefore the two densities at time are uncorrelated.
The above structure suggests that the dynamical transition can be described in a static framework by introducing a replicated version of the system Kirkpatrick and Thirumalai (1989); Monasson (1995); Mézard and Parisi (1996). In fact, the replica method allows exactly to compute averages of the form , that enter in Eq. (32), in a static framework without the need of solving the dynamics. For every particle we introduce additional particles identical to the first one. In this way we obtain copies of the original system, labeled by . The interaction potential between two particles belonging to replicas is . We set , the original potential, and we fix for to be an attractive potential that constrains the replicas to be in the same metastable state.
Let us now define our basic fields that describe the one and two point density functions
To detect the dynamical transition one has to study the two point correlation functions when for , and in the limit which reproduces the original model Monasson (1995); Mézard and Parisi (1996). We denote by the equilibrium average for the replicated system under the conditions stated above. The crucial observation is that in the limit , all replicas fall in the same state but are otherwise uncorrelated inside the state. This leads to the following rule to compute the average : one should
factorize the averages when they involve different replicas, and
remove the replica indexes.
For instance, for any spatial argument, and for , we have that following the prescription above
which is exactly the kind of average we want to compute. Similarly, assuming that different letters denote different values of the indexes:
Let us introduce a space-dependent order parameter
and the two-replica correlation function
where is once again an arbitrary short ranged function. We are interested in these functions for . Using the prescriptions above we obtain
At this point the replica indexes can be dropped because the one-replica average in a metastable state is the same for all replicas, and we get
which provides the crucial identification between replicas and the long time limit of dynamics in a metastable state.
Similar mappings can be obtained for four-point correlations. We define the correlation matrix of the order parameter as (for and ):
where the superscript is useful to keep in mind that we performed a smoothing through the function . Performing similar manipulations as for the two-point functions, we have
and in the first term the average can be factorized over different indexes. Comparing this with Eq. (33) we obtain:
iii.4 The replicated free energy
We now discuss how replica correlation functions can be computed. We introduce some standard notations of liquid theory Hansen and McDonald (1986) and we adapt them to the replicated system.
Let us start with the grand canonical partition function for a -dimensional fluid with pairwise additive potential , chemical potential , and under an external field . The logarithm of the partition function reads:
where we have used the following definitions for the fields
and the microscopic details of the system are encoded in
To study the glassy phase we will follow the method introduced in Monasson (1995); Mézard and Parisi (1996). We replicate the system introducing other copies of this original fluid, with interaction between copy and copy denoted by , so that the logarithm of the replicated partition function is given by
where the definition of the fields must be modified in order to take into account different replicas
In the following, to lighten the notations, we will sometimes (when this leads to no ambiguity):
use shorthand notations for the spatial positions, e.g. , ;
similarly, use ;
drop the replica and space indexes and simply denote , ;
and similarly for similar or more complex quantities.
To study the glassy phase we need to know the correlation functions of the two fields and . Let us underline that in this scheme, the details of the (attractive) interaction between different replicas is not important because in the end we will send it to zero. In fact if there is a glassy phase below at a certain dynamical temperature , this infinitesimal attractive potential is enough to let all the replicas fall down in the same state. Hence, it is very convenient to perform a double Legendre transform and write the free energy as a function of the averages of and , which we denote and Morita and Hiroike (1961); De Dominicis and Martin (1964); Cornwall et al. (1974). In this way, we can take directly the limit where there is no interaction between different replicas and look for a solution where the replicas remain correlated in this limit. We obtain Mézard and Parisi (1996):
where and are the solution of the two equations
Morita and Hiroike Morita and Hiroike (1961) showed that this double Legendre transform can be written as
and we have introduced
The term is the sum of all two-particle irreducible diagrams that are defined precisely in Morita and Hiroike (1961); De Dominicis and Martin (1964); Cornwall et al. (1974). We do not give more details because this term will be mostly neglected when we will perform concrete numerical computations. However, the formalism we develop below holds in full generality so one could include 2PI diagrams in future works. For instance, this can be done through a systematic expansion in powers of the off-diagonal term of . The price to pay is that the result depends on many-body correlations of the non-replicated liquid Jacquin and Zamponi (2012).
It will be useful in the following to define the direct correlation function through a replicated version of the Ornstein-Zernike equation:
whose solution can be written through a series expansion in the following way
The free-energy is computed by evaluating the functional at the physical correlator which solves the following equation Morita and Hiroike (1961); De Dominicis and Martin (1964); Cornwall et al. (1974):
The density field can be determined by a similar equations as a function of the chemical potential, however here we are interested in a solution with , hence we can directly fix the density in this way. Moreover, we are interested in a solution for the two point function which has eventually (below the dynamical temperature ) a 1RSB structure
In particular in the high temperature phase we expect that the off-diagonal part of this solution, namely , is trivial (it corresponds to uncorrelated replicas, hence it is identically equal to 1) while below the dynamical temperature we have a non trivial solution. Note that the glass transition can be crossed either by lowering temperature or by increasing density, the second strategy being the only possible one for hard spheres like systems. In the following general discussion we will typically refer to lowering temperature: but all of our results apply to any other path in the phase diagram that crosses the glass transition line. We will indeed present concrete numerical calculations both in temperature and density.
Iv Landau expansion of the free energy around the glassy solution
In this section, we will show how one can construct a Landau expansion of the free energy. There are two main differences with respect to the simple ferromagnetic example of Sec. II. First of all, even for a homogeneous system, the order parameter is and it keeps a non trivial dependence on space. Second, at the mean field level the glass transition is a random first order transition: even if it is a second order transition from a thermodynamic point of view, the order parameter has a finite jump at the critical (dynamical) temperature . This implies that we cannot approach smoothly the glass phase from the liquid one. In the following, we will assume that we are in the glass phase at , where is non-trivial, and study how the limit of is approached from positive .
In Sec. IV.1 we introduce an appropriate scalar order parameter and perform a Landau expansion of the free energy for small deviations of the order parameter around the critical point. In Sec. IV.2 we show how this expansion can be used to compute the MCT exponents, in particular the exponent , following Caltagirone et al. (2012). In Sec. IV.3 we perform a more detailed study of the mass matrix, i.e. the quadratic term of the expansion. In Sec. IV.4 we use this to show that the value of the MCT critical exponents do not depend on the details of the definition of the scalar order parameter; as a side product we obtain a much simpler expression for these exponents.
iv.1 Free energy for a uniform field
We want to define the free energy as a function of the order parameter , defined in Eq. (38), in the case in which it is uniform, i.e. independent of . The way we can produce this quantity is just by maximizing the free energy with respect to under the constraint that is given by its definition, which is enforced through a Lagrange multiplier :
We now that in absence of the constraint, , the free energy is maximum for , Eq. (58), which corresponds to some value