Dynamical arrest with zero complexity: the unusual behavior of the spherical Blume Emery Griffiths disordered model
The short- and long-time dynamics of model systems undergoing a glass transition with apparent inversion of Kauzmann and dynamical arrest glass transition lines is investigated. These models belong to the class of the spherical mean-field approximation of a spin- model with -body quenched disordered interaction, with , termed spherical Blume-Emery-Griffiths models. Depending on temperature and chemical potential the system is found in a paramagnetic or in a glassy phase and the transition between these phases can be of a different nature. In specific regions of the phase diagram coexistence of low density and high density paramagnets can occur, as well as the coexistence of spin-glass and paramagnetic phases. The exact static solution for the glassy phase is known to be obtained by the one-step replica symmetry breaking ansatz. Different scenarios arise for both the dynamic and the thermodynamic transitions. These include: (i) the usual random first- order transition (Kauzmann-like) for mean-field glasses preceded by a dynamic transition, (ii) a thermodynamic first-order transition with phase coexistence and latent heat and (iii) a regime of apparent inversion of static transition line and dynamic transition lines, the latter defined as a non-zero complexity line. The latter inversion, though, turns out to be preceded by a novel dynamical arrest line at higher temperature. Crossover between different regimes is analyzed by solving mode coupling theory equations throughout the space of external thermodynamic parameters and the relationship with the underlying statics is discussed.
In the present work we investigate the dynamic properties of a glassy system in which, under certain external conditions, both glass and fluid can coexists, yielding different scenarios for dynamical arrest and for the fluid-glass transition. These properties can be studied in statistical mechanical models with bosonic spin- variables, where the holes play the role of inactive states, that is, the so-called Blume-Capel Blume (1966); Capel (1966) or Blume-Emery Griffiths (BEG) Blume et al. (1971) models. In these models the fluid phase corresponds to a paramagnet and the solid phase is either a ferromagnet (no or weak disorder) Blume (1966); Capel (1966); Blume et al. (1971); Schupper and Shnerb (2004, 2005) or a spin glass (strong disorder) Ghatak and Sherrington (1977); Lage and de Almeida (1982); Mottishaw and Sherrington (1985); Sellitto et al. (1997); Schreiber (1999); Crisanti and Leuzzi (2002, 2004). In the present work we consider an extension to -spin interacting systems with spin-, to and continuous (spherical) variables Ferrari and Leuzzi (2011) to better represent continuous density fluctuations, alike to liquid-like compounds.
In the presence of quenched disorder the random BEG model with pairwise (), as well as its spherical counterpart, is known to display both a continuous paramagnet/spin-glass phase transition and a first-order one (first order in the thermodynamic sense, i.e. with latent heat and a region of phase coexistence). Furthermore, melting upon cooling Rastogi et al. (1999); Greer (2000); van Ruth and Rastogi (2004); Plazanet and et al. (2004); Tombari and et al. (2005); Ferrari and et al. (2007); Angelini et al. (2009) can occur, with a spin glass at high and a paramagnet at low . These properties have been observed in the mean-field approximation, where the self-consistent solution for the spin-glass phase is computed in the full replica symmetry breaking (RSB) Parisi ansatz Crisanti and Leuzzi (2005) and on the cubic 3D lattice with nearest-neighbor couplings Paoluzzi et al. (2010); Leuzzi et al. (2011). The frustrated BEG model has been studied by means of numerical renormalization group techniques, as well, with results depending on the underlying lattice and the renormalization technique adopted Ozcelik and Berker (2008); Antenucci et al. (2014a, b).
Mean-field spin-glass models with Ising Gardner (1985), soft Kirkpatrick and Thirumalai (1987a, b) or spherical Crisanti and Sommers (1992); Crisanti et al. (2003) spins with more than two-spin interactions, called -spin models, are known to yield the so-called random first-order transition, i.e., a phase transition across which no internal energy discontinuity occurs but the order parameter (the Edwards-Anderson overlap ) jumps from zero to a finite value. Their glassy phase is described by an ansatz with one RSB Parisi (1979). In a cooling procedure, the thermodynamic transition is preceded by a dynamic transition due to the onset of a very large number of metastable states separated by high barriers Leuzzi and Nieuwenhuizen (2008). “Very large” means that the number of states grows exponentially with the size of the system: where the coefficient is the configurational entropy, also called complexity in the framework of spin-glass systems (see, e.g, Refs. Müller et al. (2006); Crisanti et al. (2004) and references therein). “High barriers” means that the free energy difference between a local minimum in the free energy functional of the configurational space (also called free energy landscape) and a nearby maximum (or saddle) grows with . The phenomenology of the -spin spin-glass systems is, in many respects, very similar to the one of structural glasses. These models are, therefore, sometimes called mean-field glasses. The occurrence of non-zero is a fundamental property both in mean-field systems Kirkpatrick and Thirumalai (1987a, b); Crisanti and Sommers (1995) and outside the range of validity of mean-field theory, e.g. in computer glass models Sciortino and Tartaglia (2005); Binder and W. (2005), or, indirectly, by measuring the excess entropy of glasses in experiments, see, e.g., Ref. Leuzzi and Nieuwenhuizen (2008) and references therein. The barriers’ height turns out to diverge in the thermodynamic limit in the mean-field approximation, this being an artifact of mean-field glasses. The thus induced dynamic transition corresponds to the transition predicted by another mean-field theory for the dynamics of supercooled liquids: the mode coupling theory Götze (2009). The thermodynamic transition occurring at a lower temperature is, instead, the mean-field equivalent of the so-called Kauzmann transition in glasses, also known as the ideal glass transition Leuzzi and Nieuwenhuizen (2008). This was initially predicted by Gibbs and Di Marzio Gibbs and Di Marzio (1958) and its occurrence in real strutural glasses is still object of an ongoing debate Hecksher et al. (2008); Eckmann and Procaccia (2008); Tanaka (2003); Martinez-Garcia et al. (2014).
We are going to investigate the complex dynamic properties consequent to the combination of a kind of interaction inducing structural glass behavior and the presence of hole states (aka, spin state ) inducing phase coexistence. The latter element is, possibly, responsible for melting upon cooling Crisanti and Leuzzi (2005); Paoluzzi et al. (2010). The first of such models was brought about by Sellitto in the pairwise random orthogonal model with spin- variables Sellitto (2006). In the present dynamic work we rather consider the multi-body interaction model of Ref. Ferrari and Leuzzi (2011), where both high temperature coexistence of high- and low-density paramagnetic phases, and low temperature coexistence of (low-density) paramagnetic and spin-glass phases are displayed.
where the variables are bosonic spins (i.e., ), and the couplings are independent quenched random variables distributed with a Gaussian probability density,
the external parameter is called “crystal field” in literature, and it essentially plays the role of a chemical potential. Because of the bosonic spins, we cannot define the continous spin approximation of the model with spherical constraint in the usual way Berlin and Kac (1952). We must first rewrite the Hamiltonian as an Ising-spin problem on a lattice-gas
where the term is necessary to keep the ratio of filled-in to empty sites identical to the one of the original Hamiltonian, see Ferrari et al. (2012); Griffiths (1967) for details. We then introduce the variableCaiazzo et al. (2002)
This way, the model Hamiltonian assumes the form
where all the degrees of freedom, ’s and ’s are now Ising spins. A continuous spin model can then be constructed by imposing two independent spherical constraints
Programma/Pspinbosonic/Test/ The thermodynamic properties of the model we just defined are thoroughly studied, applying 1RSB theory, in reference Ferrari and Leuzzi (2011). For the sake of brevity, henceforth we shall refer to Ref. Ferrari and Leuzzi (2011) as the static study (that is, replica theory-based), in contrast with the dynamic study that constitutes the principal aim and subject of this paper. Before deriving and solving the dynamical equations of the model, it is suitable to briefly summarize the static results of Ferrari and Leuzzi (2011), with particular emphasis on the aspects that will be most relevant for the dynamical study that we are going to report.
ii.1 The static phase diagram
The 1RSB free energy for the model is
with the definitions
where the parameter is the ratio of filled-in to empty sites (that is, the density of the system), and the and are respectively the mutual overlap and self-overlap, as usual in a 1RSB Ansatz. The extremization of the (6) whith respect to and yields the saddle-point equations
The static lines of the phase diagram, in absence of an external magnetic field, can be determined by setting and studying the system given by the equations (8a), (8b), and the saddle-point condition for the 1RSB parameter
where is the Crisanti-Sommers functionCrisanti and Sommers (1992)
The dynamical arrest lines can be, as well, identified using replica theory, by solving the saddle-point equations for , and by imposing the following marginality condition for the solution in the parameter space:
This corresponds to looking for a spinodal point in the free energy as a function of and .
The resulting phase diagram for the model is reported in figure 1.
From it, it can be seen that the system exhibits a rich phenomenology, with both random first-order transitions (RFOT) and thermodynamic first-order phase transitions (TFOPT). Here we will just comment on them briefly to adequately introduce our dynamical study, along with statics novelties with respect to previous analysis. The interested reader can find all the details in Ferrari and Leuzzi (2011).
Random first-order transition For low enough , the system exhibits the random first-order phenomenology typical of the -spin model. Along the dynamic transition line , the system undergoes a dynamical arrest, meaning that the relaxation towards the paramagnetic, stable state is blocked by the presence of an exponentially large number of metastable SG states which trap the dynamics. Being the SG states metastable, this transition is not captured by the static saddle-point equations, and has to be studied by solving the dynamics of the model, or by using the marginality condition for the dynamics Franz and Parisi (1995); Crisanti (2008); Ferrari et al. (2012); Crisanti and Sommers (1992).
At a temperature lower than , a static transition takes place, whereupon the number of SG states becomes subexponential (their complexity, also called configurational entropy, vanishes) and they become stable with respect to the paramagnet, yielding the equilibrium SG solution Castellani and Cavagna (2005); Crisanti and Sommers (1992). This feature of the model, occurring for , is equivalent to the -spin model phenomenology.
Thermodynamic first-order phase transitions In the region of the phase diagram between the spinodal lines, but above the RFOT line, two paramagnetic phases, termed PM and PM, coexist, both with but with two different density values : and . The two paramagnets are labelled and according to their density value being, respectively, large and small. These values can be determined by solving the saddle-point Eq. (8b) in the limit, yielding the expression
For , this is a polynomial equation with three solutions for . The solution with the intermediate value of turns out to be always unstable (see Ref. [Ferrari and Leuzzi, 2011]), leaving only a high-density and a low-density solutions. Since the density has a continuous behavior along the PM/SG thermodynamic transition, Eq. (12) can be used, as well, to determine the value of the density for the SG phase at the transition point. This means that the spinodal lines can be determined in a parametric form in as
By studying those expressions, it can be readily checked that for , at high the spinodal curve is an asymptote of the axis; this means that, however large , the system can always present an high density phase, if the temperature is low enough. This fact will be of capital importance in the following of this paper.
Besides the RFOT, thus, the system also exhibits both a PM/PM and, furthermore, a PM/SG thermodynamic first order transition, that means standard first order transitions with phase coexistence and latent heat. Both transitions take place along the TFOPT line in figure 1 and coexisting phases exist between the spinodal lines. As the temperature is decreased to cross the RFOT dynamic line, we observe that only the high-density paramagnet undergoes dynamical arrest, while the low density PM phase is unperturbed. Thus, at lower than the crossing point of TFOPT and RFOT lines, the transition occurs between a low density paramagnet PM, with and , and an high-density SG with and . The intersection takes place for .
High density dynamical transition. At , the dynamical RFOT line and the spinodal SG line intersect. It can then be seen that for , the dynamical RFOT line coincides with the spinodal line of the TFOPT, which means that the dynamical arrest in the PM phase will take place as soon as phase separation occurs. From the thermodynamic point of view, we have coexistence between two paramagnets, as before. However, if we perform a quenching dynamics from the high density PM phase, a dynamical arrest into a metastable SG phase will take place.
Iii The dynamics
We are now ready to derive the dynamical equations for the model. Let us first separate the disordered part of the Hamiltonian (4) from the deterministic one
The relaxation dynamics is, then, governed by the Langevin equations
where we assume the noise fields and to be delta-correlated:
with (taking the Boltzmann constant )
Following Crisanti et al. (1993), we have inserted the Lagrange multipliers and in order to enforce the spherical constraint.
The quantities we are interested in are the correlation functions and the response functions of the system. In this case, differently from the standard spherical p-spin model, we have two different types of degrees of freedom, and thus two possible external perturbing fields, one for each of them. As a result of this, we have four different correlation functions
and four response functions
where and are time-dependent perturbing fields conjugated with the and degrees of freedom, respectively.
Our aim is to use equations (15) to obtain self-consistent
dynamical equations for the functions above. We
employ the generating functional method devised in Martin et al. (1973) by
Martin, Siggia and Rose, and already used for the -spin model in
Kirkpatrick and Thirumalai (1987c) by Kirkpatrick and Thirumalai. The MSR approach
consist essentially in defining a generating functional
We have emphasized the fact that the generating functional still depends on the quenched random couplings , and so does every quantity generated by it; so, in principle, we would have to average them over the disorder in order to obtain the correlation and response functions we want. However, as remarked by De Dominicis in Dominicis (1978), since the generating functional in absence of external currents is by definition normalized to one
it is independent from the variables of the system, and so it can be averaged over the disorder directly. This is in contrast with the static partition function for a system with quenched disorder, which is not self-averaging. As a result of this, the use of replica theory is not needed in the dynamical framework; this fact constitutes the main advantage of the dynamical approach over the static one.
Performing the average over the disorder leads to a decoupling of the lattice sites and a coupling of the configurations of the system at different times, as it happens for the -spin model in Kirkpatrick and Thirumalai (1987c). This is to be conceptually compared to the decoupling of the sites in the static approach with replicas, yielding a coupling between different replicas Gross and Mézard (1984). It is then possible, using saddle point methods, to write an effective generating functional which yields two single site dynamic equations valid for every degree of freedom in the lattice. The details of the derivation of the dynamics can be found in the appendix. We report here the site-independent dynamical equations
where the correlation matrix for the noise terms and has been renormalized in the following way
We have also defined the constant
and the two functions
The equations for the correlation and response function can then be easily obtained, as reported in appendix A.
iii.1 Symmetries, equilibrium and ergodicity
In appendix A, we derive eight coupled differential equations for eight different unknown functions. We now specify them to the particular problem we want to study, i.e. identifying dynamical arrest. In order to to this, we can restrict ourselves to an equilibrium (i.e. starting from an equilibrium initial condition) and ergodic dynamics. This implies time-translational invariance (TTI) of the correlators
and that the fluctuation-dissipation theorem (FDT) holds
where and denote any correlation and response function couple in the system, respectively, and is the Heaviside step function. These assumptions are valid in the high temperature PM phase, where ergodicity is not broken, but they are generally false when the system is cooled below the dynamical transition temperature , where a transition to a phase with broken ergodicity takes place. Second, we notice that both the model Hamiltonian (4) and the effective generating functional (38) are symmetric with respect to a switch
This means that the and evolve in the same statistical ensemble, which, in turn, implies that the correlation functions obey the relations
that can then be extended to the response functions by exploiting the FDT
Once that these relations are established, we can see that only the two correlation functions and are needed to completely describe the dynamics of the system. Thus, we can define the “total” correlation function of the system , by normalizing the to one
where we have used the spherical constraint
and the relation (3) between the product and the site occupation number
Since we assume the dynamics to be at equilibrium, the density , being an one-time quantity, is a constant of motion, equal to its equilibrium value
Using the equations for the and , we obtain the following integro-differential equation for the total correlation function
which is a closed equation, whose solution can be found once the values of the parameters , and are known. It can be seen as a Mode Coupling schematic equation, and it is worth noticing that in the limit of the parameter (the fraction of filled-in sites) going to one recovers standard MC equations. The density is assumed, at all times, to be equal to its equilibrium value, given by Eq. (12). As a result of this, in the region between spinodal lines, where phase coexistence occurs, the dynamics will have to be studied separately for each one of the two coexisting phases. In the high density limit case , thermodynamically occurring for low , the proper limit of Eq. (III.1) is considered in appendix A.3. The study of Eq. (III.1) will be one of the main focuses of this work.
An equation can be derived also for the difference between and , obtaining
which has the trivial solution
where the constant is defined as
If now we use Eq. (12) to eliminate , we get
which is always negative for any value of and . This means that the difference between the two correlators tends to zero for long enough times, and thus the dynamics of the system can be always solved using the function only.
iii.2 Solving the dynamic equations
In this section we report the results obtained by numerically solving equation (III.1) in various representative points of the phase diagram. Eq. (III.1) is an integro-differential mode-coupling like equation Götze (2009), that can be solved using the standard algorithm intruced by Fuchs et al. Fuchs et al. (1991) and extended in different ways, cf. e. g., Refs. Berthier et al. (2007); Crisanti et al. (2011); Crisanti and Leuzzi (2015). We consider three meaningful cases to illustrate the varoius occuring regimes.
: dynamic transition
At and high the system yields a single paramagnetic phase (PM), which undergoes a dynamical transition with ergodicity breaking at Ferrari and Leuzzi (2011). In figure 2 we report the total correlation function of the system as the transition is approached from above.
As we can see, our dynamic equation (III.1) yields the dynamic transition predicted in Ferrari and Leuzzi (2011), and the correlator shows the typical phenomenology of a mode-coupling like dynamic arrest. The transition temperature corresponds up to a error with the predicted value and the height of the plateau (also called the non-ergodicity parameter) is , as expected.
: phase coexistence
For , the situation is richer and more interesting; for this value of the crystal field, at low temperature the system undergoes phase coexistence, yielding two separate paramagnetic phases with high (PM) and low (PM) density. As we anticipated, the dynamics of the system has to be solved separately for each one of these two phases. Their behavior turns out, actually, to be quite different, as only the high density PM phase undergoes ergodicity breaking as the dynamical line is crossed. In figure 3 we plot the resulting correlators as the system enters the phase coexistence zone, and the high density phase undergoes the dynamic transition.
We see that, again, the expected phenomenology is reproduced by our equation. At only a single paramagnetic phase is present, but for , two different paramagnetic phases separate; at , the high density phase correlator shows the typical plateau as the dynamic transition is approached, and as the cooling continues, the high density PM phase eventually undergoes a dynamical arrest, while the low density paramagnet PM stays unchanged until the first order phase transition takes place to the thermodynamic 1RSB-stable glass phase.
: anomalous dynamical arrest
Up to the prevuoius case, our dynamic analysis confirms the results of the static one Ferrari and Leuzzi (2011). However, as we anticipated in section I, for , our dynamical equation yield dynamic arrest at finite temperature when starting from the PM phase, an effect that was not identified in the static analysis, where the thermodynamically dominant phase is low density PM. We shall now report the results obtained by solving equation (III.1), leaving the study and discussion of the new arrested phase for the next section.
In order to show the onset of this new transition, we choose , and we cool the system starting from a high density initial condition crossing the spinodal line during the procedure. The PM spinodal line, for these chemical potential values, turns to be a dynamical arrest transition line, as shown in figure 4. We can observe that, as soon as the two PM phases separate, the high density phase is already an arrested SG phase, with a nonzero overlap, while the low density phase shows no sign of dynamical arrest. This is at difference with respect to the static results of Ref. Ferrari and Leuzzi (2011), where the high density phase is supposed to be still paramagnetic for .
What is puzzling about this result is the fact that the high density phase is already deep into the SG when the separation occurs: for we have , which is already much higher than the that we would expect for a system which approaches from above an usual dynamical arrest. This means that if the high density phase existed even above the spinodal line, its dynamic transition temperature would be actually much higher than ; however, this effect is not visible since only the low density PM phase exists in that region. If we follow the same cooling procedure for (i.e., we cross the spinodal line at during the cooling), the results are not very different, and the high density arrested phase is still present. In this case, the value of the overlap at the separation line is even higher, with for .
Solving equation (III.1) for higher does not change the general situation, so we will not report any results for higher values. The point is that, since the spinodal line is an asymptote of the axis (as we have mentioned in section II), then for arbitrarily large , a high density phase exists at . According to both the marginality condition for the statics, and the dynamic results presented so far, this phase presents a dynamical arrest into a SG phase with nonzero overlap, that can be realized by selecting atypically dense initial conditions at those temperature and chemical potential.
Iv Complexity and free energy
Since the system undergoes a dynamical arrest, we would expect the RFOT phenomenology which holds in the other regions of the phase diagram to be present in this case as well, inside the high minimum which corresponds to the PM+ phase (the low density paramagnet is completely orthogonal to our discussion). In summary, we expect the metastable states which trap the dynamics (and maximize the complexity) to have a higher in-state free-energy than the one of the paramagnetic, ergodic state. We might also expect a complexity to be strictly positive for every up to a static temperature where states with null complexity are born and a static transition takes place. We computed both quantities using replica theory as in Ref. Ferrari and Leuzzi (2011) and report the corresponding curves in figures 5 and 6.
We see that the usual RFOT picture does not hold for all values of along the new dynamical arrest line. Indeed, it holds only for where , which in fact corresponds to the point where the static line touches the new dynamical arrest line. In this region, the complexity of the states is positive for every (as one can see in the upper panel of figure 6) and the free energy of the states which trap the dynamics is higher than the paramagnetic one. On lowering at constant , the complexity touches zero (corresponding to the condition Eq. (9)) and a static transition to a SG phase takes place as usual.
This picture does not hold anymore for . For , the maximum complexity condition (dynamical arrest line) and the zero complexity condition (static line) coincide, which means that the complexity is zero (i.e., the number of all states becomes subexponential) and the free energy of the trapping states becomes equal to the paramagnetic (PM+), cf. figure 5. This is alike to the occurrence of the static Kauzmann transition but in this case both SG and PM phases are metastable and dynamically occurring only for initial conditions with a density atypically high for these chemical potential values. When is increased along the high density arrest line, the free energy of the trapping states becomes lower than the paramagnetic one and continues to decrease indefinitely, while the complexity becomes more and more negative as reported in figure 6. If one looks at the phase diagram 1, it is possible to see that there is a range of such that, on cooling, the static line is met before the dynamic one Ferrari and Leuzzi (2011). This would mean that along this path in the parameter space the states with low free energy and complexity arise before the ones with maximum complexity and free energy, only eventually appearing on the maximum complexity line (that, for , is the dynamic line): the complexity of the glassy metastable states becomes again non-zero crossing the static line on cooling and reaches its maximum at the line formerly denoted as dynamic line. However, this is true for up until . From that point on, the static line touches the axis and the complexity stays negative for every value of . In summary, using both the dynamical equations and the replica approach, we find an anomalous, complexity free, dynamical transition, that occurs if the system is initially prepared at values of the density corresponding to a paramagnetic metastable state at high density. This state always coexists with a more probable and thermodynamically dominant low density paramagnetic state. It is the latter which, at lower and/or lower , undergoes a first order phase transition to a spin-glass phase.
In this work, we have studied, both dynamically and statically, a disordered model which shows both an RFOT-like phenomenology (dynamical arrest, complexity, Kauzmann transition, etc.), and an “ordinary” first order phase transition with latent heat and phase coexistence. We have derived and solved the equations for the equilibrium dynamics of the model and completed the static replica-based study of Ferrari and Leuzzi (2011) with novel results. In doing so, we have noticed the presence, in a certain region of the phase diagram, of a RFOT-like dynamical arrest line, which however shows a non-positive complexity and does not work as a precursor for a Kauzmann-like static transition, in contrast with the usual phenomenology expected in RFOT models.
The picture we propose to explain this is the following. Our model has two order parameters, the density and the 1RSB self-overlap (termed from now on). One can imagine to construct a potential function of and , by plotting the paramagnetic free energy as a function of , and then performing a Franz-Parisi like Franz and Parisi (1995) construction along the axis, for every . For the transition scenario is qualitatively alike to the random first order transition one.
For chemical potential values in the interval , instead, the scenario changes. For high only one minimum is present but, on cooling, the spinodal line of a second paramagnetic phase, at higher density, is crossed: a second, metastable minimum with and density is formed, termed . Further lowering the temperature the dynamical transition line is crossed. There, a metastable phase with and arises from the PM solution, with a higher free energy. This phase consists of many equivalent states and corresponds to the arrested glassy phase. The difference in free energy between the SG metastable minima and the free energy of the PM minimum is equal to the complexity counting the log of the number of SG metastable states. If is lowered further, we cross the static transition line, where the minimum has the same height as the paramagnetic one (null complexity) and thus it becomes stable: we have a static transition in a spin-glass phase.
To summarize, we have a three-step process on cooling for :
A secondary PM minimum with and is formed (the stable phase is the low density PM).
A SG minimum with , is formed, arising from PM.
The SG becomes stable with respect to the low density paramagnet PM and a static transition takes place.
This scenario almost corresponds the usual RFOT phenomenology. The only difference is that there are two paramagnetic phases, one stable (PM) and one metastable at higher density, PM, and that the glassy metastable states at the threshold free energy arise inside the PM.
When is raised beyond , the order of these steps above is changed. Two apart scenarios appear. For , steps and exchange. Fixing slightly above , and looking at the potential only in the direction of , the minimum will have already formed. However, this has no effect on the thermodynamics since does not yet correspond to a minimum on the axis. For that to happen, must be lowered to cross the PM spinodal. At lower the process goes on as previously.
For , step 1 becomes the last to happen: again, the system dynamics is arrested as soon as the PM spinodal is reached, but the SG minimum along not only is formed before the PM has a chance to appear, but it has even become stable with respect to the PM. We stress the fact that this unusual behavior is possible only because the system has two order parameters, differently from usual RFOT models whose behavior is governed only by . The fact that the minimum in is already formed for is, indeed, confirmed by the fact that the naive marginality condition , used in Ref. Ferrari and Leuzzi (2011), completely misses this new arrest line: in the direction of the minimum is already formed and the curvature is positive. The whole Hessian matrix of the replicated free energy with respect to and (evaluated at ) must be used to detect the new line, as reported in Sec. II.1. This picture is corroborated by the fact that this phenomenology is found in the PM phase, whose density goes up and approaches as increases, as reported in the upper panel of figure 5: this means that the PM phase of our model becomes more and more similar to the usual -spin spherical model (PSM) Crisanti and Sommers (1992), and the dynamical equations behave accordingly as explained in appendix A.3. However, we also stress that such density values are thermodynamically extremely unlikely to occur for these large values of the chemical potential .
The research leading to these results has received funding from the Italian Ministry of Education, University and Research under the Basic Research Investigation Fund (FIRB/2008) program/CINECA grant code RBFR08M3P4 and under the PRIN2010 program, grant code 2010HXAW77-008, and from the European Research Council, under the European Union’s Seventh Framework Programme FP7/2007-2013/ from the People Programme (Marie Curie Actions) under REA grant agreement n 290038, NETADIS project, and ERC grant agreement n 247328, CryPheRaSy project. U.F. thanks A. Destexhe for hosting at the European Institute for Theoretical Neuroscience.
Appendix A Derivation of the dynamical equations for the p-spin Blume-Capel spherical model
In this appendix we provide the step-by-step derivation of the dynamical equations studied in this paper.
a.1 Computation of the effective generating functional
We write down the full expression for the MSR generating functional
where denotes a functional integration measure over all lattice sites; for example
The first step is to perform the average over the disorder of expression (30) for the generating functional; since the disorder is contained only in the part of the hamiltonian, we have to perform the average only on the third line of the (30). Thus we have to compute the integral
where we have symmetrized the couplings; performing the gaussian integration yields
As we anticipated in section III, the average over the disorder has decoupled the lattice sites, at the price of generating a coupling between configurations of the system at different times.
This dynamical coupling is conceptually similar to the coupling between replicas that occurs in the static treatment of the -spin model Crisanti and Sommers (1992); Castellani and Cavagna (2005); so, following Kirkpatrick and Thirumalai (1987c), we define the dynamical overlaps between auxiliary MSR fields,
between dynamical fields,
and between auxiliary and dynamical fields