Quantitative field theory of the glass transition

Abstract

We develop a full microscopic replica field theory of the dynamical transition in glasses. By studying the soft modes that appear at the dynamical temperature, we obtain an effective theory for the critical fluctuations. This analysis leads to several results: we give expressions for the mean field critical exponents, and we analytically study the critical behavior of a set of four-points correlation functions, from which we can extract the dynamical correlation length. Finally, we can obtain a Ginzburg criterion that states the range of validity of our analysis. We compute all these quantities within the hypernetted chain approximation for the Gibbs free energy, and we find results that are consistent with numerical simulations.

Dynamical heterogeneities in structural glasses have been the object of intensive investigations in the last 15 y (1). The early Adams–Gibbs theory of glass formation was based on the concept of cooperatively rearranging regions with sizes that become larger and larger when the glass region is approached. Such large cooperatively rearranging regions imply the existence of dynamical heterogeneities characterized by a large correlation length. Large-scale dynamical heterogeneities are expected to be present in any framework where glassiness is caused by collective effects: They are, indeed, the smoking guns for these effects (1–4). Therefore, it is not a surprise that two popular approaches to glasses, mode-coupling theory (MCT) (5) and the replica method (6, 7), both agree with the Adams–Gibbs scenario and predict large-scale dynamical heterogeneities with a dynamical correlation length that diverges at the transition to the glass phase. This qualitative prediction is very interesting, but to make additional progresses, it would be important to get quantitative predictions that can be compared with numerical simulations and experiments.

At the mean field level, where both thermodynamic and dynamic aspects can be solved exactly, it is found that the replica and MCT approaches are intimately related. The study of spherical p-spin models, where dynamics are exactly described by a schematic MCT equation and equilibrium displays glassy phenomena related to replica symmetry breaking, shows how the glass transition described by MCT is related to the emergence of metastable states in equilibrium (8, 9). That basic observation, made more then 20 y ago in the work by Kirkpatrick et al. (10), opened the way to the application of the mean field theory of spin glasses to the physics of supercooled liquids and glasses (11–13). Despite this clear relation at the level of mean field schematic models, when one tries to apply the mean field theory to realistic models of simple liquids (5–7, 14, 15), approximations are mandatory, and because of the approximations, the connection between statics and dynamics becomes more difficult to establish. It has been shown in the work by Szamel (16) that, under suitable approximations (similar to the one of MCT), the long time limit of the MCT equations could be derived from a replicated liquid theory. Unfortunately, this time limit leads to expressions that are not variational, and one cannot get an approximation for the free energy from the computation. Using standard liquid theory approximations within replica theory instead (6, 7, 14, 15), one finds strong discrepancies between predictions from MCT and replicas, which become particularly pronounced in large dimensions (7, 17).

Other than this consistency problem, in finite dimensions, one would like to compute the corrections caused by fluctuations around the mean field approximation. When this program is carried out, one finds that there are two important sources of corrections to the mean field scenario. The first corrections originate from critical fluctuations that become important around the glass transition below the upper critical dimension, like in any standard critical phenomenon (18, 19). The second corrections are nonperturbative phenomena related to activated processes. They can be taken into account by a phenomenological approach, leading to a number of predictions that are in good agreement with experiment (11); however, the theoretical foundations of this approach are still controversial (20), and alternative (but possibly related) phenomenological descriptions of activated relaxation in glasses have been developed, mostly based on the concept of dynamical facilitation (21).

In this paper, we will only consider critical fluctuations around mean field, and therefore, we will not take into account activated processes. Critical fluctuations have been previously described within MCT (18, 22–24). However, field theoretical methods are not yet under complete control in the context of dynamics, and it is, therefore, extremely important to set up a static replica field theoretical description of dynamical heterogeneities in such a way that well-established equilibrium field theory methods, such as the renormalization group, can be applied to the glass transition problem. This result is what we achieve in this paper. We obtain a low-energy effective action that describes critical fluctuations on approaching the glass transition, with coupling constants that are obtained directly from the interparticle interaction potential using standard liquid theory. This process allows us to compute prefactors to the singular behavior of physical observables in the mean field approximation, such as the correlation length or the four-point correlation functions. In addition, we show that an important characterization of dynamics, the MCT exponents, can be obtained within the static replica framework. Using the well-established hypernetted chain approximation (HNC) approximation of liquid theory, we perform explicit computations for hard- and soft-sphere models and Lennard–Jones potentials, and we obtain good agreement with available numerical data. Finally, we introduce a quantitative Ginzburg criterion defining a region, where perturbative corrections to mean field theory can be neglected.

Dynamical Heterogeneities

In this section, we consider a system of N particles in a volume V interacting through a pairwise potential v(r) in a D dimensional space. The dynamical glass transition is characterized by an (apparent) divergence of the relaxation time of density fluctuations, which becomes frozen in the glass phase. If is the local density at point x and time t and is its equilibrium average, the transition can be conveniently characterized using correlation functions. Consider the density profiles at time 0 and time t, respectively, given by and . We can define a local similarity measure of these configurations as (Eq. 1)

where f(x) is an arbitrary smoothing function of the density field with some short-range A. In experiments, f(x) could describe the resolution of the detection system and can be, for instance, a Gaussian of width A.

Let us call the spatially and thermally averaged correlation function. Typically, on approaching the dynamical glass transition T_d, C(t) displays a two-step relaxation, with a fast β-relaxation occurring on shorter times down to a plateau and a much slower α-relaxation from the plateau to zero (5). Close to the plateau at C(t) = C_d, one has C(t) ∼ C_d + t^−a in the β-regime. The departure from the plateau (beginning of α-relaxation) is described by C(t) ∼ C_d − ℬt^b. One can define the α-relaxation time by C(τ_α) = C(0)/e. It displays an apparent power-law divergence at the transition, τ_α ∼ |T − T_d|^−γ. All of these behaviors are predicted by MCT (5). In low dimensions, a rapid crossover to a different regime dominated by activation is observed, and the divergence at T_d is avoided; however, the power-law regime is the more robust the higher the dimension (25, 26) or the longer the range of the interaction (27).

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 T_d is approached (1–3, 28). This result can be quantified by introducing the correlation function of (i.e., a four-point dynamical correlation) (Eq. 2):

The latter decays, because G₄(r, t) ∼ exp(−r/ξ(t)) with a dynamical correlation length that grows at the end of the β-regime and has a maximum ξ = ξ(t ∼ τ_α) that also (apparently) diverges as a power law when T_d is approached.

MCT (5) and its extensions (18, 22–24, 29, 30) give precise predictions for the critical exponents. However, as discussed in the Introduction, this dynamical transition can be also described, at the mean field level, in a static framework. The advantage is that calculations are simplified, and therefore, the theory can be pushed forward, particularly by constructing a reduced field theory and setting up a systematic loop expansion that allows us to obtain detailed predictions for the upper critical dimension and the critical exponents (19). Moreover, very accurate approximations for the static free energy of liquids have been constructed (31), and one can make use of them to obtain quantitative predictions for the physical observables. These predictions are the aim of the rest of this paper.

Connection Between Replicas and Dynamics

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. Hence, we can write (Eq. 3)

where 〈•〉_m denotes an average in a metastable state, and the over line denotes an average over the metastable states with equilibrium weights.

The dynamical transition can be described in a static framework by introducing a replicated version of the system (14, 32): for every particle, we introduce m − 1 additional particles identical to the first one. In this way, we obtain m copies of the original system, labeled by a = 1, …, m. The interaction potential between two particles belonging to replicas a, b is v_ab(r). We set v_aa(r) = v(r), the original potential, and we fix v_ab(r) for a ≠ b 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 (Eq. 4):

To detect the dynamical transition, one has to study the two-point correlation functions when v_ab(r) → 0 for a ≠ b, and in the limit m → 1, which reproduces the original model (14, 32). In this limit, the two-replica correlation function is, for a ≠ b (Eq. 5),

Because of the limit v_ab(r) → 0, the two replicas fall in the same state but are otherwise uncorrelated inside the state; therefore, we obtain , which provides the crucial identification between replicas and dynamics. Similar mappings can be obtained for four-point correlations.

Replica Field Theory for the Dynamical Transition

We introduce (for convenience) an external field ν_a(x) that derives from a space-dependent chemical potential, in such a way that the density correlation functions can be obtained by taking the derivative of the free energy with respect to it (31). The free energy is defined as the logarithm of the partition function, and its double Legendre transform defines the Gibbs free energy (31, 33) (Eq. 6):

where , and Γ_2PI is the sum of two-line irreducible diagrams (33). The average values of the fields in Eq. 4, namely and , can be obtained by solving the saddle point equation (Eq. 7),

and similarly, they can be obtained for ρ_a(x). Here, we consider a homogeneous liquid; hence, ρ_a(x) = ρ.

We have to assume, at this point, that a mean field approximation of the free energy is available, which we shall use as the starting point of our computations. Within this approximation, we want to study the behavior of in the double limit m → 1 and v_a≠b → 0, which signal the dynamical transition: if T > T_d, then , whereas if T ≤ T_d, a nontrivial off-diagonal solution persists in the limit v_a≠b → 0. At the mean field level, the appearance of the nontrivial solution is a bifurcation phenomenon, and therefore, if we come from below the transition and we define ε = T_d − T, we have, for ε → 0 (Eq. 8),

where k₀(x) is normalized as , and κ is a constant. From the saddle point (Eq. 7), we obtain that the Hessian matrix for the off-diagonal elements (i.e., for a ≠ b, c ≠ d) (Eq. 9),

which is considered as a kernel operator both in standard and replica space, develops a zero mode at T_d. This finding means that, if the transition is approached from below, the fundamental eigenvalue of this operator is proportional to because of the bifurcation-like phenomenology. Moreover, the eigenvector corresponding to it is k₀(x − y).

Exploiting the replica symmetry of the saddle point solution (Eq. 7), the most general form of the Hessian matrix is given by (Eq. 10)

where M₁, M₂, and M₃ depend on x₁, …, x₄. From this equation, one can show that, because the zero mode k₀(x − y) is independent of the replica indices, in the replica limit m → 1, it is an eigenvector of the kernel operator M₁. To study the correlation functions for the fields in Eq. 4, we can produce a power series expansion of the Gibbs free energy in terms of the fluctuation of the field from its saddle point value. Defining the field , we can expand the Gibbs free energy up to the third order. It is convenient to define p_i and q_i as the momenta conjugated to the half-sum and the difference of the spatial arguments of Δρ_ab(x_i, y_i). Using translation invariance, we write the replica action in Fourier space as (Eq. 11)

Because of the zero mode of the Hessian matrix, the connected correlation function of Δρ_ab(x, y) shows critical fluctuations at the transition.

To make the connection with the dynamical correlation, we define an overlap function among replicas, q_ab(r), as in Eq. 1, substituting the configurations at time 0 and t by replicas a and b. We expect that all of the critical fluctuations of q_ab(r) can be captured by a projection on the zero mode, leading from Eq. 11 to an effective action. We can study the fluctuations of q_ab(r) for generic functions f by performing a Legendre transform of Eq. 11. However, the results are quite involved, and here, for clarity, we will first consider the simplest case, where f(x) = k₀(x). Of course, this process is not a practical choice for numerical simulations or experiments, because k₀ is quite difficult to measure; however, the theoretical computations are much simpler in this case. Later, we will show that any other choice of f leads to the same results for the critical quantities, and it only affects the prefactor of the correlation functions. The projection onto the zero mode can be done by choosing and substituting this finding in Eq. 11. The field ϕ_ab(x) is the component of the overlap along the zero mode, and we perform a perturbative expansion at small momentum p. The effective replica field theory that arises is equivalent to a Landau-like gradient expansion along the critical modes (Eq. 12):

Eq. 12 is the effective low-energy replica field theory that we will use to compute the critical properties of the system. All of its coefficients can, in principle, be computed from the microscopic details of the systems after an approximation for Γ is available. In fact, they can be given explicit expressions as functions of derivatives of the Gibbs free energy and the zero mode, which both derive from the interaction potential (SI Text).

Correlation Functions, Correlation Length, and Critical Exponents

The effective replica field theory in Eq. 12 can be used to compute the MCT parameter λ. This quantity is related to the MCT critical exponents that control the approach to the plateau by the relation (Eq. 13)

In addition, the exponent that controls the growth of the relaxation time τ_α ∼ |T − T_d|^−γ is given by γ = 1/(2a) + 1/(2b). Although λ is a dynamical parameter, it has been explicitly shown recently in disordered mean field models, and it can be argued on general ground (34) that this parameter can be related to a ratio of six-point static correlation functions computable in the replica field theory that we have just derived. In this scheme, the exponent parameter is given by (Eq. 14)

Moreover, the field theory above can be used at the Gaussian level to obtain the correlation functions of the overlap. The analysis of the quadratic part of Eq. 12 shows that the correlation length is controlled by the diagonal part, being m₂ and m₃ finite at the transition. The result is (Eq. 15)

and it corresponds to the divergence of the dynamical correlation length ξ(t) in the β-regime (22–24).

Moreover, we can compute in detail the critical behavior of many possible dynamical four-point functions that are identified with different matrix elements of the inverse of the Hessian matrix in Eq. 9 (19). Here, we give the results for the simplest one, the so-called in-state or thermal susceptibility, which is given by (Eq. 16)

where E₀[⋅] has to be intended as the average over the initial positions of the particles, whereas 〈•〉 is an average over different trajectories (i.e., over the noise for Langevin dynamics or over the initial velocities for Newton dynamics). In the long time limit, this quantity is one of the critical contributions to the G₄(r, t) in Eq. 2, and it can be computed directly from the replica field theory above (19). Here, we had to generalize the calculation in ref. 19 to take into account the structure of the zero mode and the presence of the smoothing function f(x). The result is (Eq. 17)

We obtain that the correlation length and its prefactor are not dependent on the function f(x) and are always given by Eq. 15. The only dependence on f(x) of the four-point function is in the prefactor G₀. The full four-point correlation (Eq. 2) is known to display a doubled singularity with respect to Eq. 17. In fact, with the choice f(x) = k₀(x), one finds G₄(p) = G_th(p) − (m₂ + m₃)G_th(p)² (19). For generic f(x), the computation of the prefactor is more involved and will not be presented here.

A Ginzburg Criterion

All of the calculations above are based on the assumption that a mean field approximation of the free energy of the system is given. From these calculations, we derive the effective Landau field theory (Eq. 12). From its coefficients, we extracted all of the mean field critical exponents as well as microscopic expressions for the prefactors. Now, we can check whether loop corrections to the effective field theory strongly affect the mean field predictions by means of a Landau–Ginzburg computation. In other words, we want to see whether the loop corrections to the bare correlation function are small. In principle, we should take the field theory derived above, and then, we should compute the first nontrivial loop diagrams that give the first correction to the propagator in replica space. This computation is quite involved, because we have to deal with replica indices. However, it has been shown in ref. 19 that the leading divergent behavior of the above field theory can be mapped to the one of a scalar field in a cubic potential with a random field (Eq. 18):

where the random field has zero mean and correlation , and the coupling constants are given by g = w₂ − w₁ and Δ = −m₂ − m₃.

The terms δm(g, Δ) and δh(g, Δ) are counterterms needed to enforce that the critical point is not shifted by loop corrections. By computing the first one-loop diagram and imposing that the relative correction is small with respect to the bare quantity, we arrive to the following Landau–Ginzburg criterion (19)

where the (dimensional) Ginzburg number is given by (Eq. 20)

This computation is correct only below the upper critical dimension D_u = 8. For D ≥ D_u, the theory is divergent in the UV, and the Ginzburg number depends on the microscopic details; however, the critical exponents coincide with the mean field ones.

Results in the HNC Approximation

Up to now, the calculations were very general, and the results above hold for any given approximation of the replicated free energy functional that displays the correct mean field glassy phenomenology. One of the advantages of our static approach is, indeed, that it can be systematically improved by considering more accurate approximations of Γ.

Here, we report results obtained from the replicated HNC approach that amount to neglecting the Γ_2PI term in Eq. 6, which has been shown to give the correct glassy phenomenology at the mean field level (14, 15). Applying the formulae above, we find that, in the HNC approximation, the parameter λ is given by (Eq. 21)

where , , and the direct correlation function c_ab(q) is related to h_ab(q) by the replicated Ornstein–Zernicke relation (14). Similar expressions can be obtained for all of the other coefficients (SI Text).

To produce concrete numerical results, we have numerically solved the HNC equations by standard methods (14) for a large variety of systems in D = 3. In particular, we have considered

• Hard Spheres (HS): v(r) = 0 for r > r₀ and v(r) = ∞ otherwise.

• Harmonic Spheres (HarmS): .

• Soft Spheres (SS-n): .

• Lennard–Jones (LJ): .

• Weeks–Chandler—Andersen (WCA): .

In all cases, we fix units in such a way that r₀ = 1, ε = 1, and the Boltzmann constant k_B = 1. For HS and SS, temperature is irrelevant (for SS, the only relevant parameter is a combination of density and temperature; hence, we fix T = 1 for convenience), and we study the system as a function of density to determine the glass transition density ρ_d. For the other systems, we studied the transition as a function of both density and temperature.

To obtain numerically the zero mode, we have used the definition in Eq. 8 and estimated it by the numerical derivative of with respect to when ε → 0. A plot of the zero mode for HS is in Fig. 1. Interestingly, we find that the zero mode has the same structure in Fourier space as the static structure factor S(q) and the nonergodicity parameter f(q), which is the Fourier transform of the long time limit of Eq. 1 in the glass phase (5). This finding offers a rationalization of the common practice of concentrating on momenta of the order of the peak of S(q) in the study of glassy relaxation.

Fig. 1.

The zero mode k₀(q), the structure factor S(q), and the nonergodicity factor f(q) for HS at the dynamical transition ρ_d = 1.176 in the HNC approximation.

From the zero mode, we can compute all of the coefficients of the effective action from which we obtain the physical quantities. In particular, we can compute the prefactor ξ₀ of the growth of the correlation length and the Ginzburg number. Moreover, we have computed the prefactor G₀ of the in-state susceptibility (Eq. 17) using a box function , where θ(x) is the Heaviside step function and A = 0.1r₀. All of the results are collected in Tables 1 and 2.

Table 1.

Numerical values of the coefficients of the effective action and the physical quantities from the HNC approximation

System	T	ρd	w1	w2	m2	m3	σ	μ	λ	ξ0	G0	Gi
SS-6	1	6.691	0.121	0.0845	−0.229	0.0273	0.0484	0.130	0.697	0.601	224	0.370
SS-9	1	2.912	2.41	1.70	−1.34	0.157	0.405	1.35	0.705	0.548	34.3	0.166
SS-12	1	2.057	8.58	6.08	−2.89	0.328	0.938	3.77	0.709	0.498	14.2	0.154
LJ	0.7	1.407	33.1	23.5	−6.39	0.719	2.45	10.3	0.709	0.489	6.00	0.108
HarmS	10−3	1.335	40.4	29.1	−8.34	0.850	1.92	19.3	0.719	0.315	2.82	0.535
HarmS	10−4	1.196	51.5	38.9	−10.0	0.957	2.03	27.0	0.756	0.274	1.69	0.622
HarmS	10−5	1.170	54.3	41.5	−10.3	0.979	2.09	27.1	0.764	0.278	1.66	0.593
HS		1.169	54.5	41.5	−10.3	0.984	2.10	26.7	0.761	0.280	1.67	0.606

For each potential, lengths are given in units of r₀, and energies are given in units of ε, with k_B = 1. Data are at fixed temperature using density as a control parameter with ε = ρ_d − ρ.

Table 2.

Same as Table 1, but here, the data are at fixed density using temperature as a control parameter with ε = T_d − T

System	ρ	Td	w1	w2	m2	m3	σ	μ	λ	ξ0	G0	Gi
LJ	1.2	0.335	58.2	41.4	−8.94	0.999	3.65	14.2	0.711	0.507	4.56	0.0937
LJ	1.27	0.438	47.9	33.8	−7.96	0.916	3.18	11.1	0.705	0.536	5.74	0.102
LJ	1.4	0.683	33.7	23.9	−6.46	0.726	2.49	7.24	0.710	0.586	8.52	0.106
WCA	1.2	0.325	61.0	42.9	−9.65	1.05	3.29	15.1	0.703	0.467	4.37	0.179
WCA	1.4	0.692	34.5	24.2	−6.68	0.746	2.39	7.21	0.701	0.576	8.67	0.143

The value of λ that we find is almost the same for all investigated systems and is consistent with the result of MCT (5) and numerical results for these systems. Note, however, that the location of the critical point predicted by HNC is different from the one of MCT (e.g., for HS, HNC predicts ρ_d = 1.169, whereas MCT predicts ρ_d = 0.978) (5). This result is an example of the fact, already mentioned in the Introduction, that different approximation schemes lead to different results. Another example of this problem is obtained by comparing the results for LJ and WCA at ρ = 1.2, 1.4 (Table 2) with MCT and the numerical data reported in table 1 in ref. 35. The most interesting numerical result is the Ginzburg number. We predict that (perturbative) corrections to mean field results in D = 3 should remain small as long as the dynamical correlation length is smaller than ∼1. Note that a different Ginzburg criterion for the validity of MCT, based on a phenomenological approach, has been derived in ref. 20: the results of that analysis also suggest that corrections to mean field will appear when the correlation length is ∼1.

Unfortunately, not many data for the critical behavior of four-point correlations in the β-regime are available (36, 37). It would, thus, be very interesting to get high-precision simulation data in the β-regime.

Conclusions

We have studied the replica field theory for the dynamical transition in glasses in detail. By using the HNC approximation, we have computed many physical observables directly from the microscopic expression of the interaction potential. First, we provided a way to compute the mode-coupling exponent parameter λ. The numerical values obtained are in good agreement with the experimental and numerical estimates. Second, we have computed the prefactor of the correlation length at the transition together with the prefactor of the in-state four-point correlation function. Third, we have self-consistently closed our analysis by looking at the loop corrections to the mean field quantities to produce a Ginzburg criterion that states how close we have to be to the dynamical transition to see deviations from mean field theory. We found that the range currently accessible to numerical simulations in 3D is close to the point where such corrections should become important. Of course, nonperturbative corrections (activated processes) are not included in our analysis, but they are responsible for strong deviations from the MCT regimen when the transition is approached.

Our analysis is quite general, because it relies only on the assumption that the approximation scheme used for the Gibbs free energy shows the correct mean field glassy phenomenology. Hence, it can, in principle, be repeated in different approximation schemes to go beyond HNC and obtain more accurate expressions for physical quantities.