Fluctuations in glassy systems
Claudio Chamon1 and Leticia F. Cugliandolo2
1 Physics Department, Boston University,
590 Commonwealth Avenue, Boston, MA 02215, USA
2Laboratoire de Physique Théorique et Hautes Énergies, Jussieu,
5ème étage, Tour 25, 4 Place Jussieu, 75252 Paris Cedex 05, France
chamon@bu.edu, leticia@lpt.ens.fr
Abstract. We summarize a theoretical framework based on global time-
reparametrization invariance that explains the origin of dynamic fluctuations in glassy
systems. We introduce the main ideas without getting into much technical details.
We describe a number of consequences arising from this scenario that can be tested
numerically and experimentally distinguishing those that can also be explained by other
mechanisms from the ones that we believe, are special to our proposal. We support our
claims by presenting some numerical checks performed on the 3d Edwards-Anderson
spin-glass. Finally, we discuss up to which extent these ideas apply to super-cooled
liquids that have been studied in much more detail up to present.
http://arxiv.org/abs/0704.0684v1
CONTENTS 2
Contents
1 Why glasses? vs. universality in glassy dynamics 2
2 Time reparametrization invariance 5
2.1 Mean-field models – dynamic equations . . . . . . . . . . . . . . . . . . . 5
2.2 Structural glasses: the p ≥ 3 cases . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Short-ranged models – dynamic action . . . . . . . . . . . . . . . . . . . 10
2.4 Turning a nuisance into something useful - symmetry as a guideline . . . 11
2.5 The spherical p = 2 case or mean-field domain growth . . . . . . . . . . . 13
2.6 Quantum problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Consequences and tests 14
3.1 Two-time correlation length . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Scaling of the pdf of local two-time functions . . . . . . . . . . . . . . . 17
3.3 Effective action for local ages . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Two-time scaling of local functions . . . . . . . . . . . . . . . . . . . . . 21
3.5 Multi-time scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Local fluctuation-dissipation relation . . . . . . . . . . . . . . . . . . . . 25
3.7 Infinite susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Discussion 27
1. Why glasses? vs. universality in glassy dynamics
It is common to encounter in nature systems that resist equilibration with their
environments and display what is called glassiness. The name is derived from what
we normally know as glasses, an irregular array of silicon and oxygen atoms without
crystalline order, much as a liquid, but as hard as a solid. The molecular diffusion
within glasses is extremely sluggish, slowing down by over 10 orders of magnitude as
the temperature is slightly decreased near the operationally defined glass transition
temperature. Hence, the term glassiness became generically associated with very slow
dynamics [1].
So, why glasses? The answer to this question has been the focus of much research
effort for long. It is certainly a rather difficult question, and there have been a number
of ideas lined up for trying to understand how material systems become glassy. It is not
even clear whether in many systems there is a thermodynamic phase one can call glass,
or there is simply a dynamical crossover at low enough temperatures.
Do we need to fully answer why glasses? before we really further our understanding
of glassy dynamics? We take the point of view that, by starting from the fact that glassy
systems exist (as nature presents us with concrete examples), we can then attempt to
characterize whatever possible universal properties there are in glassy dynamics.
CONTENTS 3
To make this statement clearer, let us turn to a question that Anderson poses in
the introduction of his Concepts in solids text [2]: why solids? This question, again,
is a rather complex one, and it is not untwined from the question why glasses? if we
focus on why a regular array of atoms, as opposed to a random packing, forms in the
first place. Even if one assumes that a crystalline structure forms at low temperatures,
a detailed quantitative analysis of the energetics remains to be done so as to determine
if the packing is hexagonal, cubic, body or face centered, etc. Nonetheless, if one starts
from the existence (as observed in nature) of a state with broken translational symmetry,
one can construct theories of lattice vibrations (and quantize it) and of electronic band
structure (and discern between insulators, metals and semiconductors). Solid state
physics starts from the solid.
The approach we review in this paper relies on a similar philosophy: we do not
claim any theory of the glass transition, and we do not attempt to answer why glasses?
with this particular approach. Our theory does not allow us to make non-universal
predictions, such as what the glass transition temperature is (if one can be defined) for
a certain material, or whether the material displays glassy behavior at all. We aim at
understanding if there is a set of principles, guided by symmetry considerations, that
can be used to understand certain universal aspects of glassy dynamics once the glass
state is presented. For example, glasses age [3]. We thus expect that there exists a
unified approach to describe aging phenomena, and we seek some guiding principles,
based on dynamical symmetries, that could allow us to understand universal properties
in the aging regime, including the scaling of spatial heterogeneities.
We propose that the symmetry that captures the universal aging dynamics
of glassy systems is the invariance of an effective dynamical action under uniform
reparametrizations of the time scales [4]-[9]. Such type of invariance had been known
to exist since the early days in which the mean-field equilibrium dynamics of spin-
glasses was tried to be understood [10, 11] and it was later encountered in the better
formulated out of equilibrium dynamic formalism applied to the same systems [12, 13].
The invariance means that in the asymptotic regime of very long times a family of
solutions to the equations of motion is found. This ‘annoyance’, we claim, has actually
a physical meaning and implications in the fluctuating dynamics of real glasses. In
order to make this statement concrete, it is necessary to show that the invariance
also exists in finite dimensional glassy models. With this aim we showed that global
time reparametrization invariance emerges in the long times action of short-range spin-
glasses assuming causality and a separation of time scales [4]-[6]. Basically, the second
assumption amounts to start from a glass state, where one may claim there is a
separation between, roughly, a time regime where relaxation is fast and another where
relaxation is slow. The invariance can then be used to describe dynamic fluctuations in
spin-glasses and, we conjectured, in other glassy systems as well.
Physically, the emergence of time reparametrization invariance can be though of
in the following way. The out of equilibrium relaxation of glassy systems is well
characterized by two-time functions, either correlations or linear responses. In the slow
CONTENTS 4
and aging regime they depend on two times and time-translation invariance is lost. The
proper measure of ‘time’ inside the system is the value of the correlation itself, and not
the ‘wall clock’ in the laboratory. For instance, in a spin-glass the proper measure of
sample age is the spin-spin overlap or in particle systems it is the incoherent correlation
function. Age measures can fluctuate from point to point in the sample, what we called
heterogeneous aging, with younger and older pieces (lower and higher values of the
correlation) coexisting at the same values of the two laboratory times. The fact that
the effective dynamical action becomes invariant under global time reparametrizations,
t → h(t), everywhere in the sample, means that the action weights the fluctuations
of the proper ages, C(~r; t1, t2), directly, and the times t1 and t2 in the action are just
integrated over as dummy variables. To draw an analogy, in theories of quantum gravity
the space-time variables Xµ(τ, σ) are the proper variables, and the action is invariant
under conformal transformations of the world-sheet parameters τ and σ.
So what does global time-reparametrization invariance symmetry concretely teach
us about observables in glasses? So far we discussed a global symmetry or invariance
with respect to uniform time-reparametrizations. By looking at spatially heterogeneous
reparametrizations, we can predict the behavior of local correlations and linear
susceptibilities and the relations between them. For example, we predict that, after
a convenient normalization that we explain in the main text, the triangular relation
between the local coarse-grained correlations, C(~r; t1, t2), C(~r; t2, t3) and C(~r; t1, t3), as
a function of the intermediate time t2, t3 < t2 < t1, at all spatial points, ~r, should be
identical to the global triangular relation, in the asymptotic limit of very long absolute
times and delays between them and very large coarse-graining linear length [9]. Different
sites can be retarded or advanced with respect to the global behaviour but they should all
have the same overall type of decay. Similarly, the relation between local susceptibilities
and their associated correlations should be identical all over the sample [5, 6] leading to
a uniform effective temperature [14].
The purpose of this article is to describe our current understanding of dynamic
fluctuations (heterogeneities) in the non-equilibrium relaxation of glassy systems arising
from the time-reparametrization invariance scenario [4]-[9],[15]. We illustrate it by
presenting critical tests.
The structure of this review is the following. In Sect. 2 we explain the origin of
the time-reparametrization invariance scenario. We do not present detailed derivations
that were already published but we aim at highlighting the main ideas behind the scene.
In Sect. 3 we list several measurable consequences of the theory. We discuss how they
have been examined numerically in different glassy models. The discuss here which
consequences could also be explained by other approaches and which, we believe, are
unique to our scenario. Finally, in Sect. 4 we discuss the scenario. Since this research
project is not closed yet, we present some proposals for numeric and experimental tests
as well as some ideas about analytic calculations that could help us understanding the
limitations of our proposal.
CONTENTS 5
2. Time reparametrization invariance
In this Section we explain how global time-reparametrization invariance develops
asymptotically in the aging regime of glassy models. For the sake of simplicity we focus
on the classical formalism and at the end of this section we mention the modifications
introduced by quantum fluctuations.
2.1. Mean-field models – dynamic equations
Schematic models of spin-glasses, structural glasses and ferromagnetic clean coarsening
are encoded in the family of p-spin models defined by [16]
H = −
i1i2...ip
Ji1i2...ipsi1si2 . . . sip (1)
with quenched disordered exchanges distributed with the Gaussian law P (Ji1i2...ip) ∝
−p!J2
i1i2...ip
/(2Np−1)
. The dynamic variables si, i = 1, . . . , N , are of Ising type, si = ±1,
or satisfy a global spherical constraint
i=1 s
i = N . p is an integer parameter: with
p = 2 and Ising variables one mimics spin-glasses, with p > 2 and Ising or spherical
spins the phenomenology of structural glasses is recovered, and with p = 2 and spherical
spins one describes ferromagnetic domain-growth in clean systems. The sum runs over
all p-uplets; for this reason these models are ‘mean-field’ in the sense that the saddle-
point evaluation of the partition function or the dynamic generating functional is exact
in the thermodynamic limit, N → ∞. The Hamiltonian (1) also represents the potential
energy of a particle with position ~s = (s1, . . . , sN) on an infinite dimensional hypercube
(si = ±1) or hypersphere (
i=1 s
i = N).
Dynamics is introduced with a Langevin equation that represents the coupling of
the spins to an equilibrated thermal environment. Ising spins are then replaced by
soft variables by introducing a double-well potential energy,
i=1 V (si), with V (si) =
a(s2i − 1)
2. The initial condition is usually chosen to be random thus mimicking a rapid
quench from infinite temperature to the working temperature T . In the limit N → ∞
exact Schwinger-Dyson equations couple the global correlation and instantaneous linear
response
NC(t, tw) =
si(t)si(tw) , NR(t, tw) =
δsi(t)
δhi(tw)
i=1 si(t) = 0 for all t). The field hi couples linearly to the spin variables and, in
general, we are interested in perturbing fields that are uncorrelated with the equilibrium
states of the systems. It is not necessary to average over thermal noise or quenched
disorder since these quantities do not fluctuate in the out of equilibrium regime reached
when the infinite volume limit (N → ∞) has been taken at the outset. Here are in
what follows times are measured from an origin that corresponds to the quench to the
working temperature. Our study applies then in the order of limits
CONTENTS 6
in which the exact causal Schwinger-Dyson equations for spherical models at times t ≥ tw
read [12, 19]
(∂t − zt)C(t, tw) =
dt′ Σ(t, t′)C(t′, tw) +
dt′ D(t, t′)R(tw, t
′) , (4)
(∂t − zt)R(t, tw) =
dt′ Σ(t, t′)R(t′, tw) + δ(t− tw) , (5)
where the vertex, D, and self-energy, Σ, are functions of C and R
D(t, tw) =
Cp−1(t, tw) , Σ(t, tw) =
p(p− 1)
Cp−2(t, tw)R(t, tw) . (6)
The Lagrange multiplier zt is fixed by requiring C(t, t) = 1. For Ising problems the soft
spin constraint can be treated in the mode-coupling approximation [17] or else, one can
apply a T − Tc expansion taking advantage of the fact that the phase transition is of
second order for p = 2 (Sherrington-Kirkpatrick or SK model) [13].
In such fully-connected models all higher order correlations and linear responses
factorize and can be written as functions of these two-time functions. Self-consistent
approximate treatments of interacting particle models with realistic potentials, like the
mode-coupling approach, yield similar equations with the addition of a wave-vector
dependence that in the present context can also be taken into account by considering
models of d-dimensional random manifolds embedded in N → ∞ dimensional
spaces [18, 19].
2.2. Structural glasses: the p ≥ 3 cases
We now focus on the p ≥ 3 cases that mimic the structural glass problem. We mention
at the end of this subsection the Ising p = 2 (SK) spin-glass case that is not conceptually
different but just technically more involved. In Sect. 2.5 we discuss the p = 2 spherical
problem that yields a mean-field description of coarsening phenomena and is rather
different from the point of view of time-reparametrization transformations.
Equations (4) and (5 are causal and be solved numerically by constructing C(t, tw)
and R(t, tw) from the initial instant t = tw = 0. An analytic solution is possible in the
asymptotic limit, as we discuss below. We first present the main features of C and R
and we later explain how these are obtained from the asymptotic analytic solution.
Equations (4) and (5) have a dynamic transition at a critical temperature Td(p). At
T > Td the dynamics occurs in equilibrium and close to Td the decay of the correlations
slows down as in super-cooled liquids with the α relaxation time diverging as a power law
of T − Td [16, 20]. Below Td eqs. (4) and (5) admit a unique solution [12, 13] that is no
longer stationary. The behaviour of the low-temperature correlation and susceptibility
is sketched in Fig. 1.
The low-temperature solution presents a separation of time-scales. In the long
waiting-time limit the self-correlation and integrated linear response or susceptibility,
χ(t, tw) ≡
dt′R(t, t′), can be written as
C(t, tw) = Cst(t− tw) + Cag(t, tw) , (7)
CONTENTS 7
1e+00
1e-01
1e-02
1e+051e+031e+011e-01
rapid & stationary (C st )
aging &
(Cag)
1e+00
1e-01
1e+051e+031e+011e-01
rapid & stationary (χ st)
aging & slow (χag)
Figure 1. Sketch of the relaxation of the self-correlation and susceptibility in the
glassy regime. The separation of time-scales is clear in the figure. The Edwards-
Anderson parameter, qea the corresponding susceptibility χea and the asymptotic value
limt→∞ χ(t, tw) are indicated with horizontal lines.
χ(t, tw) = χst(t− tw) + χag(t, tw) . (8)
The first terms in the right-hand-side describe the stationary regime at short time-
differences in which the correlation and susceptibility relatively rapidly approach a
plateau at limt−tw→∞ limtw→∞C(t, tw) = qea and limt−tw→∞ limtw→∞ χ(t, tw) = (1 −
qea)/T ≡ χea. The second terms are the aging relaxation of C towards zero (in
the absence of an external field), and the aging response of the system towards the
asymptotic value χea + qea/Teff with Teff a parameter with the interpretation of an
effective temperature [14].
The stationary and aging relaxation are fast and slow in the sense that
∂tCst(t, tw) ∼ Cst(t, tw) C > qea , (9)
∂tCag(t, tw) ≪ Cag(t, tw) C < qea , (10)
∂tχst(t, tw) ∼ χst(t, tw) χ < χea , (11)
∂tχag(t, tw) ≪ χag(t, tw) χ > χea . (12)
The aging self-correlation and susceptibility scale as
Cag(t, tw) ≈ qea fc
R(tw)
, χag(t, tw) ≈ qea fχ
R(tw)
. (13)
The scaling functions satisfy the limit conditions fc(1) = 1, fc(∞) = 0, fχ(1) = 0
and fχ(∞) = 1/Teff . Using mathematical properties of monotonic two-time functions
one can show that such a scaling holds asymptotically in each (two) time-scale of the
evolution [13]. While in a system undergoing finite dimensional coarsening R(t) has
a natural interpretation as the typical domain radius, in mean-field models there is
no immediate understanding of the ‘clock’ R(t) that, in a sense, sets the macroscopic
time-scale. The numerical solution suggests that R(t) is just a power of time.
CONTENTS 8
In the asymptotic limit in which the additive separation of time-scales with the
scaling form (13) holds it is convenient to use a parametric description of the dynamics
in which times do not appear explicitly. More precisely, the approach to the asymptotic
scaling, and fc and fχ, can be put to the test by constructing ‘triangular relations’
between correlations and susceptibilities, respectively. For generic three long times
t1 ≥ t2 ≥ t3 ≫ t0 one computes the correlations C(tµ, tν), µ > ν = 1, 2, 3. If the times
are such that the ratios R(tµ)/R(tν) remain finite in the asymptotic limit, that is to
say C(tµ, tν) = Cag(tµ, tν), one has
C(t1, t3) = qea fc
f−1c [C(t1, t2)/qea] f
c [C(t2, t3)/qea]
. (14)
If, instead, t1 = t2 + τ with τ > 0 finite, and R(t2)/R(t3) finite in such a way that
C(t1, t2) = qea + Cst(t1 − t2) and C(t2, t3) = Cag(t2, t3) one has
C(t1, t3) = min [C(t1, t2), C(t2, t3) ] (15)
asymptotically. This form goes under the name of dynamic ultrametricity. In the
opposite case t3 = t2 − τ and R(t1)/R(t2) finite dynamic ultrametricity also holds.
These relations follow immediately from the additive separation of time-scales (8) and
the scaling (13) but they can be shown without assuming dynamic scaling, just by
using the monotonicity properties of temporal correlations [13]. The simplest way to
see these relations at work is to display C(t1, t2) against C(t2, t3), for a chosen value of
C(t1, t3) < qea, in a parametric plot in which t2 varies from t3 to t1. In the asymptotic
limit t3 → ∞ the construction reaches a stable master curve as displayed in Fig. 2-
left. The vertical and horizontal parts correspond to t2 such that C(t1, t2) > qea and
C(t2, t3) > qea, respectively, and dynamic ultrametricity holds. The curved part is for
t2 such that all correlations are in the aging regime and its functional form is fully
determined by fc. The ‘clock’ R yields the speed at which the data-point moves on the
parametric curve. A similar construction can be done for the susceptibilities.
The stationary correlation, Cst, and susceptibility, χst, are linked by the equilibrium
fluctuation dissipation theorem (fdt), χst = (1− Cst)/T . In the aging regime, instead,
there is a non-trivial relation between χ and C: χag = (qea − Cag)/Teff that yields
fχ(x) = (1−fc(x))/Teff . This relation is also better appreciated if shown in a parametric
construction in which times do not appear explicitly. In the long waiting-time limit the
plot χ(t, tw) against C(t, tw) with t the parameter running from tw to infinity approaches
a broken line form with the slopes −1/T (for C > qea) and −1/Teff (for C < qea). Again,
the ‘clock’ R yields the speed at which this curve is constructed upon increasing t.
An analytic solution to the Schwinger-Dyson equations was derived in the limit
of long waiting-time in which the separation of time-scales, that is to say the plateaus
in C and χ, are fully established. In the aging regime, one uses the fact that the
variation of the correlation and linear susceptibility are negligible with respect to all
terms in the right-hand-side of the equations and can thus be dropped. Furthermore,
one approximates the integrals by separating the contributions from the stationary and
CONTENTS 9
10.80.60.40.20
10.80.40.20
Figure 2. Sketch of the parametric representation of the correlation and susceptibility.
Left: triangular relation between the correlation function in the asymptotic limit
t3 → ∞. The three curves correspond to different t1’s such that C(t1, t3) takes three
values. The breaking points lie at qea. The arrow indicates the sense of the evolution
when t2 increases from t3 to t1. Right: susceptibility, χ(t2, t3) against correlation,
C(t2, t3) at fixed t3 using t2 ≥ t3 as a parameter in the long t3 limit. The breaking point
at (qea, χea) separates the stationary regime where the equilibrium fdt is satisfied from
the aging regime where it is modified. The slopes are −1/T and −1/Teff , respectively.
The arrow also indicates here the sense of the evolution when t2 increases from t3.
aging regimes [19]. Equation (5) becomes
µ∞Rag(t, tw) ∼
p(p− 1)
dt′ Cp−2ag (t, t
′)Rag(t, t
′)Rag(t
′, tw) (16)
(µ∞ is a constant with contributions from limt→∞ zt and border terms in the integrals).
The companion eq. (4) takes a similar form within the same approximation. Now, the
surprise is that the approximate equations are invariant under the transformation
t→ ht ≡ h(t) ,
Cag(t, tw) → Cag(ht, htw) ,
Rag(t, tw) → ḣtw Rag(ht, htw) ,
with ht positive and monotonic and ḣtw ≡ dhtw/dtw. While the functions fc and
fχ and their fd relation are fixed by the remaining approximate equation (16) and
its companion, the time-reparametrization invariance does not allow one to compute,
analytically, the clock R(t). This problem is similar to the velocity selection problem
present for instance in Fisher differential equation describing front propagation and the
like [21]. The exact Schwinger-Dyson equations do have a unique solution with a special
function R(t) that is selected by the short-time difference effect of the time-derivatives.
However, as time increases and time-differences increase too the effect of the time-
derivatives diminishes. In the approximate analytic solution we take advantage of this
fact to solve the equations asymptotically but we introduce in this way a symmetry that
does not allow us to fix R(t). We obtain, instead, a family of solutions parametrized by
ht. It is important to reckon that the parametric constructions in Fig. 2 are independent
of the clock and thus are fully determined by the approximate treatment.
CONTENTS 10
The case p = 2 with Ising spins or Sherrington-Kirkpatrick model has a more
complicated scaling form with a sequence of two-time scales leading to dynamic
ultrametricity for all C < qea asymptotically [13]. This behaviour is technically more
involved but, as far as the symmetry properties are concerned, it is similar to the case
treated above. The full dynamic equations have a unique solution but the approximate
ones acquire time-reparametrization invariance. We shall not discuss these cases further
in the rest of this review.
2.3. Short-ranged models – dynamic action
In a series of papers [4]-[9] we claimed that the time-reparametrization invariance thus
far introduced via the asymptotic solution to the dynamic equations in mean-field glassy
problems is indeed an asymptotic property of the dynamics of glassy systems, mean-field
and finite dimensional. The separation of time-scales stationary-aging has been observed
in a variety of glassy systems with numerical simulations and experiments [3, 22]. The
slowness of the decay in the aging regime, eqs. (10) and (12) and a weak long-term
memory of the kind (13), are the hallmark of glassy relaxation. The idea is then that
time-reparametrization invariance is the symmetry associated to the dominant dynamic
fluctuations in these sytems.
In order to pursue this idea forward one has to first prove that the symmetry of
the saddle-point equations is also a symmetry of the action in the dynamic generating
functional not only of fully-connected spin models of the mean-field type but also of finite
dimensional glassy systems. In [4] we derived and studied the symmetry properties of
the dynamic action – the so-called Martin-Siggia-Rose action associated to Langevin
stochastic dynamics – of the disorder averaged soft-spin 3d Edwards-Anderson (ea)
model of spin-glasses. (H =
〈ij〉 Jijsisj with Jij Gaussian random variables with zero
mean and taking non-zero value only on nearest neighbours on the d-dimensional lattice
and si = ±1.) In our analysis we took a number of steps that we briefly recall here.
First, we introduced four fluctuating two-time fields defined on the lattice sites,
Qabi (t, tw) , with i = 1, . . . , N and a, b = 0, 1 . (18)
Their thermal averages are the expected values of the local two-time self-correlation
(a = b = 0), the retarded linear response (a = 0, b = 1), the advanced linear response
(a = 1, b = 0), and a fourth observable (a = b = 1) that vanishes if causality is preserved.
Second, we assumed that a separation of time-scales fast-slow, of the type described in
the previous subsection, applies to these fluctuating fields too. Third, we determined the
long-time action by using a Renormalization Group (RG) scheme in the time variables.
This allowed us to write the full action as a sum of two contributions: one from the fast
regime holding at short time-differences, another one from the slow regime valid at long
time-differences. The coupling between these two vanishes asymptotically. Fourth, we
analyzed the surviving terms in the action, that are just the slow contribution. Using
advanced and retarded scaling dimensions that are just the labels a and b of the fields,
CONTENTS 11
one finds that the global time-reparametrization,
t→ ht ≡ h(t) , Q̃
i (t, tw) = (ḣt)
a(ḣtw)
b Qabi (ht, htw) , (19)
leaves all surviving terms unmodified. This step is concisely carried out as follows. Take
a generic term in the action. Under (19) it transforms as
dtν · · · →
dtν (ḣtν )
∆ν · · ·
where N is the number of time integrals, the dots represent a product of the fields
Qabi and ∆ν is the sum of all factors ḣtν arising from the transformation of the fields.
Interestingly enough, one finds that all the ∆ν equal one. Thus, with a simple change
of variables one absorbs each factor ḣtν in the corresponding integration variable and
dtν · · · →
dhν · · · (20)
Note that in order to prove invariance under the simpler and more common scale
transformation, ht = λt, it is enough to have
ν=1∆ν = N . Scale invariance is included
in the larger global time-reparametrization symmetry but it is, clearly, more restrictive.
Finally, we showed that the measure in the functional integral is also global time-
reparametrization invariant, completing the program. We refer the reader to Refs. [4]
and [6] for the technical details leading to these results.
In the disordered 3d ea model studied we carried out the disorder average. The
presence of quenched disorder gave us an analytic control of the theory but this does
not necessarily mean that such a symmetry develops only for the long time regime
of models with quenched disorder. It appears that if the essential assumptions are
causality and unitarity, and a separation of time scales that takes the action to a non-
trivial asymptotic state (some glassy state), then one expects the symmetry to exist
for systems without quenched disorder but that are glassy nevertheless. In order to
check the development of time-reparametrization invariance in problems of particles in
interaction in finite dimensions one should first obtain the relevant action to work with.
A good candidate for a starting point is the Dean-Kawasaki stochastic equation for the
evolution of the local density [23]. The idea is then to write its dynamic generating
functional and study the symmetry properties of the effective action assuming that a
separation of time-scales exists. We are currently carrying out this study.
2.4. Turning a nuisance into something useful - symmetry as a guideline
The global time-reparametrization invariance implies that the action describing the long
time slow dynamics of a spin-glass is basically a “geometric” random surface theory, with
the Q’s themselves as the natural coordinates. The original two times parametrize the
surface. Physical quantities, as the bulk integrated response χ(t1, t2) =
dt′R(t′, t2)
and correlation C(t1, t2) have scaling dimension zero under t → h(t) as well as their
local counterparts. The emergence of this gauge-like symmetry, which may appear first
CONTENTS 12
as a nuisance that relates too many solutions for just one problem. However, it may
provide a simple way to understand spatial fluctuations in systems that possess this
global time-reparametrization symmetry.
Here, a simple analogy with the problem of a ferromagnet may elucidate the point
we want to make. In a ferromagnet the action is invariant under uniform rotations of
the magnetization vector ~m. In the ordered phase, rotation symmetry is spontaneously
broken, and a certain magnetization direction ~m0 is picked. Typically, a vanishingly
small pinning field selects this direction. The low action excitations are the spin waves,
fluctuations of the uniform, symmetry broken, state. These spin waves can be described
in terms of smooth spatially fluctuating rotations around the uniform magnetization
state. The spin waves, generated by using slowly varying local rotations, are the
Goldstone modes of the ferromagnet.
Similarly, in the aging regime of the glassy systems, the action has a global
symmetry, under uniform time-reparametrizations, t → h(t), with the fields
transforming as in eq. (19). The probability weight of having certain local two-
time correlation and response, the observables, should be independent of this
reparametrization. After coarse-graining over a linear length ℓ the non-vanishing
fluctuating fields are the local coarse-grained correlation (a = b = 0) and linear response
(a = 0, b = 1),
C(~r; t, tw) =
j∈V~r
sj(t)sj(tw) , R(~r; t, tw) =
j∈V~r
δsj(t)
δhj(tw)
with the sum carried over the spins in the volume V~r = ℓ
d centered at ~r, and a is the
lattice spacing. The transformation (17) is now restated as
t→ ht ≡ h(t) ,
Cag(~r; t, tw) → Cag(~r; ht, htw) ,
Rag(~r; t, tw) → ḣtw Rag(~r; ht, htw)
, (21)
and it is an asymptotic symmetry of the action for the slow coarse-grained degrees of
freedom. Indeed, the symmetry breaking terms, that have their origin in the short-
time dynamics and short-time difference dynamics, are not identical to zero but become
vanishing small asymptotically. The particular scaling function R(t) selected by the
system is determined by matching the fast and the slow dynamics. It depends on several
details – the existence of external forcing, the nature of the microscopic interactions, etc.
In other words, the fast modes which are absent in the slow dynamics act as symmetry
breaking fields for the slow modes.
In analogy with the spin-wave fluctuations in magnetic systems, that are dictated by
the rotational symmetry, we proposed that the smooth fluctuations in the glassy phase
can be obtained by studying the slow varying, position dependent reparametrizations
t→ h(~r, t) around the one reparametrization R(t) selected by the short-time dynamics.
In other words, we basically proposed that there are Goldstone modes for the
glassy action which can be written as slowly varying, spatially inhomogeneous time
reparametrizations. This suggests that the slow part of the coarse-grained local
CONTENTS 13
correlations and susceptibilities should scale as
Cag(~r; t, tw) ≈ qea fc
h(~r, t)
h(~r, tw)
, χag(~r; t, tw) ∼ fχ
h(~r, t)
h(~r, tw)
, (22)
with fc and fχ the same functions describing the global correlation and susceptibility,
respectively [eqs. (13) and (23)] and the same function h(~r, t) scaling the two-time
correlation and susceptibility on each site ~r [4, 5, 6]. The sum rules Cag(t, tw) =
ddr Cag(r; t, tw) and χag(t, tw) = V
ddr χag(r; t, tw) apply.
The reason for this proposal is that the global reparametrization invariance in time
of the dynamic action in this two-time regime leads to low action excitations (Goldstone
modes) for smoothly varying spatial fluctuations in the reparametrization of time, but
not in the external form of the scaling functions. As in a sigma model (for example, to
describe the ferromagnet), the external functions fc and fχ fix the manifold of states,
and the local time reparametrizations correspond to fluctuations restricted to this fixed
manifold of states (in the ferromagnet, the tilting of direction but not the magnitude of
the magnetization vector).
2.5. The spherical p = 2 case or mean-field domain growth
The spherical SK model (p = 2) can be solved exactly by analyzing the Langevin
equation in the basis of eigenvectors of the random matrix Jij [24]. While the correlation
has a very similar behaviour to the one of the p ≥ 3 cases (see Fig. 1-left), the
susceptibility is quite different. One can mention that the stationary and aging regimes
in the linear response are not so sharply separated in this case. If one uses an additive
separation as in (8) the aging contribution to the integrated linear response, χag(t, tw),
vanishes asymptotically. More precisely,
χag(t, tw) ∼ t
. (23)
Importantly enough, even though the inequalities (10) and (12) are still valid, a
careful inspection of all terms in the Schwinger-Dyson equations shows that they are of
the same order asymptotically. Moreover, the stationary contribution to the equations
in the aging regime is not just a constant: the corrections associated to the asymptotic
approach to the plateau in the correlation cannot be neglected. As a result one cannot
simply drop the time derivatives and the Schwinger-Dyson equations in the aging regime
are not time-reparametrization invariant but just scale invariant, that is to say, they are
unchanged by the transformation t→ h(t) = λt, with C and R transforming as in (17)
and λ a positive constant. A similar mechanism, though even harder to prove, applies
to the dynamic equations for the correlation and linear response of the non-conserved
dynamics of the O(N) ferromagnetic model in the large N limit [8].
In line with what we explained above, the effective action for the slow degrees
of freedom of the p = 2 spherical model and, more generally, the d-dimensional
ferromagnetic O(N) model in the large N limit, are not invariant under global time-
reparametrizations but only under global rescaling of time, t → λt. This marks
CONTENTS 14
an important difference between models with a finite aging response and these quasi
quadratic models with a vanishing aging response [8].
This result is important for a number of reasons: first, it suggests that the
susceptibility, or even the effective temperature, might be intimately related to the
symmetry properties of the dynamics and consequently of the fluctuations; second, it
suggests that the mechanism for fluctuations in coarsening systems might be different
from the one of other glassy problems with finite and well-defined effective temperatures.
In order to justify the latter statement it remains to be checked whether the reduction of
time-reparametrization invariance to just scale invariance also holds in finite dimensional
non-field coarsening.
2.6. Quantum problems
In the case of quantum models one introduces the effect of dissipation by coupling the
system to an environment represented, typically, by an infinite ensemble of quantum
harmonic oscillators. One then uses the Schwinger-Keldysh formalism to write a
generating functional and from it, in the fully-connected or infinite dimensional cases,
one derives Schwinger-Dyson equations similar to the ones above. The asymptotic
analysis of these equations follows the same steps as in the classical limit [25, 17] (at
least in the case of a weak coupling to the bath [26]) and the time-reparametrization
invariance also applies.
The appearance of an asymptotic invariance under time-reparatrizations in the
mean-field dynamic equations was related to the reparametrization invariance of the
replica treatment of the statics of the same models [10, 27]. The latter remains
rather abstract. Brézin and de Dominicis [27] studied the consequences of twisting the
reparametrizations in the replica approach. Interestingly enough, this can be simply
done in a dynamic treatment either by applying shear forces or by applying heat-
baths with different inherent dynamics to different parts of the system. More precisely,
using a model with open boundary conditions one could apply a thermal bath with
a characteristic time-scale on one end and a different thermal bath with a different
characteristic time-scale on the opposite end and see how a time-reparametrization
‘flow’ establishes in the model.
3. Consequences and tests
In this Section we discuss how one can put these ideas to the test by presenting a number
of consequences of global time-reparametrization invariance that are directly measurable
numerically and experimentally. The properties that we discuss explicitly are:
1. A growing dynamical correlation length.
2. Scaling of the pdf of local two-time functions.
3. Functional form of the pdf of local two-time functions.
4. Triangular relations between two-time functions.
CONTENTS 15
5. Scaling relations for general multi-time functions.
6. Local fluctuation-dissipation relations.
7. Infinite susceptibilities.
In considering these predictions, we shall separate them into two distinct
classes. The first class contains predictions that are consistent with other theoretical
scenarios as well as with the presence of reparametrization invariance. Hence, while
reparametrization invariance leads to these predictions, this class alone cannot be used
to argue for the role of the symmetry over other mechanisms. The second class, on the
other hand, contains predictions that are not natural within other frameworks, and to
the date of this report have no obvious explanation within other frameworks. Properties
1 and 2 belong to the first class, properties 3-7 to the second. We shall present these
properties in detail below.
3.1. Two-time correlation length
In equilibrium statistical models one defines the static correlation length, ξeq, from the
spatial decay of the correlation between the fluctuations of the order parameter measured
at two space points, 〈 [φ(~r) − 〈 φ(~r) 〉][φ(~r′) − 〈 φ(~r′) 〉] 〉
|~r−~r′|=∆
∼ ∆−d+2−η e−∆/ξeq ,
where the angular brackets denote an average over the Gibbs-Boltzmann measure.
ξeq depends on temperature and, in second order phase transitions, it diverges at Tc
leaving only an algebraically decaying correlation. A dynamic equilibrium correlation
length characterizes the spatial decay of equal-time correlations in the equilibrium
relaxation of ‘usual’ systems. Similarly to the static case one defines ξ(t) via
〈 [φ(~r, t) − 〈 φ(~r, t) 〉][φ(~r′, t) − 〈 φ(~r′, t) 〉] 〉
|~r−~r′|=∆
∼ e−∆/ξ(t); the angular brackets
indicate here an average over thermal histories and, for simplicity, we omitted the
algebraic correction to the exponential decay. The average over thermal noise can
be traded for an integration over the reference space-point ~r and one then obtains
ξ(t) from V −1
ddr δφ(~r, t)δφ(~r′, t) |
|~r−~r′|=∆
∼ e−∆/ξ(t) where δφ(~r, t) ≡ φ(~r, t) −
ddr′′ φ(~r′′, t). The correlation length, ξ(t), depends now on temperature and
total time.
In systems with slow dynamics in which the order parameter is a two-time entity,
a two-time correlation length can be defined in analogy to what we described in the
previous paragraph:
[φ(~r, t)φ(~r, tw)− 〈 φ(~r, t)φ(~r, tw) 〉]
× [φ(~r′, t)φ(~r′, tw)− 〈 φ(~r′, t)φ(~r′, tw) 〉]
|~r−~r′|=∆
∼ e−∆/ξ(t,tw) . (24)
Once again by trading the thermal average by a spatial average 〈 φ(~r, t)φ(~r, tw) 〉 becomes
the global correlation C(t, tw) and ξ(t, tw) is derived from the four-point correlation
C4(∆; t, tw) ≡
ddr δ[φ(~r, t)φ(~r, tw)]δ[φ(~r′, t)φ(~r′, tw)]
|~r−~r′|=∆
CONTENTS 16
1e+081e+061e+041e+021e+00
tw=1k
tw=10k
tw=100k  0
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k  0
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k
Figure 3. The correlation length in the 3d EA model at T = 0.6 < Tc and L = 100.
(a) As a function of t − tw; (b) as a function of 1 − C; (c) in the scaling form t
against 1 − C. These results are taken from [9]. The vertical arrow in panels (b) and
(c) indicates the value of qea.
with δ[φ(~r, t)φ(~r, tw)] ≡ φ(~r, t)φ(~r, tw) − C(t, tw). The quantity C4(∆; t, tw) measures
the probability that a fluctuation of the two-time composite field φ(~r, t)φ(~r, tw) with
respect to its global average C(t, tw) in the spatial position ~r affects a fluctuation of the
same composite field at a different site ~r′ located at a distance ∆ from ~r and averaged
over the whole ensemble of reference points ~r in the sample.
The numerical analysis of C4 in the low temperature out of equilibrium dynamics
of the 3d ea model [6, 9] (see Fig. 3), soft sphere [28] and Lennard-Jones [29] mixtures
yield
ξ(t, tw) ∼
ξst(t− tw) C > qea
ξag(t, tw) C < qea
ξag(t, tw) ∼ t
w g(C) and a a small power (26)
(a logarithmic growth is also possible). g(C) is a monotonically decreasing function of C.
This is a two-time monotonically growing function even at time-lags that are longer than
the waiting-time dependent α relaxation time. Note the difference with what is observed
in the super-cooled phase where a number of numerical and experimental measurements
point at a dynamic correlation length that reaches a maximum at the α relaxation time
and later diminishes to zero [30]. The divergence has a clear interpretation within the
global time-reparametrization scenario: it is due to the generation and development of
the zero mode. The global reparametrization invariance symmetry develops only in the
limit of very long times, so that at intermediate times the irrelevant terms that are scale
down to zero are still manifest. These irrelevant terms, symmetry breaking ones, give
a finite length scale (or a finite ‘mass’) to the soft reparametrization modes. Because
they are irrelevant, the correlation length increases or, equivalently, the mass decreases
asymptotically.
Even though the numerically accessible times are sufficiently long so as to see
the separation of time-scales in the relaxation of the global correlation, the correlation
length is still very short. In numerical simulations ξ reaches of the order of 4 lattice
spacings or inter-particle distances in the spin-glass problem or soft sphere and Lennard-
Jones system, respectively. ξ just increases by, say, a factor 4 when the waiting-time is
CONTENTS 17
increased by nearly 4 orders of magnitude. The expected limit ξ(t, tw) → ∞ is thus far
from being attained.
As we already mentioned in the introduction to this Section, the growing length
scale is consistent with the development of time-reparametrization invariance, but it
is also consistent with other mechanisms. Hence, it alone cannot be used to argue
unambiguously in favor of the symmetry-based approach. For example, theories based
on the mode-coupling approach or random first order scenario [33] and its refinement
including the effect of entropic droplets [34], dynamical criticality controlled by a zero-
temperature critical point [35] and frustration limited domains [36] are used to justify
the growth of a dynamic length-scale, at least in the super-cooled liquid. Hence, the
existence of a growing length scale belongs to the first class of predictions we mentioned
in the introduction to this section.
Finally, let us note that the fluctuations in the susceptibility, or in multi-time
correlations, see eq. (35), and associated susceptibilities, can be used to derive other
correlation lengths. It would be interesting to check whether all these behave in the
same manner.
3.2. Scaling of the pdf of local two-time functions
The most direct way of testing the mere existence of local fluctuations is to measure the
probability distribution function (pdf) of local coarse-grained correlators, C(~r; t, tw),
and linear susceptibilities, χ(~r; t, tw), at different pairs of times, t and tw. In such
a measurement one is forced to use finite coarse-graining lengths. ℓ then becomes a
parameters that has to be taken into account in the scaling analysis of the results.
In Ref. [9] we showed that, quite generally, the pdf of local coarse-grained
correlators can be scaled onto universal curves as long as the global correlation, C(t, tw),
is the same, and the ratio of the coarse graining length over the dynamical correlation
length, ℓ/ξ(t, tw), is held fixed (see [31] for a similar discussion applied to the super-
cooled liquid). Such scaling can be easily understood as follows. At fixed temperature
the pdf ρ[Cr; t, tw, ℓ, L] depends on four parameters: two times, t and tw, and two
lengths, the coarse-graining length, ℓ, and the size of the system, L. As in the aging
regime C(t, tw) is a monotonic function of the two times and ξ(t, tw) ∼ t
wg(C), one
can trade the two times by C and ξ in complete generality. The next step is a scaling
assumption: that in the long times limit the pdfs depend on the coarse graining length
ℓ, the total size L and the scale ξ only through the ratios ℓ/ξ and ξ/L. This last
step, we should stress, is really a scaling assumption, and not a trivial requirement
from dimensional analysis. The lengths ℓ, L and ξ are already dimensionless as they
are measured in units of the lattice spacing. The end result from the rewriting of the
parameters in terms of the global correlation and the scaling hypothesis is that the
pdfs characterizing the heterogeneous constant temperature aging of the system can be
written as
ρ[Cr;C(t, tw), ℓ/ξ(t, tw), ξ(t, tw)/L] . (27)
CONTENTS 18
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k
Figure 4. pdf of local coarse-grained correlations Cr at different times t and tw in
the 3d Edwards-Anderson model with L = 100 at T = 0.6 < Tc. The waiting-times
are given in the key and the global correlation is fixed to C = 0.4 < qea. (a) The
coarse-graining boxes have linear size ℓ = 9 in all cases. The curves do not collapse, a
slow drift with increasing tw is clear in the figure. (b) Variable coarse-graining length
ℓ chosen so as to held ℓ/ξ approximately constant. The collapse improves considerably
with respect to panel (a). These results are taken from [9].
In Fig. 4 we show the scaling of the distribution of local coarse-grained correlations
in the 3d ea model and the effect of the scaling variable ℓ/ξ (the size of the system, L,
is sufficiently large so that ξ/L vanishes in practice). It is noteworthy that a reasonable
scaling with the global correlation held fixed and not taking into account the effect of
the second scaling variable has been already achieved, approximately, in the Edwards-
Anderson model [5, 6], as well as in the kinetically constrained models studied in [7]
and Lennard-Jones systems [32]. This is justified by the fact that ξ varies very slowly
with tw. However, it is clear that at long though finite times one has to hold the ratio
ℓ/ξ constant to obtain a full collapse in all cases.
The scaling variable ℓ/ξ allows one to study the change in the functional form of the
pdfs upon modifying the coarse-graining volume. Indeed, the pdfs should crossover
from a non-trivial form to a simple Gaussian when ℓ goes through the value ξ. In
summary, for finite ξ one identifies three ℓ-dependent regimes with different functional
forms of the pdfs:
• ℓ ≪ ξ. For finite ξ this means ℓ of the order of the lattice spacing, ℓ ∼ a. In this
case the pdfs do not have any particular structure.
• ℓ ∼ ξ. For finite but large ξ this case is non-trivial and indeed the one that is
accessed with numerical simulations and experiments. One finds that the statistics
is non-Gaussian for all C. The skewness decreases from zero at C = 1 to
reach a minimum and then increase again at small values of C. As regards the
functional form, one finds that a Gumbel-like functional form, characterized by a
real parameter that depends on C and ℓ/ξ describes the data rather well for, say,
qea/2
∼ qea.
• ℓ ≫ ξ. In this limit one matches the central-limit theorem conditions and the
statistics becomes Gaussian for all C.
CONTENTS 19
The analysis of the ℓ-dependent pdfs thus provides an independent way to estimate ξ.
3.3. Effective action for local ages
The argument leading to eq. (27) is a scaling hypothesis and it does not rely on time-
reparametrization invariance. Indeed, there exist models in which the scaling (28) for
the pdf of local coarse-grained correlations is found, e.g. the O(N) model in the large
N limit [8], and global time-reparametrization does not apply.
The implications of time-reparametrization invariance appear later, as a prediction
for the functional form of the asymptotic limit
ρ∞(Cr;C) ≡ lim
t,tw→∞
C(t,tw)=C
ρ[Cr;C(t, tw), ℓ/ξ(t, tw), ξ(t, tw)/L] . (28)
In order to study the functional form that ρ∞ can take let us now use the symmetry
argument to analyse the statistics of the fluctuations of the local correlations. So far
we have not yet determined how much the h(~r, t) vary in space and time. To this end
we need to derive an effective action for these functions that will tell us how costly it
is to deviate from the average clock R(t). Ideally, one would like to derive this action
from the microscopic one. This should be possible in quasi mean-field models such as
the p-spin model with Kac long (but finite) range interactions [37]. For the moment
we have just proposed the simplest action that serves our purposes in the ideal limit in
which the zero mode is fully developed ξ → ∞ and the local quantities are measured in
the infinite coarse-graining volume limit with ℓ/ξ → 0 [7]. Otherwise the parameter ℓ
should be taken into account.
To start with we worked with the more convenient transformed variable h(~r, t) ≡
e−ϕ(~r,t) that implies
Cag(r; t, tw) ≈ qea fc
h(~r, t)
h(~r, tw)
= qea fc
dt′∂t′ ϕ(~r,t
and we searched for the simplest action that satisfies the constraints due to the
symmetries. These are:
i. The action must be invariant under a global time reparametrization t→ h(t).
ii. If our interest is in short-ranged problems, the action must be written using local
terms. The action can thus contain products evaluated at a single time and point in
space of terms such as ϕ(~r, t), ∂tϕ(~r, t), ∇ϕ(~r, t), ∇∂tϕ(~r, t), and similar derivatives.
iii. The scaling form in eq. (29) is invariant under ϕ(~r, t) → ϕ(~r, t) + Φ(~r), with Φ(~r)
independent of time. Thus, the action must also have this symmetry.
iv. The action must be positive definite.
These requirements largely restrict the possible actions. The one with the smallest
number of spatial derivatives (most relevant terms) is
S[ϕ] =
(∇∂tϕ(~r, t))
∂tϕ(~r, t)
, (30)
CONTENTS 20
with K a stiffness. A term M ∂tϕ(~r, t) is also allowed by symmetry but since its space-
time integral is constant we drop it. The action solely depends on the time derivatives
∂tϕ(~r, t) and it is simple to check that it satisfies all the four constraints enumerated
above (the last requirement follows from the fact that h(~r, t) are monotonically
increasing functions of time) [7].
Due to the simple form (30) the ∂tϕ(~r, t) are uncorrelated at any two different times
t1 and t2. Thus the expression ∆ϕ~r|
dt′ ∂t′ϕ(~r, t
′) entering the exponential in
the scaling form in eq. (29) is a sum of uncorrelated random variables in time. One
can interpret such expression as the displacement of a random walker with position
dependent velocities. Alternatively, one can think of the space-dependent differences
∆ϕ~r|
as the net space-dependent height (labeled by t) of a stack of spatially fluctuating
layers dt ∂tϕ(~r, t). The action for the fluctuating surfaces of each layer is given by
eq. (30).
The statistics of the ∆ϕ~r|
are completely determined as follows. The action
(30) transforms into one of a Gaussian surface after the introduction of a ‘proper’ time
τ ≡ lnR(t), and the change of variables, ψ2(~r, t) = ∂τϕ(~r, τ). Indeed,
Cag(~r; t, tw) ≈ fc
R lnR(t)
lnR(tw)
dτ ′ ψ2(~r,τ ′)
S[ψ] = K
dτ ′ [∇ψ(~r, τ ′)]2 . (32)
Due to the Gaussian statistics of the ψ, it is simple to show that connected N -point
correlations of ∆
varphi~r1 |
satisfy
〈∆ϕr1|
∆ϕr2 |
· · ·∆ϕrN |
〉c = [τ(t)− τ(tw)] F(~r1, ~r2, . . . , ~rN) , (33)
where the function F can be obtained from Wick’s theorem, summing over all graphs
that visit all sites (connected) with two lines (because of ψ2) for each vertex i
corresponding to a position ri. The reparametrized times appear only in the prefactor
τ(t) − τ(tw) = ln[R(t)/R(tw)]. The probabilistic features of the fluctuations of local
correlations C(~r, t, tw) depend on times only through R(t)/R(tw), and hence only
through the global correlation itself C(t, tw). In consequence, the action (32) implies
the scaling (28). The fact that the time-dependencies of the statistical properties of the
two-time local coarse-grained correlations are fully determined by the global correlation
is a very welcome property of action (30) since it was not obvious a priori.
Having the forms in eqs. (31) and (32) allows us to make some quantitative
predictions about the form of the pdfs. With some algebraic manipulations one shows
the following generic features [7]:
• The distribution is non-Gaussian for all C.
• For C
∼ qea the pdf is negatively skewed and once put into normal form it is
very close to the distribution of the global equilibrium magnetization in the 2d xy
model [38]. It can then be approximately described by a generalized Gumbel form
with real parameter.
CONTENTS 21
• In the opposite limit C
∼ 0 the pdf is positively skewed and it does not take any
recognizable form.
If one is interested in testing the action further one can simply use eq. (32) to generate,
numerically, the ψ(~r, t), construct the C(~r, t, tw) from these functions, and then compare
the pdfs thus obtained to the ones measured, say, in a numerical simulation of a given
problem. Note that the scaling function fc also plays a role in the functional form of
the pdf of local correlations. The same argument applies to the susceptibilities.
The local coarse-grained correlations are, by construction, sums of correlated
random variables (unless ℓ ≫ ξ). With numerical simulations of the 3d ea model [9]
and kinetically facilitated lattice gases [7] we found that the pdfs of correlations coarse-
grained over finite lengths ℓ have a functional form that resembles a generalized Gumbel
distribution characterized by a continuous parameter that depends on ℓ/ξ and the value
of the global correlation, C, when C
∼ qea (see also [38]). This fact is consistent with the
discussion above and also with the observation of Bertin and Clusel that Gumbel-like
pdfs with real parameter characterize the statistics of sums of random variables with
particular correlations between the elements [39]. The fact that we obtain Gumbel-like
distributions then means that the correlations between the terms in the sum are of the
form needed to get this type of pdf.
In short, the time-reparametrization scenario predicts, in its simplest setting, that
eqs. (31) and (32) fully characterise the fluctuation of the local correlations in the large
times and coarse-graining volume limits.
3.4. Two-time scaling of local functions
As we argued in Sect. 2.4, the global time-reparametrization invariance suggests that
in the ideal asymptotic limit the slow part of the coarse-grained local correlations and
susceptibilities should scale as in eq. (22) in the ideal limit a ≪ ℓ ≪ ξ. In practice
the ideal limit is not reached and one is forced to work with finite correlation lengths
and thus finite coarse-graining lengths too. The finite ℓ will then play a role and has
to be taken into account. We now present some tests of eq. (22) that are based on the
parametric representation of the dynamics explained in Sect. 2 and take into account
the finite value of ℓ.
Let us then imagine that we compute three local coarse-grained two-time
correlations, C~r, at three space points ~r1, ~r2 and ~r3, using a given coarse-graining length,
a≪ ℓ, and that we obtain functional forms that are characterized by eq. (22) with, say,
h(~r1, t) = ln (t/t0), h(~r2, t) = t/t0, and h(~r3, t) = e
ln2(t/t0), in the aging regime. In Fig. 5
we sketch the decay of these correlations for the same tw as a function of time-delay.
The plateau is at the same height since qea as well as the the full stationary decay are
not expected to fluctuate. The external function fc is the same in all curves. It is clear
that the decay of the three correlations follows a different pace, the one at ~r3 is the
fastest while the relaxation at ~r2 is the slowest.
The simplest way to put the proposal (22) to the test is to analyze the implications
CONTENTS 22
1e+00
1e-01
1e-02
1e+061e+041e+021e+00
h3  0
 0.25
 0.75
 0  0.25  0.5  0.75  1
Figure 5. Left: sketch of the decay of the correlation with the same stationary decay
to qea – shown with a horizontal dashed line – and three choices of the scaling function
h(r1, t) = ln(t/t0), h(r2, t) = t/t0, and h(r3, t) = e
lna(t/t0). The waiting-time is the
same in all curves. Right: the relation between the integrated linear response against
the correlation. With a solid line, the parametric plot for fixed and long tw, using t
as a parameter that increases from tw at C = 1 to ∞ at C = 0. With symbols, the
three pairs (Cj(t, tw), χj(t, tw)) for the same tw, a fixed value of t and hj(t) as in the
left panel.
of eq. (22) on local triangular relations. In Sect. 2 we showed that two-time functions
with a separation of time-scales as in eq. (8) and an aging scaling as in eq. (13) are
related in a parametric way in which times disappear, see the sketch in Fig. 2-left.
Equation (22) implies that the local (fluctuating) two-time functions should verify the
same relation
Cag(~r; t1, t3) = qea fc
f−1c [Cag(~r; t1, t2)/qea] f
c [Cag(~r; t2, t3)/qea]
. (34)
This is a result of the fact that time-reparametrization invariance restricts the
fluctuations to appear only in the local functions h(~r, t) while the function fc is locked
to be the global one everywhere in the sample.
A pictorial inspection of this relation should take into account the fact that while
the stationary decay is not expected to fluctuate, the full aging relaxation and, in
particular, the minimal value of the local two-time functions, C(~r; t1, t3), are indeed
fluctuating quantities. The parametric construction on different spatial regions should
yield ‘parallel translated’ curves with respect to the global one, as displayed in Fig. 2-
left. Fluctuations in the function fc would yield different functional forms in the
curved part of the parametric construction. A more quantitative analysis can be
done by using the knowledge of fc that can be extracted from the global correlation
decay. Indeed, if fc is known, the parametric plot f
c (C~r12/qea)/
f−1c (C~r13/qea) against
f−1c (C~r23/qea)/
f−1c (C~r13/qea) should yield a master curve identical to the global one
with different sites just being advanced or retarded with respect to the global value.
This is another way of stating that the sample ages in a heterogeneous manner, with
some regions being younger (other older) than the global average. (For simplicity
we used a chort-hand notation, C~rµν = C(~r; tµ, tν) with µ, ν = 1, 2, 3.) If the time-
CONTENTS 23
 0  0.2  0.4  0.6  0.8  1
Ccg(t3,t2)
 0  0.2  0.4  0.6  0.8  1
Cr 23 (Cg 13/Cr 13)
 0  0.2  0.4  0.6  0.8  1
Cr 23 (Cg 13/Cr 13)
 0  0.2  0.4  0.6  0.8  1
Cr 23 (Cg 13/Cr 13)
Figure 6. The triangular relation in the 3d ea model. Upper left panel: the thick
(black) line represents the global C(t1, t2) against C(t2, t3) using t2 as a parameter
varying between t3 = 5 × 10
4 MCs and t1 = 9 × 10
6 MCs, C(t1, t3) ∼ 0.35 and
qea ∼ 0.8. The curved part is well represented by the hyperbolic form C(t1, t2) ∼
qeaC(t1, t3)/C(t2, t3) ∼ 0.79 × 0.35/C(t2, t3) that corresponds to fc(x) ∼ x
−b. With
different points joined with thin lines we show the triangular relations between the
local coarse-grained correlations on five randomly chosen sites on the lattice (ℓ = 30).
Upper right panel and lower left and right panels: 2d projection of the joint probability
density of C(r; t1, t2)
C(t1, t3)/C(r; t1, t3) and C(r; t2, t3)
C(t1, t3)/C(r; t1, t3) at
fixed three values of the intermediate time, t2 = 1.5 × 10
5 MCs, 8 × 105 MCs and
5× 106 MCs, respectively and ℓ = 10. The global C(t1, t2) against C(t2, t3) using t2 as
a parameter is shown with a thick green line. The green points indicate the location of
C(t1, t2) and C(t2, t3) for the chosen t2’s. Each point in the scatter plot corresponds
to a site, r. The lines indicated the boundary surrounding 25%, 50% and 75% of the
probability density. The cloud extends mostly along the global relation as predicted
by time-reparametrization invariance. These results are taken from [9].
CONTENTS 24
reparametrization mode is indeed flat the local values should lie all along this master
curve in the aging regime.
The conclusions drawn above apply in the strict a ≪ ℓ ≪ ξ limit. In simulations
and experiments ξ is finite and even rather short. Thus, ℓ is forced to also be a rather
small parameter, in which case ‘finite size’ fluctuations in fc are also expected to exist.
The claim is that the latter should scale down to zero faster (in ℓ) than the fluctuations
that are related to the zero mode.
We have tested these claims in the non-equilibrium dynamics of the 3d ea spin-
glass [9]. The results are shown in Fig. 6. In the upper left panel we show the global
triangular relation (thick black line) as well as the local one on four chosen sites. The
separation of time-scales is clear in the plot. The aging part is rather well described by
fc(x) ∼ x
−b and the local curves are quite parallel indeed. In the remaining panels in
Fig. 6 we show the 2dprojection of the joint probability density of the site fluctuations
in the local coarse-grained correlations at different chosen times t2, t2 = 1.5× 10
5 MCs
(upper right), t2 = 8 × 10
5 MCs (lower left), and t2 = 5 × 10
6 MCs (lower right).
Taking advantage of the fact that fc(x) ∼ x
−b we use a very convenient normalization
in which we multiply the horizontal and vertical axes by [C13/C~r13]
1/2. Global time-
reparametrization invariance, expressed in eq. (34), implies that the data points should
spread along the global curve indicated with a thick green line in the figure. Some sites
could be advanced, others retarded, with respect to the global value – shown with a
point on the green curve – but all should lie mainly on the same master curve. This is
indeed quite well reproduced by the simulation data in the three cases, C(t1, t2) close to
C(t1, t3) (upper right panel), C(t1, t2) close to qea (lower right panel), and C(t1, t2) far
from both (lower left panel). Most of the data points tend to follow the master curve
though some fall away from it. The reason for this is that eq. (34) should be strictly
satisfied only in the very large coarse-graining volume limit (ℓ≫ a) with ℓ/ξ ≪ 1 while
we are here using ℓ = 10a ∼ ξ, see the discussion in Sect. 3.1.
The triangular relation can be used to test the fluctuations of the local
susceptibilities too. Indeed, if the separation of time scales (8) and the scaling (13)
apply to the global susceptibility, the local ones, after the convenient normalization
by the maximum value, should follow another master curve, identical to the global one.
Note that time-reparametrization invariance as we discuss it here implies that both local
correlations and susceptibilities should be fluctuating quantities.
Finally, notice that, in contrast to the growing correlation length scale, there is
no blatantly obvious explanation of these triangular relations within other theoretical
scenarios. These relations are perhaps the most direct consequence of the time
reparametrization symmetry arguments, and so this prediction falls within the second
class we discussed in the introduction to this section.
CONTENTS 25
3.5. Multi-time scaling
In general muti-time correlations are non-trivially related to two-time ones. One can
take as an example a generic coarse-grained connected four-time correlation. If this
function is monotonic with respect to all times, and the two-point correlations scale as
in (13) for all pairs (tµ, tν) with µ, ν = 1, 2, 3, 4, the four-time correlation should behave
C(~r; t1, t2, t3, t4) = g
h(~r, t1)
h(~r, t2)
h(~r, t2)
h(~r, t3)
h(~r, t3)
h(~r, t4)
with the same external function g for all r, in the asymptotic limit in which all times are
widely separated and the corresponding two-time correlations fall below qea. Parametric
constructions could be envisaged to test this relation.
3.6. Local fluctuation-dissipation relation
The asymptotic relation between the global correlation and susceptibility
tw→∞,C(t,tw)=C
χ(t, tw) = χ̂(C) (36)
was first obtained in mean-field disordered models [12, 13] and later observed in
simulations of many realistic systems (spins and particles in interaction on finite
dimensional spaces). −(dCχ̂(C))
−1 defines an effective temperature [14]. In the aging
regime, that is to say for C < qea, three behaviours have been observed in mean-field
systems: in structural glass models χ̂(C) is linear in C (solid line in the right-panel in
Fig. 5); in spin-glass models χ̂(C) is a non-linear function of C; in coarsening systems
χ̂(C) is a constant equal to (1− qea)/T .
The scaling in eq. (22) implies that the parametric construction ‘local susceptibility
against local correlation’ should fall on the master curve for the global quantities but
can be advanced or retarded with respect to the global value; again in the theoretical
limit a ≪ ℓ ≪ ξ. This behaviour is sketched in Fig. 5-right for the three sites
whose correlations are displayed in the left panel. The restricted relation between local
susceptibility and local correlation in eq. (22) arise from the fact that the fluctuations
are due to local reparametrizations alone and not to changes in the external functions
fc and fχ (much as in transverse vs. longitudinal fluctuations in a non-linear σ-model).
An important property of the interpretation of the fluctuation dissipation relation
in terms of effective temperatures is that one expects all observables evolving in the same
time-scale to equilibrate and hence have the same value of the effective temperature [14]
in an asymptotic regime with slow dynamics and small entropy production. Within
the time-reparametrization invariance approach the local effective temperature, defined
from the slope of the χ̂(C) plot, is automatically the same in the whole sample within
a correlation scale, just because the functions fc and fχ do not fluctuate.
In Fig. 7 we show the joint pdf of local correlations and susceptibilities of the
3d ea spin-glass in its glassy phase; the accord with the analytic prediction is very
CONTENTS 26
satisfactory [5, 6] with the additional spreading away from the master curve ascribed to
the fact that ℓ is finite and not very different from ξ.
+++ Bulk
0 0.5 1
Figure 7. (a) The joint pdf ρ(Cr , χr) at two times (tw, t) such that the global
correlation is C(t, tw) = 0.7 < qea in the 3d ea model. (b) Projection of three contour
levels. The crosses are the parametric construction χ̂(C) for several values of the total
time t larger than tw. The dotted straight line is fdt at the working temperature T .
These results are taken from [6].
3.7. Infinite susceptibilities
Zero modes are intimately related to infinite susceptibilities. Indeed, systems with
continuous symmetries are sensitive to arbitrarily weak perturbations. In the present
context the approximate global time-reparametrization invariance implies that one
can easily change the ‘clock’ R(t) characterizing the scaling of the global correlation
and linear response by applying infinitely weak perturbations that couple to the zero
mode. An illustration of this property is the fact that the aging relaxation dynamics
of glassy systems is rendered stationary by a weak perturbing force that does not
derive from a potential while the χ̂(C) relation in the slow regime is not modified [40].
One could envisage more refined tests such as applying a perturbation that imposes
different scalings on two macroscopic borders of the system and see how the time-
reparametrization wave develops in the bulk.
3.8. Conclusions
In conclusion, the global time-reparametrization invariance scenario gives a mechanism
for the divergence of the correlation length ξ though others have also been proposed in
the literature. There are a number of predictions, as the parametric relations between
local coarse-grained correlations measured at different times and the local fluctuation-
dissipation relations that, to our knowledge, are not explained by other scenarios. As
regards to the easier to measure pdfs of local correlations the framework is not only
CONTENTS 27
consistent with the scaling (28) – that also arises from simple scaling assumptions –
but it also justifies the non-Gaussian and Gumbel-like functional form of the pdfs
that follows from the proposed effective action for local ‘ages’. Moreover, systems with
global time-reparametrization invariance should have as important fluctuations in the
local susceptibilities as in the local coarse-grained responses.
4. Discussion
We presented a summary of studies of dynamical fluctuations in glassy systems that
are based on the idea that, in the long time regime, a global time-reparametrization
invariance emerges in the effective action describing the aging dynamics. We discussed
how this symmetry concretely appears in mean-field spin models, and how it can
be shown to emerge at the level of the action for short-ranged spin glass models
with quenched disorder. Two assumptions are used to prove the global time-
reparametrization of the action for the short-range spin glass model i) causality and
unitarity, and ii) a separation of time scales between a fast (or stationary) and a slow
(or aging) relaxation, where in the latter time translation invariance is broken.
That the dynamical action is symmetric under uniform, i.e., spatially independent
reparametrizations of time variables (t → h(t)) suggests that the dynamic fluctuations
that cost little action should be describable in terms of position dependent, long
wavelength, reparametrizations of the form t→ h(~r, t). These should be the Goldstone
modes associated with breaking time-reparametrization invariance symmetry.
We presented predictions of this theoretical framework and tests that we performed
in the 3d Edwards-Anderson model to falsify these predictions. Among the consequences
of our theoretical framework are those listed in Sect. 3. Some of them find an
explanation within other theories as well; others are particularly related to the time
reparametrization invariance scenario. For example, a correlation length that grows
in time is associated with the asymptotic approach to the long-time regime in which
the symmetry is fully developped, and the long wavelength modes eventually become
massless. The existence of a growing length is also predicted by other models. The
functional form of the triangular relations relating local two-time correlations between
three different times is, instead, particular to our framework.
In this review we showed tests of the predictions of the global time-
reparametrization invariance scenario performed on a finite dimensional spin-glass
model [4]-[9]. In the past we also studied the dynamics of kinetically facilitated models,
without quenched disorder, along the same lines [7]. We believe that, if aging dynamics
is a universal property of glassy systems, then these ideas should also apply to interacting
particle systems without quenched disorder. The rationale is that the two assumptions
leading to the global time-reparametrization invariance of the dynamical action, namely
causality and unitarity, and a separation of time scales, should also hold for other glassy
systems. The former assumption we can take as a fact. The second is, in a way, the
assumption that a glassy phase exists, even though we say nothing as of why it does. As
CONTENTS 28
we stated in the introduction, we do not aim at the question why glasses?, but instead we
focus on the possible universal dynamical properties once the glassy state is presented by
nature. Some of the consequences of the symmetry have already been tested numerically
in Lennard-Jones systems [29, 32] but there is still much room for more detailed studies,
including the analysis of local triangular relation between correlation and susceptibilities
and tests of their joint behaviour.
We thus propose that the asymptotic global time-reparametrization invariance,
and the associated low action long wavelength local reparametrizations, constitute the
mechanism by which dynamic fluctuations, that is to say heterogeneities, are generated
in the glassy state. Moreover, this mechanism may also apply, in an approximate form,
to the super-cooled liquid regime. It should just be an approximation because in super-
cooled liquids the symmetry is not fully developed and there is then a cut-off setting
the limit of the spatial and temporal extent of the heterogeneities, in sharp constrast
to the low temperature glassy regime in which the symmetry is realized asymptotically
and fluctuations of all sizes exist. The growth and divergence of the (two-time) dynamic
correlation length defined from the study of the space correlation of the two-time order
parameter is a manifestation of the growth and divergence of these fluctuations in the
glassy state; in contrast, such growth is interrupted in the super-cooled liquid.
To better understand the distiction between the glassy state with its asymptotic
symmetry and the super-cooled state without the fully developing symmetry, consider
the phenomenology of dynamic fluctuations as a function of temperature. Dynamic
heterogeneities in the super-cooled liquid phase have been identified numerically and
experimentally [52]. These are in a number of ways more important than what is
observed in a simple liquid or a solid. In the super-cooled liquid phase while the
full relaxation is stationary, there is still a time-scale separation with the correlations
decaying as a function of time-difference first rapidly to a temperature-independent
plateau and then slowly towards zero. The latter is the structural or α-relaxation. The
α relaxation time, tα, is finite in the super-cooled liquid regime and it increases upon
decreasing temperature. The global parametrization chosen by the symmetry breaking
terms in the slow regime is R(t) ∝ e−t/tα and C(t, tw) = qeafc(e
−(t−tw)/tα) in this case.
The mode-coupling approach to super-cooled liquids is based on approximate
dynamic equations for the relevant correlators of realistic systems that are very similar
to the p-spin Schwinger-Dyson equations in the high temperature phase [16, 18]. In
these equations the correlators are already expressed as functions of the time-difference,
τ ≡ t − tw. Close to the critical temperature the separation of time-scales develops in
the mode-coupling equations. The approximate analysis of the α relaxation predicted
by these equations also relies on dropping the τ derivatives and approximating the
integrals by assuming a sharp time-scale separation. The remaining asymptotic (large
τ) equations are invariant under reparametrizations of τ .
We then expect the local coarse-grained correlations and integrated linear responses
in the super-cooled liquid to be, to a first approximation, stationary (after a sufficiently
long waiting-time that goes beyond the equilibration time) but with different finite
CONTENTS 29
structural relaxation times, fluctuating about the value that characterizes the decay
of the global correlations. This is consistent with the experimental observation that
dynamic heterogeneities in supercooled liquids seem to have a lifetime of the order of
the relaxation time. Deviations from stationarity are not completely excluded for finite
ℓ but they should be less important than in the aging low-temperature regime.
There is, however, an important difference with respect to the aging regime, in
which the equilibration time diverges and local relaxation times or, better stated, local
ages can fluctuate without limit when tw → ∞. At high temperatures one does
not expect to find heterogeneities with arbitrary long relaxation time. Furthermore,
heterogeneities have a finite spatial extent and one can then suppress them by using
sufficiently large coarse-graining volumes. The correlation length is stationary, ξ(τ),
and it saturates in the limit of long-time differences, τ → ∞. The saturation value,
though, increases for decreasing temperature. From a theoretical point of view, this
picture is, in a sense, similar to the one that describes the equilibrium paramagnetic
phase in the O(N) model, just above the ordering transition temperature.
When lowering the temperature the size and life-time of the heterogeneities
increases. A p-spin or mode-coupling approach predicts that their typical size and thus
limτ→∞ ξ(τ) diverge at the mode coupling transition temperature [33]. In real systems
the divergence at Tc is rendered smooth and ξ does not strictly diverge asymptotically.
At still lower temperatures the bulk quantities age and we expect then to observe
heterogeneous aging dynamics of the kind described in this review, with a two-time
dependent correlation length for the local fluctuations. The heterogeneities age as well,
in a ‘dynamic’ way. By this we mean that if a region looks older than another one when
observed on a given time-window, it can reverse its status and look younger than the
same other region when observed on a different time-window.
The infinite susceptibility with respect to perturbations that couple to the zero
mode are illustrated by the fact that the clock of the bulk quantities that is selected
dynamically is very easy to modify with external perturbations. A small force that does
not derive from a potential and is applied on every spin in the model renders an aging p
spin model stationary [40] while the model maintains a separation of time-scales in which
the fast scale follows the temperature of the bath, T , while the slow scale is controlled
by an an effective temperature, Teff > T . In this case, the aging system selects a
time-reparametrization R(t) = t while in the perturbed model R(t) = e−t/tα . Similarly,
the aging of a Lennard-Jones mixture is stopped by an homogeneous shear [41]. A
different way to modify the time-reparametrization that characterizes the decay of the
correlations is by using complex thermal baths [42].
The picture that we have described applies to long times but not as long as to enter
the activated regime that we still do not know how to characterize theoretically, not
even at the bulk level. This regime corresponds to times that scale with the system size.
The success of mean-field models, or the mode-coupling approach, in describing the
bulk dynamics of glassy systems, at least not to close to the crossover glass temperature
and at a qualitative level, allows us to claim that these extremely long times scales are
CONTENTS 30
unrealistic if not too close to the glass transition Tg even as far as dynamic fluctuations
are concerned.
The ideas discussed in this paper should not only apply to systems that relax
in a non-equilibrium manner as glasses but also to systems with slow dynamics and a
separation of time-scales that are kept out of equilibrium with a (weak) external forcing.
Recently, there has been much interest in the appearance of shear localization, in the
form of shear bands, in the rheology of complex fluids. Along the lines here described
it would be very interesting to analyze the fluctuations in the local reparametrizations
in the fluidized shear band and the ‘jammed’ glassy band.
The analytic treatment of mean-field quantum glassy systems follows similar steps
to the ones presented here. The Schwinger-Keldysh approach replaces the Martin-
Siggia-Rose one but these formalisms are very similar indeed. The analytic solution to
the dynamic equations in the limit in which the coupling to the environment is weak
also uses the fact that the dynamics in the aging regime is very slow. The approximate
equations then become time-reparametrization invariant. One can then expect that
similar dynamic fluctuations arise in glassy problems in which quantum fluctuations
are important. Moreover the proof of global time-reparametrization invariance for spin-
glasses has been presented directly in the quantum formalism. Novel experimental
techniques may be apt to study dynamic heterogeneities in glasses when quantum
fluctuations are important.
We expect to find similar fluctuations using finite size systems and examining the
behaviour of the mesoscopic run-to-run fluctuations of the global correlations. These
may be easier to measure experimentally using mesoscopic systems.
Importantly enough, global time reparametrization invariance does not develop in
all models with slow and aging correlation functions. The O(N) ferromagnetic model in
the large N limit is a case in which global time-reparametrization invariance is reduced
to just scale invariance [8].
Last, but not least, the approach based on reparametrization invariance suggests
that it may be possible to search for universality in glassiness. A Ginzburg-Landau
theory for phase transitions captures universal properties that are independent of the
details of the material. It is symmetry that defines the universality classes. For example,
one requires rotational invariance of the Ginzburg-Landau action when describing
ferromagnets. Time reparametrization invariance may be the underlying symmetry that
must be satisfied by the Ginzburg-Landau action of all glasses. What would determine
if a system is glassy or not? We are tempted to say the answer is if the symmetry
is generated or not at long times. Knowing how to describe the universal behavior
may tell us all the common properties of glasses, but surely it will not allow us to
make non-universal predictions, such as what is the glass transition temperature for a
certain material, or whether the material displays glassy behavior at all. This quest for
universality is a very interesting theoretical scenario that needs to be confronted.
We have tried to state as clearly as possible the implications of our proposal. The
CONTENTS 31
full project is not yet complete since several questions about its limitations remain open.
A number of issues should be addressed are:
(i) From a phenomenological point of view, to perform strong tests of the global
time-reparametrization invariance scenario in molecular dynamic simulations [28, 44, 45]
of realistic glassy systems and experiments [46]-[51]. More precisely, the triangular
relations between local coarse-grained correlations and the local fluctuation dissipation
relations should be analysed to give support or else falsify this conjecture.
(ii) From an analytic point of view, to derive the effective action for local
reparametrizations for glassy models with and without quenched disorder. We are
currently working on this project in collaboration with S. Franz. One idea is to study
the p spin disordered model with Kac long-range interaction. ANothe one is to study
the symmetries of the dynamic action associated with the Dean-Kawasaki equation for
the evolution of the density of a system of particles in interaction.
(iv) From a mixed analytic and numerical point of view to analyse fluctuations
in models with global aging dynamics of different type. To this end, one can study
dynamic fluctuations in simple coarsening systems in finite dimensions. In these cases
the morphology of domains can be characterized in great detail [43]. We could, in
principle, understand the fluctuations in the local correlations and linear responses from
a microscopic point of view. Whether these are similar or different to the ones in other
glassy problems is still to be established and the outcome of this study could clarify the
relevance of the value of the effective temperature in determining the characteristics of
the dynamic fluctuations.
(v) In the same line as the above, the analysis of fluctuations of elastic lines in the
presence of impurities could help us understanding the coarsening phenomenon but also
the role of diffusion that superimposes in these cases to the aging dynamics [15].
These are just a few questions posed by this proposal that are still not answered.
Acknowledgments
We thank our collaborators J. J. Arenzon, C. Aron, A. J. Bray, S. Bustingorry,
H. E. Castillo, P. Charbonneau, D. Domı́nguez, J. L. Iguain, L. D. C. Jaubert, M. P.
Kennett, M. Picco, D. R. Reichman, M. Sellitto, A. Sicilia and H. Yoshino.
We also wish to especially thank L. Berthier, G. Biroli, J-P Bouchaud, D. S. Dean,
G. Fabricius, T. Grigera, E. Fradkin, S. Franz, J. Kurchan, H. Makse, D. Stariolo and L.
Valluzzi for very helpful discussions. We acknowledge financial support from NSF-CNRS
INT-0128922, NSF DMR-0305482, DMR 0403997 and PICS 3172. LFC is a member of
Institut Universitaire de France. LFC thanks the Newton Institute at the University of
Cambridge, ICTP at Trieste, and Universidad Nacional de Mar del Plata, Argentina, CC
the LPTHE at Jussieu, Paris, France, and LFC and CC the Aspen Center for Physics
CONTENTS 32
for hospitality where part of this work has been carried out.
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http://arxiv.org/abs/cond-mat/0610789
http://arxiv.org/abs/cond-mat/0610857
http://arxiv.org/abs/cond-mat/0605733
http://arxiv.org/abs/cond-mat/0509127
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http://arxiv.org/abs/cond-mat/0610195
	Why glasses? vs. universality in glassy dynamics
	Time reparametrization invariance
	Mean-field models – dynamic equations
	Structural glasses: the p3 cases
	Short-ranged models – dynamic action
	Turning a nuisance into something useful - symmetry as a guideline
	The spherical p=2 case or mean-field domain growth
	Quantum problems
	Consequences and tests
	Two-time correlation length
	Scaling of the pdf of local two-time functions
	Effective action for local ages
	Two-time scaling of local functions
	Multi-time scaling
	Local fluctuation-dissipation relation
	Infinite susceptibilities
	Conclusions
	Discussion

  We summarize a theoretical framework based on global time-reparametrization
invariance that explains the origin of dynamic fluctuations in glassy systems.
We introduce the main ideas without getting into much technical details. We
describe a number of consequences arising from this scenario that can be tested
numerically and experimentally distinguishing those that can also be explained
by other mechanisms from the ones that we believe, are special to our proposal.
We support our claims by presenting some numerical checks performed on the 3d
Edwards-Anderson spin-glass. Finally, we discuss up to which extent these ideas
apply to super-cooled liquids that have been studied in much more detail up to
present.

Fluctuations in glassy systems
Claudio Chamon1 and Leticia F. Cugliandolo2
1 Physics Department, Boston University,
590 Commonwealth Avenue, Boston, MA 02215, USA
2Laboratoire de Physique Théorique et Hautes Énergies, Jussieu,
5ème étage, Tour 25, 4 Place Jussieu, 75252 Paris Cedex 05, France
chamon@bu.edu, leticia@lpt.ens.fr
Abstract. We summarize a theoretical framework based on global time-
reparametrization invariance that explains the origin of dynamic fluctuations in glassy
systems. We introduce the main ideas without getting into much technical details.
We describe a number of consequences arising from this scenario that can be tested
numerically and experimentally distinguishing those that can also be explained by other
mechanisms from the ones that we believe, are special to our proposal. We support our
claims by presenting some numerical checks performed on the 3d Edwards-Anderson
spin-glass. Finally, we discuss up to which extent these ideas apply to super-cooled
liquids that have been studied in much more detail up to present.
http://arxiv.org/abs/0704.0684v1
CONTENTS 2
Contents
1 Why glasses? vs. universality in glassy dynamics 2
2 Time reparametrization invariance 5
2.1 Mean-field models – dynamic equations . . . . . . . . . . . . . . . . . . . 5
2.2 Structural glasses: the p ≥ 3 cases . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Short-ranged models – dynamic action . . . . . . . . . . . . . . . . . . . 10
2.4 Turning a nuisance into something useful - symmetry as a guideline . . . 11
2.5 The spherical p = 2 case or mean-field domain growth . . . . . . . . . . . 13
2.6 Quantum problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Consequences and tests 14
3.1 Two-time correlation length . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Scaling of the pdf of local two-time functions . . . . . . . . . . . . . . . 17
3.3 Effective action for local ages . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Two-time scaling of local functions . . . . . . . . . . . . . . . . . . . . . 21
3.5 Multi-time scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Local fluctuation-dissipation relation . . . . . . . . . . . . . . . . . . . . 25
3.7 Infinite susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Discussion 27
1. Why glasses? vs. universality in glassy dynamics
It is common to encounter in nature systems that resist equilibration with their
environments and display what is called glassiness. The name is derived from what
we normally know as glasses, an irregular array of silicon and oxygen atoms without
crystalline order, much as a liquid, but as hard as a solid. The molecular diffusion
within glasses is extremely sluggish, slowing down by over 10 orders of magnitude as
the temperature is slightly decreased near the operationally defined glass transition
temperature. Hence, the term glassiness became generically associated with very slow
dynamics [1].
So, why glasses? The answer to this question has been the focus of much research
effort for long. It is certainly a rather difficult question, and there have been a number
of ideas lined up for trying to understand how material systems become glassy. It is not
even clear whether in many systems there is a thermodynamic phase one can call glass,
or there is simply a dynamical crossover at low enough temperatures.
Do we need to fully answer why glasses? before we really further our understanding
of glassy dynamics? We take the point of view that, by starting from the fact that glassy
systems exist (as nature presents us with concrete examples), we can then attempt to
characterize whatever possible universal properties there are in glassy dynamics.
CONTENTS 3
To make this statement clearer, let us turn to a question that Anderson poses in
the introduction of his Concepts in solids text [2]: why solids? This question, again,
is a rather complex one, and it is not untwined from the question why glasses? if we
focus on why a regular array of atoms, as opposed to a random packing, forms in the
first place. Even if one assumes that a crystalline structure forms at low temperatures,
a detailed quantitative analysis of the energetics remains to be done so as to determine
if the packing is hexagonal, cubic, body or face centered, etc. Nonetheless, if one starts
from the existence (as observed in nature) of a state with broken translational symmetry,
one can construct theories of lattice vibrations (and quantize it) and of electronic band
structure (and discern between insulators, metals and semiconductors). Solid state
physics starts from the solid.
The approach we review in this paper relies on a similar philosophy: we do not
claim any theory of the glass transition, and we do not attempt to answer why glasses?
with this particular approach. Our theory does not allow us to make non-universal
predictions, such as what the glass transition temperature is (if one can be defined) for
a certain material, or whether the material displays glassy behavior at all. We aim at
understanding if there is a set of principles, guided by symmetry considerations, that
can be used to understand certain universal aspects of glassy dynamics once the glass
state is presented. For example, glasses age [3]. We thus expect that there exists a
unified approach to describe aging phenomena, and we seek some guiding principles,
based on dynamical symmetries, that could allow us to understand universal properties
in the aging regime, including the scaling of spatial heterogeneities.
We propose that the symmetry that captures the universal aging dynamics
of glassy systems is the invariance of an effective dynamical action under uniform
reparametrizations of the time scales [4]-[9]. Such type of invariance had been known
to exist since the early days in which the mean-field equilibrium dynamics of spin-
glasses was tried to be understood [10, 11] and it was later encountered in the better
formulated out of equilibrium dynamic formalism applied to the same systems [12, 13].
The invariance means that in the asymptotic regime of very long times a family of
solutions to the equations of motion is found. This ‘annoyance’, we claim, has actually
a physical meaning and implications in the fluctuating dynamics of real glasses. In
order to make this statement concrete, it is necessary to show that the invariance
also exists in finite dimensional glassy models. With this aim we showed that global
time reparametrization invariance emerges in the long times action of short-range spin-
glasses assuming causality and a separation of time scales [4]-[6]. Basically, the second
assumption amounts to start from a glass state, where one may claim there is a
separation between, roughly, a time regime where relaxation is fast and another where
relaxation is slow. The invariance can then be used to describe dynamic fluctuations in
spin-glasses and, we conjectured, in other glassy systems as well.
Physically, the emergence of time reparametrization invariance can be though of
in the following way. The out of equilibrium relaxation of glassy systems is well
characterized by two-time functions, either correlations or linear responses. In the slow
CONTENTS 4
and aging regime they depend on two times and time-translation invariance is lost. The
proper measure of ‘time’ inside the system is the value of the correlation itself, and not
the ‘wall clock’ in the laboratory. For instance, in a spin-glass the proper measure of
sample age is the spin-spin overlap or in particle systems it is the incoherent correlation
function. Age measures can fluctuate from point to point in the sample, what we called
heterogeneous aging, with younger and older pieces (lower and higher values of the
correlation) coexisting at the same values of the two laboratory times. The fact that
the effective dynamical action becomes invariant under global time reparametrizations,
t → h(t), everywhere in the sample, means that the action weights the fluctuations
of the proper ages, C(~r; t1, t2), directly, and the times t1 and t2 in the action are just
integrated over as dummy variables. To draw an analogy, in theories of quantum gravity
the space-time variables Xµ(τ, σ) are the proper variables, and the action is invariant
under conformal transformations of the world-sheet parameters τ and σ.
So what does global time-reparametrization invariance symmetry concretely teach
us about observables in glasses? So far we discussed a global symmetry or invariance
with respect to uniform time-reparametrizations. By looking at spatially heterogeneous
reparametrizations, we can predict the behavior of local correlations and linear
susceptibilities and the relations between them. For example, we predict that, after
a convenient normalization that we explain in the main text, the triangular relation
between the local coarse-grained correlations, C(~r; t1, t2), C(~r; t2, t3) and C(~r; t1, t3), as
a function of the intermediate time t2, t3 < t2 < t1, at all spatial points, ~r, should be
identical to the global triangular relation, in the asymptotic limit of very long absolute
times and delays between them and very large coarse-graining linear length [9]. Different
sites can be retarded or advanced with respect to the global behaviour but they should all
have the same overall type of decay. Similarly, the relation between local susceptibilities
and their associated correlations should be identical all over the sample [5, 6] leading to
a uniform effective temperature [14].
The purpose of this article is to describe our current understanding of dynamic
fluctuations (heterogeneities) in the non-equilibrium relaxation of glassy systems arising
from the time-reparametrization invariance scenario [4]-[9],[15]. We illustrate it by
presenting critical tests.
The structure of this review is the following. In Sect. 2 we explain the origin of
the time-reparametrization invariance scenario. We do not present detailed derivations
that were already published but we aim at highlighting the main ideas behind the scene.
In Sect. 3 we list several measurable consequences of the theory. We discuss how they
have been examined numerically in different glassy models. The discuss here which
consequences could also be explained by other approaches and which, we believe, are
unique to our scenario. Finally, in Sect. 4 we discuss the scenario. Since this research
project is not closed yet, we present some proposals for numeric and experimental tests
as well as some ideas about analytic calculations that could help us understanding the
limitations of our proposal.
CONTENTS 5
2. Time reparametrization invariance
In this Section we explain how global time-reparametrization invariance develops
asymptotically in the aging regime of glassy models. For the sake of simplicity we focus
on the classical formalism and at the end of this section we mention the modifications
introduced by quantum fluctuations.
2.1. Mean-field models – dynamic equations
Schematic models of spin-glasses, structural glasses and ferromagnetic clean coarsening
are encoded in the family of p-spin models defined by [16]
H = −
i1i2...ip
Ji1i2...ipsi1si2 . . . sip (1)
with quenched disordered exchanges distributed with the Gaussian law P (Ji1i2...ip) ∝
−p!J2
i1i2...ip
/(2Np−1)
. The dynamic variables si, i = 1, . . . , N , are of Ising type, si = ±1,
or satisfy a global spherical constraint
i=1 s
i = N . p is an integer parameter: with
p = 2 and Ising variables one mimics spin-glasses, with p > 2 and Ising or spherical
spins the phenomenology of structural glasses is recovered, and with p = 2 and spherical
spins one describes ferromagnetic domain-growth in clean systems. The sum runs over
all p-uplets; for this reason these models are ‘mean-field’ in the sense that the saddle-
point evaluation of the partition function or the dynamic generating functional is exact
in the thermodynamic limit, N → ∞. The Hamiltonian (1) also represents the potential
energy of a particle with position ~s = (s1, . . . , sN) on an infinite dimensional hypercube
(si = ±1) or hypersphere (
i=1 s
i = N).
Dynamics is introduced with a Langevin equation that represents the coupling of
the spins to an equilibrated thermal environment. Ising spins are then replaced by
soft variables by introducing a double-well potential energy,
i=1 V (si), with V (si) =
a(s2i − 1)
2. The initial condition is usually chosen to be random thus mimicking a rapid
quench from infinite temperature to the working temperature T . In the limit N → ∞
exact Schwinger-Dyson equations couple the global correlation and instantaneous linear
response
NC(t, tw) =
si(t)si(tw) , NR(t, tw) =
δsi(t)
δhi(tw)
i=1 si(t) = 0 for all t). The field hi couples linearly to the spin variables and, in
general, we are interested in perturbing fields that are uncorrelated with the equilibrium
states of the systems. It is not necessary to average over thermal noise or quenched
disorder since these quantities do not fluctuate in the out of equilibrium regime reached
when the infinite volume limit (N → ∞) has been taken at the outset. Here are in
what follows times are measured from an origin that corresponds to the quench to the
working temperature. Our study applies then in the order of limits
CONTENTS 6
in which the exact causal Schwinger-Dyson equations for spherical models at times t ≥ tw
read [12, 19]
(∂t − zt)C(t, tw) =
dt′ Σ(t, t′)C(t′, tw) +
dt′ D(t, t′)R(tw, t
′) , (4)
(∂t − zt)R(t, tw) =
dt′ Σ(t, t′)R(t′, tw) + δ(t− tw) , (5)
where the vertex, D, and self-energy, Σ, are functions of C and R
D(t, tw) =
Cp−1(t, tw) , Σ(t, tw) =
p(p− 1)
Cp−2(t, tw)R(t, tw) . (6)
The Lagrange multiplier zt is fixed by requiring C(t, t) = 1. For Ising problems the soft
spin constraint can be treated in the mode-coupling approximation [17] or else, one can
apply a T − Tc expansion taking advantage of the fact that the phase transition is of
second order for p = 2 (Sherrington-Kirkpatrick or SK model) [13].
In such fully-connected models all higher order correlations and linear responses
factorize and can be written as functions of these two-time functions. Self-consistent
approximate treatments of interacting particle models with realistic potentials, like the
mode-coupling approach, yield similar equations with the addition of a wave-vector
dependence that in the present context can also be taken into account by considering
models of d-dimensional random manifolds embedded in N → ∞ dimensional
spaces [18, 19].
2.2. Structural glasses: the p ≥ 3 cases
We now focus on the p ≥ 3 cases that mimic the structural glass problem. We mention
at the end of this subsection the Ising p = 2 (SK) spin-glass case that is not conceptually
different but just technically more involved. In Sect. 2.5 we discuss the p = 2 spherical
problem that yields a mean-field description of coarsening phenomena and is rather
different from the point of view of time-reparametrization transformations.
Equations (4) and (5 are causal and be solved numerically by constructing C(t, tw)
and R(t, tw) from the initial instant t = tw = 0. An analytic solution is possible in the
asymptotic limit, as we discuss below. We first present the main features of C and R
and we later explain how these are obtained from the asymptotic analytic solution.
Equations (4) and (5) have a dynamic transition at a critical temperature Td(p). At
T > Td the dynamics occurs in equilibrium and close to Td the decay of the correlations
slows down as in super-cooled liquids with the α relaxation time diverging as a power law
of T − Td [16, 20]. Below Td eqs. (4) and (5) admit a unique solution [12, 13] that is no
longer stationary. The behaviour of the low-temperature correlation and susceptibility
is sketched in Fig. 1.
The low-temperature solution presents a separation of time-scales. In the long
waiting-time limit the self-correlation and integrated linear response or susceptibility,
χ(t, tw) ≡
dt′R(t, t′), can be written as
C(t, tw) = Cst(t− tw) + Cag(t, tw) , (7)
CONTENTS 7
1e+00
1e-01
1e-02
1e+051e+031e+011e-01
rapid & stationary (C st )
aging &
(Cag)
1e+00
1e-01
1e+051e+031e+011e-01
rapid & stationary (χ st)
aging & slow (χag)
Figure 1. Sketch of the relaxation of the self-correlation and susceptibility in the
glassy regime. The separation of time-scales is clear in the figure. The Edwards-
Anderson parameter, qea the corresponding susceptibility χea and the asymptotic value
limt→∞ χ(t, tw) are indicated with horizontal lines.
χ(t, tw) = χst(t− tw) + χag(t, tw) . (8)
The first terms in the right-hand-side describe the stationary regime at short time-
differences in which the correlation and susceptibility relatively rapidly approach a
plateau at limt−tw→∞ limtw→∞C(t, tw) = qea and limt−tw→∞ limtw→∞ χ(t, tw) = (1 −
qea)/T ≡ χea. The second terms are the aging relaxation of C towards zero (in
the absence of an external field), and the aging response of the system towards the
asymptotic value χea + qea/Teff with Teff a parameter with the interpretation of an
effective temperature [14].
The stationary and aging relaxation are fast and slow in the sense that
∂tCst(t, tw) ∼ Cst(t, tw) C > qea , (9)
∂tCag(t, tw) ≪ Cag(t, tw) C < qea , (10)
∂tχst(t, tw) ∼ χst(t, tw) χ < χea , (11)
∂tχag(t, tw) ≪ χag(t, tw) χ > χea . (12)
The aging self-correlation and susceptibility scale as
Cag(t, tw) ≈ qea fc
R(tw)
, χag(t, tw) ≈ qea fχ
R(tw)
. (13)
The scaling functions satisfy the limit conditions fc(1) = 1, fc(∞) = 0, fχ(1) = 0
and fχ(∞) = 1/Teff . Using mathematical properties of monotonic two-time functions
one can show that such a scaling holds asymptotically in each (two) time-scale of the
evolution [13]. While in a system undergoing finite dimensional coarsening R(t) has
a natural interpretation as the typical domain radius, in mean-field models there is
no immediate understanding of the ‘clock’ R(t) that, in a sense, sets the macroscopic
time-scale. The numerical solution suggests that R(t) is just a power of time.
CONTENTS 8
In the asymptotic limit in which the additive separation of time-scales with the
scaling form (13) holds it is convenient to use a parametric description of the dynamics
in which times do not appear explicitly. More precisely, the approach to the asymptotic
scaling, and fc and fχ, can be put to the test by constructing ‘triangular relations’
between correlations and susceptibilities, respectively. For generic three long times
t1 ≥ t2 ≥ t3 ≫ t0 one computes the correlations C(tµ, tν), µ > ν = 1, 2, 3. If the times
are such that the ratios R(tµ)/R(tν) remain finite in the asymptotic limit, that is to
say C(tµ, tν) = Cag(tµ, tν), one has
C(t1, t3) = qea fc
f−1c [C(t1, t2)/qea] f
c [C(t2, t3)/qea]
. (14)
If, instead, t1 = t2 + τ with τ > 0 finite, and R(t2)/R(t3) finite in such a way that
C(t1, t2) = qea + Cst(t1 − t2) and C(t2, t3) = Cag(t2, t3) one has
C(t1, t3) = min [C(t1, t2), C(t2, t3) ] (15)
asymptotically. This form goes under the name of dynamic ultrametricity. In the
opposite case t3 = t2 − τ and R(t1)/R(t2) finite dynamic ultrametricity also holds.
These relations follow immediately from the additive separation of time-scales (8) and
the scaling (13) but they can be shown without assuming dynamic scaling, just by
using the monotonicity properties of temporal correlations [13]. The simplest way to
see these relations at work is to display C(t1, t2) against C(t2, t3), for a chosen value of
C(t1, t3) < qea, in a parametric plot in which t2 varies from t3 to t1. In the asymptotic
limit t3 → ∞ the construction reaches a stable master curve as displayed in Fig. 2-
left. The vertical and horizontal parts correspond to t2 such that C(t1, t2) > qea and
C(t2, t3) > qea, respectively, and dynamic ultrametricity holds. The curved part is for
t2 such that all correlations are in the aging regime and its functional form is fully
determined by fc. The ‘clock’ R yields the speed at which the data-point moves on the
parametric curve. A similar construction can be done for the susceptibilities.
The stationary correlation, Cst, and susceptibility, χst, are linked by the equilibrium
fluctuation dissipation theorem (fdt), χst = (1− Cst)/T . In the aging regime, instead,
there is a non-trivial relation between χ and C: χag = (qea − Cag)/Teff that yields
fχ(x) = (1−fc(x))/Teff . This relation is also better appreciated if shown in a parametric
construction in which times do not appear explicitly. In the long waiting-time limit the
plot χ(t, tw) against C(t, tw) with t the parameter running from tw to infinity approaches
a broken line form with the slopes −1/T (for C > qea) and −1/Teff (for C < qea). Again,
the ‘clock’ R yields the speed at which this curve is constructed upon increasing t.
An analytic solution to the Schwinger-Dyson equations was derived in the limit
of long waiting-time in which the separation of time-scales, that is to say the plateaus
in C and χ, are fully established. In the aging regime, one uses the fact that the
variation of the correlation and linear susceptibility are negligible with respect to all
terms in the right-hand-side of the equations and can thus be dropped. Furthermore,
one approximates the integrals by separating the contributions from the stationary and
CONTENTS 9
10.80.60.40.20
10.80.40.20
Figure 2. Sketch of the parametric representation of the correlation and susceptibility.
Left: triangular relation between the correlation function in the asymptotic limit
t3 → ∞. The three curves correspond to different t1’s such that C(t1, t3) takes three
values. The breaking points lie at qea. The arrow indicates the sense of the evolution
when t2 increases from t3 to t1. Right: susceptibility, χ(t2, t3) against correlation,
C(t2, t3) at fixed t3 using t2 ≥ t3 as a parameter in the long t3 limit. The breaking point
at (qea, χea) separates the stationary regime where the equilibrium fdt is satisfied from
the aging regime where it is modified. The slopes are −1/T and −1/Teff , respectively.
The arrow also indicates here the sense of the evolution when t2 increases from t3.
aging regimes [19]. Equation (5) becomes
µ∞Rag(t, tw) ∼
p(p− 1)
dt′ Cp−2ag (t, t
′)Rag(t, t
′)Rag(t
′, tw) (16)
(µ∞ is a constant with contributions from limt→∞ zt and border terms in the integrals).
The companion eq. (4) takes a similar form within the same approximation. Now, the
surprise is that the approximate equations are invariant under the transformation
t→ ht ≡ h(t) ,
Cag(t, tw) → Cag(ht, htw) ,
Rag(t, tw) → ḣtw Rag(ht, htw) ,
with ht positive and monotonic and ḣtw ≡ dhtw/dtw. While the functions fc and
fχ and their fd relation are fixed by the remaining approximate equation (16) and
its companion, the time-reparametrization invariance does not allow one to compute,
analytically, the clock R(t). This problem is similar to the velocity selection problem
present for instance in Fisher differential equation describing front propagation and the
like [21]. The exact Schwinger-Dyson equations do have a unique solution with a special
function R(t) that is selected by the short-time difference effect of the time-derivatives.
However, as time increases and time-differences increase too the effect of the time-
derivatives diminishes. In the approximate analytic solution we take advantage of this
fact to solve the equations asymptotically but we introduce in this way a symmetry that
does not allow us to fix R(t). We obtain, instead, a family of solutions parametrized by
ht. It is important to reckon that the parametric constructions in Fig. 2 are independent
of the clock and thus are fully determined by the approximate treatment.
CONTENTS 10
The case p = 2 with Ising spins or Sherrington-Kirkpatrick model has a more
complicated scaling form with a sequence of two-time scales leading to dynamic
ultrametricity for all C < qea asymptotically [13]. This behaviour is technically more
involved but, as far as the symmetry properties are concerned, it is similar to the case
treated above. The full dynamic equations have a unique solution but the approximate
ones acquire time-reparametrization invariance. We shall not discuss these cases further
in the rest of this review.
2.3. Short-ranged models – dynamic action
In a series of papers [4]-[9] we claimed that the time-reparametrization invariance thus
far introduced via the asymptotic solution to the dynamic equations in mean-field glassy
problems is indeed an asymptotic property of the dynamics of glassy systems, mean-field
and finite dimensional. The separation of time-scales stationary-aging has been observed
in a variety of glassy systems with numerical simulations and experiments [3, 22]. The
slowness of the decay in the aging regime, eqs. (10) and (12) and a weak long-term
memory of the kind (13), are the hallmark of glassy relaxation. The idea is then that
time-reparametrization invariance is the symmetry associated to the dominant dynamic
fluctuations in these sytems.
In order to pursue this idea forward one has to first prove that the symmetry of
the saddle-point equations is also a symmetry of the action in the dynamic generating
functional not only of fully-connected spin models of the mean-field type but also of finite
dimensional glassy systems. In [4] we derived and studied the symmetry properties of
the dynamic action – the so-called Martin-Siggia-Rose action associated to Langevin
stochastic dynamics – of the disorder averaged soft-spin 3d Edwards-Anderson (ea)
model of spin-glasses. (H =
〈ij〉 Jijsisj with Jij Gaussian random variables with zero
mean and taking non-zero value only on nearest neighbours on the d-dimensional lattice
and si = ±1.) In our analysis we took a number of steps that we briefly recall here.
First, we introduced four fluctuating two-time fields defined on the lattice sites,
Qabi (t, tw) , with i = 1, . . . , N and a, b = 0, 1 . (18)
Their thermal averages are the expected values of the local two-time self-correlation
(a = b = 0), the retarded linear response (a = 0, b = 1), the advanced linear response
(a = 1, b = 0), and a fourth observable (a = b = 1) that vanishes if causality is preserved.
Second, we assumed that a separation of time-scales fast-slow, of the type described in
the previous subsection, applies to these fluctuating fields too. Third, we determined the
long-time action by using a Renormalization Group (RG) scheme in the time variables.
This allowed us to write the full action as a sum of two contributions: one from the fast
regime holding at short time-differences, another one from the slow regime valid at long
time-differences. The coupling between these two vanishes asymptotically. Fourth, we
analyzed the surviving terms in the action, that are just the slow contribution. Using
advanced and retarded scaling dimensions that are just the labels a and b of the fields,
CONTENTS 11
one finds that the global time-reparametrization,
t→ ht ≡ h(t) , Q̃
i (t, tw) = (ḣt)
a(ḣtw)
b Qabi (ht, htw) , (19)
leaves all surviving terms unmodified. This step is concisely carried out as follows. Take
a generic term in the action. Under (19) it transforms as
dtν · · · →
dtν (ḣtν )
∆ν · · ·
where N is the number of time integrals, the dots represent a product of the fields
Qabi and ∆ν is the sum of all factors ḣtν arising from the transformation of the fields.
Interestingly enough, one finds that all the ∆ν equal one. Thus, with a simple change
of variables one absorbs each factor ḣtν in the corresponding integration variable and
dtν · · · →
dhν · · · (20)
Note that in order to prove invariance under the simpler and more common scale
transformation, ht = λt, it is enough to have
ν=1∆ν = N . Scale invariance is included
in the larger global time-reparametrization symmetry but it is, clearly, more restrictive.
Finally, we showed that the measure in the functional integral is also global time-
reparametrization invariant, completing the program. We refer the reader to Refs. [4]
and [6] for the technical details leading to these results.
In the disordered 3d ea model studied we carried out the disorder average. The
presence of quenched disorder gave us an analytic control of the theory but this does
not necessarily mean that such a symmetry develops only for the long time regime
of models with quenched disorder. It appears that if the essential assumptions are
causality and unitarity, and a separation of time scales that takes the action to a non-
trivial asymptotic state (some glassy state), then one expects the symmetry to exist
for systems without quenched disorder but that are glassy nevertheless. In order to
check the development of time-reparametrization invariance in problems of particles in
interaction in finite dimensions one should first obtain the relevant action to work with.
A good candidate for a starting point is the Dean-Kawasaki stochastic equation for the
evolution of the local density [23]. The idea is then to write its dynamic generating
functional and study the symmetry properties of the effective action assuming that a
separation of time-scales exists. We are currently carrying out this study.
2.4. Turning a nuisance into something useful - symmetry as a guideline
The global time-reparametrization invariance implies that the action describing the long
time slow dynamics of a spin-glass is basically a “geometric” random surface theory, with
the Q’s themselves as the natural coordinates. The original two times parametrize the
surface. Physical quantities, as the bulk integrated response χ(t1, t2) =
dt′R(t′, t2)
and correlation C(t1, t2) have scaling dimension zero under t → h(t) as well as their
local counterparts. The emergence of this gauge-like symmetry, which may appear first
CONTENTS 12
as a nuisance that relates too many solutions for just one problem. However, it may
provide a simple way to understand spatial fluctuations in systems that possess this
global time-reparametrization symmetry.
Here, a simple analogy with the problem of a ferromagnet may elucidate the point
we want to make. In a ferromagnet the action is invariant under uniform rotations of
the magnetization vector ~m. In the ordered phase, rotation symmetry is spontaneously
broken, and a certain magnetization direction ~m0 is picked. Typically, a vanishingly
small pinning field selects this direction. The low action excitations are the spin waves,
fluctuations of the uniform, symmetry broken, state. These spin waves can be described
in terms of smooth spatially fluctuating rotations around the uniform magnetization
state. The spin waves, generated by using slowly varying local rotations, are the
Goldstone modes of the ferromagnet.
Similarly, in the aging regime of the glassy systems, the action has a global
symmetry, under uniform time-reparametrizations, t → h(t), with the fields
transforming as in eq. (19). The probability weight of having certain local two-
time correlation and response, the observables, should be independent of this
reparametrization. After coarse-graining over a linear length ℓ the non-vanishing
fluctuating fields are the local coarse-grained correlation (a = b = 0) and linear response
(a = 0, b = 1),
C(~r; t, tw) =
j∈V~r
sj(t)sj(tw) , R(~r; t, tw) =
j∈V~r
δsj(t)
δhj(tw)
with the sum carried over the spins in the volume V~r = ℓ
d centered at ~r, and a is the
lattice spacing. The transformation (17) is now restated as
t→ ht ≡ h(t) ,
Cag(~r; t, tw) → Cag(~r; ht, htw) ,
Rag(~r; t, tw) → ḣtw Rag(~r; ht, htw)
, (21)
and it is an asymptotic symmetry of the action for the slow coarse-grained degrees of
freedom. Indeed, the symmetry breaking terms, that have their origin in the short-
time dynamics and short-time difference dynamics, are not identical to zero but become
vanishing small asymptotically. The particular scaling function R(t) selected by the
system is determined by matching the fast and the slow dynamics. It depends on several
details – the existence of external forcing, the nature of the microscopic interactions, etc.
In other words, the fast modes which are absent in the slow dynamics act as symmetry
breaking fields for the slow modes.
In analogy with the spin-wave fluctuations in magnetic systems, that are dictated by
the rotational symmetry, we proposed that the smooth fluctuations in the glassy phase
can be obtained by studying the slow varying, position dependent reparametrizations
t→ h(~r, t) around the one reparametrization R(t) selected by the short-time dynamics.
In other words, we basically proposed that there are Goldstone modes for the
glassy action which can be written as slowly varying, spatially inhomogeneous time
reparametrizations. This suggests that the slow part of the coarse-grained local
CONTENTS 13
correlations and susceptibilities should scale as
Cag(~r; t, tw) ≈ qea fc
h(~r, t)
h(~r, tw)
, χag(~r; t, tw) ∼ fχ
h(~r, t)
h(~r, tw)
, (22)
with fc and fχ the same functions describing the global correlation and susceptibility,
respectively [eqs. (13) and (23)] and the same function h(~r, t) scaling the two-time
correlation and susceptibility on each site ~r [4, 5, 6]. The sum rules Cag(t, tw) =
ddr Cag(r; t, tw) and χag(t, tw) = V
ddr χag(r; t, tw) apply.
The reason for this proposal is that the global reparametrization invariance in time
of the dynamic action in this two-time regime leads to low action excitations (Goldstone
modes) for smoothly varying spatial fluctuations in the reparametrization of time, but
not in the external form of the scaling functions. As in a sigma model (for example, to
describe the ferromagnet), the external functions fc and fχ fix the manifold of states,
and the local time reparametrizations correspond to fluctuations restricted to this fixed
manifold of states (in the ferromagnet, the tilting of direction but not the magnitude of
the magnetization vector).
2.5. The spherical p = 2 case or mean-field domain growth
The spherical SK model (p = 2) can be solved exactly by analyzing the Langevin
equation in the basis of eigenvectors of the random matrix Jij [24]. While the correlation
has a very similar behaviour to the one of the p ≥ 3 cases (see Fig. 1-left), the
susceptibility is quite different. One can mention that the stationary and aging regimes
in the linear response are not so sharply separated in this case. If one uses an additive
separation as in (8) the aging contribution to the integrated linear response, χag(t, tw),
vanishes asymptotically. More precisely,
χag(t, tw) ∼ t
. (23)
Importantly enough, even though the inequalities (10) and (12) are still valid, a
careful inspection of all terms in the Schwinger-Dyson equations shows that they are of
the same order asymptotically. Moreover, the stationary contribution to the equations
in the aging regime is not just a constant: the corrections associated to the asymptotic
approach to the plateau in the correlation cannot be neglected. As a result one cannot
simply drop the time derivatives and the Schwinger-Dyson equations in the aging regime
are not time-reparametrization invariant but just scale invariant, that is to say, they are
unchanged by the transformation t→ h(t) = λt, with C and R transforming as in (17)
and λ a positive constant. A similar mechanism, though even harder to prove, applies
to the dynamic equations for the correlation and linear response of the non-conserved
dynamics of the O(N) ferromagnetic model in the large N limit [8].
In line with what we explained above, the effective action for the slow degrees
of freedom of the p = 2 spherical model and, more generally, the d-dimensional
ferromagnetic O(N) model in the large N limit, are not invariant under global time-
reparametrizations but only under global rescaling of time, t → λt. This marks
CONTENTS 14
an important difference between models with a finite aging response and these quasi
quadratic models with a vanishing aging response [8].
This result is important for a number of reasons: first, it suggests that the
susceptibility, or even the effective temperature, might be intimately related to the
symmetry properties of the dynamics and consequently of the fluctuations; second, it
suggests that the mechanism for fluctuations in coarsening systems might be different
from the one of other glassy problems with finite and well-defined effective temperatures.
In order to justify the latter statement it remains to be checked whether the reduction of
time-reparametrization invariance to just scale invariance also holds in finite dimensional
non-field coarsening.
2.6. Quantum problems
In the case of quantum models one introduces the effect of dissipation by coupling the
system to an environment represented, typically, by an infinite ensemble of quantum
harmonic oscillators. One then uses the Schwinger-Keldysh formalism to write a
generating functional and from it, in the fully-connected or infinite dimensional cases,
one derives Schwinger-Dyson equations similar to the ones above. The asymptotic
analysis of these equations follows the same steps as in the classical limit [25, 17] (at
least in the case of a weak coupling to the bath [26]) and the time-reparametrization
invariance also applies.
The appearance of an asymptotic invariance under time-reparatrizations in the
mean-field dynamic equations was related to the reparametrization invariance of the
replica treatment of the statics of the same models [10, 27]. The latter remains
rather abstract. Brézin and de Dominicis [27] studied the consequences of twisting the
reparametrizations in the replica approach. Interestingly enough, this can be simply
done in a dynamic treatment either by applying shear forces or by applying heat-
baths with different inherent dynamics to different parts of the system. More precisely,
using a model with open boundary conditions one could apply a thermal bath with
a characteristic time-scale on one end and a different thermal bath with a different
characteristic time-scale on the opposite end and see how a time-reparametrization
‘flow’ establishes in the model.
3. Consequences and tests
In this Section we discuss how one can put these ideas to the test by presenting a number
of consequences of global time-reparametrization invariance that are directly measurable
numerically and experimentally. The properties that we discuss explicitly are:
1. A growing dynamical correlation length.
2. Scaling of the pdf of local two-time functions.
3. Functional form of the pdf of local two-time functions.
4. Triangular relations between two-time functions.
CONTENTS 15
5. Scaling relations for general multi-time functions.
6. Local fluctuation-dissipation relations.
7. Infinite susceptibilities.
In considering these predictions, we shall separate them into two distinct
classes. The first class contains predictions that are consistent with other theoretical
scenarios as well as with the presence of reparametrization invariance. Hence, while
reparametrization invariance leads to these predictions, this class alone cannot be used
to argue for the role of the symmetry over other mechanisms. The second class, on the
other hand, contains predictions that are not natural within other frameworks, and to
the date of this report have no obvious explanation within other frameworks. Properties
1 and 2 belong to the first class, properties 3-7 to the second. We shall present these
properties in detail below.
3.1. Two-time correlation length
In equilibrium statistical models one defines the static correlation length, ξeq, from the
spatial decay of the correlation between the fluctuations of the order parameter measured
at two space points, 〈 [φ(~r) − 〈 φ(~r) 〉][φ(~r′) − 〈 φ(~r′) 〉] 〉
|~r−~r′|=∆
∼ ∆−d+2−η e−∆/ξeq ,
where the angular brackets denote an average over the Gibbs-Boltzmann measure.
ξeq depends on temperature and, in second order phase transitions, it diverges at Tc
leaving only an algebraically decaying correlation. A dynamic equilibrium correlation
length characterizes the spatial decay of equal-time correlations in the equilibrium
relaxation of ‘usual’ systems. Similarly to the static case one defines ξ(t) via
〈 [φ(~r, t) − 〈 φ(~r, t) 〉][φ(~r′, t) − 〈 φ(~r′, t) 〉] 〉
|~r−~r′|=∆
∼ e−∆/ξ(t); the angular brackets
indicate here an average over thermal histories and, for simplicity, we omitted the
algebraic correction to the exponential decay. The average over thermal noise can
be traded for an integration over the reference space-point ~r and one then obtains
ξ(t) from V −1
ddr δφ(~r, t)δφ(~r′, t) |
|~r−~r′|=∆
∼ e−∆/ξ(t) where δφ(~r, t) ≡ φ(~r, t) −
ddr′′ φ(~r′′, t). The correlation length, ξ(t), depends now on temperature and
total time.
In systems with slow dynamics in which the order parameter is a two-time entity,
a two-time correlation length can be defined in analogy to what we described in the
previous paragraph:
[φ(~r, t)φ(~r, tw)− 〈 φ(~r, t)φ(~r, tw) 〉]
× [φ(~r′, t)φ(~r′, tw)− 〈 φ(~r′, t)φ(~r′, tw) 〉]
|~r−~r′|=∆
∼ e−∆/ξ(t,tw) . (24)
Once again by trading the thermal average by a spatial average 〈 φ(~r, t)φ(~r, tw) 〉 becomes
the global correlation C(t, tw) and ξ(t, tw) is derived from the four-point correlation
C4(∆; t, tw) ≡
ddr δ[φ(~r, t)φ(~r, tw)]δ[φ(~r′, t)φ(~r′, tw)]
|~r−~r′|=∆
CONTENTS 16
1e+081e+061e+041e+021e+00
tw=1k
tw=10k
tw=100k  0
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k  0
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k
Figure 3. The correlation length in the 3d EA model at T = 0.6 < Tc and L = 100.
(a) As a function of t − tw; (b) as a function of 1 − C; (c) in the scaling form t
against 1 − C. These results are taken from [9]. The vertical arrow in panels (b) and
(c) indicates the value of qea.
with δ[φ(~r, t)φ(~r, tw)] ≡ φ(~r, t)φ(~r, tw) − C(t, tw). The quantity C4(∆; t, tw) measures
the probability that a fluctuation of the two-time composite field φ(~r, t)φ(~r, tw) with
respect to its global average C(t, tw) in the spatial position ~r affects a fluctuation of the
same composite field at a different site ~r′ located at a distance ∆ from ~r and averaged
over the whole ensemble of reference points ~r in the sample.
The numerical analysis of C4 in the low temperature out of equilibrium dynamics
of the 3d ea model [6, 9] (see Fig. 3), soft sphere [28] and Lennard-Jones [29] mixtures
yield
ξ(t, tw) ∼
ξst(t− tw) C > qea
ξag(t, tw) C < qea
ξag(t, tw) ∼ t
w g(C) and a a small power (26)
(a logarithmic growth is also possible). g(C) is a monotonically decreasing function of C.
This is a two-time monotonically growing function even at time-lags that are longer than
the waiting-time dependent α relaxation time. Note the difference with what is observed
in the super-cooled phase where a number of numerical and experimental measurements
point at a dynamic correlation length that reaches a maximum at the α relaxation time
and later diminishes to zero [30]. The divergence has a clear interpretation within the
global time-reparametrization scenario: it is due to the generation and development of
the zero mode. The global reparametrization invariance symmetry develops only in the
limit of very long times, so that at intermediate times the irrelevant terms that are scale
down to zero are still manifest. These irrelevant terms, symmetry breaking ones, give
a finite length scale (or a finite ‘mass’) to the soft reparametrization modes. Because
they are irrelevant, the correlation length increases or, equivalently, the mass decreases
asymptotically.
Even though the numerically accessible times are sufficiently long so as to see
the separation of time-scales in the relaxation of the global correlation, the correlation
length is still very short. In numerical simulations ξ reaches of the order of 4 lattice
spacings or inter-particle distances in the spin-glass problem or soft sphere and Lennard-
Jones system, respectively. ξ just increases by, say, a factor 4 when the waiting-time is
CONTENTS 17
increased by nearly 4 orders of magnitude. The expected limit ξ(t, tw) → ∞ is thus far
from being attained.
As we already mentioned in the introduction to this Section, the growing length
scale is consistent with the development of time-reparametrization invariance, but it
is also consistent with other mechanisms. Hence, it alone cannot be used to argue
unambiguously in favor of the symmetry-based approach. For example, theories based
on the mode-coupling approach or random first order scenario [33] and its refinement
including the effect of entropic droplets [34], dynamical criticality controlled by a zero-
temperature critical point [35] and frustration limited domains [36] are used to justify
the growth of a dynamic length-scale, at least in the super-cooled liquid. Hence, the
existence of a growing length scale belongs to the first class of predictions we mentioned
in the introduction to this section.
Finally, let us note that the fluctuations in the susceptibility, or in multi-time
correlations, see eq. (35), and associated susceptibilities, can be used to derive other
correlation lengths. It would be interesting to check whether all these behave in the
same manner.
3.2. Scaling of the pdf of local two-time functions
The most direct way of testing the mere existence of local fluctuations is to measure the
probability distribution function (pdf) of local coarse-grained correlators, C(~r; t, tw),
and linear susceptibilities, χ(~r; t, tw), at different pairs of times, t and tw. In such
a measurement one is forced to use finite coarse-graining lengths. ℓ then becomes a
parameters that has to be taken into account in the scaling analysis of the results.
In Ref. [9] we showed that, quite generally, the pdf of local coarse-grained
correlators can be scaled onto universal curves as long as the global correlation, C(t, tw),
is the same, and the ratio of the coarse graining length over the dynamical correlation
length, ℓ/ξ(t, tw), is held fixed (see [31] for a similar discussion applied to the super-
cooled liquid). Such scaling can be easily understood as follows. At fixed temperature
the pdf ρ[Cr; t, tw, ℓ, L] depends on four parameters: two times, t and tw, and two
lengths, the coarse-graining length, ℓ, and the size of the system, L. As in the aging
regime C(t, tw) is a monotonic function of the two times and ξ(t, tw) ∼ t
wg(C), one
can trade the two times by C and ξ in complete generality. The next step is a scaling
assumption: that in the long times limit the pdfs depend on the coarse graining length
ℓ, the total size L and the scale ξ only through the ratios ℓ/ξ and ξ/L. This last
step, we should stress, is really a scaling assumption, and not a trivial requirement
from dimensional analysis. The lengths ℓ, L and ξ are already dimensionless as they
are measured in units of the lattice spacing. The end result from the rewriting of the
parameters in terms of the global correlation and the scaling hypothesis is that the
pdfs characterizing the heterogeneous constant temperature aging of the system can be
written as
ρ[Cr;C(t, tw), ℓ/ξ(t, tw), ξ(t, tw)/L] . (27)
CONTENTS 18
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k
 0  0.2  0.4  0.6  0.8  1
tw=1k
tw=10k
tw=100k
Figure 4. pdf of local coarse-grained correlations Cr at different times t and tw in
the 3d Edwards-Anderson model with L = 100 at T = 0.6 < Tc. The waiting-times
are given in the key and the global correlation is fixed to C = 0.4 < qea. (a) The
coarse-graining boxes have linear size ℓ = 9 in all cases. The curves do not collapse, a
slow drift with increasing tw is clear in the figure. (b) Variable coarse-graining length
ℓ chosen so as to held ℓ/ξ approximately constant. The collapse improves considerably
with respect to panel (a). These results are taken from [9].
In Fig. 4 we show the scaling of the distribution of local coarse-grained correlations
in the 3d ea model and the effect of the scaling variable ℓ/ξ (the size of the system, L,
is sufficiently large so that ξ/L vanishes in practice). It is noteworthy that a reasonable
scaling with the global correlation held fixed and not taking into account the effect of
the second scaling variable has been already achieved, approximately, in the Edwards-
Anderson model [5, 6], as well as in the kinetically constrained models studied in [7]
and Lennard-Jones systems [32]. This is justified by the fact that ξ varies very slowly
with tw. However, it is clear that at long though finite times one has to hold the ratio
ℓ/ξ constant to obtain a full collapse in all cases.
The scaling variable ℓ/ξ allows one to study the change in the functional form of the
pdfs upon modifying the coarse-graining volume. Indeed, the pdfs should crossover
from a non-trivial form to a simple Gaussian when ℓ goes through the value ξ. In
summary, for finite ξ one identifies three ℓ-dependent regimes with different functional
forms of the pdfs:
• ℓ ≪ ξ. For finite ξ this means ℓ of the order of the lattice spacing, ℓ ∼ a. In this
case the pdfs do not have any particular structure.
• ℓ ∼ ξ. For finite but large ξ this case is non-trivial and indeed the one that is
accessed with numerical simulations and experiments. One finds that the statistics
is non-Gaussian for all C. The skewness decreases from zero at C = 1 to
reach a minimum and then increase again at small values of C. As regards the
functional form, one finds that a Gumbel-like functional form, characterized by a
real parameter that depends on C and ℓ/ξ describes the data rather well for, say,
qea/2
∼ qea.
• ℓ ≫ ξ. In this limit one matches the central-limit theorem conditions and the
statistics becomes Gaussian for all C.
CONTENTS 19
The analysis of the ℓ-dependent pdfs thus provides an independent way to estimate ξ.
3.3. Effective action for local ages
The argument leading to eq. (27) is a scaling hypothesis and it does not rely on time-
reparametrization invariance. Indeed, there exist models in which the scaling (28) for
the pdf of local coarse-grained correlations is found, e.g. the O(N) model in the large
N limit [8], and global time-reparametrization does not apply.
The implications of time-reparametrization invariance appear later, as a prediction
for the functional form of the asymptotic limit
ρ∞(Cr;C) ≡ lim
t,tw→∞
C(t,tw)=C
ρ[Cr;C(t, tw), ℓ/ξ(t, tw), ξ(t, tw)/L] . (28)
In order to study the functional form that ρ∞ can take let us now use the symmetry
argument to analyse the statistics of the fluctuations of the local correlations. So far
we have not yet determined how much the h(~r, t) vary in space and time. To this end
we need to derive an effective action for these functions that will tell us how costly it
is to deviate from the average clock R(t). Ideally, one would like to derive this action
from the microscopic one. This should be possible in quasi mean-field models such as
the p-spin model with Kac long (but finite) range interactions [37]. For the moment
we have just proposed the simplest action that serves our purposes in the ideal limit in
which the zero mode is fully developed ξ → ∞ and the local quantities are measured in
the infinite coarse-graining volume limit with ℓ/ξ → 0 [7]. Otherwise the parameter ℓ
should be taken into account.
To start with we worked with the more convenient transformed variable h(~r, t) ≡
e−ϕ(~r,t) that implies
Cag(r; t, tw) ≈ qea fc
h(~r, t)
h(~r, tw)
= qea fc
dt′∂t′ ϕ(~r,t
and we searched for the simplest action that satisfies the constraints due to the
symmetries. These are:
i. The action must be invariant under a global time reparametrization t→ h(t).
ii. If our interest is in short-ranged problems, the action must be written using local
terms. The action can thus contain products evaluated at a single time and point in
space of terms such as ϕ(~r, t), ∂tϕ(~r, t), ∇ϕ(~r, t), ∇∂tϕ(~r, t), and similar derivatives.
iii. The scaling form in eq. (29) is invariant under ϕ(~r, t) → ϕ(~r, t) + Φ(~r), with Φ(~r)
independent of time. Thus, the action must also have this symmetry.
iv. The action must be positive definite.
These requirements largely restrict the possible actions. The one with the smallest
number of spatial derivatives (most relevant terms) is
S[ϕ] =
(∇∂tϕ(~r, t))
∂tϕ(~r, t)
, (30)
CONTENTS 20
with K a stiffness. A term M ∂tϕ(~r, t) is also allowed by symmetry but since its space-
time integral is constant we drop it. The action solely depends on the time derivatives
∂tϕ(~r, t) and it is simple to check that it satisfies all the four constraints enumerated
above (the last requirement follows from the fact that h(~r, t) are monotonically
increasing functions of time) [7].
Due to the simple form (30) the ∂tϕ(~r, t) are uncorrelated at any two different times
t1 and t2. Thus the expression ∆ϕ~r|
dt′ ∂t′ϕ(~r, t
′) entering the exponential in
the scaling form in eq. (29) is a sum of uncorrelated random variables in time. One
can interpret such expression as the displacement of a random walker with position
dependent velocities. Alternatively, one can think of the space-dependent differences
∆ϕ~r|
as the net space-dependent height (labeled by t) of a stack of spatially fluctuating
layers dt ∂tϕ(~r, t). The action for the fluctuating surfaces of each layer is given by
eq. (30).
The statistics of the ∆ϕ~r|
are completely determined as follows. The action
(30) transforms into one of a Gaussian surface after the introduction of a ‘proper’ time
τ ≡ lnR(t), and the change of variables, ψ2(~r, t) = ∂τϕ(~r, τ). Indeed,
Cag(~r; t, tw) ≈ fc
R lnR(t)
lnR(tw)
dτ ′ ψ2(~r,τ ′)
S[ψ] = K
dτ ′ [∇ψ(~r, τ ′)]2 . (32)
Due to the Gaussian statistics of the ψ, it is simple to show that connected N -point
correlations of ∆
varphi~r1 |
satisfy
〈∆ϕr1|
∆ϕr2 |
· · ·∆ϕrN |
〉c = [τ(t)− τ(tw)] F(~r1, ~r2, . . . , ~rN) , (33)
where the function F can be obtained from Wick’s theorem, summing over all graphs
that visit all sites (connected) with two lines (because of ψ2) for each vertex i
corresponding to a position ri. The reparametrized times appear only in the prefactor
τ(t) − τ(tw) = ln[R(t)/R(tw)]. The probabilistic features of the fluctuations of local
correlations C(~r, t, tw) depend on times only through R(t)/R(tw), and hence only
through the global correlation itself C(t, tw). In consequence, the action (32) implies
the scaling (28). The fact that the time-dependencies of the statistical properties of the
two-time local coarse-grained correlations are fully determined by the global correlation
is a very welcome property of action (30) since it was not obvious a priori.
Having the forms in eqs. (31) and (32) allows us to make some quantitative
predictions about the form of the pdfs. With some algebraic manipulations one shows
the following generic features [7]:
• The distribution is non-Gaussian for all C.
• For C
∼ qea the pdf is negatively skewed and once put into normal form it is
very close to the distribution of the global equilibrium magnetization in the 2d xy
model [38]. It can then be approximately described by a generalized Gumbel form
with real parameter.
CONTENTS 21
• In the opposite limit C
∼ 0 the pdf is positively skewed and it does not take any
recognizable form.
If one is interested in testing the action further one can simply use eq. (32) to generate,
numerically, the ψ(~r, t), construct the C(~r, t, tw) from these functions, and then compare
the pdfs thus obtained to the ones measured, say, in a numerical simulation of a given
problem. Note that the scaling function fc also plays a role in the functional form of
the pdf of local correlations. The same argument applies to the susceptibilities.
The local coarse-grained correlations are, by construction, sums of correlated
random variables (unless ℓ ≫ ξ). With numerical simulations of the 3d ea model [9]
and kinetically facilitated lattice gases [7] we found that the pdfs of correlations coarse-
grained over finite lengths ℓ have a functional form that resembles a generalized Gumbel
distribution characterized by a continuous parameter that depends on ℓ/ξ and the value
of the global correlation, C, when C
∼ qea (see also [38]). This fact is consistent with the
discussion above and also with the observation of Bertin and Clusel that Gumbel-like
pdfs with real parameter characterize the statistics of sums of random variables with
particular correlations between the elements [39]. The fact that we obtain Gumbel-like
distributions then means that the correlations between the terms in the sum are of the
form needed to get this type of pdf.
In short, the time-reparametrization scenario predicts, in its simplest setting, that
eqs. (31) and (32) fully characterise the fluctuation of the local correlations in the large
times and coarse-graining volume limits.
3.4. Two-time scaling of local functions
As we argued in Sect. 2.4, the global time-reparametrization invariance suggests that
in the ideal asymptotic limit the slow part of the coarse-grained local correlations and
susceptibilities should scale as in eq. (22) in the ideal limit a ≪ ℓ ≪ ξ. In practice
the ideal limit is not reached and one is forced to work with finite correlation lengths
and thus finite coarse-graining lengths too. The finite ℓ will then play a role and has
to be taken into account. We now present some tests of eq. (22) that are based on the
parametric representation of the dynamics explained in Sect. 2 and take into account
the finite value of ℓ.
Let us then imagine that we compute three local coarse-grained two-time
correlations, C~r, at three space points ~r1, ~r2 and ~r3, using a given coarse-graining length,
a≪ ℓ, and that we obtain functional forms that are characterized by eq. (22) with, say,
h(~r1, t) = ln (t/t0), h(~r2, t) = t/t0, and h(~r3, t) = e
ln2(t/t0), in the aging regime. In Fig. 5
we sketch the decay of these correlations for the same tw as a function of time-delay.
The plateau is at the same height since qea as well as the the full stationary decay are
not expected to fluctuate. The external function fc is the same in all curves. It is clear
that the decay of the three correlations follows a different pace, the one at ~r3 is the
fastest while the relaxation at ~r2 is the slowest.
The simplest way to put the proposal (22) to the test is to analyze the implications
CONTENTS 22
1e+00
1e-01
1e-02
1e+061e+041e+021e+00
h3  0
 0.25
 0.75
 0  0.25  0.5  0.75  1
Figure 5. Left: sketch of the decay of the correlation with the same stationary decay
to qea – shown with a horizontal dashed line – and three choices of the scaling function
h(r1, t) = ln(t/t0), h(r2, t) = t/t0, and h(r3, t) = e
lna(t/t0). The waiting-time is the
same in all curves. Right: the relation between the integrated linear response against
the correlation. With a solid line, the parametric plot for fixed and long tw, using t
as a parameter that increases from tw at C = 1 to ∞ at C = 0. With symbols, the
three pairs (Cj(t, tw), χj(t, tw)) for the same tw, a fixed value of t and hj(t) as in the
left panel.
of eq. (22) on local triangular relations. In Sect. 2 we showed that two-time functions
with a separation of time-scales as in eq. (8) and an aging scaling as in eq. (13) are
related in a parametric way in which times disappear, see the sketch in Fig. 2-left.
Equation (22) implies that the local (fluctuating) two-time functions should verify the
same relation
Cag(~r; t1, t3) = qea fc
f−1c [Cag(~r; t1, t2)/qea] f
c [Cag(~r; t2, t3)/qea]
. (34)
This is a result of the fact that time-reparametrization invariance restricts the
fluctuations to appear only in the local functions h(~r, t) while the function fc is locked
to be the global one everywhere in the sample.
A pictorial inspection of this relation should take into account the fact that while
the stationary decay is not expected to fluctuate, the full aging relaxation and, in
particular, the minimal value of the local two-time functions, C(~r; t1, t3), are indeed
fluctuating quantities. The parametric construction on different spatial regions should
yield ‘parallel translated’ curves with respect to the global one, as displayed in Fig. 2-
left. Fluctuations in the function fc would yield different functional forms in the
curved part of the parametric construction. A more quantitative analysis can be
done by using the knowledge of fc that can be extracted from the global correlation
decay. Indeed, if fc is known, the parametric plot f
c (C~r12/qea)/
f−1c (C~r13/qea) against
f−1c (C~r23/qea)/
f−1c (C~r13/qea) should yield a master curve identical to the global one
with different sites just being advanced or retarded with respect to the global value.
This is another way of stating that the sample ages in a heterogeneous manner, with
some regions being younger (other older) than the global average. (For simplicity
we used a chort-hand notation, C~rµν = C(~r; tµ, tν) with µ, ν = 1, 2, 3.) If the time-
CONTENTS 23
 0  0.2  0.4  0.6  0.8  1
Ccg(t3,t2)
 0  0.2  0.4  0.6  0.8  1
Cr 23 (Cg 13/Cr 13)
 0  0.2  0.4  0.6  0.8  1
Cr 23 (Cg 13/Cr 13)
 0  0.2  0.4  0.6  0.8  1
Cr 23 (Cg 13/Cr 13)
Figure 6. The triangular relation in the 3d ea model. Upper left panel: the thick
(black) line represents the global C(t1, t2) against C(t2, t3) using t2 as a parameter
varying between t3 = 5 × 10
4 MCs and t1 = 9 × 10
6 MCs, C(t1, t3) ∼ 0.35 and
qea ∼ 0.8. The curved part is well represented by the hyperbolic form C(t1, t2) ∼
qeaC(t1, t3)/C(t2, t3) ∼ 0.79 × 0.35/C(t2, t3) that corresponds to fc(x) ∼ x
−b. With
different points joined with thin lines we show the triangular relations between the
local coarse-grained correlations on five randomly chosen sites on the lattice (ℓ = 30).
Upper right panel and lower left and right panels: 2d projection of the joint probability
density of C(r; t1, t2)
C(t1, t3)/C(r; t1, t3) and C(r; t2, t3)
C(t1, t3)/C(r; t1, t3) at
fixed three values of the intermediate time, t2 = 1.5 × 10
5 MCs, 8 × 105 MCs and
5× 106 MCs, respectively and ℓ = 10. The global C(t1, t2) against C(t2, t3) using t2 as
a parameter is shown with a thick green line. The green points indicate the location of
C(t1, t2) and C(t2, t3) for the chosen t2’s. Each point in the scatter plot corresponds
to a site, r. The lines indicated the boundary surrounding 25%, 50% and 75% of the
probability density. The cloud extends mostly along the global relation as predicted
by time-reparametrization invariance. These results are taken from [9].
CONTENTS 24
reparametrization mode is indeed flat the local values should lie all along this master
curve in the aging regime.
The conclusions drawn above apply in the strict a ≪ ℓ ≪ ξ limit. In simulations
and experiments ξ is finite and even rather short. Thus, ℓ is forced to also be a rather
small parameter, in which case ‘finite size’ fluctuations in fc are also expected to exist.
The claim is that the latter should scale down to zero faster (in ℓ) than the fluctuations
that are related to the zero mode.
We have tested these claims in the non-equilibrium dynamics of the 3d ea spin-
glass [9]. The results are shown in Fig. 6. In the upper left panel we show the global
triangular relation (thick black line) as well as the local one on four chosen sites. The
separation of time-scales is clear in the plot. The aging part is rather well described by
fc(x) ∼ x
−b and the local curves are quite parallel indeed. In the remaining panels in
Fig. 6 we show the 2dprojection of the joint probability density of the site fluctuations
in the local coarse-grained correlations at different chosen times t2, t2 = 1.5× 10
5 MCs
(upper right), t2 = 8 × 10
5 MCs (lower left), and t2 = 5 × 10
6 MCs (lower right).
Taking advantage of the fact that fc(x) ∼ x
−b we use a very convenient normalization
in which we multiply the horizontal and vertical axes by [C13/C~r13]
1/2. Global time-
reparametrization invariance, expressed in eq. (34), implies that the data points should
spread along the global curve indicated with a thick green line in the figure. Some sites
could be advanced, others retarded, with respect to the global value – shown with a
point on the green curve – but all should lie mainly on the same master curve. This is
indeed quite well reproduced by the simulation data in the three cases, C(t1, t2) close to
C(t1, t3) (upper right panel), C(t1, t2) close to qea (lower right panel), and C(t1, t2) far
from both (lower left panel). Most of the data points tend to follow the master curve
though some fall away from it. The reason for this is that eq. (34) should be strictly
satisfied only in the very large coarse-graining volume limit (ℓ≫ a) with ℓ/ξ ≪ 1 while
we are here using ℓ = 10a ∼ ξ, see the discussion in Sect. 3.1.
The triangular relation can be used to test the fluctuations of the local
susceptibilities too. Indeed, if the separation of time scales (8) and the scaling (13)
apply to the global susceptibility, the local ones, after the convenient normalization
by the maximum value, should follow another master curve, identical to the global one.
Note that time-reparametrization invariance as we discuss it here implies that both local
correlations and susceptibilities should be fluctuating quantities.
Finally, notice that, in contrast to the growing correlation length scale, there is
no blatantly obvious explanation of these triangular relations within other theoretical
scenarios. These relations are perhaps the most direct consequence of the time
reparametrization symmetry arguments, and so this prediction falls within the second
class we discussed in the introduction to this section.
CONTENTS 25
3.5. Multi-time scaling
In general muti-time correlations are non-trivially related to two-time ones. One can
take as an example a generic coarse-grained connected four-time correlation. If this
function is monotonic with respect to all times, and the two-point correlations scale as
in (13) for all pairs (tµ, tν) with µ, ν = 1, 2, 3, 4, the four-time correlation should behave
C(~r; t1, t2, t3, t4) = g
h(~r, t1)
h(~r, t2)
h(~r, t2)
h(~r, t3)
h(~r, t3)
h(~r, t4)
with the same external function g for all r, in the asymptotic limit in which all times are
widely separated and the corresponding two-time correlations fall below qea. Parametric
constructions could be envisaged to test this relation.
3.6. Local fluctuation-dissipation relation
The asymptotic relation between the global correlation and susceptibility
tw→∞,C(t,tw)=C
χ(t, tw) = χ̂(C) (36)
was first obtained in mean-field disordered models [12, 13] and later observed in
simulations of many realistic systems (spins and particles in interaction on finite
dimensional spaces). −(dCχ̂(C))
−1 defines an effective temperature [14]. In the aging
regime, that is to say for C < qea, three behaviours have been observed in mean-field
systems: in structural glass models χ̂(C) is linear in C (solid line in the right-panel in
Fig. 5); in spin-glass models χ̂(C) is a non-linear function of C; in coarsening systems
χ̂(C) is a constant equal to (1− qea)/T .
The scaling in eq. (22) implies that the parametric construction ‘local susceptibility
against local correlation’ should fall on the master curve for the global quantities but
can be advanced or retarded with respect to the global value; again in the theoretical
limit a ≪ ℓ ≪ ξ. This behaviour is sketched in Fig. 5-right for the three sites
whose correlations are displayed in the left panel. The restricted relation between local
susceptibility and local correlation in eq. (22) arise from the fact that the fluctuations
are due to local reparametrizations alone and not to changes in the external functions
fc and fχ (much as in transverse vs. longitudinal fluctuations in a non-linear σ-model).
An important property of the interpretation of the fluctuation dissipation relation
in terms of effective temperatures is that one expects all observables evolving in the same
time-scale to equilibrate and hence have the same value of the effective temperature [14]
in an asymptotic regime with slow dynamics and small entropy production. Within
the time-reparametrization invariance approach the local effective temperature, defined
from the slope of the χ̂(C) plot, is automatically the same in the whole sample within
a correlation scale, just because the functions fc and fχ do not fluctuate.
In Fig. 7 we show the joint pdf of local correlations and susceptibilities of the
3d ea spin-glass in its glassy phase; the accord with the analytic prediction is very
CONTENTS 26
satisfactory [5, 6] with the additional spreading away from the master curve ascribed to
the fact that ℓ is finite and not very different from ξ.
+++ Bulk
0 0.5 1
Figure 7. (a) The joint pdf ρ(Cr , χr) at two times (tw, t) such that the global
correlation is C(t, tw) = 0.7 < qea in the 3d ea model. (b) Projection of three contour
levels. The crosses are the parametric construction χ̂(C) for several values of the total
time t larger than tw. The dotted straight line is fdt at the working temperature T .
These results are taken from [6].
3.7. Infinite susceptibilities
Zero modes are intimately related to infinite susceptibilities. Indeed, systems with
continuous symmetries are sensitive to arbitrarily weak perturbations. In the present
context the approximate global time-reparametrization invariance implies that one
can easily change the ‘clock’ R(t) characterizing the scaling of the global correlation
and linear response by applying infinitely weak perturbations that couple to the zero
mode. An illustration of this property is the fact that the aging relaxation dynamics
of glassy systems is rendered stationary by a weak perturbing force that does not
derive from a potential while the χ̂(C) relation in the slow regime is not modified [40].
One could envisage more refined tests such as applying a perturbation that imposes
different scalings on two macroscopic borders of the system and see how the time-
reparametrization wave develops in the bulk.
3.8. Conclusions
In conclusion, the global time-reparametrization invariance scenario gives a mechanism
for the divergence of the correlation length ξ though others have also been proposed in
the literature. There are a number of predictions, as the parametric relations between
local coarse-grained correlations measured at different times and the local fluctuation-
dissipation relations that, to our knowledge, are not explained by other scenarios. As
regards to the easier to measure pdfs of local correlations the framework is not only
CONTENTS 27
consistent with the scaling (28) – that also arises from simple scaling assumptions –
but it also justifies the non-Gaussian and Gumbel-like functional form of the pdfs
that follows from the proposed effective action for local ‘ages’. Moreover, systems with
global time-reparametrization invariance should have as important fluctuations in the
local susceptibilities as in the local coarse-grained responses.
4. Discussion
We presented a summary of studies of dynamical fluctuations in glassy systems that
are based on the idea that, in the long time regime, a global time-reparametrization
invariance emerges in the effective action describing the aging dynamics. We discussed
how this symmetry concretely appears in mean-field spin models, and how it can
be shown to emerge at the level of the action for short-ranged spin glass models
with quenched disorder. Two assumptions are used to prove the global time-
reparametrization of the action for the short-range spin glass model i) causality and
unitarity, and ii) a separation of time scales between a fast (or stationary) and a slow
(or aging) relaxation, where in the latter time translation invariance is broken.
That the dynamical action is symmetric under uniform, i.e., spatially independent
reparametrizations of time variables (t → h(t)) suggests that the dynamic fluctuations
that cost little action should be describable in terms of position dependent, long
wavelength, reparametrizations of the form t→ h(~r, t). These should be the Goldstone
modes associated with breaking time-reparametrization invariance symmetry.
We presented predictions of this theoretical framework and tests that we performed
in the 3d Edwards-Anderson model to falsify these predictions. Among the consequences
of our theoretical framework are those listed in Sect. 3. Some of them find an
explanation within other theories as well; others are particularly related to the time
reparametrization invariance scenario. For example, a correlation length that grows
in time is associated with the asymptotic approach to the long-time regime in which
the symmetry is fully developped, and the long wavelength modes eventually become
massless. The existence of a growing length is also predicted by other models. The
functional form of the triangular relations relating local two-time correlations between
three different times is, instead, particular to our framework.
In this review we showed tests of the predictions of the global time-
reparametrization invariance scenario performed on a finite dimensional spin-glass
model [4]-[9]. In the past we also studied the dynamics of kinetically facilitated models,
without quenched disorder, along the same lines [7]. We believe that, if aging dynamics
is a universal property of glassy systems, then these ideas should also apply to interacting
particle systems without quenched disorder. The rationale is that the two assumptions
leading to the global time-reparametrization invariance of the dynamical action, namely
causality and unitarity, and a separation of time scales, should also hold for other glassy
systems. The former assumption we can take as a fact. The second is, in a way, the
assumption that a glassy phase exists, even though we say nothing as of why it does. As
CONTENTS 28
we stated in the introduction, we do not aim at the question why glasses?, but instead we
focus on the possible universal dynamical properties once the glassy state is presented by
nature. Some of the consequences of the symmetry have already been tested numerically
in Lennard-Jones systems [29, 32] but there is still much room for more detailed studies,
including the analysis of local triangular relation between correlation and susceptibilities
and tests of their joint behaviour.
We thus propose that the asymptotic global time-reparametrization invariance,
and the associated low action long wavelength local reparametrizations, constitute the
mechanism by which dynamic fluctuations, that is to say heterogeneities, are generated
in the glassy state. Moreover, this mechanism may also apply, in an approximate form,
to the super-cooled liquid regime. It should just be an approximation because in super-
cooled liquids the symmetry is not fully developed and there is then a cut-off setting
the limit of the spatial and temporal extent of the heterogeneities, in sharp constrast
to the low temperature glassy regime in which the symmetry is realized asymptotically
and fluctuations of all sizes exist. The growth and divergence of the (two-time) dynamic
correlation length defined from the study of the space correlation of the two-time order
parameter is a manifestation of the growth and divergence of these fluctuations in the
glassy state; in contrast, such growth is interrupted in the super-cooled liquid.
To better understand the distiction between the glassy state with its asymptotic
symmetry and the super-cooled state without the fully developing symmetry, consider
the phenomenology of dynamic fluctuations as a function of temperature. Dynamic
heterogeneities in the super-cooled liquid phase have been identified numerically and
experimentally [52]. These are in a number of ways more important than what is
observed in a simple liquid or a solid. In the super-cooled liquid phase while the
full relaxation is stationary, there is still a time-scale separation with the correlations
decaying as a function of time-difference first rapidly to a temperature-independent
plateau and then slowly towards zero. The latter is the structural or α-relaxation. The
α relaxation time, tα, is finite in the super-cooled liquid regime and it increases upon
decreasing temperature. The global parametrization chosen by the symmetry breaking
terms in the slow regime is R(t) ∝ e−t/tα and C(t, tw) = qeafc(e
−(t−tw)/tα) in this case.
The mode-coupling approach to super-cooled liquids is based on approximate
dynamic equations for the relevant correlators of realistic systems that are very similar
to the p-spin Schwinger-Dyson equations in the high temperature phase [16, 18]. In
these equations the correlators are already expressed as functions of the time-difference,
τ ≡ t − tw. Close to the critical temperature the separation of time-scales develops in
the mode-coupling equations. The approximate analysis of the α relaxation predicted
by these equations also relies on dropping the τ derivatives and approximating the
integrals by assuming a sharp time-scale separation. The remaining asymptotic (large
τ) equations are invariant under reparametrizations of τ .
We then expect the local coarse-grained correlations and integrated linear responses
in the super-cooled liquid to be, to a first approximation, stationary (after a sufficiently
long waiting-time that goes beyond the equilibration time) but with different finite
CONTENTS 29
structural relaxation times, fluctuating about the value that characterizes the decay
of the global correlations. This is consistent with the experimental observation that
dynamic heterogeneities in supercooled liquids seem to have a lifetime of the order of
the relaxation time. Deviations from stationarity are not completely excluded for finite
ℓ but they should be less important than in the aging low-temperature regime.
There is, however, an important difference with respect to the aging regime, in
which the equilibration time diverges and local relaxation times or, better stated, local
ages can fluctuate without limit when tw → ∞. At high temperatures one does
not expect to find heterogeneities with arbitrary long relaxation time. Furthermore,
heterogeneities have a finite spatial extent and one can then suppress them by using
sufficiently large coarse-graining volumes. The correlation length is stationary, ξ(τ),
and it saturates in the limit of long-time differences, τ → ∞. The saturation value,
though, increases for decreasing temperature. From a theoretical point of view, this
picture is, in a sense, similar to the one that describes the equilibrium paramagnetic
phase in the O(N) model, just above the ordering transition temperature.
When lowering the temperature the size and life-time of the heterogeneities
increases. A p-spin or mode-coupling approach predicts that their typical size and thus
limτ→∞ ξ(τ) diverge at the mode coupling transition temperature [33]. In real systems
the divergence at Tc is rendered smooth and ξ does not strictly diverge asymptotically.
At still lower temperatures the bulk quantities age and we expect then to observe
heterogeneous aging dynamics of the kind described in this review, with a two-time
dependent correlation length for the local fluctuations. The heterogeneities age as well,
in a ‘dynamic’ way. By this we mean that if a region looks older than another one when
observed on a given time-window, it can reverse its status and look younger than the
same other region when observed on a different time-window.
The infinite susceptibility with respect to perturbations that couple to the zero
mode are illustrated by the fact that the clock of the bulk quantities that is selected
dynamically is very easy to modify with external perturbations. A small force that does
not derive from a potential and is applied on every spin in the model renders an aging p
spin model stationary [40] while the model maintains a separation of time-scales in which
the fast scale follows the temperature of the bath, T , while the slow scale is controlled
by an an effective temperature, Teff > T . In this case, the aging system selects a
time-reparametrization R(t) = t while in the perturbed model R(t) = e−t/tα . Similarly,
the aging of a Lennard-Jones mixture is stopped by an homogeneous shear [41]. A
different way to modify the time-reparametrization that characterizes the decay of the
correlations is by using complex thermal baths [42].
The picture that we have described applies to long times but not as long as to enter
the activated regime that we still do not know how to characterize theoretically, not
even at the bulk level. This regime corresponds to times that scale with the system size.
The success of mean-field models, or the mode-coupling approach, in describing the
bulk dynamics of glassy systems, at least not to close to the crossover glass temperature
and at a qualitative level, allows us to claim that these extremely long times scales are
CONTENTS 30
unrealistic if not too close to the glass transition Tg even as far as dynamic fluctuations
are concerned.
The ideas discussed in this paper should not only apply to systems that relax
in a non-equilibrium manner as glasses but also to systems with slow dynamics and a
separation of time-scales that are kept out of equilibrium with a (weak) external forcing.
Recently, there has been much interest in the appearance of shear localization, in the
form of shear bands, in the rheology of complex fluids. Along the lines here described
it would be very interesting to analyze the fluctuations in the local reparametrizations
in the fluidized shear band and the ‘jammed’ glassy band.
The analytic treatment of mean-field quantum glassy systems follows similar steps
to the ones presented here. The Schwinger-Keldysh approach replaces the Martin-
Siggia-Rose one but these formalisms are very similar indeed. The analytic solution to
the dynamic equations in the limit in which the coupling to the environment is weak
also uses the fact that the dynamics in the aging regime is very slow. The approximate
equations then become time-reparametrization invariant. One can then expect that
similar dynamic fluctuations arise in glassy problems in which quantum fluctuations
are important. Moreover the proof of global time-reparametrization invariance for spin-
glasses has been presented directly in the quantum formalism. Novel experimental
techniques may be apt to study dynamic heterogeneities in glasses when quantum
fluctuations are important.
We expect to find similar fluctuations using finite size systems and examining the
behaviour of the mesoscopic run-to-run fluctuations of the global correlations. These
may be easier to measure experimentally using mesoscopic systems.
Importantly enough, global time reparametrization invariance does not develop in
all models with slow and aging correlation functions. The O(N) ferromagnetic model in
the large N limit is a case in which global time-reparametrization invariance is reduced
to just scale invariance [8].
Last, but not least, the approach based on reparametrization invariance suggests
that it may be possible to search for universality in glassiness. A Ginzburg-Landau
theory for phase transitions captures universal properties that are independent of the
details of the material. It is symmetry that defines the universality classes. For example,
one requires rotational invariance of the Ginzburg-Landau action when describing
ferromagnets. Time reparametrization invariance may be the underlying symmetry that
must be satisfied by the Ginzburg-Landau action of all glasses. What would determine
if a system is glassy or not? We are tempted to say the answer is if the symmetry
is generated or not at long times. Knowing how to describe the universal behavior
may tell us all the common properties of glasses, but surely it will not allow us to
make non-universal predictions, such as what is the glass transition temperature for a
certain material, or whether the material displays glassy behavior at all. This quest for
universality is a very interesting theoretical scenario that needs to be confronted.
We have tried to state as clearly as possible the implications of our proposal. The
CONTENTS 31
full project is not yet complete since several questions about its limitations remain open.
A number of issues should be addressed are:
(i) From a phenomenological point of view, to perform strong tests of the global
time-reparametrization invariance scenario in molecular dynamic simulations [28, 44, 45]
of realistic glassy systems and experiments [46]-[51]. More precisely, the triangular
relations between local coarse-grained correlations and the local fluctuation dissipation
relations should be analysed to give support or else falsify this conjecture.
(ii) From an analytic point of view, to derive the effective action for local
reparametrizations for glassy models with and without quenched disorder. We are
currently working on this project in collaboration with S. Franz. One idea is to study
the p spin disordered model with Kac long-range interaction. ANothe one is to study
the symmetries of the dynamic action associated with the Dean-Kawasaki equation for
the evolution of the density of a system of particles in interaction.
(iv) From a mixed analytic and numerical point of view to analyse fluctuations
in models with global aging dynamics of different type. To this end, one can study
dynamic fluctuations in simple coarsening systems in finite dimensions. In these cases
the morphology of domains can be characterized in great detail [43]. We could, in
principle, understand the fluctuations in the local correlations and linear responses from
a microscopic point of view. Whether these are similar or different to the ones in other
glassy problems is still to be established and the outcome of this study could clarify the
relevance of the value of the effective temperature in determining the characteristics of
the dynamic fluctuations.
(v) In the same line as the above, the analysis of fluctuations of elastic lines in the
presence of impurities could help us understanding the coarsening phenomenon but also
the role of diffusion that superimposes in these cases to the aging dynamics [15].
These are just a few questions posed by this proposal that are still not answered.
Acknowledgments
We thank our collaborators J. J. Arenzon, C. Aron, A. J. Bray, S. Bustingorry,
H. E. Castillo, P. Charbonneau, D. Domı́nguez, J. L. Iguain, L. D. C. Jaubert, M. P.
Kennett, M. Picco, D. R. Reichman, M. Sellitto, A. Sicilia and H. Yoshino.
We also wish to especially thank L. Berthier, G. Biroli, J-P Bouchaud, D. S. Dean,
G. Fabricius, T. Grigera, E. Fradkin, S. Franz, J. Kurchan, H. Makse, D. Stariolo and L.
Valluzzi for very helpful discussions. We acknowledge financial support from NSF-CNRS
INT-0128922, NSF DMR-0305482, DMR 0403997 and PICS 3172. LFC is a member of
Institut Universitaire de France. LFC thanks the Newton Institute at the University of
Cambridge, ICTP at Trieste, and Universidad Nacional de Mar del Plata, Argentina, CC
the LPTHE at Jussieu, Paris, France, and LFC and CC the Aspen Center for Physics
CONTENTS 32
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	Why glasses? vs. universality in glassy dynamics
	Time reparametrization invariance
	Mean-field models – dynamic equations
	Structural glasses: the p3 cases
	Short-ranged models – dynamic action
	Turning a nuisance into something useful - symmetry as a guideline
	The spherical p=2 case or mean-field domain growth
	Quantum problems
	Consequences and tests
	Two-time correlation length
	Scaling of the pdf of local two-time functions
	Effective action for local ages
	Two-time scaling of local functions
	Multi-time scaling
	Local fluctuation-dissipation relation
	Infinite susceptibilities
	Conclusions
	Discussion
