[Show abstract][Hide abstract] ABSTRACT: Motivated by the connection between the dynamical transition predicted by the
mean-field theory of glass-forming liquids and the spinodal of an Ising model
in a quenched random field (RFIM) beyond mean-field, we revisit the phenomenon
of spinodals in the presence of quenched disorder and develop a complete theory
for it. By working at zero temperature in the quasi-statically driven RFIM,
thermal fluctuations are eliminated and one can give a rigorous content to the
notion of spinodal. We show that the spinodal transition is due to the
depinning and the subsequent expansion of rare droplets. We work out the
critical behavior, which, in any finite dimension, is very different from the
mean-field one: the characteristic length diverges exponentially and the
thermodynamic quantities display very mild non-analyticities much like in a
Griffith phenomenon. On the basis of our results we assess the physical content
and the status of the dynamical transition predicted by the mean-field theory
of glassy dynamics.
[Show abstract][Hide abstract] ABSTRACT: Slow dynamics in glassy systems is often interpreted as due to thermally
activated events between "metastable" states. This emphasizes the role of
nonperturbative fluctuations, which is especially dramatic when these
fluctuations destroy a putative phase transition predicted at the mean-field
level. To gain insight into such hard problems, we consider the implementation
of a generic back-and-forth process, between microscopic theory and observable
behavior via effective theories, in a toy model that is simple enough to allow
for a thorough investigation: the one-dimensional $\varphi^4$ theory at low
temperature. We consider two ways of restricting the extent of the
fluctuations, which both lead to a nonconvex effective potential (or free
energy) : either through a finite-size system or by means of a running infrared
cutoff within the nonperturbative Renormalization Group formalism. We discuss
the physical insight one can get and the ways to treat strongly nonperturbative
fluctuations in this context.
[Show abstract][Hide abstract] ABSTRACT: This work provide a thorough study of L\'evy or heavy-tailed random matrices
(LM). By analysing the self-consistent equation on the probability distribution
of the diagonal elements of the resolvent we establish the equation determining
the localisation transition and obtain the phase diagram of LMs. Using
arguments based on super-symmetric field theory and Dyson Brownian motion we
show that the eigenvalue statistics is the same one of the Gaussian Orthogonal
Ensemble in the whole delocalised phase and is Poisson in the localised phase.
Our numerics confirms these findings, valid in the limit of infinitely large
LMs, but also reveals that the characteristic scale governing finite size
effects diverges much faster than a power law approaching the transition and is
already very large far from it. This leads to a very wide cross-over region in
which the system looks as if it were in a mixed phase. Our results, together
with the ones obtained previously, provide now a complete theory of L\'evy
matrices.
[Show abstract][Hide abstract] ABSTRACT: We develop a theory of amorphous interfaces in glass-forming liquids. We show
that the statistical properties of these surfaces, which separate regions
characterized by different amorphous arrangements of particles, coincide with
the ones of domain walls in the random field Ising model. A major consequence
of our results is that super-cooled liquids are characterized by two different
static lengths: the point-to-set $\xi_{PS}$ which is a measure of the spatial
extent of cooperative rearranging regions and the wandering length $\xi_\perp$
which is related to the fluctuations of their shape. We find that $\xi_\perp$
grows when approaching the glass transition but slower than $\xi_{PS}$. The
wandering length increases as $s_c^{-1/2}$, where $s_c$ is the configurational
entropy. Our results strengthen the relationship with the random field Ising
model found in recent works. They are in agreement with previous numerical
studies of amorphous interfaces and provide a theoretical framework for
explaining numerical and experimental findings on pinned particle systems and
static lengths in glass-forming liquids.
[Show abstract][Hide abstract] ABSTRACT: Recent works on hard spheres in the limit of infinite dimensions revealed
that glass states, envisioned as meta-basins in configuration space, can break
up in a multitude of separate basins at low enough temperature or high enough
pressure, leading to the emergence of new kinds of soft-modes and unusual
properties. In this paper we study by perturbative renormalisation group
techniques the fate of this transition, which has been discovered in disordered
mean-field models in the '80s. We find that the upper critical dimension d_u
above which mean-field results hold is strictly larger than six and apparently
non-universal, i.e. system dependent. Below d_u, we do not find any
perturbative attractive fixed point (except for a tiny region of the 1RSB
breaking parameter), thus showing that the transition in three dimensions
either does not exist or changes nature from its mean-field counterpart. We
also discuss some issues related to the low temperature full replica symmetry
breaking phase found in infinite dimensions, as well as a possible relationship
with the behavior of spin glasses in a field.
Physical Review B 10/2014; 91(10). DOI:10.1103/PhysRevB.91.100202 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: By using real space renormalisation group (RG) methods we show that
spin-glasses in a field display a new kind of transition in high dimensions.
The corresponding critical properties and the spin-glass phase are governed by
two non-perturbative zero temperature fixed points of the RG flow. We compute
the critical exponents, discuss the RG flow and its relevance for three
dimensional systems. The new spin-glass phase we discovered has unusual
properties, which are intermediate between the ones conjectured by droplet and
full replica symmetry breaking theories. These results provide a new
perspective on the long-standing debate about the behaviour of spin-glasses in
a field.
[Show abstract][Hide abstract] ABSTRACT: We introduce a new disordered system, the Super-Potts model, which is a more
frustrated version of the Potts glass. Its elementary degrees of freedom are
variables that can take M values and are coupled via pair-wise interactions.
Its exact solution on a completely connected lattice demonstrates that for
large enough M it belongs to the class of mean-field systems solved by a one
step replica symmetry breaking Ansatz. Numerical simulations by the parallel
tempering technique show that in three dimensions it displays a
phenomenological behaviour similar to the one of glass-forming liquids. The
Super-Potts glass is therefore the first long-sought disordered model allowing
one to perform extensive and detailed studies of the Random First Order
Transition in finite dimensions. We also discuss its behaviour for small values
of M, which is similar to the one of spin-glasses in a field.
Physical Review B 07/2014; 90(22). DOI:10.1103/PhysRevB.90.220201 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We analyse, using Inhomogenous Mode-Coupling Theory, the critical scaling
behaviour of the dynamical susceptibility at a distance epsilon from continuous
second-order glass transitions. We find that the dynamical correlation length
xi behaves generically as epsilon^{-1/3} and that the upper critical dimension
is equal to six. More surprisingly, we find activated dynamic scaling, where xi
grows with time as [ln(t)]^2 exactly at criticality. All these results suggest
a deep analogy between the glassy behaviour of attractive colloids or randomly
pinned supercooled liquids and that of the Random Field Ising Model.
[Show abstract][Hide abstract] ABSTRACT: The dramatic dynamic slowing down associated with the glass transition is considered by many to be related to the existence of a static length scale that grows when temperature decreases. Defining, identifying, and measuring such a length is a subtle problem. Recently, two proposals, based on very different insights regarding the relevant physics, were put forward. One approach is based on the point-to-set correlation technique and the other on the scale where the lowest eigenvalue of the Hessian matrix becomes sensitive to disorder. Here we present numerical evidence that the two approaches might result in the same identical length scale. This provides mutual support for their relevance and, at the same time, raises interesting theoretical questions, discussed in the conclusion.
[Show abstract][Hide abstract] ABSTRACT: We introduce an approach to derive an effective scalar field theory for the
glass transition; the fluctuating field is the overlap between equilibrium
configurations. We apply it to the case of constrained liquids for which the
introduction of a conjugate source to the overlap field was predicted to lead
to an equilibrium critical point. We show that the long-distance physics in the
vicinity of this critical point is in the same universality class as that of a
paradigmatic disordered model: the random-field Ising model. The quenched
disorder is provided here by a reference equilibrium liquid configuration. We
discuss to what extent this field-theoretical description and the mapping to
the random field Ising model hold in the whole supercooled liquid regime, in
particular near the glass transition.
[Show abstract][Hide abstract] ABSTRACT: In this work, we numerically investigate a new method for the characterization of growing length scales associated with spatially heterogeneous dynamics of glass-forming liquids. This approach, motivated by the formulation of the inhomogeneous mode-coupling theory (IMCT) [Biroli, G.; et al. Phys. Rev. Lett. 2006 97, 195701], utilizes inhomogeneous molecular dynamics simulations in which the system is perturbed by a spatially modulated external potential. We show that the response of the two-point correlation function to the external field allows one to probe dynamic correlations. We examine the critical properties shown by this function, in particular, the associated dynamic correlation length, that is found to be comparable to the one extracted from standardly employed four-point correlation functions. Our numerical results are in qualitative agreement with IMCT predictions but suggest that one has to take into account fluctuations not included in this mean-field approach to reach quantitative agreement. Advantages of our approach over the more conventional one based on four-point correlation functions are discussed.
The Journal of Physical Chemistry B 07/2013; 117(42). DOI:10.1021/jp4035419 · 3.30 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study the effect of confinement on glassy liquids using Random First Order
Transition theory as framework. We show that the characteristic length-scale
above which confinement effects become negligible is related to the
point-to-set length-scale introduced to measure the spatial extent of amorphous
order in super-cooled liquids. By confining below this characteristic size, the
system becomes a glass. Eventually, for very small sizes, the effect of the
boundary is so strong that any collective glassy behavior is wiped out. We
clarify similarities and differences between the physical behaviors induced by
confinement and by pinning particles outside a spherical cavity (the protocol
introduced to measure the point-to-set length). Finally, we discuss possible
numerical and experimental tests of our predictions.
[Show abstract][Hide abstract] ABSTRACT: We characterize vibrational motion occurring at low temperatures in dense suspensions of soft repulsive spheres over a broad range of volume fractions encompassing the jamming transition at (T = 0, ϕ = ϕJ). We find that characteristic time and length scales of thermal vibrations obey critical scaling in the vicinity of the jamming transition. We show in particular that the amplitude and the time scale of dynamic fluctuations diverge symmetrically on both sides of the transition, and directly reveal a diverging correlation length. The critical region near ϕJ is divided in three different regimes separated by a characteristic temperature scale T(⋆)(ϕ) that vanishes quadratically with the distance to ϕJ. While two of them, (T < T(⋆)(ϕ), ϕ > ϕJ) and (T < T(⋆)(ϕ), ϕ < ϕJ), are described by harmonic theories developed in the zero temperature limit, the third one for T > T(⋆)(ϕ) is inherently anharmonic and displays new critical properties. We find that the quadratic scaling of T(⋆)(ϕ) is due to nonperturbative anharmonic contributions, its amplitude being orders of magnitude smaller than the perturbative prediction based on the expansion to quartic order in the interactions. Our results show that thermal vibrations in colloidal assemblies directly reveal the critical nature of the jamming transition. The critical region, however, is very narrow and has not yet been attained experimentally, even in recent specifically-dedicated experiments.
The Journal of Chemical Physics 03/2013; 138(12):12A507. DOI:10.1063/1.4769251 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We provide here a brief perspective on the glass transition field. It is an assessment, written from the point of view of theory, of where the field is and where it seems to be heading. We first give an overview of the main phenomenological characteristics, or "stylised facts," of the glass transition problem, i.e., the central observations that a theory of the physics of glass formation should aim to explain in a unified manner. We describe recent developments, with a particular focus on real space properties, including dynamical heterogeneity and facilitation, the search for underlying spatial or structural correlations, and the relation between the thermal glass transition and athermal jamming. We then discuss briefly how competing theories of the glass transition have adapted and evolved to account for such real space issues. We consider in detail two conceptual and methodological approaches put forward recently, that aim to access the fundamental critical phenomenon underlying the glass transition, be it thermodynamic or dynamic in origin, by means of biasing of ensembles, of configurations in the thermodynamic case, or of trajectories in the dynamic case. We end with a short outlook.
The Journal of Chemical Physics 03/2013; 138(12):12A301. DOI:10.1063/1.4795539 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a detailed analysis of glass transitions induced by pinning particles at random from an equilibrium configuration. We first develop a mean-field analysis based on the study of p-spin spherical disordered models and then obtain the three-dimensional critical behavior by the Migdal-Kadanoff real space renormalization group method. We unveil the important physical differences with the case in which particles are pinned from a random (or very high temperature) configuration. We contrast the pinning particles approach to the ones based on biasing dynamical trajectories with respect to their activity and on coupling to equilibrium configurations. Finally, we discuss numerical and experimental tests.
The Journal of Chemical Physics 03/2013; 138(12):12A547. DOI:10.1063/1.4790400 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give an overview of our recent works in which the a.c. nonlinear
dielectric response of an archetypical glassformer (glycerol) was
measured close to its glass transition temperature T g . The
purpose was to investigate the prediction that the nonlinear
susceptibility is directly related to the number of dynamically
correlated molecules N { corr} (T). We explain that two
nonlinear susceptibilities are available, namely χ3
(3) and χ3 (1), which correspond
respectively to the nonlinear cubic response at the third harmonics and
at the first harmonics. We describe how to measure these nonlinear
responses, even if they yield signals much smaller than that of the
linear response. We show that both \vert {χ
}3^{(3)}(ω,T)\vert and \vert {χ
}3^{(1)}(ω,T)\vert are peaked as a function of the
angular frequency ω and mainly obeys critical scaling as a
function of ωτα(T), where
τα(T) is the relaxation time of the liquid. Both
χ3 (3) and χ3 (1)
decay with the same power-law of ω beyond the peak. The height of
the peak increases as the temperature approaches T g : This
yields an accurate determination of the temperature dependence of N
{ corr} (T), once the contribution of saturation of dipoles
is disentangled from that of dynamical glassy correlations.
NATO Science for Peace and Security Series B: Physics and Biophysics 01/2013; DOI:10.1007/978-94-007-5012-8_7
[Show abstract][Hide abstract] ABSTRACT: We show that non-interacting disordered electrons on a Bethe lattice display
a new intermediate phase which is delocalized but non-ergodic, i.e. it is
characterized by Poisson instead of GOE statistics. The physical signature of
this phase is a very heterogenous transport that proceeds over a few disorder
dependent paths only. We show that the transition to the usual ergodic
delocalized phase, which takes place for a disorder strength smaller than the
one leading to the localization transition, is related to the freezing-glass
transition of directed polymers in random media. The numerical study of level
and eigenstate statistics, and of the singular properties of the probability
distribution of the local density of states all support the existence of this
new intermediate phase. Our results suggest that the localization transition
may change nature in high dimensional systems.
[Show abstract][Hide abstract] ABSTRACT: Several mean-field computations have revealed the existence of an out of
equilibrium dynamical transition induced by quantum quenching an isolated
system starting from its symmetry broken phase. In this work we focus on the
quantum phi^4 N-component field theory. By taking into account dynamical
fluctuations at the Hartree-Fock level, corresponding to the leading order of
the 1/N expansion, we derive the critical properties of the dynamical
transition beyond mean-field theory (including at finite temperature). We find
diverging time and length-scales, dynamic scaling and aging. Finally, we unveil
a relationship with coarsening, an off-equilibrium dynamical regime that can be
induced by quenching from the symmetric toward the symmetry broken phase.
Physical Review B 11/2012; 88(20). DOI:10.1103/PhysRevB.88.201110 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We have measured, as a function of the age $t_a$, the aging of the nonlinear
dielectric susceptibility $\chi_3$ of glycerol below the glass transition.
Whereas the linear susceptibility can be accurately accounted for in terms of
an age dependent relaxation time $\tau_{\alpha}(t_a)$, this scaling breaks down
for $\chi_3$, suggesting an increase of the amplitude of $\chi_3$. This is a
strong indication that the number $N_{corr}$ of molecules involved in
relaxation events increases with $t_a$. For $T=0.96 \times T_g$, we find that
$N_{corr}$ increases by $\sim 10%$ when $t_a$ varies from $1\mathrm{ks}$ to
$100\mathrm{ks}$. This sheds new light on the relation between length scales
and time scales in glasses.