We consider the problem of slow activation dynamics in glassy systems undergoing a random first order phase transition. Using an effective potential approach to supercooled liquids, we determine the spectrum of activation barriers for entropic droplets. We demonstrate that fluctuations of the configurational entropy and of the liquid glass surface tension are crucial to achieve an understanding of the barrier fluctuations in glassy systems and thus are ultimatively responsible for the broad spectrum of excitations and heterogeneous dynamics in glasses. In particular we derive a relation between the length scale for dynamic heterogeneity and the related barrier fluctuations. Diluted entropic droplets are shown to have a Gaussian distribution of barriers, strongly suggesting that non-Gaussian behavior results from droplet-droplet interactions.
The critical behaviour of the dynamical transition of glassy system is controlled by a Replica Symmetric action with n=1 replicas. The most divergent diagrams in the loop expansion correspond at all orders to the solutions of a stochastic equation leading to perturbative dimensional reduction. The theory describe accurately numerical simulations of mean-field models.
We study the stability of the glassy equilibrium state of a directed polymer in random media against weak perturbations including the intriguing problem of temperature-chaos. The statistical correlation of two real replica systems under perturbations, such as small difference of temperature, is studied using the mapping to a quantum problem of interacting bosons. Explicit replica symmetry breaking takes place being induced by the infinitesimally weak perturbations in the $n \to 0$ limit followed by the thermodynamic limit which leads to complete decorrelation of the two real replicas. The over…
It is shown that the limit $t-t'\to\infty$ of the equilibrium dynamic self-energy can be computed from the $n\to 1$ limit of the static self-energy of a $n$-times replicated system with one step replica symmetry breaking structure. It is also shown that the Dyson equation of the replicated system leads in the $n\to 1$ limit to the bifurcation equation for the glass ergodicity breaking parameter computed from dynamics. The equivalence of the replica formalism to the long time limit of the equilibrium relaxation dynamics is proved to all orders in perturbation for a scalar theory.
We discuss the slow relaxation phenomenon in glassy systems by means of replicas by constructing a static field theory approach to the problem. At the mean field level we study how criticality in the four point correlation functions arises because of the presence of soft modes and we derive an effective replica field theory for these critical fluctuations. By using this at the Gaussian level we obtain many physical quantities: the correlation length, the exponent parameter that controls the Mode-Coupling dynamical exponents for the two-point correlation functions, and the prefactor of the crit…
Gallager codes are the best error-correcting codes to-date. In this paper we study them by using the tools of statistical mechanics. The corresponding statistical mechanics model is a spin model on a sparse random graph. The model can be solved by elementary methods (i.e. without replicas) in a large connectivity limit. For low enough temperatures it presents a completely frozen glassy phase (q_{EA}=1). The same scenario is shown to hold for finite connectivities. In this case we adopt the replica approach and exhibit a one-step replica symmetry breaking order parameter. We argue that our ansa…
The behavior of a newly introduced overlap parameter is analyzed, measuring the correlation between intensity fluctuations of waves in random media in different physical regimes, with varying amount of disorder and non-linearity. Its relationship is established to the standard Parisi overlap order parameter in replica theory for spin-glasses. In the complex spherical spin-glass model, describing the onset and behavior of random lasers, replica symmetry breaking in the intensity fluctuation overlap is shown to occur at high pumping or low temperature. This order parameter identifies the laser t…
We endow a system of interacting particles with two distinct, local, Markovian and reversible microscopic dynamics. Using common field-theoretic techniques used to investigate the presence of a glass transition, we find that while the first, standard, dynamical rules lead to glassy behavior, the other one leads to a simple exponential relaxation towards equilibrium. This finding questions the intrinsic link that exists between the underlying, thermodynamical, energy landscape, and the dynamical rules with which this landscape is explored by the system. Our peculiar choice of dynam- ical rules …
We consider off-equilibrium dynamics at the critical temperature in a class of glassy system. The off-equilibrium correlation and response functions obey a precise scaling form in the aging regime. The structure of the {\it equilibrium} replicated Gibbs free energy fixes the corresponding {\it off-equilibrium} scaling functions implicitly through two functional equations. The details of the model enter these equations only through the ratio $w_2/w_1$ of the cubic coefficients (proper vertexes) of the replicated Gibbs free energy. Therefore the off-equilibrium dynamical exponents are controlled…
In this talk we review our theoretical understanding of spin glasses paying a particular attention to the basic physical ideas. We introduce the replica method and we describe its probabilistic consequences (we stress the recently discovered importance of stochastic stability). We show that the replica method is not restricted to systems with quenched disorder. We present the consequences on the dynamics of the system when it slows approaches equilibrium are presented: they are confirmed by large scale simulations, while we are still awaiting for a direct experimental verification.
In these lectures I will study some properties that are shared by many glassy systems. I will shown how some of these properties can be understood in the framework of the mean field approach based on the replica method and I will discuss which are the difficulties which we have to face when we apply the replica method to realistic short range models. We finally present a partially successfully application of the replica method to soft sphere glasses.
Insulating granular aluminum is one of the proto-typical disordered insulators whose low temperature electrical conductance exhibits ubiquitous non-equilibrium phenomena. These include slow responses to temperature or gate voltage changes, characteristic field effect anomalies and ageing phenomena typical of a glass. In this system the influence of temperature on the glassy dynamics has remained elusive, leading to the belief that the slow relaxations essentially proceed via elastic quantum tunneling. A similar situation was met in insulating indium oxide and it was concluded that in high carr…
We develop a quantum replica method for interacting particle systems and use it to estimate the location of the glass transition line in Helium 4. Although we do not fully succeed in taking into account all quantum effects, we make a thorough semiclassical analysis. We confirm previous suggestions that quantum fluctuations promote the formation of the glass and give a quantitative estimate of this effect at high density. Finally, we discuss the difficulties that are met when one tries to extend the calculation to the region of low densities and low temperatures, where quantum effects are stron…
Replica Field Theory for a Generalized Franz--Parisi Potential of Inhomogeneous Glassy Systems: New Closure and the Associated Self-Consistent Equation
The spherical mean field approximation of a spin-1 model with p-body quenched disordered interaction is investigated. Depending on temperature and chemical potential the system is found in a paramagnetic or in a glassy phase and the transition between these phases can be of different nature. In given conditions inverse freezing occurs. As $p=2$ the glassy phase is replica symmetric and the transition is always continuous in the phase diagram. For $p>2$ the exact solution for the glassy phase is obtained by the one step replica symmetry breaking Ansatz. Different scenarios arise for both the dy…
We compute the Functional Renormalization Group (FRG) disorder- correlator function R(v) for d-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space dimension N. It measures the equilibrium response of the manifold in a quadratic potential well as the center of the well is varied from 0 to v. We find two distinct scaling regimes: (i) a "single shock" regime, v^2 ~ 1/L^d where L^d is the system volume and (ii) a "thermodynamic" regime, v^2 ~ N. In regime (i) all the equivalent replica symmetry breaking (RSB) saddle points within the Gaussian variati…
Slow dynamics in disordered materials prohibits direct simulation of their rich nonequilibrium behavior at large scales. "Patchwork dynamics" is introduced to mimic relaxation over a very broad range of time scales by equilibrating or optimizing directly on successive length scales. This dynamics is used to study coarsening and to replicate memory effects for spin glasses and random ferromagnets. It is also used to find, with high confidence, exact ground states in large or toroidal samples.
The state-following technique allows the study of metastable glassy states under external perturbations. Here we show how this construction can be used to study the behavior of glassy states of Hard Spheres in infinite dimensions under compression or shear strain. In a preceding work it has been shown that in both cases, when the external perturbation is sufficiently strong, glassy states undergo a second-order transition, called the Gardner transition, whereupon a hierarchical structure of marginal micro-states manifests within the original glass state. The purpose of this work is to study th…
We consider an infinite-range spherical p-spin glass model with an additional r-spin ferromagnetic interaction, both statically using a replica analysis and dynamically via a generating functional method. For r>2 we find that there are first order transitions to ferromagnetic phases. For r<p there are two ferromagnetic phases, one non-glassy replica symmetric and one exhibiting glassy one-step replica symmetry breaking and aging, whereas for r>=p only the replica symmetric phase exists.
Activated dynamics in a glassy system undergoing steady shear deformation is studied by numerical simulations. Our results show that the external driving force has a strong influence on the barrier crossing rate, even though the reaction coordinate is only weakly coupled to the nonequilibrium system. This "driven activation" can be quantified by introducing in the Arrhenius expression an effective temperature, which is close to the one determined from the fluctuation-dissipation relation. This conclusion is supported by analytical results for a simplified model system.
Critical dynamics in various glass models including those described by mode coupling theory is described by scale-invariant dynamical equations with a single non-universal quantity, i.e. the so-called parameter exponent that determines all the dynamical critical exponents. We show that these equations follow from the structure of the static replicated Gibbs free energy near the critical point. In particular the exponent parameter is given by the ratio between two cubic proper vertexes that can be expressed as six-point cumulants measured in a purely static framework.
We investigate the classical Renyi entropy S_n and the associated mutual information I_n in the Sherrington-Kirkpatrick (S-K) model, which is the paradigm model of mean-field spin glasses. Using classical Monte Carlo simulations and analytical tools we investigate the S-K model on the n-sheets booklet. This is obtained by gluing together n independent copies of the model, and it is the main ingredient to construct the Renyi entanglement-related quantities. We find a glassy phase at low temperature, whereas at high temperature the model exhibits paramagnetic behavior, consistent with the regula…
A theoretical analysis [Angelani et al., Phys. Rev. Lett. 96, 065702 (2006)] predicts glassy behaviour of light in a nonlinear random medium. This implies slow dynamics related to the presence of many metastable states. We consider very general equations (that also apply to other systems, like Bose-Condensed gases) describing light in a disordered non-linear medium and through some approximations we relate them to a mean-field spin-glass-like model. The model is solved by the replica method, and replica-symmetry breaking phase transition is predicted. The transition describes a mode-locking pr…
Using an effective potential approach, we present a replica instanton theory for the dynamics of entropic droplets in glassy systems. Replica symmetry breaking in the droplet interface leads to a length scale dependent reduction of the droplet surface tension and changes the character of the dynamical heterogeneity and activated dynamics in glasses.
We consider the critical properties of points of continuous glass transition as one can find in liquids in presence of constraints or in liquids in porous media. Through a one loop analysis we show that the critical Replica Field Theory describing these points can be mapped in the $ϕ^4$-Random Field Ising Model. We confirm our analysis studying the finite size scaling of the $p$-spin model defined on sparse random graph, where a fraction of variables is frozen such that the phase transition is of a continuous kind.
We study the nonlinear dynamics of a multi-mode random laser using the methods of statistical physics of disordered systems. A replica-symmetry breaking phase transition is predicted as a function of the pump intensity. We thus show that light propagating in a random non-linear medium displays glassy behavior, i.e. the photon gas has a multitude of metastable states and a non vanishing complexity, corresponding to mode-locking processes in random lasers. The present work reveals the existence of new physical phenomena, and demonstrates how nonlinear optics and random lasers can be a benchmark …
Here we review the approach to glassy systems based on the replica method and we introduce the main ingredients of replica symmetry breaking. We explain why the replica method has been successful in spin glass and why it should be successful for real glasses.
We consider a Lattice Gas model in which the sites interact via infinite-ranged random couplings independently distributed with a Gaussian probability density. This is the Lattice Gas analogue of the well known Sherrington-Kirkpatrick Ising Spin Glass. We present results of replica approach in the Replica Symmetric approximation. Even with zero-mean of the couplings a line of first order liquid-gas transitions occurs. Replica Symmetry Breaking should give up to a glassy transition inside the liquid phase.
We investigate the replica symmetry broken (RSB) phase of spin glass (SG) models in a random field defined on Bethe lattices at zero temperature. From the properties of the RSB solution we deduce a closed equation for the extreme values of the cavity fields. This equation turns out not to depend on the parameters defining the RSB, and it predicts that the spontaneous RSB does not take place homogeneously on the whole system. Indeed, there exist spins having the same effective local field in all local ground states, exactly as in the replica symmetric (RS) phase, while the spontaneous RSB manif…
We show that the replica approach to glassy dynamics provides, in spite of its static nature, a characterization of critical dynamics in the $β$-regime of super-cooled liquids that is equivalent to the one of Mode-Coupling-Theory, both qualitatively and quantitatively. The nature and extent of this equivalence is discussed in connection to the main open problems of the current theory.
We compare dynamical heterogeneities in equilibrated supercooled liquids and in the nonequilibrium glassy state within the framework of the random first order transition theory. Fluctuating mobility generation and transport in the glass are treated by numerically solving stochastic continuum equations for mobility and fictive temperature fields that arise from an extended mode coupling theory containing activated events. Fluctuating spatiotemporal structures in aging and rejuvenating glasses lead to dynamical heterogeneity in glasses with characteristics distinct from those found in the equili…
We provide a compact derivation of the static and dynamic equations for infinite-dimensional particle systems in the liquid and glass phases. The static derivation is based on the introduction of an "auxiliary" disorder and the use of the replica method. The dynamic derivation is based on the general analogy between replicas and the supersymmetric formulation of dynamics. We show that static and dynamic results are consistent, and follow the Random First Order Transition scenario of mean field disordered glassy systems.
We study the glassy transition for simple liquids in the hypernetted chain (HNC) approximation by means of an effective potential recently introduced. Integrating the HNC equations for hard spheres, we find a transition scenario analogous to that of the long range disordered systems with ``one step replica symmetry breaking''. Our result agree qualitatively with Monte Carlo simulations of three dimensional hard spheres.
The barriers between metastable states near the glass transition of a random heteropolymer are studied using replicas by describing inhomogeneous states. The instanton solution for a replica space free energy functional is found numerically to estimate the size of activation barriers and of the critical nuclei themselves between the dynamic and the static glass transition temperatures.
Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks, the purpose of this paper is two-fold: at first we develop rigorous mathematical approaches to address properly a statistical mechanical picture of the phenomenon of {\em replica symmetry breaking} (RSB) in these networks, then -- deepening results stemmed via these routes -- we aim to inspect the {\em glassiness} that they hide. In particular, regarding the meth…
We study the role of fluctuations on the thermodynamic glassy properties of plaquette spin models, more specifically on the transition involving an overlap order parameter in the presence of an attractive coupling between different replicas of the system. We consider both short-range fluctuations associated with the local environment on Bethe lattices and long-range fluctuations that distinguish Euclidean from Bethe lattices with the same local environment. We find that the phase diagram in the temperature-coupling plane is very sensitive to the former but, at least for the $3$-dimensional (sq…
We present a first principle scheme to compute the rigidity, i. e. the shear-modulus of structural glasses at finite temperatures using the cloned liquid theory, which combines the replica theory and the liquid theory. With the aid of the replica method which enables disentanglement of thermal fluctuations in liquids into intra-state and inter-state fluctuations, we extract the rigidity of metastable amorphous solid states in the supercooled liquid and glass phases. The result can be understood intuitively without replicas. As a test case, we apply the scheme to the supercooled and glassy stat…
Using molecular dynamics simulation, we have calculated the pressure dependence of the diffusion constant in a binary Lennard-Jones Glass. We observe four temperature regimes. The apparent activation volume drops from high values in the hot liquid to a plateau value. Near the critical temperature of the mode coupling theory it rises steeply, but in the glassy state we find again small values, similar to the ones in the liquid. The peak of the activation volume at the critical temperature is in agreement with the prediction of mode coupling theory.
We explore several potential issues that have been raised over the years regarding the "entropic droplet" scenario of activated transport in liquids, due to Wolynes and coworkers, with the aim of clarifying the status of various approximations of the random first order transition theory (RFOT) of the structural glass transition. In doing so, we estimate the mismatch penalty between alternative aperiodic structures, above the glass transition; the penalty is equal to the typical magnitude of free energy fluctuations in the liquid. The resulting expressions for the activation barrier and the coo…
