In this paper we develop a method introduced by one of us to study metastable states in spin glasses. We consider a 'potential function' defined as the free energy of a system at a given temperature T constrained to have a fixed overlap with a reference configuration of equilibrium at temperature T ′ . We apply the method to the spherical p-spin glass and to some generalization of this model in the range of temperatures between the dynamic and the static transition. The analysis suggests a correspondence among local minima of the potential and metastable states. This correspondence is confirmed studying the relaxation dynamics at temperature T of a system starting from an initial configuration equilibrated at a different temperature T ′ .

We study the mean-field phase diagram of glassy systems in a field pointing in the direction of a metastable state. We find competition among a "magnetized" and a "disordered" phase, that are separated by a coexistence line as in ordinary first order phase transitions. The coexistence line terminates in a critical point, which in principle can be observed in numerical simulations of glassy models.

We corroborate the idea of a close connection between replica-symmetry breaking and aging in the linear response function for a large class of finite-dimensional systems with short-range interactions. In these systems, which are characterized by a continuity condition with respect to weak random perturbations of the Hamiltonian, the "fluctuation dissipation ratio" in off-equilibrium dynamics should be equal to the static cumulative distribution function of the overlaps. This allows for an experimental measurement of the equilibrium order parameter function. [S0031-9007 (98)06959-2] PACS numbers: 05.20. -y, 05.40. + j, 75.10.NrThe glassy state of matter can appear in systems with quenched disorder (like spin-glasses), or in nondisordered systems. Ergodicity breaking takes a special form in these systems. A rather generic situation is the existence of many solid, "glass," phases, which are very different from one another, and unrelated among themselves by symmetry transformations. Hence the Gibbs equilibrium measure decomposes into a mixture of many pure states. This phenomenon was first studied in detail in the mean field theory of spin-glasses, where it received the name of replica-symmetry breaking [1]. But it can be defined in a straightforward way and easily extended to other systems, by considering an order parameter function, the overlap distribution function. This function measures the probability that two configurations of the system, picked up independently with the Gibbs measure, lie at a given distance from each other [2]. Replica-symmetry breaking is made manifest when this function is nontrivial.The existence of nontrivial overlap distributions, first found in mean field systems, has been shown unambiguously, through numerical simulations, in finite-dimensional spin-glass systems with short-range interactions [3]. This order parameter function is a very important tool for the mathematical description of the Gibbs state. Unfortunately it seems impossible to access it experimentally for two reasons: (1) Large glassy systems never reach equilibrium at low temperatures; (2) The measurement of the distance between configurations requires a detailed observation at the microscopic-atomic-level, which is impossible. (In simulations, the second objection disappears, and one can get around the first one by working with smart algorithms and small enough systems.)The first objection is a very basic one: experimentally, glassy systems exhibit a nonequilibrium behavior, which requires a dynamical description. Quite often, they exhibit a special type of dynamical behavior called aging, i.e., the property that extensive one-time quantities like the energy, magnetization, etc., are asymptotically close to time-independent values, whereas two-time quantities, like the autocorrelation functions and their associated linear response functions, continue to depend on the time elapsed after the quench even for long times. Aging, defined in this way, appears in mean field spin-glasses and has been exhibited in spin-glass expe...

We report an analytical study of the vibrational spectrum of the simplest model of jamming, the soft perceptron. We identify two distinct classes of soft modes. The first kind of modes are related to isostaticity and appear only in the close vicinity of the jamming transition. The second kind of modes instead are present everywhere in the glass phase and are related to the hierarchical structure of the potential energy landscape. Our results highlight the universality of the spectrum of normal modes in disordered systems, and open the way toward a detailed analytical understanding of the vibrational spectrum of low-temperature glasses.glasses | jamming | normal modes | boson peak L ow-energy excitations in disordered glassy systems have received a great deal of attention because of their multiple interesting features and their importance for thermodynamic and transport properties of low-temperature glasses. Much debate has been concentrated around the deviation of the spectrum from the Debye law for solids, due to an excess of low-energy excitations, known as the "boson peak" (1).The vibrational spectrum of glasses is a natural problem of random matrix theory. In fact, the Hessian of a disordered system is a random matrix due to the random position of particles in the sample. The distribution of the particles induces nontrivial correlations between the matrix elements. Many attempted to explain the observed spectrum of eigenvalues by replacing the true statistical ensemble with some simpler ones, in which correlations are neglected or treated in approximate ways (2-11). However, most of these models are not microscopically grounded, thus making it difficult to assess which of the proposed mechanisms are the most relevant and understand their interplay.In this work we will focus on two ways of inducing a boson peak in random matrix models. First, it has been suggested that the boson peak is due to the vicinity to the jamming transition where glasses are isostatic (12, 13). Isostaticity means that the number of degrees of freedom is exactly equal to the number of interactions. Isostaticity implies marginal mechanical stability (MMS): cutting one particle contact induces an unstable soft mode that allows particles to slide without paying any energy cost (14,15). From this hypothesis, scaling laws have been derived that characterize the spectrum as a function of the distance from an isostatic point (11,12,16). Second, it has been proposed that low-temperature glasses have a complex energy landscape with a hierarchical distribution of energy minima and barriers (17). Minima are marginally stable (18) and display anomalous soft modes (11,19) related to the lowest energy barriers (20-22). We will denote this second kind of marginality as landscape marginal stability (LMS).Both mechanisms described above are highly universal. LMS is a generic property of mean-field strongly disordered models (18). MMS holds for a broad class of simple random matrix models (6,10,11,16) and for realistic glass models (12,23,24) at the iso...

In this paper, we discuss theoretically the behaviour of the four-point non-linear susceptibility and its associated correlation length for supercooled liquids close to the modecoupling instability temperature T c . We work in the theoretical framework of the glass transition as described by mean-field theory of disordered systems, and the hypernetted-chain approximation. Our results give an interpretation framework for recent numerical findings on heterogeneities in supercooled liquid dynamics.