Glasses are amorphous solids whose constituent particles are caged by their neighbours and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a 'roughness transition' to fractal basins, which brings about isostaticity and marginal stability on approaching jamming. Critical exponents for the basin width, the weak force distribution and the spatial spread of quasi-contacts near jamming can be analytically determined. Their value is found to be compatible with numerical observations. This advance incorporates the jamming transition of granular materials into the framework of glass theory. Because temperature and pressure control what features of the landscape are experienced, glass mechanics and transport are expected to reflect the features of the topology we discuss here.

Despite decades of work, gaining a first-principle understanding of amorphous materials remains an extremely challenging problem. However, recent theoretical breakthroughs have led to the formulation of an exact solution of a microscopic model in the mean-field limit of infinite spatial dimension, and numerical simulations have remarkably confirmed the dimensional robustness of some of the predictions. This review describes these latest advances. More specifically, we consider the dynamical and thermodynamic descriptions of hard spheres around the dynamical, Gardner and jamming transitions. Comparing meanfield predictions with the finite-dimensional simulations, we identify robust aspects of the theory and uncover its more sensitive features. We conclude with a brief overview of ongoing research.

In the first part of this paper, we derive the general replica equations that describe infinitedimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a strong sense in the mean field limit. We also suggest how the computation could be generalized in an approximate way to finite dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results. Contents

We consider the theory of the glass phase and jamming of hard spheres in the large space dimension limit. Building upon the exact expression for the free-energy functional obtained previously, we find that the random first order transition (RFOT) scenario is realized here with two thermodynamic transitions: the usual Kauzmann point associated with entropy crisis and a further transition at higher pressures in which a glassy structure of microstates is developed within each amorphous state. This kind of glass-glass transition into a phase dominating the higher densities was described years ago by Elisabeth Gardner, and may well be a generic feature of RFOT. Microstates that are small excitations of an amorphous matrix-separated by low entropic or energetic barriers-thus emerge naturally, and modify the high pressure (or low temperature) limit of the thermodynamic functions.

We consider the adiabatic evolution of glassy states under external perturbations. Although the formalism we use is very general, we focus here on infinite-dimensional hard spheres where an exact analysis is possible. We consider perturbations of the boundary, i.e. compression or (volume preserving) shear-strain, and we compute the response of glassy states to such perturbations: pressure and shear-stress. We find that both quantities overshoot before the glass state becomes unstable at a spinodal point where it melts into a liquid (or yields). We also estimate the yield stress of the glass. Finally, we study the stability of the glass basins towards breaking into sub-basins, corresponding to a Gardner transition. We find that close to the dynamical transition, glasses undergo a Gardner transition after an infinitesimal perturbation.Introduction -Glasses are long lived metastable states of matter, in which particles are confined around an amorphous structure [1,2]. For a given sample of a material, the glass state is not unique: depending on the preparation protocol, the material can be trapped in different glasses, each displaying different thermodynamic properties. For example, the specific volume of a glass prepared by cooling a liquid depends strongly on the cooling rate [1,2]. Other procedures, such as vapor deposition, produce very stable glasses, with higher density than those obtained by simple cooling [3,4]. When heated up, glasses show hysteresis: their energy (specific volume) remains below the liquid one, until a "spinodal" point is reached, at which they melt into the liquid (see e.g. [2, Fig.1] and [4, Fig.2]).The behavior of glasses under shear-strain also shows similarly complex phenomena. Suppose to prepare a glass by cooling a liquid at a given rate until some low temperature T is reached. After cooling, a strain γ is applied and the stress σ is recorded. At small γ, an elastic (linear) regime where σ ∼ µγ is found. At larger γ, the stress reaches a maximum and then decreases until an instability is reached, where the glass yields and starts to flow (see e.g. [5, Fig.3c] and [6, Fig.2]). The amplitude of the shear modulus µ and of the stress overshoot increase when the cooling rate is decreased, and more stable glasses are reached.Computing these observables theoretically is a difficult challenge, because glassy states are always prepared through non-equilibrium dynamical protocols. Firstprinciple dynamical theories such as Mode-Coupling Theory (MCT) [7] are successful in describing properties of supercooled liquids close to the glass state (including the stress overshoot [8]), but they fail to describe glasses at low temperatures and high pressures [9]. The dynamical facilitation picture can successfully describe calorimetric properties of glasses [10], but for the moment it does not allow one to perform first-principles calculations starting from the microscopic interaction potential. To bypass the difficulty of describing all the dynamical details of glass formation, one can exploit a standa...

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...

Random constraint satisfaction problems (CSP) have been studied extensively using statistical physics techniques. They provide a benchmark to study average case scenarios instead of the worst case one. The interplay between statistical physics of disordered systems and computer science has brought new light into the realm of computational complexity theory, by introducing the notion of clustering of solutions, related to replica symmetry breaking. However, the class of problems in which clustering has been studied often involve discrete degrees of freedom: standard random CSPs are random K-SAT (aka disordered Ising models) or random coloring problems (aka disordered Potts models). In this work we consider instead problems that involve continuous degrees of freedom. The simplest prototype of these problems is the perceptron. Here we discuss in detail the full phase diagram of the model. In the regions of parameter space where the problem is non-convex, leading to multiple disconnected clusters of solutions, the solution is critical at the SAT/UNSAT threshold and lies in the same universality class of the jamming transition of soft spheres. We show how the critical behavior at the satisfiability threshold emerges, and we compute the critical exponents associated to the approach to the transition from both the SAT and UNSAT phase. We conjecture that there is a large universality class of non-convex continuous CSPs whose SAT-UNSAT threshold is described by the same scaling solution.

What characterizes a solid is the way that it responds to external stresses. Ordered solids, such as crystals, exhibit an elastic regime followed by a plastic regime, both understood microscopically in terms of lattice distortion and dislocations. For amorphous solids the situation is instead less clear, and the microscopic understanding of the response to deformation and stress is a very active research topic. Several studies have revealed that even in the elastic regime the response is very jerky at low temperature, resembling very much the response of disordered magnetic materials 1-6 . Here we show that in a very large class of amorphous solids this behaviour emerges upon decreasing temperature, as a phase transition, where standard elastic behaviour breaks down. At the transition all nonlinear elastic moduli diverge and standard elasticity theory no longer holds. Below the transition, the response to deformation becomes history-and time-dependent.Our work connects two different lines of research on amorphous solids such as structural, colloidal and granular glasses. The first focuses on their behaviour at low temperature. With the aim of understanding the response of glasses to deformations, there have been extensive numerical studies of stress versus strain curves obtained by quenching model systems at zero temperature. One of the main outcomes is that the increase of the stress is punctuated by sudden drops related to avalanche-like rearrangements both before and after the yielding point [1][2][3][4][5][6] . This behaviour makes the measurements, and even the definition of elastic moduli fairly involved. In a series of works, Procaccia et al. have given evidence that in some models of glasses, such as Lennard-Jones mixtures (and variants), nonlinear elastic moduli exhibit diverging fluctuations, and linear elastic moduli differ depending on the way they are defined from the stress-strain curve 7,8 . Another independent research stream has focused on gaining an understanding of the jamming and glass transitions of hard spheres both from real-space and mean-field theory perspectives 9,10 . The exact solution obtained in the limit of infinite dimensions revealed that by increasing the pressure a hard sphere glass exhibits a transition within the solid phase, where multiple arrangements emerge as different competing solid phases 11,12 . This is called the Gardner transition, in analogy with previous results in disordered spin models 13,14 . Recent simulations have confirmed that in three dimensions these different arrangements indeed become increasingly long-lived, possibly leading to ergodicity breaking 15 . These mean-field analyses complement and strengthen all the remarkable results found in the past two decades on jammed hard spheres glasses. The major outcome of these real-space studies was the discovery that amorphous jammed solids are marginally stable-that is, characterized by soft modes and critical behaviour, and in consequence by properties which are very different from those of usual crystalline ...