One of the most remarkable predictions to emerge out of the exact infinite-dimensional solution of the glass problem is the Gardner transition. Although this transition was first theoretically proposed a generation ago for certain mean-field spin glass models, its materials relevance was only realized when a systematic effort to relate glass formation and jamming was undertaken. A number of nontrivial physical signatures associated with the Gardner transition have since been considered in various areas, from models of structural glasses to constraint satisfaction problems. This perspective surveys these recent advances and discusses the novel research opportunities that arise from them.

As recently as five years ago, the Gardner transition was but an exotic feature of abstract models known only by a cabal of statistical physicists. The recent surge in interest follows from the realization that mean-field models of structural glasses, which are exact in the limit of infinite dimension, d, can also undergo such a transition. A coordinated effort to better understand the putative materials properties of this transition in real(istic) physical systems has since ensued. Because these advances have taken place fairly rapidly and over a range of subfields, it can be challenging to piece them together into a forward vision. This perspective aims to contextualize the many theoretical, numerical, and experimental opportunities that lie ahead. In contrast to a recent review of hard sphere glasses,1 the scope of this perspective is thus both narrower and broader. We here focus exclusively on Gardner physics but do so for a richer variety of glass formers and from a broader variety of viewpoints. We also include an array of recent results that help identify upcoming research challenges.

To understand what a Gardner transition might look like in a materials context, we need to recall that a hallmark of solids is their elastic response to small deformations, i.e., their rigidity. In (perfect) crystals, rigidity is due to the spontaneous breaking of translational invariance that follows a first-order phase transition. The resulting crystalline solid is “thermodynamically” stable. In glasses and other amorphous solids, by contrast, rigidity emerges from the appearance of a multitude of long-lived metastable states. These solids also spontaneously break translational invariance, but only for a finite—albeit long—time. Rigidity is then a “dynamical” phenomenon. Once a system is confined to a metastable glass state, it responds elastically to an applied deformation, as would a crystal. Upon crossing the Gardner transition, however, that response changes spectacularly (Fig. 1). At the Gardner transition and in the Gardner phase that follows, the materials response becomes strongly nonlinear and is characterized by intermittent plastic events, all while the system remains globally solid. The breakdown of solidity happens at larger deformations, where the metastable glass state disappears entirely.

FIG. 1.

Schematic illustration of the evolution of a glass free energy basin under an applied perturbation γ, be it mechanical, thermal, or otherwise. At low γ, in the simple glass phase, the glass responds elastically and its free energy, F, is quadratic (green arrows). Beyond a Gardner transition γG, the free energy basin becomes rugged and correspondingly the response becomes intermittent, due to jumps between sub-basins (red arrows). At even larger deformations, γY, the basin might disappear or yield (black arrow). Adapted from Y. Jin, P. Urbani, F. Zamponi, and H. Yoshino, Sci. Adv. 4, eaat6387 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License.

FIG. 1.

Schematic illustration of the evolution of a glass free energy basin under an applied perturbation γ, be it mechanical, thermal, or otherwise. At low γ, in the simple glass phase, the glass responds elastically and its free energy, F, is quadratic (green arrows). Beyond a Gardner transition γG, the free energy basin becomes rugged and correspondingly the response becomes intermittent, due to jumps between sub-basins (red arrows). At even larger deformations, γY, the basin might disappear or yield (black arrow). Adapted from Y. Jin, P. Urbani, F. Zamponi, and H. Yoshino, Sci. Adv. 4, eaat6387 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License.

Close modal

This perspective elaborates on this point from a variety of outlooks. Section II revisits the original discovery of the Gardner transition in spin-glass models and presents its phase space description. Section III recapitulates its rediscovery in hard-sphere models, and Sec. IV recapitulates the limitations of its applicability in systems with softer pair interactions. Sections V A and VI extend Gardner physics to the rheology of amorphous solids and to polydisperse crystals, respectively. Renormalization group (RG) considerations of the Gardner criticality are presented in Sec. VII, and the connection between Gardner physics and constraint satisfaction problems (CSPs) is discussed in Sec. VIII. Section IX briefly concludes.

In order to understand how the Gardner transition emerges in the mean-field description of structural glasses, it is useful to revisit its original discovery in a class of mean-field spin glasses. We thus start this perspective with a brief historical and conceptual detour through these abstract statistical mechanics models.

In 1979, after years of theoretical struggle, the low temperature mean-field solution of the Sherrington-Kirkpatrick (SK) fully connected spin glass model3 was identified by Parisi.4 This feat relied on the invention of the replica symmetry breaking (RSB) scheme. It took a few years to interpret physically,5,6 and decades to secure mathematically,7,8 even just parts of the approach. As part of these efforts, in the 1980s, several statistical physicists turned their attention to other fully-connected models amenable to the RSB scheme, such as Derrida’s Random Energy Model (REM), and spin glasses with p-body,9 Heisenberg,10 or Potts11 interactions. The REM family of models was especially important in elucidating the meaning of replicas because these models could also be solved fairly straightforwardly using the Markov inequality.12 

The decoding of RSB revealed its intimate relationship to ergodicity breaking. Spin glass models were then broadly sorted into two classes, depending on their type of ergodicity breaking and hence of RSB transition (see Ref. 5 for more details):

  • REM-like models, or discontinuous spin glasses, exhibit a thermodynamic transition of mixed first- and second-order character. Like at a first-order transition, the relevant (Edwards-Anderson overlap) order parameter jumps discontinuously and the associated susceptibility does not diverge, but like at a second-order transition no latent heat is released. The ergodic components that dominate the measure at low temperatures are far in phase space from each other and are absolutely stable; different components have small Edwards-Anderson overlaps and are separated by extensive barriers. In the technical replica jargon, these models require a single step of RSB and are thus called 1RSB models (Fig. 2).

  • SK-like or continuous spin glass models exhibit a thermodynamic second-order spin-glass transition that is accompanied with a divergent susceptibility. The whole spin-glass phase is then marginal, ergodic components (or states) are critical and close to each other, and barriers between states are subextensive. Moreover, the organization of these states is both hierarchical and ultrametric, which corresponds to a maximally rich organization of phase space. In the replica formalism, marginality is associated with the vanishing of the “replicon eigenvalue,” which in technical jargon corresponds to “continuous” or full RSB (fullRSB).

FIG. 2.

The phase space of a simple glass (REM-like) phase is formed of many distinct metastable clusters that are internally ergodic (blue regions), as in a perfect crystal, but separated by extensive barriers (gray regions). Upon crossing a Gardner transition, the internal structure of these clusters changes dramatically. Each metastable glass state then fractures into an infinite hierarchy of sub-basins (green, yellow, and red regions). Republished with permission from P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Annu. Rev. Condens. Matter. Phys. 8, 265-288 (2017). Copyright 2017 Annual Reviews, Inc.

FIG. 2.

The phase space of a simple glass (REM-like) phase is formed of many distinct metastable clusters that are internally ergodic (blue regions), as in a perfect crystal, but separated by extensive barriers (gray regions). Upon crossing a Gardner transition, the internal structure of these clusters changes dramatically. Each metastable glass state then fractures into an infinite hierarchy of sub-basins (green, yellow, and red regions). Republished with permission from P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Annu. Rev. Condens. Matter. Phys. 8, 265-288 (2017). Copyright 2017 Annual Reviews, Inc.

Close modal

In 1985, two papers further enriched this classification. Gross, Kantor, and Sompolinsky, considering Potts glasses,11 and Gardner, considering Ising p-spin glasses,13 both noted that after a first phase transition to a 1RSB phase, these two REM-like models exhibit a fullRSB transition at a lower temperature (see Fig. 2). These reports of what was later called a “Gardner transition” were followed by others. Yet for most of the ensuing three decades, these findings were only mentioned anecdotally and treated as exotic features of exotic models.

Meanwhile, the physical understanding of REM-like models markedly improved when Kirkpatrick, Thirumalai, and Wolynes14 recognized that their phenomenology is akin to that of structural glasses. They then realized that a first nonergodicity, dynamical transition analogous to the transition in the mode-coupling theory of glasses,15,16 at temperature Td, precedes the actual thermodynamic transition.17 At Td, the phase space of these mean-field models becomes composed of an exponential (in the number of spins or particles) multiplicity of ergodic components that are dynamically inaccessible from one another. This multiplicity is measured by the complexity, which is comparable to the entropy of a solid in excess of its vibrational contribution, i.e., its configurational entropy.18,19 Beyond the dynamical transition, the complexity shrinks with temperature and vanishes at the thermodynamic transition at a finite temperature TK, as the configurational entropy seems to do in certain glass forming liquids, leading to an entropy crisis, as first noted by Kauzmann.20 On the basis of this analogy, the Random first order transition (RFOT) theory of REM-like models, which was understood to include structural glasses, was proposed.14,21,22

Over the following decades, the spherical p-spin model became the central focus of the statistical mechanics of disordered systems, as a prototypical model of the RFOT universality class. Because this model does not exhibit a Gardner transition, the putative importance of this transition on structural glasses was left completely unexplored for some time. Further consideration of Gardner transitions slowly re-emerged in the late 1990s. Barrat, Franz, and Parisi23 showed them to be fairly ubiquitous (in essentially all spin glass models except the spherical p-spin), and Montanari and Ricci-Tersenghi24 revisited Gardner’s work with more sophisticated theoretical tools—naming the transition by the same token. These studies were by then, however, quite far from the structural glass mainstream.

Although the putative relevance of the Gardner transition to structural glasses was first suggested decades ago,25,26 only recently was Gardner physics fully brought to the field of amorphous solids. This advance resulted from the remarkable convergence of three concurrent yet independent lines of research, separately inspired by the seminal Liu-Nagel proposal for a unified description of amorphous solids.27 

  • Studies of dense sphere packings around jamming in two and three dimensions established through numerical simulations and analytical arguments that such packings are marginally stable,28–38 in that they display an excess of low-energy modes that gives rise to anomalous scalings of physical quantities around jamming.

  • Numerical studies of hard sphere fluids showed that glass formability increases with dimension,39–42 which paved the way for numerical studies of the jamming transition in higher dimensions.39,42–44

  • In the footsteps of Ref. 25, an analytical exploration of the glass phase of hard spheres in the mean-field, d limit provided an analytical phase diagram within the simplest RS assumption.45–47 

Comparing these three sets of results gave rise to a conundrum. While the RS assumption predicts hyperstatic packings and simple critical exponents at jamming in d,46 numerical simulations32,33,39,43 and analytical arguments36 showed that isostatic packings and a nontrivial criticality are actually observed in all accessible d.

In order to solve this puzzle, an extended theoretical study of infinite-dimensional hard sphere glasses was undertaken.48 The ensuing discovery of a Gardner transition49 and of the Gardner phase that lies beyond it, up to and including jamming,50,51 put forward a physically coherent description of jamming. The fullRSB solution associated with the Gardner phase indeed predicts that jammed sphere packings are isostatic and critically nontrivial in d as well. Building on this finding, the rest of the infinite-dimensional phase diagram of hard spheres was completed1,52–55,166 (see Fig. 3). Additional scaling arguments37,56,57 and analytical calculations58 provided the critical scaling of the jammed phase and of finite-temperature systems.

FIG. 3.

Phase diagram for hard spheres in the limit d. At equilibrium (solid black line), for φ < φd the liquid is ergodic, while for φ > φd a large number of ergodic clusters are separated from one another, which gives rise to equilibrium glass states. Upon slowly compressing any such glass state (dotted lines), a Gardner transition (blue line) is encountered prior to reaching jamming (red line) at infinite (reduced) pressure, p. Republished with permission from P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Annu. Rev. Condens. Matter. Phys. 8, 265-288 (2017). Copyright 2017 Annual Reviews, Inc.

FIG. 3.

Phase diagram for hard spheres in the limit d. At equilibrium (solid black line), for φ < φd the liquid is ergodic, while for φ > φd a large number of ergodic clusters are separated from one another, which gives rise to equilibrium glass states. Upon slowly compressing any such glass state (dotted lines), a Gardner transition (blue line) is encountered prior to reaching jamming (red line) at infinite (reduced) pressure, p. Republished with permission from P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Annu. Rev. Condens. Matter. Phys. 8, 265-288 (2017). Copyright 2017 Annual Reviews, Inc.

Close modal

In hindsight, the existence of a mean-field Gardner transition upon approaching jamming should not have been so surprising, because that same transition is found in essentially all REM-like models,11,13,23,24,59–62 which includes infinite-dimensional hard spheres. (Hard spheres are nevertheless the first structural glass model without quenched disorder in which a Gardner transition was explicitly found.49) What was rightfully surprising is the dimensional robustness of the jamming criticality, as discussed below. In the rest of this section, we separately consider the Gardner transition, the Gardner phase, and jamming and conclude by discussing possible paths toward experimental validations of the theory.

As further discussed in Sec. VII, the very existence of a Gardner transition in d < 6 was (and in some ways is still) far from obvious, based on renormalization group considerations. Hence, the first numerical study of this transition targeted a system from a family of idealized models in which particles interact in an abstract space with the topology of a random graph. This topology guarantees that long-range spatial fluctuations are fully suppressed and thus cannot extinguish mean-field critical points.63 Note that in the mean-field phase diagram of Fig. 3, the Gardner transition is very close to the dynamical transition at low densities.52 In that region, the system cannot easily be confined within a glass basin because of activated and hopping processes, and thus the Gardner transition cannot be properly detected. Luckily enough, in the chosen Mari-Kurchan (MK) variant of these models (originally proposed by Kraichnan64), equilibrium configurations can be generated at negligible computational cost65–67 via the planting technique.68 This feature was essential to properly study the Gardner transition in this system.69 

With the MK results in hand as reference and with (contemporary yet unrelated) stunning improvements in sampling techniques that reach deep into the glass regime via the swap Monte Carlo algorithm,70,71 the Gardner transition for hard sphere models in d = 2 and d = 3 could then be explored.72 This work first evinced that a growing correlation length, susceptibility, and correlation time qualitatively consistent with the mean-field Gardner transition can be observed in model glass formers. The numerical results obtained so far however only qualitatively hints at a transition.

A proper finite-size scaling study should more reliably test for the existence of a critical transition, but this project has not yet been undertaken. An extraction of the corresponding critical exponents is thus also missing. Such a study would certainly be computationally challenging, but the relative robustness of the Gardner phenomenology in hard spheres compared to the equivalent transition in spin glasses73,74 (see Sec. VII) suggests that the analysis might not be as painful as it first seems. Thanks to the recent extension of enhanced sampling techniques to higher dimensions,75 similar considerations in d = 4 and 5 might also be within reach. The complex criticality scenarios proposed for this transition (see Sec. VII) could thus be clarified. It may well be that the situations in dimensions d = 2, 3, or 4 are all different, and future work should address this question.

The criticality of the Gardner phase beyond the transition is theoretically less controlled than the transition itself, and the corresponding numerical simulations are also less developed. Because equilibrating configurations in the Gardner phase requires prohibitively long simulation times, one possibility is to study the off-equilibrium dynamics following quenches, as is routinely done in spin glasses.6,76 Recent numerical works in this direction in d = 3 in Ref. 77 and d = 2 in Ref. 78 are consistent with a complex organization of vibrational states separated by a hierarchy of barriers that gives rise to a rich aging behavior.

Other than the complex aging behavior associated with a distribution of relaxation barriers, however, few insights have been clearly formulated. From a mean-field standpoint, solving the out-of-equilibrium dynamics in a fullRSB phase is a daunting challenge.6,79,80 It should, nonetheless, be possible to probe numerically and experimentally the state structure in that regime.

What has been robustly established is that after a direct quench from the simple glass phase past the estimated location of the Gardner transition, single-particle displacements exhibit a slow aging dynamics. Even after very large times, the system does not reach any kind of steady state. Dynamics inside a glassy basin must therefore proceed via a very slow exploration of a large number of states, separated by large barriers. Empirically, this implies that quenching two copies of the same hard sphere configuration to large pressures produces two distinct packings that are dynamically inaccessible from one another over the numerical time window. It has further been demonstrated that large correlation lengths develop and grow slowly with time after a quench.77,78,81 The barriers that separate distinct packings thus correspond to highly collective and correlated particle displacements (as already suggested in Ref. 33). Furthermore, by performing multiple quenches from a given hard sphere glass configuration, it is found that a very large number of distinct packings can be produced, and that these different packings are organized hierarchically,78 in a way that is strongly reminiscent of the fullRSB description. Alternatively, one can sample the nearby jammed minima and find that they are both hierarchical and ultrametric.81 It is interesting to emphasize that the same phenomenology is found in d = 2 hard disks, where a sharp Gardner transition might not take place (see Sec. VII). The phenomenology and complexity of the phase space of hard spheres are therefore robust findings that should have direct experimental consequences, independently of the Gardner transition.

In contrast to the above two regimes, jamming criticality presents strong agreement between theory and numerics. The mean-field, d solution51 and scaling arguments36 predict that at jamming the distributions of both small forces and small interparticle gaps exhibit nontrivial power-law scalings. Remarkably, numerical results for even d = 3 and d = 2 seemingly scale with those of the same nontrivial power-law exponents,37,39,43,44 although without a clear theoretical basis for this superuniversality (see Sec. VII). The asymptotic scaling of the particle cage diameter upon approaching jamming33,50,56 and of the excitation spectrum82–85 matches theoretical predictions similarly closely. This agreement is not only remarkable for the Gardner phase specifically but is also the most notable materials prediction ever to emerge from a fullRSB analysis. Here again, no proper study of finite-size corrections yet validates these findings, but the robustness of existing numerical results leaves little ambiguity.

Experimental efforts to detect evidence of the Gardner transition are ongoing. The first such evidence came in the form of a study of a quasithermalized two-dimensional hard-sphere-like granular system.86,87 However, more precise measurements in this vein may be difficult to achieve due to the need for systems to be well thermalized in order to clearly identify signatures of a Gardner transition (see Fig. 3). The associated structural relaxation time is indeed well beyond experimental reach for the millimeter-size granular particles typically used to model glasses. In this respect, colloidal systems offer a more fruitful avenue because it is possible to combine smaller-sized particles (allowing proper thermalization over experimental timescales) with light-scattering or microscopy techniques that are in principle able to resolve single-particle dynamics even at very small length scales. A second path to direct verification of a Gardner phase in physical systems is offered by the short-time dynamics of the glass. Colloids within a stable glass experience a fixed cage structure; hence, the crossover from ballistic motion to caged behavior is expected to be simple and result in an exponential crossover in the mean squared displacement. By contrast, in a Gardner phase, cages are themselves ever shifting as the system explores the hierarchical phase space. Colloids should thus be constantly leaving small cages only to find themselves in slightly larger cages ad infinitum. One then expects a logarithmic growth of the mean squared displacement with lag time. Intriguingly, a first study in this direction finds that glassy colloidal suspensions exhibit just such a change in short-time dynamics.88 

The critical scaling of the mean squared displacement upon approaching jamming has also been measured in a colloidal glass,89 and the results are here directly compatible with theoretical predictions. Comparable validations of the force and gap criticality at jamming are, however, unlikely. The truly minute length and force scales over which power-law scalings are predicted lie well below the resolution of any existing experimental setup. Indirect measures of the phase space structure are thus more likely to provide a real-world connection between amorphous solids and Gardner physics in the foreseeable future.

Since the initial analysis of the glass phase of hard spheres, a much larger class of interaction potentials has been analyzed within the realm of mean-field theory. A Gardner phase has been identified for a number of pair potentials and a broad range of physical conditions.58,90,91 In short, the mean-field approach predicts that all types of pair potentials—including soft repulsive spheres, Lennard-Jones, and others—can undergo a Gardner transition in some parts of their phase diagrams, as we now detail.

A representative example (Fig. 4) is the mean-field theoretical prediction for separately cooling and compressing an equilibrium glass state of soft repulsive spheres. The glass is of course unstable toward melting if the transformation takes it back to the fluid phase. More interestingly, it can undergo a Gardner transition to a marginally stable glass phase either by cooling or by compression. Generally, mean-field theory thus states that cooling a glass or compressing a colloidal glass with repulsive interactions could lead to a marginally stable Gardner phase.

FIG. 4.

Mean-field phase diagram for soft repulsive WCA particles. The glass at point g1 is prepared under equilibrium conditions at temperature Tg1 and density ρg1 and is then adiabatically followed up to different temperatures and densities, measured as w=ln[ρ/ρg1]. The glass state then either fluidizes (unstable region, beyond the dashed line), remains within an ergodic cluster (simple glass phase), or undergoes a Gardner transition (dark blue lines) to enter the Gardner phase (light blue areas). This example illustrates that in d soft glasses can generically undergo a Gardner transition under a variety of physical conditions, including around the jamming point wj of the glass prepared at g1. Reprinted with permission from C. Scalliet, L. Berthier, and F. Zamponi, Phys. Rev. E 99, 012107 (2019). Copyright 2019 American Physical Society.

FIG. 4.

Mean-field phase diagram for soft repulsive WCA particles. The glass at point g1 is prepared under equilibrium conditions at temperature Tg1 and density ρg1 and is then adiabatically followed up to different temperatures and densities, measured as w=ln[ρ/ρg1]. The glass state then either fluidizes (unstable region, beyond the dashed line), remains within an ergodic cluster (simple glass phase), or undergoes a Gardner transition (dark blue lines) to enter the Gardner phase (light blue areas). This example illustrates that in d soft glasses can generically undergo a Gardner transition under a variety of physical conditions, including around the jamming point wj of the glass prepared at g1. Reprinted with permission from C. Scalliet, L. Berthier, and F. Zamponi, Phys. Rev. E 99, 012107 (2019). Copyright 2019 American Physical Society.

Close modal

The theoretical suggestion that all flavors of glass formers might undergo a Gardner transition initially raised the exciting possibility that the physical properties of a broad class of amorphous materials could be characterized by the low-lying collective excitations that characterize Gardner phases. This theoretical path could then potentially unify the physical properties of systems as diverse as granular materials near their jamming transition and atomistic glass-forming materials using the concept of marginal stability.38,49,50

A key step on this path aimed to confirm that a Gardner transition and the physics associated with the Gardner phase can be found in systems other than hard spheres. To this end, it is instructive to analyze models intermediate between hard spheres and standard Lennard-Jones glass formers, such as Weeks-Chandler-Andersen (WCA), harmonic, or Hertzian particles. In these models of soft repulsive spheres, particles can overlap at a finite energetic cost, controlled by an energy scale ϵ, if the interparticle distance is smaller than the interaction range, σ. The control parameters of the model are thus temperature and density, as in the mean-field analysis presented in Fig. 4. In the limit of ϵ/kBT → 0 (which experimentally corresponds to relatively large colloids or stiff particles), these particles are still thermally agitated but are also close to their jamming transition.92 Computer simulations performed in this region of the phase diagram93 confirm that all signatures of a Gardner transition (aging, slow dynamics, and spatial correlations) can be observed in d = 3, thus reflecting the emergence of a complex phase space structure. These results show that the mean-field analysis remains qualitatively correct for three-dimensional models, at least in a region of the phase diagram close to the jamming transition.

Moving further away from this transition, computer simulations of Lennard-Jones particles in d = 2 and soft repulsive spheres in d = 3 have, however, revealed that a Gardner phase is not necessarily universally observed.94,95 Even when the transition is absent, the free energy basin of the simple glass phase breaks into several sub-basins at low temperatures,94,95 consistently with what has been observed in many numerical studies of the potential energy landscape of model glasses.96–99 A more refined analysis of the associated time and length scales, however, revealed that these sub-basins in fact correspond to a rare population of highly localized “defects.”94 The physics of this phenomenon thus strongly contrasts with the mean-field prediction, which suggests that excitations should involve system-wide correlations. While this finding quashed theoretical hopes of universality, it also opened a way for a more systematic study of the interplay between extended defects, related to the mean-field Gardner phase, and localized defects, expected to be omnipresent in low-dimensional glasses.96–100 A promising line of research is the discovery that localized defects may themselves possess some universal properties.101–103 The link between localized defects found in dense liquids far from jamming to collective excitations that proliferate close to jamming in marginally stable systems also remains to be understood.104 

Interestingly, the phenomenology of these localized defects is strongly reminiscent of the elementary excitations implicitly assumed by the empirical two-level system descriptions, which were first proposed more than 40 years ago but are still without a clear microscopic origin.105,106 The search for a Gardner phase in low-temperature glasses107 may thus provide novel computational and experimental approaches to understand localized excitations in low-temperature glasses, and their interplay with more extended ones. Future work should aim at quantifying and generalizing these findings, to couple them with quantum mechanical descriptions that could explain the tunneling characteristics of two-level systems, and to quantitatively determine whether localized defects are indeed responsible for the cryogenic properties of amorphous solids.

Sections III and IV considered the emergence of a Gardner transition in a glass state that is either compressed or cooled, which are both isotropic transformations. Here, we consider putative signatures of this same physics, but for the anisotropic transformations that emerge in the context of rheology. More specifically, we examine the stress-strain curves and the elastic response of glass states.

The study of the rheology of amorphous solids by means of mean-field methods was initiated in Refs. 108 and 109, and additional results for d systems were obtained in Refs. 52, 54, 58, and 110. Motivated by these results, numerical studies in d = 3 have considered elasticity, dilatancy, plasticity, marginal stability, yielding, and shear jamming.2,111 For dynamically arrested equilibrium liquid configurations, the initial response to shear is found to be perfectly elastic, and these materials can be characterized by a shear modulus μ = dσ/dγ and dilatancy R = dP/d(γ2), where pressure P and shear stress σ are conjugate observables (responses) to φ and γ, respectively. Stress and pressure thus both increase under an applied strain at constant density. At larger strains, however, the response becomes nonlinear and intermittent. In the mean-field description, the Gardner transition gives rise to this behavior, but in finite dimensions both collective and local excitations likely contribute.112–122 The glass then either yields or jams, as illustrated in the phase diagram of Fig. 5. Yielding appears in d as a spinodal point at which the glass solution is lost via a saddle-node bifurcation, leading to characteristic square-root singularities of physical observables,52 in particular, σ and P. In d = 3, however, this singularity is smoothed by fluctuations, leading to a more complex behavior.2,111,123,124 Jamming under shear corresponds to the joint divergence of σ and P, and its phenomenology is similar to its isotropic counterpart.2,110

FIG. 5.

(a) Theoretical shearing phase diagram of d hard spheres prepared in an equilibrium glass state at scaled packing fraction φ^g=8. Reprinted with permission from P. Urbani and F. Zamponi, Phys. Rev. Lett. 118, 038001 (2017). Copyright 2017 by the American Physical Society. The glass is first compressed at scaled packing fraction φ^ and then strained at shear strain γ, resulting in a (φ^,γ) stability map. The simple glass (stable) phase is delimited by the yielding (brown) line at lower density and by the shear jamming (blue) line at higher density. A Gardner transition (red) line delimits the Gardner (marginal) phase at high (φ^,γ). (b) Corresponding numerical phase diagram of d = 3 hard spheres prepared in equilibrium at φg = 0.655. Adapted from Y. Jin, P. Urbani, F. Zamponi, and H. Yoshino, Sci. Adv. 4, eaat6387 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License. The qualitative agreement between the mean-field prediction and the d = 3 numerical simulations is remarkable, except in the vicinity of the critical point (gray diamond), where the shear jamming and shear yielding lines merge. The qualitative difference might be explained either in the presence of localized plastic excitations in d = 3 systems (in which the role of the Gardner transition has yet to be clarified) or by the different nature of the yielding transition in d = 3 (shear band nucleation vs spinodal instability).

FIG. 5.

(a) Theoretical shearing phase diagram of d hard spheres prepared in an equilibrium glass state at scaled packing fraction φ^g=8. Reprinted with permission from P. Urbani and F. Zamponi, Phys. Rev. Lett. 118, 038001 (2017). Copyright 2017 by the American Physical Society. The glass is first compressed at scaled packing fraction φ^ and then strained at shear strain γ, resulting in a (φ^,γ) stability map. The simple glass (stable) phase is delimited by the yielding (brown) line at lower density and by the shear jamming (blue) line at higher density. A Gardner transition (red) line delimits the Gardner (marginal) phase at high (φ^,γ). (b) Corresponding numerical phase diagram of d = 3 hard spheres prepared in equilibrium at φg = 0.655. Adapted from Y. Jin, P. Urbani, F. Zamponi, and H. Yoshino, Sci. Adv. 4, eaat6387 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License. The qualitative agreement between the mean-field prediction and the d = 3 numerical simulations is remarkable, except in the vicinity of the critical point (gray diamond), where the shear jamming and shear yielding lines merge. The qualitative difference might be explained either in the presence of localized plastic excitations in d = 3 systems (in which the role of the Gardner transition has yet to be clarified) or by the different nature of the yielding transition in d = 3 (shear band nucleation vs spinodal instability).

Close modal

Glasses prepared beyond the Gardner transition display a markedly different elastic response. The simplest observable in this context is the shear modulus, μ. In the stable solid phase, μ is given by a single number, i.e., the elasticity of the stable glass basin. In the Gardner phase, by contrast, the mean-field, d solution predicts that the linear response of the system takes a long time to relax1 because the ultrametric structure of glass states is associated with an infinite hierarchy of time scales. After a short transient, the system equilibrates in the individual glass state in which it was prepared initially, the mean squared displacement approaches a constant value ΔEA, and correspondingly the stress decays, σ(t) ∼ μEA)γ. Waiting longer times (that diverge with the system size) allows the system to explore a wider portion of the ultrametric tree of states. Correspondingly, the mean squared displacement increases, Δ > ΔEA, and the stress further decays to σ(t) ∼ μ(Δ)γ. The Gardner phase is then characterized by a function μ(Δ) that gives the effective shear modulus of the system,53 when it is allowed to explore phase space up to a mean squared displacement Δ. The function μ(Δ) can be computed analytically in d, see Ref. 53. In particular, around jamming it is found that μEA) ∼ Pκ, while μ(Δ < ΔEA) ∼ PμEA); hence, a dramatic softening should be observed if the system is allowed to leave the glass state in which it was prepared. This d prediction has been verified numerically111,125 in d = 3 and might be related to the experimental findings of Refs. 126 and 127.

Another key prediction of the mean-field, d solution concerns the nonlinear elastic response. Upon approaching the Gardner transition, all nonlinear elastic moduli are expected to diverge.58 More precisely, consider the expansion of the elastic stress in powers of γ,

σ=μ1γ+13!μ3γ3+,  μn=dnσdγnγ=0,
(1)

where μ1 = μ is the elastic shear modulus and μ3, μ5, … are nonlinear response coefficients. These quantities are random variables because they depend on the particular glass sample under investigation. Upon approaching the Gardner transition from the stable phase, the average coefficient, μn, remains finite, but sample-to-sample fluctuations diverge as δμn2ε32n/V, where ε is the distance from the transition. Although fluctuations scale proportionally to 1/V, the prefactor diverges for all n > 2. The linear response therefore remains well defined, but sample-to-sample fluctuations of the nonlinear moduli grow to be extremely large.

The divergence of the fluctuations of elastic moduli is related to plasticity, i.e., the piecewise linear behavior of the stress-strain curves in finite samples.116 A true divergence in thermodynamically equilibrated solids requires a divergent correlation length at the Gardner transition; the mean-field d solution thus predicts plasticity to be extensive in the Gardner phase.58 An infinitesimal variation of density or shear strain in that phase indeed leads to extensive rearrangements, i.e., avalanches.38,128,129 The size distribution of these avalanches then scales as a power law with universal critical exponents, as has been confirmed numerically.129 In summary, while the response of the solid in the stable phase is elastic and reversible, in the Gardner marginally stable phase the response is plastic and irreversible, leading to divergent nonlinear susceptibilities. It is important however to stress that in finite dimensions, localized plastic excitations involving a few particles are always present, independently of any Gardner transition;116 these localized excitations can induce large fluctuations of the elastic moduli that should disappear in the thermodynamic limit, if equilibration within the glass basin can be reached. See Ref. 2 for a more complete discussion of irreversibility and plasticity in this context.

Recent extension of the theory to the thermal regime using soft harmonic spheres90,91 will likely be followed by numerical work in finite dimension. An exciting possibility would be for these studies to provide a sharper understanding of the rheological role of localized excitations,112–122 as a counterpart of the comparable excitations observed in isotropically cooled or compressed systems.

Although the Gardner transition has only been predicted for systems that are fully disordered, such as spin and structural glasses, the minimal conditions for its existence might not be so stringent. Several authors have indeed suggested that slightly disordered crystal packings display a marginal stability akin to that of fully disordered jammed packings.63,131,132 In this section, we consider minimally disordered crystals that exhibit a behavior reminiscent of the Gardner physics.

It is straightforward to see how the disorder inherent to amorphous solids might give rise to a Gardner transition. The ultrametric structure of phase space follows from the hierarchy of distinct vibrational states that emerge as the system is deformed (Fig. 6). In perfectly crystalline materials, by contrast, every particle is contained within an identical and mechanically stable vibrational environment. Crystals are thus marked both by highly degenerate overconstraints and the absence of any mechanism to break this degeneracy. For instance, all particles in a face-centered cubic (fcc) crystal of hard spheres in d = 3 have exactly Z = 12 contacting neighbors at close packing, far in excess of the minimal isostatic criterion, Z = 6, for marginal stability. An extensive number of contacts can thus be broken without destabilizing the system. However, this manifold degeneracy is easily destroyed by the introduction of even just a little dispersion in the particle size (polydispersity). Such a perturbation explodes the high crystal symmetry into an enormous number of nearly identical, but distinct close-packed structures, each ultimately resulting from a unique set of correlated structural choices in the local environment.

FIG. 6.

Schematic illustration of a Gardner-like scenario for polydisperse crystals. A perfect crystal (purple particles) has only one well-separated densest packing. A crystal of polydisperse particles (colored from red to purple, according to their growing particle radius), by contrast, has a large number of nearly equivalent packings. Beyond the pressure onset of Gardner-like physics, at PG, such systems are forced into irreversible constraints. For the sake of illustration, consider the particle outlined in black. It is free to collide with all its nearest neighbors at low pressures, but forced to have one or another set of collisional contacts (green stars) as pressure increases. This overall process is cooperative and thus cannot be reduced to such a simple local description, but this illustration can nonetheless be useful to understand and explain the physics at play. Reprinted with permission from P. Charbonneau, E. I. Corwin, L. Fu, G. Tsekenis, and M. van der Naald, Phys. Rev. E 99, 020901(R) (2019). Copyright 2019 American Physical Society.

FIG. 6.

Schematic illustration of a Gardner-like scenario for polydisperse crystals. A perfect crystal (purple particles) has only one well-separated densest packing. A crystal of polydisperse particles (colored from red to purple, according to their growing particle radius), by contrast, has a large number of nearly equivalent packings. Beyond the pressure onset of Gardner-like physics, at PG, such systems are forced into irreversible constraints. For the sake of illustration, consider the particle outlined in black. It is free to collide with all its nearest neighbors at low pressures, but forced to have one or another set of collisional contacts (green stars) as pressure increases. This overall process is cooperative and thus cannot be reduced to such a simple local description, but this illustration can nonetheless be useful to understand and explain the physics at play. Reprinted with permission from P. Charbonneau, E. I. Corwin, L. Fu, G. Tsekenis, and M. van der Naald, Phys. Rev. E 99, 020901(R) (2019). Copyright 2019 American Physical Society.

Close modal

Recent computer simulations of slightly polydisperse three-dimensional crystals130 have identified Gardner-like and critical jamming signatures that are functionally identical to those found in fully amorphous systems in d = 3: power-law scaling of forces near jamming, a growing relaxation time as the crystal basin breaks into smaller domains, followed by a rich aging behavior.

The simplicity of these systems and of their underlying crystal order helps clarify the physical interpretation of the Gardner physics more generally. In particular, we learn from polydisperse crystals of hard particles that the position of the Gardner-like transition is controlled by the ratio of the spread of the polydispersity, δ, to the typical interparticle spacing set by pressure, h ∼ 1/P. For P ≫ 1/δ, anomalous vibrational states are expected because the cage symmetry is then destroyed (Fig. 6). In amorphous hard sphere solids, by contrast, the position of the Gardner transition is known from the mean-field, d solution to be controlled by the distance of the equilibrium glass state from φd (see Fig. 3), but this relationship does not provide an intuitive underlying microscopic mechanism. The analogy with polydisperse crystals, however, suggests that the Gardner transition emerges in amorphous solids from the geometric heterogeneity inherent to self-induced caging. Because this heterogeneity (as probed, for instance, by the long-time value of the dynamical susceptibility, χ4) monotonically decreases as the distance of the equilibrium glass state from the dynamical transition increases, we thereby obtain a more intuitive understanding of the depression of the Gardner transition with increasing equilibrium glass density (see Fig. 3).

The study of polydisperse crystals promises to further illuminate Gardner physics. From a theoretical standpoint, no explanation for their Gardner-like behavior exists beyond the intuitive analogies presented above. Developing a framework that could explain the similarities and differences between fully amorphous solids and polydisperse crystals would likely provide additional insight into the microscopic processes at play. From a computational standpoint, the simplicity of polydisperse crystals makes them ideal for conducting finite-size scaling analysis of the Gardner criticality. Such studies could prove determinative in assessing whether a thermodynamic phase transition or a crossover is actually at play. A systematic consideration of the jamming criticality could similarly be undertaken. From an experimental standpoint, these systems offer great promise for validating Gardner physics in the laboratory. As we discussed in Sec. III, thermalizing colloidal or granular amorphous solids to the regime of interest to observe the Gardner transition is a very challenging experimental project. By contrast, crystals of hard grains can be manually prepared close to equilibrium, and colloidal crystallization of hard polydisperse particles is reasonably facile.133 Because both families of materials are commercially available with a sufficiently low polydispersity (typically, δ ∼ 1%–10%) to distinguish crystal formation from the Gardner phase, there should be no significant barriers to a thorough experimental test of this phenomenon. In fact, an excess of low-energy excitations around jamming was reported in simulations63 and in experiments134 of polydisperse crystals, before any proposal for a Gardner transition was even made. Revisiting these systems with a more fully developed Gardner physics in mind should therefore be reasonably productive. For instance, one could cyclically compress (without melting) crystals of agitated polydisperse, photoelastic disks and measure the change of the variability of the force network at jamming, depending on the decompression amplitude. Qualitatively different memory of previous force networks would be expected on either sides of the transition.

Gardner physics in mean-field theory is associated with the existence of soft, long-range excitations. One should thus expect that a full understanding of its criticality should be accessible from a renormalization group (RG) approach. Three different types of criticalities are then to be considered: (i) the jamming transition, (ii) the Gardner phase, and (iii) the Gardner transition. Because our understanding of the Gardner phase away from jamming is still fairly rudimentary, in this section, we only consider types (i) and (iii).

As we have seen in Sec. III, jamming criticality in finite d is extremely well described by the mean-field, d solution. In particular, all power-law scalings are essentially independent of d for d ≥ 2. This superuniversal behavior is unusual. Most critical phenomena exhibit a strong dimensional dependence at low d. A possible explanation for this phenomenon has been suggested in Refs. 135–137. By comparing the fluctuations of the contact number Z to its average—as is commonly done in critical phenomena for the order parameter in a correlated volume—one concludes that the upper critical dimension for jamming is du = 2. Adding fluctuations on top of mean-field results and developing an RG treatment that would confirm this interpretation without input from numerical simulations remain, however, an open challenge. A first-principle understanding of this remarkable phenomenon is therefore still lacking. Meanwhile, its lower critical dimension is still unexplored.

The Gardner transition in finite d is under better theoretical control. The mean-field description of the Gardner transition is indeed analogous to the de Almeida-Thouless transition observed for spin glasses in an external magnetic field.142 Both transitions display the same kind of order parameter and the same kind of collective behavior (or more precisely the same kind of instability of the disordered phase) and, hence, they share a same field theory of critical fluctuations.143 This theoretical relationship and the activity on the Gardner transition of the last few years have triggered several new RG investigations of the problem.

The modern starting point is the work of Moore and Bray144 (see also Ref. 145), which showed that the basin of attraction of the Gaussian fixed point (FP) associated with the mean-field description shrinks to zero when d approaches the upper critical dimension, du = 6, from above. This result confirmed and detailed the original 1980 work of Bray and Roberts146 that the Gaussian FP disappears below du and that no other perturbative FP emerges in d < du. For many decades, this disappearance was interpreted as a signature of the nonexistence of a critical transition in this context. However, that outcome is not the only possible one; a nonperturbative FP, and hence a critical transition, could instead replace it. This possibility has recently been investigated with different techniques: nonperturbative renormalization group,139 high-order resummations,147 and real space RG.138,140,141 Although the predictions of these different analyses differ quantitatively, they all show that a nonperturbative FP can actually exist in relatively low d.

These different RG scenarios are schematized in Fig. 7 for the flow parameters of a spin-glass model: the variance of the magnetic fields and of the magnetic couplings. (Although their counterparts for structural glasses give rise to an equivalent physics, they are somewhat less intuitive. They involve the difference in energy between different local particle arrangements, which leads to a generalized field and to elastic interactions related to these local strains.148) Figures 7(a) and 7(b) depict flows for nonperturbative FP with a zero-temperature138 and a finite-temperature139 FP, respectively. Figure 7(c) depicts the flow in the absence of a FP, and hence of a transition. Figures 7(a) and 7(b) correspond to a case in which a Gardner transition can take place but has different properties: a zero-temperature FP implies activated dynamical scaling,138 whereas a finite temperature one is associated with the usual power-law relationship between time and length scales.139Figure 7(c) instead corresponds to a case without a bona fide Gardner transition. An important issue to note is that even when a nonperturbative FP exists, the existence of a critical transition in the physical system at hand is not guaranteed. Its existence depends on whether the initial condition of the RG flow, i.e., the specific model considered, lies within the basin of attraction of the FP. As a result, although the properties of the critical transition are universal, its existence is not. For a given d, the transition could thus be present for amorphous solids but absent for certain spin-glass models, or present in finite-dimensional hard spheres but not in all soft sphere models. Probing the relationship between time and length scales is a way to ascertain in experiments and in simulations to which basin of attraction, and to which scenario [(a)–(c)], a given model belongs to. From a more theoretical point of view, one could try to map the model at hand into an effective theory, e.g., lattice spin models, that could then be studied more easily by RG and numerical simulations. Neither has yet been reported.

FIG. 7.

Schematics of possible renormalization flow scenarios for the variances of the coupling constants, σJ, and magnetic fields, σh, of a spin-glass model in d < du. Note that even if the RG scheme starts with a uniform external field, a random field emerges after a few iterations.138 The points SG, SGH, SGc, and SGHc denote the fixed points (FPs) associated with the spin-glass phase, the spin-glass phase in a field, the critical point for spin glasses without field, and the critical point for spin glasses in a field, respectively. (a) In this scenario, the transition in a field is present and its associated FP likes at T = 0, i.e., the couplings and the fields become much larger than the temperature under RG.138 (b) In this scenario, the transition in a field is present and the associated FP is standard, although nonperturbative.139 (c) In this scenario, there is no FP in a field and, hence, no transition.140,141

FIG. 7.

Schematics of possible renormalization flow scenarios for the variances of the coupling constants, σJ, and magnetic fields, σh, of a spin-glass model in d < du. Note that even if the RG scheme starts with a uniform external field, a random field emerges after a few iterations.138 The points SG, SGH, SGc, and SGHc denote the fixed points (FPs) associated with the spin-glass phase, the spin-glass phase in a field, the critical point for spin glasses without field, and the critical point for spin glasses in a field, respectively. (a) In this scenario, the transition in a field is present and its associated FP likes at T = 0, i.e., the couplings and the fields become much larger than the temperature under RG.138 (b) In this scenario, the transition in a field is present and the associated FP is standard, although nonperturbative.139 (c) In this scenario, there is no FP in a field and, hence, no transition.140,141

Close modal

The lower critical dimension of the transition, if present, is not known at this stage. There are, however, results for the spin-glass transition without a field. In this case, numerical simulations,149 real space renormalization group,138 and analysis of the low temperature phase150 have converged on the value dl ≃ 2.5. Assuming that adding an external field makes the transition more fragile—a reasonable but not fully motivated proposal—then one concludes that the lower critical dimension of the Gardner transition should be equal or higher than 2.5.

In short, the current understanding is that a Gardner transition could take place in d = 3 and be related to a nonperturbative FP. Future investigations using more refined RG treatments, combined with new numerical simulations, will hopefully help close this long and intricate line of inquiry.

In an interesting intellectual twist, the Gardner transition, which emerged from the study of spin glass models on fully connected (all sites interacting with all others) graphs, re-emerged in the study of spin glass models on diluted (i.e., locally tree-like) graphs, in parallel with its consideration in structural glasses. Theoretical interest in diluted spin glass models picked up speed in the early 2000s, as interest for random versions of constraint satisfaction problems (CSP), to which they are intimately are related,152,153 boomed. In the most generic terms, CSP consist of finding assignments of variable that are subject to a set of constraints. Examples include graph coloring (Figs. 8 and 9), constrained optimization, supervised learning, packing of hard objects, etc. CSP also provide the building blocks of computational complexity theory.152,153

FIG. 8.

Sample phase diagram of the q-coloring CSP on a random regular graph with q = 4, as a function of graph connectivity c = 2α and temperature T. The zero-temperature limit of this model illustrates the various types of instabilities discussed in the text. At low connectivity, c < cd [and more generally above the line Td(c)], the space of solutions is ergodic. At cd, a dynamical transition gives rise to a clustered 1RSB phase. At cK [and more generally at TK(c)], a Kauzmann transition gives rise to condensed clusters of solutions. Beyond the coloring-uncoloring (SAT-UNSAT) transition, cq, the zero-temperature energy becomes positive, hence not all constraints can typically be satisfied (red line, analogous to the jamming line in Fig. 3). Finally, at cG [and more generally at TG(c)], a Gardner transition gives rise to a Gardner phase, within which the typical states are marginally stable. Adapted from F. Krząkała and L. Zdeborová, Europhys. Lett. 81, 57005 (2008). Copyright 2008 IOP Publishing.

FIG. 8.

Sample phase diagram of the q-coloring CSP on a random regular graph with q = 4, as a function of graph connectivity c = 2α and temperature T. The zero-temperature limit of this model illustrates the various types of instabilities discussed in the text. At low connectivity, c < cd [and more generally above the line Td(c)], the space of solutions is ergodic. At cd, a dynamical transition gives rise to a clustered 1RSB phase. At cK [and more generally at TK(c)], a Kauzmann transition gives rise to condensed clusters of solutions. Beyond the coloring-uncoloring (SAT-UNSAT) transition, cq, the zero-temperature energy becomes positive, hence not all constraints can typically be satisfied (red line, analogous to the jamming line in Fig. 3). Finally, at cG [and more generally at TG(c)], a Gardner transition gives rise to a Gardner phase, within which the typical states are marginally stable. Adapted from F. Krząkała and L. Zdeborová, Europhys. Lett. 81, 57005 (2008). Copyright 2008 IOP Publishing.

Close modal

While complexity theory is only concerned with the worst cases of a problem, interest in more representative typical cases gave birth to a natural synergy between computer science and statistical physics. The tools developed for disordered systems were found to be especially useful to answer questions, such as If we draw a CSP randomly from an ensemble of problems what is its typical complexity? Does it admit a solution? What is the structure of the space of solutions? This association emerges from the fact that random CSP problems can be formulated as spin-glass models with a Hamiltonian function being zero for satisfying (SAT) assignments and quantifying constraint violation for unsatisfying (UNSAT) ones. The key parameter is then the ratio of the number of constraints to the number of variables, α. Generically, problems are SAT at small α and UNSAT at large α, with a sharp transition between the two regimes in the thermodynamic limit of many constraints and variables.152,153 As a further twist, the statistical physics of random CSP was also pioneered by Gardner, in her analysis of the single-layer neural network (perceptron).154,155

The set of SAT assignment of the spherical perceptron, if not empty, is convex and hence connected, glassy phases are not possible and studying the SAT-UNSAT transition does not require RSB.154,155 For other prototypical CSP, such as the coloring problem, however, the space of SAT assignments is not necessarily connected and only the implementation of 1RSB-based techniques156 enabled the exact computation of the SAT-UNSAT threshold.157–159 In fact, many dilute discrete random CSP belong to the class of discontinuous (REM-like) spin glasses. The route from the SAT to the UNSAT phase as α increases is thus marked by several phase transitions, roughly following the RFOT scenario:160 at low α, the space of solutions (SAT assignments) is linked by small changes in variable assignments, i.e., ergodic; for α > αd, this space first undergoes a dynamical (clustering) transition at which solutions become organized in a large number of ergodic components but separated from one another by changes to an extensive number of variable assignments; for α > αK, the space of solutions is composed of a small number of clusters, and the SAT-UNSAT transition is reached at α > αq.

The 1RSB ansatz used to analyze these models describes families of statistically identical states that lie at the same mutual distance. As in the case of spin glasses described in Sec. II, such a description can become unstable toward a more complex organization of states, both discussed in Refs. 159 and 161. In particular, a Gardner transition has been found in the space of solutions of a specific model of coloring of a random graph.59 

Using insights from the study of structural glasses,162 it was also realized that the SAT-UNSAT transition of CSP with continuous variables (cCSP) may also naturally display a critical scaling. Because continuous variables make the space of solutions continuous, the volume of that space can smoothly shrink to zero at the SAT-UNSAT transition, as it does at the jamming transition (see, e.g., Fig. 9). The analogy between cCSP and jamming was first noticed in the MK model of spheres63 (discussed in Sec. III). Although it was noted at the time that low-frequency excitations around jammed minima63 are very similar to those observed near jamming in more realistic models,28–31,33 this hint of a possible relation between jamming criticality and the Gardner phase was overlooked.

FIG. 9.

Illustration of the difference between continuous and discrete variables in the graph coloring CSP. In the standard, discrete setting, q colors are available to color the vertices of a given graph such that no two adjacent vertices have the same color. (In the language of statistical physics, such a solution is the ground state of a Potts antiferromagnet.) In the continuous setting, colors are continuous variables on a circle and can therefore be defined by an angle θ. The constraint is that the distance between colors on contiguous vertexes should be larger than a given threshold. For any finite graph, in the discrete problem, the set of solutions is discrete and numerable. In the continuous setting, this set is continuous, and its volume goes to zero at a jamming transition. Adapted from H. Yoshino, SciPost Phys. 4, 040 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License.

FIG. 9.

Illustration of the difference between continuous and discrete variables in the graph coloring CSP. In the standard, discrete setting, q colors are available to color the vertices of a given graph such that no two adjacent vertices have the same color. (In the language of statistical physics, such a solution is the ground state of a Potts antiferromagnet.) In the continuous setting, colors are continuous variables on a circle and can therefore be defined by an angle θ. The constraint is that the distance between colors on contiguous vertexes should be larger than a given threshold. For any finite graph, in the discrete problem, the set of solutions is discrete and numerable. In the continuous setting, this set is continuous, and its volume goes to zero at a jamming transition. Adapted from H. Yoshino, SciPost Phys. 4, 040 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution 4.0 License.

Close modal

It is rather in a nonconvex generalization of Gardner’s perceptron problem that the relation between jamming and the Gardner phase in cCSP was first fully developed.162 Remarkably, isostatic jamming in this model is found in the Gardner phase and displays the same critical properties as jamming in hard spheres. It has thus been conjectured that a unique universality class encapsulates all of critical jamming. A variety of models, including the high-dimensional vector spin models with excluded volume interaction,151 feed-forward neural networks with a single hidden layer and fixed hidden-to-output weights,163 and deeper neural networks,164 further support this claim. A more complete theory of critical jamming is, however, sorely needed. The study of jamming on diluted random cCSP, which has not yet been tackled, might also provide insights into the role of finite-dimensional effects.

The last few years have seen an explosion of research motivated by the discovery of a Gardner transition in the mean-field, d solution of hard spheres. Although the initial hope of having a single transition unify the description of low-temperature glasses has been tempered, the insights it has provided are of no less significance. As mentioned in this perspective, a number of questions about the finite-dimensional Gardner transition remain actively pursued in model glass formers, in cCSP, and in RG. Of these, we are especially excited by the experimental tests made possible by recent theoretical and computational advances, and by the theoretical puzzle of the superuniversal jamming criticality. In addition, recent attempts at describing the jamming transition of slightly aspherical particles,165 which are surprisingly a lot more complex than simple spherical glass formers, suggest that the accompanying Gardner physics might be richer still.

On the 30th anniversary of the untimely passing of Elizabeth Gardner, it is a sort of intellectual justice that her deep physical insights should still carry statistical mechanics forward.

Results discussed in this perspective article have been obtained within the Simons Collaboration “Cracking the Glass Problem,” in a large collaborative research effort involving several PIs other than the authors (J. Kurchan, A. Liu, S. Nagel, G. Parisi, and M. Wyart) as well as many affiliates, postdocs, and Ph.D. students (A. Altieri, C. Brito, E. DeGiuli, D. Hexner, S. Hwang, H. Ikeda, J. Kundu, E. Lerner, Q. Liao, M. Müller, M. Ozawa, T. Rizzo, C. Scalliet, B. Seoane, J. P. Sethna, S. Spigler, G. Tsekenis, P. Urbani, S. Yaida, and H. Yoshino). We thank all of them as well as all the other colleagues with whom we have had the pleasure to carry significant parts of the work described above, in particular, M. C. Angelini, K. Daniels, O. Dauchot, Y. Jin, F. Ladieu, and C. Rainone. We further acknowledge funding from the Simons Foundation (Grant Nos. 454933 L.B., 454935 G.B., 454937 P.C., 454939 E.I.C., 454941 S.F., and 454955 F.Z.) and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 723955—GlassUniversality). The writing of this publication was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. S.F. is a member of the Institut Universitaire de France.

1.
P.
Charbonneau
,
J.
Kurchan
,
G.
Parisi
,
P.
Urbani
, and
F.
Zamponi
, “
Glass and jamming transitions: From exact results to finite-dimensional descriptions
,”
Annu. Rev. Condens. Matter Phys.
8
,
265
288
(
2017
).
2.
Y.
Jin
,
P.
Urbani
,
F.
Zamponi
, and
H.
Yoshino
, “
A stability-reversibility map unifies elasticity, plasticity, yielding and jamming in hard sphere glasses
,”
Sci. Adv.
4
,
eaat6387
(
2018
).
3.
D.
Sherrington
and
S.
Kirkpatrick
, “
Solvable model of a spin-glass
,”
Phys. Rev. Lett.
35
,
1792
(
1975
).
4.
G.
Parisi
, “
Infinite number of order parameters for spin-glasses
,”
Phys. Rev. Lett.
43
,
1754
(
1979
).
5.
M.
Mézard
,
G.
Parisi
, and
M. A.
Virasoro
,
Spin Glass Theory and Beyond
(
World Scientific
,
1987
).
6.
L. F.
Cugliandolo
and
J.
Kurchan
, “
On the out-of-equilibrium relaxation of the Sherrington-Kirkpatrick model
,”
J. Phys. A: Math. Gen.
27
,
5749
(
1994
).
7.
M.
Talagrand
,
Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models
(
Springer
,
2003
).
8.
D.
Panchenko
,
The Sherrington-Kirkpatrick Model
(
Springer Science & Business Media
,
2013
).
9.
B.
Derrida
, “
Random-energy model: An exactly solvable model of disordered systems
,”
Phys. Rev. B
24
,
2613
(
1981
).
10.
G.
Toulouse
and
M.
Gabay
, “
Mean field theory for Heisenberg spin glasses
,”
J. Phys. Lett.
42
,
103
106
(
1981
).
11.
D. J.
Gross
,
I.
Kanter
, and
H.
Sompolinsky
, “
Mean-field theory of the Potts glass
,”
Phys. Rev. Lett.
55
,
304
307
(
1985
).
12.
B.
Derrida
, “
A generalization of the random energy model which includes correlations between energies
,”
J. Phys. Lett.
46
,
401
407
(
1985
).
13.
E.
Gardner
, “
Spin glasses with p-spin interactions
,”
Nucl. Phys. B
257
,
747
765
(
1985
).
14.
T. R.
Kirkpatrick
,
D.
Thirumalai
, and
P. G.
Wolynes
, “
Scaling concepts for the dynamics of viscous liquids near an ideal glassy state
,”
Phys. Rev. A
40
,
1045
1054
(
1989
).
15.
W.
Götze
, “
Recent tests of the mode-coupling theory for glassy dynamics
,”
J. Phys.: Condens. Matter
11
,
A1
A45
(
1999
).
16.
W.
Götze
,
Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory
(
Oxford University Press
,
2008
).
17.
T. R.
Kirkpatrick
and
D.
Thirumalai
, “
Dynamics of the structural glass transition and the p-spin-interaction spin-glass model
,”
Phys. Rev. Lett.
58
,
2091
2094
(
1987
).
18.
L.
Berthier
and
G.
Biroli
, “
Theoretical perspective on the glass transition and amorphous materials
,”
Rev. Mod. Phys.
83
,
587
645
(
2011
).
19.
L.
Berthier
,
M.
Ozawa
, and
C.
Scalliet
, “
Configurational entropy of glass-forming liquids
,”
J. Chem. Phys.
150
,
160902
(
2019
).
20.
W.
Kauzmann
, “
The glassy state and the behaviour of liquids at low temperature
,”
Chem. Rev.
43
,
219
256
(
1948
).
21.
V.
Lubchenko
and
P. G.
Wolynes
, “
Theory of structural glasses and supercooled liquids
,”
Annu. Rev. Phys. Chem.
58
,
235
266
(
2007
); e-print arXiv:cond-mat/0607349.
22.
Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications
, edited by
P.
Wolynes
and
V.
Lubchenko
(
Wiley & Sons
,
2012
).
23.
A.
Barrat
,
S.
Franz
, and
G.
Parisi
, “
Temperature evolution and bifurcations of metastable states in mean-field spin glasses, with connections with structural glasses
,”
J. Phys. A: Math. Gen.
30
,
5593
5612
(
1997
).
24.
A.
Montanari
and
F.
Ricci-Tersenghi
, “
On the nature of the low-temperature phase in discontinuous mean-field spin glasses
,”
Eur. Phys. J. B
33
,
339
346
(
2003
).
25.
T. R.
Kirkpatrick
and
P. G.
Wolynes
, “
Connections between some kinetic and equilibrium theories of the glass transition
,”
Phys. Rev. A
35
,
3072
3080
(
1987
).
26.
T. R.
Kirkpatrick
and
P. G.
Wolynes
, “
Stable and metastable states in mean-field Potts and structural glasses
,”
Phys. Rev. B
36
,
8552
8564
(
1987
).
27.
A. J.
Liu
and
S. R.
Nagel
, “
Nonlinear dynamics: Jamming is not just cool any more
,”
Nature
396
,
21
22
(
1998
).
28.
C. S.
O’Hern
,
S. A.
Langer
,
A. J.
Liu
, and
S. R.
Nagel
, “
Random packings of frictionless particles
,”
Phys. Rev. Lett.
88
,
075507
(
2002
).
29.
C. S.
O’Hern
,
L. E.
Silbert
,
A. J.
Liu
, and
S. R.
Nagel
, “
Jamming at zero temperature and zero applied stress: The epitome of disorder
,”
Phys. Rev. E
68
,
011306
(
2003
).
30.
M.
Wyart
,
S.
Nagel
, and
T.
Witten
, “
Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids
,”
Europhys. Lett.
72
,
486
492
(
2005
).
31.
M.
Wyart
,
L.
Silbert
,
S.
Nagel
, and
T.
Witten
, “
Effects of compression on the vibrational modes of marginally jammed solids
,”
Phys. Rev. E
72
,
051306
(
2005
).
32.
C.
Brito
and
M.
Wyart
, “
On the rigidity of a hard-sphere glass near random close packing
,”
Europhys. Lett.
76
,
149
155
(
2006
).
33.
C.
Brito
and
M.
Wyart
, “
Geometric interpretation of previtrification in hard sphere liquids
,”
J. Chem. Phys.
131
,
024504
(
2009
).
34.
A. J.
Liu
and
S. R.
Nagel
, “
The jamming transition and the marginally jammed solid
,”
Annu. Rev. Condens. Matter Phys.
1
,
347
369
(
2010
).
35.
A.
Liu
,
S.
Nagel
,
W.
Van Saarloos
, and
M.
Wyart
, “
The jamming scenario—An introduction and outlook
,” in
Dynamical Heterogeneities and Glasses
, edited by
L.
Berthier
,
G.
Biroli
,
J.-P.
Bouchaud
,
L.
Cipelletti
, and
W.
van Saarloos
(
Oxford University Press
,
2011
).
36.
M.
Wyart
, “
Marginal stability constrains force and pair distributions at random close packing
,”
Phys. Rev. Lett.
109
,
125502
(
2012
).
37.
E.
Lerner
,
G.
During
, and
M.
Wyart
, “
Low-energy non-linear excitations in sphere packings
,”
Soft Matter
9
,
8252
8263
(
2013
).
38.
M.
Müller
and
M.
Wyart
, “
Marginal stability in structural, spin, and electron glasses
,”
Annu. Rev. Condens. Matter Phys.
6
,
177
(
2015
).
39.
M.
Skoge
,
A.
Donev
,
F. H.
Stillinger
, and
S.
Torquato
, “
Packing hyperspheres in high-dimensional Euclidean spaces
,”
Phys. Rev. E
74
,
041127
(
2006
).
40.
J. A.
van Meel
,
B.
Charbonneau
,
A.
Fortini
, and
P.
Charbonneau
, “
Hard-sphere crystallization gets rarer with increasing dimension
,”
Phys. Rev. E
80
,
061110
(
2009
).
41.
P.
Charbonneau
,
A.
Ikeda
,
J. A.
van Meel
, and
K.
Miyazaki
, “
Numerical and theoretical study of a monodisperse hard-sphere glass former
,”
Phys. Rev. E
81
,
040501
(
2010
).
42.
P.
Charbonneau
,
A.
Ikeda
,
G.
Parisi
, and
F.
Zamponi
, “
Glass transition and random close packing above three dimensions
,”
Phys. Rev. Lett.
107
,
185702
(
2011
).
43.
P.
Charbonneau
,
E. I.
Corwin
,
G.
Parisi
, and
F.
Zamponi
, “
Universal microstructure and mechanical stability of jammed packings
,”
Phys. Rev. Lett.
109
,
205501
(
2012
).
44.
P.
Charbonneau
,
E. I.
Corwin
,
G.
Parisi
, and
F.
Zamponi
, “
Jamming criticality revealed by removing localized buckling excitations
,”
Phys. Rev. Lett.
114
,
125504
(
2015
).
45.
G.
Parisi
and
F.
Zamponi
, “
The ideal glass transition of hard spheres
,”
J. Chem. Phys.
123
,
144501
(
2005
).
46.
G.
Parisi
and
F.
Zamponi
, “
Mean-field theory of hard sphere glasses and jamming
,”
Rev. Mod. Phys.
82
,
789
845
(
2010
).
47.
H.
Jacquin
,
L.
Berthier
, and
F.
Zamponi
, “
Microscopic mean-field theory of the jamming transition
,”
Phys. Rev. Lett.
106
,
135702
(
2011
).
48.
J.
Kurchan
,
G.
Parisi
, and
F.
Zamponi
, “
Exact theory of dense amorphous hard spheres in high dimension. I. The free energy
,”
J. Stat. Mech.
2012
,
P10012
.
49.
J.
Kurchan
,
G.
Parisi
,
P.
Urbani
, and
F.
Zamponi
, “
Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition
,”
J. Phys. Chem. B
117
,
12979
12994
(
2013
).
50.
P.
Charbonneau
,
J.
Kurchan
,
G.
Parisi
,
P.
Urbani
, and
F.
Zamponi
, “
Fractal free energies in structural glasses
,”
Nat. Commun.
5
,
3725
(
2014
).
51.
P.
Charbonneau
,
J.
Kurchan
,
G.
Parisi
,
P.
Urbani
, and
F.
Zamponi
, “
Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution
,”
J. Stat. Mech.
2014
,
P10009
.
52.
C.
Rainone
,
P.
Urbani
,
H.
Yoshino
, and
F.
Zamponi
, “
Following the evolution of hard sphere glasses in infinite dimensions under external perturbations: Compression and shear strain
,”
Phys. Rev. Lett.
114
,
015701
(
2015
).
53.
H.
Yoshino
and
F.
Zamponi
, “
Shear modulus of glasses: Results from the full replica-symmetry-breaking solution
,”
Phys. Rev. E
90
,
022302
(
2014
).
54.
C.
Rainone
and
P.
Urbani
, “
Following the evolution of glassy states under external perturbations: The full replica symmetry breaking solution
,”
J. Stat. Mech.
2016
,
053302
.
55.

An extension of the mean-field phase diagram shown in Fig. 3 to account for finite-dimensional fluctuations and activated processes was also discussed in Ref. 166.

56.
E.
DeGiuli
,
E.
Lerner
,
C.
Brito
, and
M.
Wyart
, “
Force distribution affects vibrational properties in hard-sphere glasses
,”
Proc. Natl. Acad. Sci. U. S. A.
111
,
17054
17059
(
2014
).
57.
E.
DeGiuli
,
E.
Lerner
, and
M.
Wyart
, “
Theory of the jamming transition at finite temperature
,”
J. Chem. Phys.
142
,
164503
(
2015
).
58.
G.
Biroli
and
P.
Urbani
, “
Breakdown of elasticity in amorphous solids
,”
Nat. Phys.
12
,
1130
1133
(
2016
).
59.
F.
Krzakała
and
L.
Zdeborová
, “
Potts glass on random graphs
,”
Europhys. Lett.
81
,
57005
(
2008
).
60.
F.
Krzakała
and
L.
Zdeborová
, “
Following Gibbs states adiabatically–the energy landscape of mean-field glassy systems
,”
Europhys. Lett.
90
,
66002
(
2010
).
61.
Y.
Sun
,
A.
Crisanti
,
F.
Krzakała
,
L.
Leuzzi
, and
L.
Zdeborová
, “
Following states in temperature in the spherical s + p-spin glass model
,”
J. Stat. Mech.
2012
,
P07002
.
62.
T.
Rizzo
, “
Replica-symmetry-breaking transitions and off-equilibrium dynamics
,”
Phys. Rev. E
88
,
032135
(
2013
).
63.
R.
Mari
,
F.
Krzakała
, and
J.
Kurchan
, “
Jamming versus glass transitions
,”
Phys. Rev. Lett.
103
,
025701
(
2009
).
64.
R. H.
Kraichnan
, “
Stochastic models for many-body systems. i. infinite systems in thermal equilibrium
,”
J. Math. Phys.
3
,
475
495
(
1962
).
65.
R.
Mari
and
J.
Kurchan
, “
Dynamical transition of glasses: From exact to approximate
,”
J. Chem. Phys.
135
,
124504
(
2011
).
66.
M.
Mézard
,
G.
Parisi
,
M.
Tarzia
, and
F.
Zamponi
, “
On the solution of a ‘solvable’ model of an ideal glass of hard spheres displaying a jamming transition
,”
J. Stat. Mech.
2011
,
P03002
.
67.
P.
Charbonneau
,
Y.
Jin
,
G.
Parisi
, and
F.
Zamponi
, “
Hopping and the Stokes–Einstein relation breakdown in simple glass formers
,”
Proc. Natl. Acad. Sci. U. S. A.
111
,
15025
15030
(
2014
).
68.
F.
Krzakała
and
L.
Zdeborová
, “
Hiding quiet solutions in random constraint satisfaction problems
,”
Phys. Rev. Lett.
102
,
238701
(
2009
).
69.
P.
Charbonneau
,
Y.
Jin
,
G.
Parisi
,
C.
Rainone
,
B.
Seoane
, and
F.
Zamponi
, “
Numerical detection of the Gardner transition in a mean-field glass former
,”
Phys. Rev. E
92
,
012316
(
2015
).
70.
L.
Berthier
,
D.
Coslovich
,
A.
Ninarello
, and
M.
Ozawa
, “
Equilibrium sampling of hard spheres up to the jamming density and beyond
,”
Phys. Rev. Lett.
116
,
238002
(
2016
).
71.
A.
Ninarello
,
L.
Berthier
, and
D.
Coslovich
, “
Models and algorithms for the next generation of glass transition studies
,”
Phys. Rev. X
7
,
021039
(
2017
).
72.
L.
Berthier
,
P.
Charbonneau
,
Y.
Jin
,
G.
Parisi
,
B.
Seoane
, and
F.
Zamponi
, “
Growing timescales and lengthscales characterizing vibrations of amorphous solids
,”
Proc. Natl. Acad. Sci. U. S. A.
113
,
8397
8401
(
2016
).
73.
L.
Leuzzi
,
G.
Parisi
,
F.
Ricci-Tersenghi
, and
J. J.
Ruiz-Lorenzo
, “
Ising spin-glass transition in a magnetic field outside the limit of validity of mean-field theory
,”
Phys. Rev. Lett.
103
,
267201
(
2009
).
74.
R. A.
Banos
,
A.
Cruz
,
L. A.
Fernandez
,
J. M.
Gil-Narvion
,
A.
Gordillo-Guerrero
,
M.
Guidetti
,
D.
Iniguez
,
A.
Maiorano
,
E.
Marinari
,
V.
Martin-Mayor
,
J.
Monforte-Garcia
,
A.
Munoz Sudupe
,
D.
Navarro
,
G.
Parisi
,
S.
Perez-Gaviro
,
J. J.
Ruiz-Lorenzo
,
S. F.
Schifano
,
B.
Seoane
,
A.
Tarancon
,
P.
Tellez
,
R.
Tripiccione
, and
D.
Yllanes
, “
Thermodynamic glass transition in a spin glass without time-reversal symmetry
,”
Proc. Natl. Acad. Sci. U. S. A.
109
,
6452
6456
(
2012
).
75.
L.
Berthier
,
P.
Charbonneau
, and
J.
Kundu
, “
Bypassing sluggishness: Swap algorithm and glassiness in high dimensions
,”
Phys. Rev. E
99
,
031301
(
2019
).
76.
E.
Vincent
,
J.
Hammann
,
M.
Ocio
,
J.-P.
Bouchaud
, and
L. F.
Cugliandolo
, “
Slow dynamics and aging in spin glasses
,” in
Complex Behaviour of Glassy Systems
(
Springer
,
1997
), pp.
184
219
.
77.
B.
Seoane
and
F.
Zamponi
, “
Spin-glass-like aging in colloidal and granular glasses
,”
Soft Matter
14
,
5222
5234
(
2018
).
78.
Q.
Liao
and
L.
Berthier
, “
Hierarchical landscape of hard disk glasses
,”
Phys. Rev. X
9
,
011049
(
2019
).
79.
T.
Maimbourg
,
J.
Kurchan
, and
F.
Zamponi
, “
Solution of the dynamics of liquids in the large-dimensional limit
,”
Phys. Rev. Lett.
116
,
015902
(
2016
).
80.
E.
Agoritsas
,
T.
Maimbourg
, and
F.
Zamponi
, “
Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. I. The isotropic case
,”
J. Phys. A: Math. Theor.
52
,
144002
(
2019
).
81.
R. C.
Dennis
and
E. I.
Corwin
, “
The jamming energy landscape is hierarchical and ultrametric
” (unpublished).
82.
E.
DeGiuli
,
A.
Laversanne-Finot
,
G.
Düring
,
E.
Lerner
, and
M.
Wyart
, “
Effects of coordination and pressure on sound attenuation, boson peak and elasticity in amorphous solids
,”
Soft Matter
10
,
5628
5644
(
2014
).
83.
S.
Franz
,
G.
Parisi
,
P.
Urbani
, and
F.
Zamponi
, “
Universal spectrum of normal modes in low-temperature glasses
,”
Proc. Natl. Acad. Sci. U. S. A.
112
,
14539
14544
(
2015
).
84.
P.
Charbonneau
,
E. I.
Corwin
,
G.
Parisi
,
A.
Poncet
, and
F.
Zamponi
, “
Universal non-Debye scaling in the density of states of amorphous solids
,”
Phys. Rev. Lett.
117
,
045503
(
2016
).
85.
H.
Mizuno
,
H.
Shiba
, and
A.
Ikeda
, “
Continuum limit of the vibrational properties of amorphous solids
,”
Proc. Natl. Acad. Sci. U. S. A.
114
,
E9767
(
2017
).
86.
A.
Seguin
and
O.
Dauchot
, “
Experimental evidence of the Gardner phase in a granular glass
,”
Phys. Rev. Lett.
117
,
228001
(
2016
).
87.
C. S.
O’Hern
, “
Viewpoint: Signs of a Gardner transition in a granular glass
,”
Physics
9
,
133
(
2016
).
88.
A. P.
Hammond
and
E. I.
Corwin
, “
Direct observation of the Gardner/marginal glass transition within a colloidal glass
” (unpublished).
89.
R.
Zargar
,
E.
DeGiuli
, and
D.
Bonn
, “
Scaling for hard-sphere colloidal glasses near jamming
,”
Europhys. Lett.
116
,
68004
(
2017
).
90.
G.
Biroli
and
P.
Urbani
, “
Liu-Nagel phase diagrams in infinite dimension
,”
SciPost Phys.
4
,
020
(
2018
).
91.
C.
Scalliet
,
L.
Berthier
, and
F.
Zamponi
, “
Marginally stable phases in mean-field structural glasses
,”
Phys. Rev. E
99
,
012107
(
2019
).
92.
L.
Berthier
and
T. A.
Witten
, “
Glass transition of dense fluids of hard and compressible spheres
,”
Phys. Rev. E
80
,
021502
(
2009
).
93.
C.
Scalliet
and
L.
Berthier
, “
Rejuvenation and memory effects in a structural glass
,” e-print arXiv:1902.08082 (
2019
).
94.
C.
Scalliet
,
L.
Berthier
, and
F.
Zamponi
, “
Absence of marginal stability in a structural glass
,”
Phys. Rev. Lett.
119
,
205501
(
2017
).
95.
B.
Seoane
,
D. R.
Reid
,
J. J.
de Pablo
, and
F.
Zamponi
, “
Low-temperature anomalies of a vapor deposited glass
,”
Phys. Rev. Mater.
2
,
015602
(
2018
).
96.
M.
Goldstein
, “
Viscous liquids and the glass transition: A potential energy barrier picture
,”
J. Chem. Phys.
51
,
3728
3739
(
1969
).
97.
F. H.
Stillinger
and
T. A.
Weber
, “
Hidden structure in liquids
,”
Phys. Rev. A
25
,
978
989
(
1982
).
98.
D.
Wales
,
Energy Landscapes: Applications to Clusters, Biomolecules and Glasses
(
Cambridge University Press
,
2003
).
99.
A.
Heuer
, “
Exploring the potential energy landscape of glass-forming systems: From inherent structures via metabasins to macroscopic transport
,”
J. Phys.: Condens. Matter
20
,
373101
(
2008
).
100.
V.
Lubchenko
and
P. G.
Wolynes
, “
The origin of the boson peak and thermal conductivity plateau in low-temperature glasses
,”
Proc. Natl. Acad. Sci. U. S. A.
100
,
1515
1518
(
2003
).
101.
E.
Lerner
,
G.
Düring
, and
E.
Bouchbinder
, “
Statistics and properties of low-frequency vibrational modes in structural glasses
,”
Phys. Rev. Lett.
117
,
035501
(
2016
).
102.
G.
Kapteijns
,
E.
Bouchbinder
, and
E.
Lerner
, “
Universal nonphononic density of states in 2d, 3d, and 4d glasses
,”
Phys. Rev. Lett.
121
,
055501
(
2018
).
103.
L.
Wang
,
A.
Ninarello
,
P.
Guan
,
L.
Berthier
,
G.
Szamel
, and
E.
Flenner
, “
Low-frequency vibrational modes of stable glasses
,”
Nat. Commun.
10
,
26
(
2019
).
104.
M.
Shimada
,
H.
Mizuno
,
M.
Wyart
, and
A.
Ikeda
, “
Spatial structure of quasilocalized vibrations in nearly jammed amorphous solids
,”
Phys. Rev. E
98
,
060901
(
2018
).
105.
P. W.
Anderson
,
B.
Halperin
, and
C. M.
Varma
, “
Anomalous low-temperature thermal properties of glasses and spin glasses
,”
Philos. Mag.
25
,
1
9
(
1972
).
106.
W. A.
Phillips
, “
Two-level states in glasses
,”
Rep. Prog. Phys.
50
,
1657
(
1987
).
107.
K.
Geirhos
,
P.
Lunkenheimer
, and
A.
Loidl
, “
Johari-goldstein relaxation far below Tg: Experimental evidence for the Gardner transition in structural glasses?
,”
Phys. Rev. Lett.
120
,
085705
(
2018
).
108.
H.
Yoshino
and
M.
Mézard
, “
Emergence of rigidity at the structural glass transition: A first-principles computation
,”
Phys. Rev. Lett.
105
,
015504
(
2010
).
109.
H.
Yoshino
, “
Replica theory of the rigidity of structural glasses
,”
J. Chem. Phys.
136
,
214108
(
2012
).
110.
P.
Urbani
and
F.
Zamponi
, “
Shear yielding and shear jamming of dense hard sphere glasses
,”
Phys. Rev. Lett.
118
,
038001
(
2017
).
111.
Y.
Jin
and
H.
Yoshino
, “
Exploring the complex free-energy landscape of the simplest glass by rheology
,”
Nat. Commun.
8
,
14935
(
2017
).
112.
C. E.
Maloney
and
A.
Lemaître
, “
Amorphous systems in athermal, quasistatic shear
,”
Phys. Rev. E
74
,
016118
(
2006
).
113.
M.
Tsamados
,
A.
Tanguy
,
C.
Goldenberg
, and
J.-L.
Barrat
, “
Local elasticity map and plasticity in a model Lennard-Jones glass
,”
Phys. Rev. E
80
,
026112
(
2009
).
114.
D.
Rodney
,
A.
Tanguy
, and
D.
Vandembroucq
, “
Modeling the mechanics of amorphous solids at different length scale and time scale
,”
Modell. Simul. Mater. Sci. Eng.
19
,
083001
(
2011
).
115.
M. L.
Manning
and
A. J.
Liu
, “
Vibrational modes identify soft spots in a sheared disordered packing
,”
Phys. Rev. Lett.
107
,
108302
(
2011
).
116.
H.
Hentschel
,
S.
Karmakar
,
E.
Lerner
, and
I.
Procaccia
, “
Do athermal amorphous solids exist?
,”
Phys. Rev. E
83
,
061101
(
2011
).
117.
A.
Zaccone
and
E.
Scossa-Romano
, “
Approximate analytical description of the nonaffine response of amorphous solids
,”
Phys. Rev. B
83
,
184205
(
2011
).
118.
J.
Lin
,
E.
Lerner
,
A.
Rosso
, and
M.
Wyart
, “
Scaling description of the yielding transition in soft amorphous solids at zero temperature
,”
Proc. Natl. Acad. Sci. U. S. A.
111
,
14382
14387
(
2014
).
119.
S. S.
Schoenholz
,
A. J.
Liu
,
R. A.
Riggleman
, and
J.
Rottler
, “
Understanding plastic deformation in thermal glasses from single-soft-spot dynamics
,”
Phys. Rev. X
4
,
031014
(
2014
).
120.
S.
Patinet
,
D.
Vandembroucq
, and
M. L.
Falk
, “
Connecting local yield stresses with plastic activity in amorphous solids
,”
Phys. Rev. Lett.
117
,
045501
(
2016
).
121.
H. J.
Hwang
,
R. A.
Riggleman
, and
J. C.
Crocker
, “
Understanding soft glassy materials using an energy landscape approach
,”
Nat. Mater.
15
,
1031
(
2016
).
122.
Z.
Schwartzman-Nowik
,
E.
Lerner
, and
E.
Bouchbinder
, “
Anisotropic structural predictor in glassy materials
,” e-print arXiv:1901.05202 (
2019
).
123.
G.
Parisi
,
I.
Procaccia
,
C.
Rainone
, and
M.
Singh
, “
Shear bands as manifestation of a criticality in yielding amorphous solids
,”
Proc. Natl. Acad. Sci. U. S. A.
114
,
5577
5582
(
2017
).
124.
M.
Ozawa
,
L.
Berthier
,
G.
Biroli
,
A.
Rosso
, and
G.
Tarjus
, “
Random critical point separates brittle and ductile yielding transitions in amorphous materials
,”
Proc. Natl. Acad. Sci. U. S. A.
115
,
6656
(
2018
).
125.
D.
Nakayama
,
H.
Yoshino
, and
F.
Zamponi
, “
Protocol-dependent shear modulus of amorphous solids
,”
J. Stat. Mech.
2016
,
104001
.
126.
M.
Otsuki
and
H.
Hayakawa
, “
Avalanche contribution to shear modulus of granular materials
,”
Phys. Rev. E
90
,
042202
(
2014
).
127.
C.
Coulais
,
A.
Seguin
, and
O.
Dauchot
, “
Shear modulus and dilatancy softening in granular packings above jamming
,”
Phys. Rev. Lett.
113
,
198001
(
2014
).
128.
P.
Le Doussal
,
M.
Müller
, and
K. J.
Wiese
, “
Avalanches in mean-field models and the Barkhausen noise in spin-glasses
,”
Europhys. Lett.
91
,
57004
(
2010
).
129.
S.
Franz
and
S.
Spigler
, “
Mean-field avalanches in jammed spheres
,”
Phys. Rev. E
95
,
022139
(
2017
).
130.
P.
Charbonneau
,
E. I.
Corwin
,
L.
Fu
,
G.
Tsekenis
, and
M.
van der Naald
, “
Glassy, Gardner-like phenomenology in minimally polydisperse crystalline systems
,”
Phys. Rev. E
99
,
020901(R)
(
2019
).
131.
C. P.
Goodrich
,
A. J.
Liu
, and
S. R.
Nagel
, “
Solids between the mechanical extremes of order and disorder
,”
Nat. Phys.
10
,
578
(
2014
).
132.
R.
Milkus
and
A.
Zaccone
, “
Local inversion-symmetry breaking controls the boson peak in glasses and crystals
,”
Phys. Rev. B
93
,
094204
(
2016
).
133.
P.
Pusey
and
W.
van Megen
, “
Phase behavior of concentrated suspensions of nearly hard colloidal spheres
,”
Nature
320
,
340
342
(
1986
).
134.
D.
Kaya
,
N. L.
Green
,
C. E.
Maloney
, and
M. F.
Islam
, “
Normal modes and density of states of disordered colloidal solids
,”
Science
329
,
656
(
2010
).
135.
M.
Wyart
, “
On the rigidity of amorphous solids
,”
Ann. Phys.
30
,
1
96
(
2005
).
136.
C. P.
Goodrich
,
A. J.
Liu
, and
S. R.
Nagel
, “
Finite-size scaling at the jamming transition
,”
Phys. Rev. Lett.
109
,
095704
(
2012
).
137.
D.
Hexner
,
P.
Urbani
, and
F.
Zamponi
, “
Can a large packing be assembled from smaller ones?
,” e-print arXiv:1902.00630 (
2019
).
138.
M. C.
Angelini
and
G.
Biroli
, “
Spin glass in a field: A new zero-temperature fixed point in finite dimensions
,”
Phys. Rev. Lett.
114
,
095701
(
2015
).
139.
P.
Charbonneau
and
S.
Yaida
, “
Nontrivial critical fixed point for replica-symmetry-breaking transitions
,”
Phys. Rev. Lett.
118
,
215701
(
2017
).
140.
M.
Moore
and
B.
Drossel
, “
p-spin model in finite dimensions and its relation to structural glasses
,”
Phys. Rev. Lett.
89
,
217202
(
2002
).
141.
J.
Yeo
and
M.
Moore
, “
Origin of the growing length scale in m-p-spin glass models
,”
Phys. Rev. E
86
,
052501
(
2012
).
142.
J. R. L.
de Almeida
and
D. J.
Thouless
, “
Stability of the Sherrington-Kirkpatrick solution of a spin glass model
,”
J. Phys. A: Math. Gen.
11
,
983
990
(
1978
).
143.

We here consider the Gardner transition obtained by compressing or annealing an amorphous solid, not the transition that could take place for the ideal glass. The field theory of the latter differs slightly.145 Given that the main conclusions do not depend on this technicality and that the former is in any event more experimentally relevant, we here neglect this subtle issue.

144.
M.
Moore
and
A. J.
Bray
, “
Disappearance of the de Almeida-Thouless line in six dimensions
,”
Phys. Rev. B
83
,
224408
(
2011
).
145.
P.
Urbani
and
G.
Biroli
, “
Gardner transition in finite dimensions
,”
Phys. Rev. B
91
,
100202
(
2015
).
146.
A.
Bray
and
S.
Roberts
, “
Renormalisation-group approach to the spin glass transition in finite magnetic fields
,”
J. Phys. C: Solid State Phys.
13
,
5405
(
1980
).
147.
P.
Charbonneau
,
Y.
Hu
,
A.
Raju
,
J. P.
Sethna
, and
S.
Yaida
, “
Morphology of renormalization-group flow for the de Almeida-Thouless–Gardner universality class
,”
Phys. Rev. E
99
,
022132
(
2019
).
148.
C. J.
Fullerton
and
M.
Moore
, “
The growing correlation length in glasses
,” e-print arXiv:1304.4420 (
2013
).
149.
S.
Boettcher
, “
Low-temperature excitations of dilute lattice spin glasses
,”
Europhys. Lett.
67
,
453
(
2004
).
150.
V.
Astuti
,
S.
Franz
, and
G.
Parisi
, “
New analysis of the free energy cost of interfaces in spin glasses
,” e-print arXiv:1810.13013 (
2018
).
151.
H.
Yoshino
, “
Disorder-free spin glass transitions and jamming in exactly solvable mean-field models
,”
SciPost Phys.
4
,
040
(
2018
).
152.
M.
Mezard
and
A.
Montanari
,
Information, Physics, and Computation
(
Oxford University Press
,
2009
).
153.
A.
Biere
,
M.
Heule
, and
H.
van Maaren
,
Handbook of Satisfiability
(
IOS Press
,
2009
), Vol. 185.
154.
E.
Gardner
, “
The space of interactions in neural network models
,”
J. Phys. A: Math. Gen.
21
,
257
(
1988
).
155.
E.
Gardner
and
B.
Derrida
, “
Optimal storage properties of neural network models
,”
J. Phys. A: Math. Gen.
21
,
271
(
1988
).
156.
M.
Mézard
and
G.
Parisi
, “
The Bethe lattice spin glass revisited
,”
Eur. Phys. J. B
20
,
217
233
(
2001
).
157.
G.
Biroli
,
R.
Monasson
, and
M.
Weigt
, “
A variational description of the ground state structure in random satisfiability problems
,”
Eur. Phys. J. B
14
,
551
568
(
2000
).
158.
M.
Mézard
,
G.
Parisi
, and
R.
Zecchina
, “
Analytic and algorithmic solution of random satisfiability problems
,”
Science
297
,
812
815
(
2002
).
159.
F.
Krzakała
,
A.
Pagnani
, and
M.
Weigt
, “
Threshold values, stability analysis, and high-q asymptotics for the coloring problem on random graphs
,”
Phys. Rev. E
70
,
046705
(
2004
).
160.
F.
Krzakała
,
A.
Montanari
,
F.
Ricci-Tersenghi
,
G.
Semerjian
, and
L.
Zdeborová
, “
Gibbs states and the set of solutions of random constraint satisfaction problems
,”
Proc. Natl. Acad. Sci. U. S. A.
104
,
10318
10323
(
2007
).
161.
A.
Montanari
,
G.
Parisi
, and
F.
Ricci-Tersenghi
, “
Instability of one-step replica-symmetry-broken phase in satisfiability problems
,”
J. Phys. A: Math. Gen.
37
,
2073
(
2004
).
162.
S.
Franz
and
G.
Parisi
, “
The simplest model of jamming
,”
J. Phys. A: Math. Theor.
49
,
145001
(
2016
).
163.
S.
Franz
,
S.
Hwang
, and
P.
Urbani
, “
Jamming in multilayer supervised learning models
,” e-print arXiv:1809.09945 (
2018
).
164.
M.
Geiger
,
S.
Spigler
,
S.
d’Ascoli
,
L.
Sagun
,
M.
Baity-Jesi
,
G.
Biroli
, and
M.
Wyart
, “
The jamming transition as a paradigm to understand the loss landscape of deep neural networks
,” e-print arXiv:1809.09349(
2018
).
165.
C.
Brito
,
H.
Ikeda
,
P.
Urbani
,
M.
Wyart
, and
F.
Zamponi
, “
Universality of jamming of nonspherical particles
,”
Proc. Natl. Acad. Sci. U. S. A.
115
,
11736
11741
(
2018
).
166.
V.
Lubchenko
and
P. G.
Wolynes
, “
Aging, jamming, and the limits of stability of amorphous solids
,”
J. Phys. Chem. B
122
,
3280
3295
(
2017
).