The exact mean-field (infinite-dimensional) solution of the hard-sphere glass problem predicts that, deep inside the glass phase, each metastable glass basin fractures into a hierarchy of marginally stable sub-basins at a Gardner transition. Berthier, Charbonneau, Jin, Parisi, Seoane, and Zamponi (PNAS 2016) reported the first direct numerical evidence that this physics is relevant in three dimensions: upon compressing hard-sphere glasses, the caging susceptibility grows by orders of magnitude, spatial correlations of the caging order parameter become system-spanning, and relaxation times grow sharply, in systems of up to N = 8000 particles. The authors explicitly left the fate of this crossover in the thermodynamic limit as an open question: is it a genuine second-order phase transition of three-dimensional amorphous solids with determinate critical exponents, or an avoided transition with a large but finite correlation length? A decade of subsequent work — one-, two-, and three-loop renormalization-group analyses, machine-learning-assisted finite-size scaling, simulations of soft spheres far from jamming, quasi-one-dimensional exact model studies, and granular, colloidal, and molecular experiments — leaves the question genuinely contested on both sides and unresolved. The problem is to resolve it: establish whether the Gardner transition survives in three dimensions and determine its critical behavior if it does.

Background

Amorphous solids — window glasses, gels, foams, and granular packings — lack crystalline order, and their low-temperature properties depart sharply from the Debye theory that describes crystals: they show an excess of vibrational states over the Debye prediction (the boson peak), anomalous heat capacity and thermal conductivity attributed to tunneling two-level systems (TLS), and slow, history-dependent responses to weak perturbations. A unified microscopic explanation of these anomalies is a long-standing goal of condensed-matter physics.

The modern theoretical framework comes from the replica method applied to hard spheres in the limit of infinite spatial dimension d, solved exactly by Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (arXiv:1310.2549). In this mean-field picture, a glass is a metastable basin of configuration space described by one-step replica symmetry breaking (1RSB), the scheme in which the overlap between replicas of the system takes a single nontrivial value. The solution predicts that deeper inside the glass phase each basin undergoes the Gardner transition — first discovered by Gardner in 1985 in p-spin spin-glass models — below which the basin fractures into an infinite hierarchy of sub-basins, described by full replica symmetry breaking (FRSB), and the glass becomes marginally stable: riddled with vanishing free-energy barriers, it responds strongly to arbitrarily weak perturbations. Marginal stability is the mechanism by which the theory proposes to unify the low-temperature anomalies of amorphous solids.

Berthier, Charbonneau, Jin, Parisi, Seoane, and Zamponi (PNAS 113, 8397 (2016)) tested this prediction numerically in three-dimensional polydisperse hard-sphere glasses (spheres whose diameters are drawn from a continuous distribution of sizes, which prevents crystallization), prepared by Monte-Carlo compression to a preparation density that plays the role of the glass transition temperature. Their key observable is the caging order parameter: the mean-square inter-particle displacement ΔAB\Delta_{AB} between two identical copies (replicas) of the same glass evolved independently, which measures how far the glass state explores phase space under perturbation. Upon compression beyond a crossover density ϕG\phi_G, ΔAB\Delta_{AB} develops large fluctuations, the associated caging susceptibility χAB\chi_{AB} grows by orders of magnitude, spatial correlations of the local caging order parameter become long-ranged (comparable to the system size), and relaxation times grow sharply — clear finite-size signatures of criticality, seen for N = 1000 and N = 8000 particles.

The unresolved issue is what these signatures become as N→∞N \to \infty. For a genuine second-order phase transition, the susceptibility and correlation length must diverge in the thermodynamic limit, with determinate critical exponents; in a finite system any peak is finite, so deciding divergence versus saturation requires systematic finite-size scaling at much larger sizes, which the source work explicitly deferred beyond its scope. The question is sharpened by a deep analogy: symmetry arguments place the Gardner transition in the same universality class as the de Almeida–Thouless (dAT) transition of spin glasses in a magnetic field — whether such replica-symmetry-breaking transitions survive below their upper critical dimension du=6d_u = 6 has been debated for over four decades despite intensive numerical scrutiny. Theory gives conflicting indications (one-loop and higher-loop renormalization-group analyses disagree), simulations split by model class (hard spheres near jamming — jamming being the density at which hard particles first form a mechanically rigid, load-bearing contact network — versus soft spheres far from jamming), and the transition is a property of out-of-equilibrium metastable states, so its location and sharpness depend on the preparation protocol, with better-annealed glasses predicted to show a sharper crossover at lower temperature. Experimentally, the inter-replica order parameter cannot be measured directly, so detection must rely on indirect proxies (nonlinear susceptibilities, logarithmic growth of the mean-square displacement, protocol-dependent shear moduli, structural signatures in the distribution of inter-particle gaps), and the granular, colloidal, and molecular systems probed so far give conflicting indications.

Problem Statement

Determine the fate of the Gardner-like crossover observed in three-dimensional amorphous solids in the thermodynamic limit. Concretely: (a) establish whether the caging susceptibility and its correlation length diverge as the particle number N→∞N \to \infty at a sharp Gardner transition point in three dimensions — a genuine second-order (out-of-equilibrium) phase transition with determinate critical exponents, expected on symmetry grounds to share the universality class of the de Almeida-Thouless transition of spin glasses in a field — or whether finite-dimensional fluctuations, such as activated dynamics (thermally driven crossing of finite energy barriers between glass configurations) and localized defects, reduce it to an avoided transition with a large but finite correlation length; and (b) determine the extent to which the mean-field predictions (full replica symmetry breaking, marginal stability, hierarchical free-energy landscape) apply to three-dimensional systems, including how the answer depends on the glass-forming model (hard spheres near jamming versus soft spheres far from jamming) and on the preparation protocol (preparation density or temperature, annealing depth), which the source study of Berthier, Charbonneau, Jin, Parisi, Seoane, and Zamponi (PNAS 2016) predicted should sharpen or shift the crossover. An answer must resolve the transition-versus-crossover question in the thermodynamic limit; determining the critical exponents, the nature of the divergent lengths and times if the transition exists, or proving their absence, is an integral part of the same objective, not a separate problem. Part (b) carries no independent completion condition: it is required only insofar as it bears on part (a), and a resolving answer need not, for its own sake, settle the extent to which the mean-field predictions apply in every model class and under every preparation protocol.

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