The fate of the Gardner transition in finite-dimensional structural glasses: existence, finite-size extrapolation, and experimental detection beyond the mean-field Mari-Kurchan model

Mean-field theory, exact for hard spheres in the limit of infinite spatial dimension $d\to\infty$, predicts that deep in the glass phase each metastable glass basin undergoes a Gardner transition into a hierarchical, marginally stable set of sub-basins (full replica symmetry breaking). Charbonneau, Jin, Parisi, Rainone, Seoane, and Zamponi (Phys. Rev. E 92, 012316, 2015) detected this transition numerically in the Mari-Kurchan (MK) model — a hard-sphere glass former that is mean-field by construction yet can be simulated in finite dimensions — in $d=3$, using planting, state following, and cloning, and identified four unresolved fronts: (i) whether the transition survives as a sharp thermodynamic transition in truly finite, low dimensions; (ii) quantitative finite-$d$ corrections to the $d\to\infty$ state-following predictions (non-Gaussian cage shapes, spatial cage heterogeneity, hopping); (iii) reliable $N\to\infty$ extrapolations of susceptibilities and order-parameter distributions at and above the Gardner density $\varphi_G$; (iv) transfer of the detection protocols to realistic models and experiments, where preparing well-equilibrated glasses without planting and replacing cloning with many experimental replicates are the key hurdles. Later work (2016-2024) established strong Gardner phenomenology in three-dimensional hard-sphere glasses, found a nontrivial renormalization-group fixed point, and produced encouraging granular and colloidal experiments, but also found avoided-transition and crossover behavior in lower dimensions and soft spheres, and no Gardner divergence in molecular glasses. Whether a sharp Gardner transition survives in finite dimensions, together with its quantitative finite-$d$ corrections, thermodynamic-limit characterization, and unambiguous experimental detection, remains open.

Background

When a liquid is supercooled or compressed fast enough, it falls out of equilibrium and forms a glass. In the random first-order transition (RFOT) theory of the glass transition, the exact solution of hard spheres in the limit of infinite spatial dimension $d\to\infty$ describes the glass phase through the replica method: one prepares many copies ('replicas') of the system pinned in the same metastable region of configuration space. A glass basin is then characterized by a cage order parameter, the typical mean-squared displacement $\Delta$ of a particle about its equilibrium position. Below the dynamical (mode-coupling) transition, which occurs at packing fraction $\varphi_d$ (the fraction of system volume occupied by the particles), the free-energy landscape splits into exponentially many metastable glass basins ('metabasins'); the exact $d\to\infty$ solution further predicts that, upon compression or cooling, each such basin undergoes a second transition — the Gardner transition — below which it fractures into a fractal hierarchy of marginally stable sub-basins. In replica language this is full replica symmetry breaking (RSB): the symmetry under permutation of the replica copies is broken in a continuous hierarchy of steps, in the same universality class as spin glasses in a magnetic field; the transition was discovered by Gardner in mean-field spin-glass models. Marginal stability means the hierarchy of sub-basins is bordered by a continuum of soft modes, so the system responds anomalously to infinitesimal perturbations. This scenario underlies the mean-field theory of jamming of hard particles and the anomalous low-temperature properties of glasses (see, e.g., the fractal free-energy-landscape analysis of Charbonneau, Kurchan, Parisi, Urbani, and Zamponi, Nat. Commun. 5, 3725 (2014)).

The Mari-Kurchan (MK) model is a hard-sphere glass former in which each pair interaction is evaluated at a distance shifted by a quenched random vector $\Lambda_{ij}$ uniformly distributed over the system volume. The random shifts suppress crystallization and nucleation between basins, so the model is mean-field by construction, yet it can be simulated in any finite dimension $d$ and shares the qualitative phase diagram of infinite-dimensional hard spheres. The most important genuinely finite-dimensional effect is hopping: particles occasionally jump between neighboring cages through narrow pathways, which washes out the $d\to\infty$ divergence of the structural relaxation time at $\varphi_d$, so that in finite $d$ the dynamical transition is 'avoided' — rounded into a crossover with no genuine singularity (hopping was analyzed in the MK model by Charbonneau, Jin, Parisi, and Zamponi, Proc. Natl. Acad. Sci. USA 111, 15025 (2014)).

The source paper (Charbonneau, Jin, Parisi, Rainone, Seoane, and Zamponi, Phys. Rev. E 92, 012316 (2015)) developed numerical procedures for detecting the Gardner transition in the MK model in $d=3$. Equilibrated liquid configurations at density $\varphi_0$ above $\varphi_d$ were 'planted' (all replicas started from the same equilibrated configuration), after which the selected glass state was followed in density by state following (SF): tracking a chosen glass metabasin under slow compression of the many replicas simultaneously. The transition density $\varphi_G$ was located by three independent diagnostics: the divergence of the relaxation time inside the cage, the divergence of the caging susceptibility (the fluctuation of the cage order parameter, whose mean-field critical behavior is governed by the vanishing of a soft mode called the replicon), and the emergence of a non-Gaussian tail in the probability distribution of the mutual displacement $\Delta_{AB}$ between independently equilibrated replicas (with $\Delta_{AB}>\Delta_{\mathrm{EA}}$, the self-overlap, signaling RSB). Some protocols additionally used cloning, the maintenance of a large population of replicas with selection, to enforce equilibrium within the metastable state. The results were found consistent with the $d\to\infty$ SF predictions, whose full replica symmetry breaking solution was completed by Rainone and Urbani (J. Stat. Mech. (2016) 053302).

The authors explicitly identified what remains unresolved. First, whether the Gardner transition exists at all in finite low dimensions is open: the one-loop renormalization-group analysis of Urbani and Biroli (Phys. Rev. B 91, 100202 (2015)) finds an upper critical dimension above which mean-field theory holds that is strictly larger than six and system-dependent, and no perturbative attractive fixed point below it, so that in $d=3$ the transition is either governed by a non-perturbative fixed point, becomes first order, or disappears. Second, the quantitative finite-$d$ corrections to the $d\to\infty$ SF predictions — from non-Gaussian cage shapes, spatial heterogeneity of cage sizes, and hopping — were not computed and require more elaborate cavity or field-theoretical calculations. Third, for densities $\varphi\gtrsim\varphi_G$ finite-size effects dominated the studied system sizes, so reliable $N\to\infty$ extrapolations of susceptibilities and distributions could not be achieved. Fourth, transferring the methodology to realistic glass models and experiments faces two operational hurdles: preparing well-equilibrated glasses (without planting) sufficiently deep above the avoided dynamical transition, and replacing cloning with a large number of equivalent experimental replicates.

Problem Statement

Ascertain the presence, nature, and measurable consequences of the Gardner transition in finite-dimensional, experimentally relevant glass formers, using the mean-field-like Mari-Kurchan model — in which the transition was first detected numerically in a structural glass — as the reference testbed for the finite-dimensional and finite-size fronts, and reaching beyond it to realistic, non-mean-field glass formers. This single objective decomposes into four scientifically linked fronts stated by the source work. (i) Finite-dimensional fate: determine whether the Gardner transition survives as a sharp thermodynamic phase transition in the $N\to\infty$ limit across low spatial dimensions generally, taking $d=2$ and $d=3$ as the primary cases, with diverging caging susceptibility and correlation length and well-defined critical exponents, or whether it is avoided and appears only as a crossover with a large but finite length scale (as suggested by one-loop renormalization group and by avoided-transition analyses), possibly with a lower critical dimension below which it disappears entirely. (ii) Quantitative finite-dimensional corrections: compute the corrections to the $d\to\infty$ state-following predictions for the Gardner line and the surrounding observables that arise in finite $d$ from non-Gaussian cage shapes, spatial heterogeneity of cage sizes, and hopping. (iii) Thermodynamic-limit extrapolation: achieve reliable $N\to\infty$ extrapolations of the caging susceptibility and of the order-parameter distributions at and above the Gardner density $\varphi_G$, where finite-size effects dominate currently accessible system sizes. (iv) Transfer to realistic systems and experiments: adapt the detection protocols (state following and its experimental annealing analogues, cloning or equivalent many-replicate protocols) to realistic glass models and to experimental systems such as colloidal, granular, and molecular glasses — including glasses equilibrated deep in the phase diagram without planting — so that the presence or absence of the transition and its observable consequences (protocol-dependent responses, anomalous aging, marginal-stability scalings) can be established there. The four fronts differ in kind: front (i) is a decisional question — the transition either survives as a sharp transition in the required dimensions or it is avoided — whereas fronts (ii)-(iv) are constructive, open-ended directions (finite-$d$ corrections, thermodynamic-limit extrapolations, and transfer to realistic systems and experiments). The objective is therefore settled by a definite verdict on its central decisional question, front (i), together with the critical behaviour it entails; a substantive result on any of the supporting directions (ii)-(iv) is a genuine partial contribution, but completion of fronts (ii)-(iv) is not required to settle the objective.

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