Does a true Gardner transition exist in three-dimensional glasses
In plain words
Mean-field theory predicts that a glass compressed deep into its solid state undergoes a further transition, the Gardner transition, where each glassy state splits into many nearly equivalent sub-states. Whether this transition survives in real three-dimensional glasses or is only a smooth crossover is unknown.
Precise statement
In the $d \to \infty$ solution for hard spheres, a stable glass becomes marginally stable at a pressure $p_G$ through full replica symmetry breaking (a hierarchy of sub-basins). For $d = 3$ hard- or soft-sphere glasses, determine whether the caging susceptibility $\chi_{AB}$ (fluctuation of the mean-square distance between two copies of the same glass) diverges in the thermodynamic limit at finite $p_G$, or stays finite. Perturbative arguments suggest the transition may be destroyed below $d = 6$, while a nonperturbative fixed point found in $d < 6$ would allow it.
What would settle it
Finite-size scaling of $\chi_{AB}$ in well-equilibrated $d = 3$ glasses showing divergence or saturation, or a renormalization-group proof of existence or absence.
Status in the literature
Simulations in $d = 2 \text{ and } 3$ report Gardner-like crossovers (2016-2023), including a generic dynamic Gardner crossover in compressed hard disks (Liao et al., PNAS 2023); field theory is split between absence of the transition below $d = 6$ and a nonperturbative fixed point that would permit it.