Gardner transition in finite dimensions

Recent works on hard spheres in the limit of infinite dimensions revealed that glass states, envisioned as metabasins in configuration space, can break up in a multitude of separate basins at low enough temperature or high enough pressure, leading to the emergence of new kinds of soft-modes and unusual properties. In this paper we study by perturbative renormalization group techniques the critical properties of this transition, which has been discovered in disordered mean-field models in the 1980s. We find that the upper-critical dimension d(u), above whichmean-field results hold, is strictly larger than six and apparently nonuniversal, i.e., system dependent. Below d(u), we do not find any perturbative attractive fixed point (except for a tiny region of the one-step replica symmetry breaking parameter), thus showing that the transition in three dimensions either is governed by a nonperturbative fixed point unrelated to the Gaussian mean-field one or becomes first order or does not exist. We also discuss possible relationships with the behavior of spin glasses in a field.