The Gardner transition in finite dimensions

TL;DR

This paper investigates the critical properties of the Gardner transition in finite dimensions using renormalization group techniques, revealing a non-universal upper critical dimension and implications for three-dimensional systems.

Contribution

It provides a perturbative analysis of the Gardner transition in finite dimensions, showing the upper critical dimension exceeds six and is system-dependent, and discusses its nature in three dimensions.

Findings

Upper critical dimension $d_u$ > 6 and system-dependent

No perturbative fixed point in 3D suggests non-perturbative effects

Possible link between Gardner transition and spin glass behavior

Abstract

Recent works on hard spheres in the limit of infinite dimensions revealed that glass states, envisioned as meta-basins 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 renormalisation group techniques the critical properties of this transition, which has been discovered in disordered mean-field models in the '80s. We find that the upper critical dimension $d_u$ above which mean-field results hold is strictly larger than six and apparently non-universal, i.e. system dependent. Below $d_u$, we do not find any perturbative attractive fixed point (except for a tiny region of the 1RSB breaking parameter), thus showing that the transition in three dimensions either is governed by a non-perturbative 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.