The idea that glasses can become marginally stable at a Gardner transition has attracted significant interest among the glass community. Yet, the situation is confusing: even at the theoretical level, renormalization group approaches provide contradictory results on whether the transition can exist in three dimensions. The Gardner transition was searched in only two experimental studies and few specific numerical models. These works lead to different conclusions for the existence of the transition, resulting in a poor understanding of the conditions under which a marginally stable phase can be observed. The very relevance of the Gardner transition for experimental glasses is at stake.
We study analytically and numerically the Weeks-Chandler-Andersen model. By changing external parameters, we continuously explore the phase diagram and regimes relevant to granular, colloidal, and molecular glasses. We revisit previous numerical studies and confirm their conclusions. We reconcile previous results and rationalise under which conditions a Gardner phase can be observed. We find that systems in the vicinity of a jamming transition possess a Gardner phase. Our findings confirm the relevance of a Gardner transition for colloidal and granular glasses, and encourage future experimental work in this direction. For molecular glasses, we find that no Gardner phase is present, but our studies reveal instead the presence of localised excitations presumably relevant for mechanical and vibrational properties of glasses.