We study Harmonic Soft Spheres as a model of thermal structural glasses in the limit of infinite dimensions. We show that cooling, compressing and shearing a glass lead to a Gardner transition and, hence, to a marginally stable amorphous solid as found for Hard Spheres systems. A general outcome of our results is that a reduced stability of the glass favors the appearance of the Gardner transition. Therefore using strong perturbations, e.g. shear and compression, on standard glasses or using weak perturbations on weakly stable glasses, e.g. the ones prepared close to the jamming point, are the generic ways to induce a Gardner transition. The formalism that we discuss allows to study general perturbations, including strain deformations that are important to study soft glassy rheology at the mean field level.
I. INTRODUCTIONWhen a liquid is cooled fast enough in such a way that crystallization is avoided, it enters in a supercooled phase where upon further decreasing the temperature it freezes in an amorphous solid phase, a glass [1]. This phenomenon has been investigated since the works by Adam and Gibbs [2] that were aimed to clarify the thermodynamical nature of the glass transition. Starting from the pioneering works by Kirkpatrick, Thirumalai and Wolynes [3][4][5][6], physicists have realized that there exists a deep connection between the Adam-Gibbs picture of the glass formation and the statistical physics of disordered systems such as spin glasses. The works of Franz, Parisi, Mézard and Monasson [7-15] showed how to adapt the replica method, a very powerful tool to study the properties of system with quenched disorder, to disorder-free Hamiltonians. This stream of ideas led finally to the exact description of the amorphous phases of hard spheres in the limit of infinite dimensions [16,17]. Hard sphere systems are good theoretical models of colloidal glasses and have been studied in recent years to understand the critical properties of the jamming transition. Remarkably, the mean field theory of hard sphere glasses is able to correctly describe the criticality of jammed packings in three dimensions giving a very accurate prediction of the critical exponents that appear at the jamming point. Furthermore, the infinite dimensional solution of the hard sphere model has suggested that colloidal glasses at very high pressure could undergo a new phase transition, the Gardner transition, that was firstly found in models of spin glasses [18][19][20][21] and whose consequence in the structural glass case are the object of a very intense research activity [22][23][24][25][26]. Beyond the Gardner point, hard sphere glasses are predicted to be marginally stable: their properties are deeply affected by non-trivial soft modes that drive strong non-linear elastic responses [25,[27][28][29][30][31][32][33][34][35][36]. The aim of the present work is to extend the analysis done for hard spheres to thermal glasses. We shall present the phase diagram for elastic spheres in infinite dimensions and thoroughly study the properties...