Since it was clear that traditional approaches did not see the Gardner transition, our main initial task was to explore several protocols and observables in a well controlled model where the Gardner transition existed for sure. With this idea, we studied in simulations a mean-field model for hard-spheres that allows an analytical description at any dimension. Using this model, and trying old spin glasses ideas, we where able to design a successful numerical protocol based on two observables: the traditional mean-square displacement, and the distance between two clones, where clones corresponded to independent compressions of the system which all started from the same configuration [1]. After testing the approach in the mean-field model, we moved to real hard spheres. We repeated the same protocol in D=2 and 3 in Ref. [2]. We were able to show, by the first time, the existence of a sharp threshold in density above which vibrations become highly correlated. Furthermore, we could also associate this threshold to a divergence of time and length-scales, as expected nearby a second order transition and a non-trivial change of the probability distribution function of the order parameter. Quite remarkably, the protocol proposed in our works [1] and [2] was suitable to be directly repeated in experiments of frictional hard disks [3], thus providing the first experimental test of the Gardner transition.
After testing the theory in hard-sphere systems, the canonical model for granulars or colloids, we explored the existence of a Gardner transition
in temperature in a simple model for molecular glasses in prepared via the vapor deposition procedure. We concluded that, although a Gardner-like threshold is observed at low temperatures, and that this threshold depends on the stability of the original glass, the transition changes its nature when thermal fluctuations are introduced. The threshold is not longer sharp, and it is not related to the apparition of collective excitations but localized ones. In this sense, we find a threshold in temperature but it does not lead to a marginal phase but it is related to localized defects in the sample [4]. Still, many of the anomalies remain visible.
The protocol followed in Refs. [1,2,4] can only be repeated in experiments if spatial resolution is available (like in granular disks), which is not generally the case in most of glassy systems. For this reason, I have also worked in developing new alternative protocols based only on macroscopic observables. During the project, I tested these ideas first in spin glasses [5,6] and I have recently applied to hard-sphere problems, but this work is still ongoing.
[1] Charbonneau, P., Jin, Y., Parisi, G., Rainone, C., Seoane, B., & Zamponi, F. (2015). Numerical detection of the Gardner transition in a mean-field glass former. Physical Review E, 92(1), 012316.
[2] Berthier, L., Charbonneau, P., Jin, Y., Parisi, G., Seoane, B., & Zamponi, F. (2016). Growing timescales and lengthscales characterizing vibrations of amorphous solids. Proceedings of the National Academy of Sciences, 201607730.
[3] Seguin, A., & Dauchot, O. (2016). Experimental Evidence of the Gardner Phase in a Granular Glass. Physical review letters, 117(22), 228001.
[4] Seoane, B., Reid, D. R., de Pablo, J. J., & Zamponi, F. (2017). Low-temperature anomalies of a vapor deposited glass. arXiv preprint arXiv:1709.04930.
[5] Baity-Jesi, M., Calore, E., Cruz, A., Fernandez, L. A., Gil-Narvión, J. M., Gordillo-Guerrero, A., ... & Monforte-Garcia, J. (2017). A statics-dynamics equivalence through the fluctuation–dissipation ratio provides a window into the spin-glass phase from nonequilibrium measurements. Proceedings of the National Academy of Sciences, 201621242.
[6] Baity-Jesi, M., Calore, E., Cruz, A., Fernandez, L. A., Gil-Narvion, J. M., Gordillo-Guerrero, A., ... & Monforte-Garcia, J. (2017). Matching microscopic and macroscopic responses in glasses. Physical Review Letters, 118(15), 157202.
