Shear modulus of glasses: Results from the full replica-symmetry-breaking solution

We compute the shear modulus of amorphous hard and soft spheres, using the exact solution in infinite spatial dimensions that has been developed recently. We characterize the behavior of this observable in the whole phase diagram, and in particular around the glass and jamming transitions. Our results are consistent with other theoretical approaches, which are unified within this general picture, and they are also consistent with numerical and experimental results. Furthermore, we discuss some properties of the out-of-equilibrium dynamics after a deep quench close to the jamming transition, and we show that a combined measure of the shear modulus and of the mean square displacement allows one to probe experimentally the complex structure of phase space predicted by the full replica-symmetry-breaking solution.

Figure 1

An illustration of the susceptibilities associated with different levels of exploration of phase space in a 2RSB structure (here with two metabasins each split into three subbasins). A system confined in a single glass basin is characterized by the intrastate susceptibility ${\chi }_2={\tilde{\chi }}_2$. A system that can explore the first level of metabasins is characterized by ${\chi }_1={\tilde{\chi }}_2+m_1{\tilde{\chi }}_1$. Finally, a system that can explore the full equilibrium structure of all metabasins has the equilibrium susceptibility $\chi ={\chi }_0={\tilde{\chi }}_2+m_1{\tilde{\chi }}_1+m_0{\tilde{\chi }}_0$.

Figure 2

(Left) Schematic evolution of the shear modulus $\mu (t,t_w)$ and the mean square displacement $\Delta (t,t_w)$, for a large time $t_w$ after a quench, as a function of $t-t_w$. (Right) The function $\frac{\beta \widehat{\mu }\left(\Delta \right)}{p}\sim 1/\gamma \left(\Delta \right)$ that characterizes the reparametrization-invariant regime, obtained from the numerical solution of the fullRSB equations [36]. The dots mark the intrastate point $({\widehat{\Delta }}_{\mathrm{EA}},\beta {\widehat{\mu }}_{\mathrm{EA}}/p)$. The three curves correspond to different cutoffs $y_{\max }\propto p$. The slight bending of the curves around the dots is an artifact of the numerical computation [36].