Absence of Marginal Stability in a Structural Glass

Marginally stable solids have peculiar physical properties that were first analyzed in the context of the jamming transition. We theoretically investigate the existence of marginal stability in a prototypical model for structural glass formers, combining analytical calculations in infinite dimensions to computer simulations in three dimensions. While mean-field theory predicts the existence of a Gardner phase transition towards a marginally stable glass phase at low temperatures, simulations show no hint of diverging time scales or length scales, but reveal instead the presence of sparse localized defects. Our results suggest that the Gardner transition is deeply affected by finite dimensional fluctuations, and raise issues about the relevance of marginal stability in structural glasses far away from jamming.

Figure 1

(a) Mean-field phase diagram. Energy of the equilibrium liquid (black line), and dynamical transition temperature $T_d$ (black square). Various glasses prepared at $T_g<T_d$ are followed out of equilibrium (full lines). When $T_g\gtrsim 0.5T_d$, these glasses undergo a Gardner transition (bullets) into marginally stable glasses (dashed lines). (b) Numerical phase diagram with equivalent representations of the liquid, dynamic transition and glass lines. Bullets locate the temperature crossover below which localized excitations appear, but do not correspond to a Gardner transition. In both panels, axes are rescaled by the value at $T_d$: $E_d=E(T_d)$.

Figure 2

(a) Mean-squared displacement $\mathrm{\Delta }(t,t_w)$ and mean-squared distance ${\mathrm{\Delta }}_{AB}(t)$ as a function of $t-t_w$ and $t$ for $T_g=0.062$ and $T={10}^{-4}$ rapidly converge to their long-time limits (dashed lines). (b) Long-time limits of ${\mathrm{\Delta }}^{\infty }$ and ${\mathrm{\Delta }}_{AB}^{\infty }$ as a function of temperature, for different $T_g$. Both quantities differ below $T^{*}(T_g)$, indicated by vertical segments.

Figure 3

(a) The susceptibility ${\chi }_{AB}$ for different $T_g$ raises mildly above the floor level (dashed line) across $T^{*}$ (vertical segments). The error bars are computed using the jackknife method. (b) The van Hove function of relative displacements has a narrow Gaussian core of width given by ${\mathrm{\Delta }}^{\infty }$ (solid blue), and exponential tails (solid red) at $T_g=0.062$ and $T={10}^{-4}$. (c) A corresponding snapshot with $N=12000$ showing the few particles having displacements outside the Gaussian range (red spheres) among a majority of particles undergoing small amplitude Gaussian vibrations (blue dots).