Numerical detection of the Gardner transition in a mean-field glass former

Recent theoretical advances predict the existence, deep into the glass phase, of a novel phase transition, the so-called Gardner transition. This transition is associated with the emergence of a complex free energy landscape composed of many marginally stable sub-basins within a glass metabasin. In this study, we explore several methods to detect numerically the Gardner transition in a simple structural glass former, the infinite-range Mari-Kurchan model. The transition point is robustly located from three independent approaches: (i) the divergence of the characteristic relaxation time, (ii) the divergence of the caging susceptibility, and (iii) the abnormal tail in the probability distribution function of cage order parameters. We show that the numerical results are fully consistent with the theoretical expectation. The methods we propose may also be generalized to more realistic numerical models as well as to experimental systems.

Figure 1

Phase diagram of the HS and MK models in the limit $d\to \infty$. Results are partially derived from Ref. [17], complemented by results obtained for the Gardner phase [36]. The liquid equation of state (EOS) given by Eq. (6) (full black line) gives $p/d=(2^d\phi /d)/2$ in the limit $d\to \infty$. The black dot denotes the dynamical glass transition at $2^d{\phi }_{\mathrm{d}}/d=4.8$. Glass EOSs from state following (SF) for $2^d{\phi }_0/d=5,6,6.667,7,8$ are also reported (thin colored lines, from left to right). The envelope for the Gardner transition for each SF (colored dot) is given by the dashed line.

Figure 2

(a) Organization of glass states (blue dots) for ${\phi }_0>{\phi }_{\mathrm{d}}$; the typical intrabasin MSD is ${\mathrm{\Delta }}_1$, while the interbasin MSD is ${\mathrm{\Delta }}_0=\infty$. (b) In the SF-full-RSB phase, ${\phi }_{\mathrm{G}}<\phi <{\phi }_{\mathrm{J}}$, glass metabasins subdivide into a hierarchical structure of sub-basins with typical innermost MSD ${\mathrm{\Delta }}_{\mathrm{EA}}$ and outermost (meta)basin MSD ${\mathrm{\Delta }}_1$. Schematics of the equilibrium $P_{\mathrm{eq}}\left(\mathrm{\Delta }\right)$ and restricted equilibrium $P_{\mathrm{SF}}\left(\mathrm{\Delta }\right)$ (blue area) distributions are given for (c) the SF-RS (or 1RSB) and (d) the full-RSB phases.

Figure 3

(a) Compressions $\left(\gamma >0\right)$ and decompressions $\left(\gamma <0\right)$ of an initial equilibrium state at ${\phi }_0=2.5$ and 4.0. The results are averaged over $N_{\mathrm{s}}=100$ samples. The theoretical curves uses data from Ref. [17] and from this work [36]. (b) Compression data are indistinguishable for $\gamma \le 0.01$.

Figure 4

SF from ${\phi }_0=2.50$ gives (a) ${\mathrm{\Delta }}_{AB}\left(t\right)$, (b) $\mathrm{\Delta }(t,t_w=0)$, and (c) $\delta \mathrm{\Delta }(t,t_w=0)$ at different $\phi$. Solid lines are fits to Eq. (21) (bottom panel).

Figure 5

(a) Long-time results for ${\mathrm{\Delta }}_{AB}\left(t\right)$. The data show that, for $\phi <{\phi }_{\mathrm{G}},{\mathrm{\Delta }}_{AB}\left(t\right)$ saturates at its asymptotic value (dotted line) in the long-time limit (see, e.g., $\phi =2.88,2.90)$. Panels (b) $\mathrm{\Delta }(t,t_w)$ and (c) $\delta \mathrm{\Delta }(t,t_w)$ are plotted as a function of $t$ for various $\phi$ and $t_w$.

Figure 6

(a) Time evolution of the caging susceptibility $\chi (t,t_w=0)$ (scatters) and ${\chi }_{AB}\left(t\right)$ (lines). (b) For three selected densities, we report the evolution of ${\chi }_{AB}\left(t\right)$ for much longer times (dotted lines) and the dependence of $\chi (t,t_w)$ on both $t$ and $t_w$ (see legend).

Figure 7

(a) Growth of ${\tau }_{\beta }^{\prime }$ with ${\phi }_{\mathrm{G}}^{\tau }-\phi$ for ${\phi }_{\mathrm{G}}^{\tau }$ from Table 2. (b) The two estimates of ${\tau }_{\beta },{\tau }_{\beta }^{\prime }$, and ${\tau }_{\beta }^{\prime \prime }$, as a function of ${\phi }_{\mathrm{G}}^{\tau }-\phi$ for ${\phi }_0=2.5$. (c) Evolution of $a$ and $b$ with $\phi$ for ${\phi }_0=2.5$.

Figure 8

Logarithmic decay of $\delta \mathrm{\Delta }(t,t_w=0)$ with time for ${\phi }_0=2.5$. The results are fitted to Eq. (23) (dashed lines).

Figure 9

Density evolution of the averages of $\mathrm{\Delta }$ and ${\mathrm{\Delta }}_{AB}$ for ${\phi }_0=2.50$. Numerical results (points) are compared with theoretical predictions for ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_{\mathrm{EA}}$, and ${\langle \mathrm{\Delta }\rangle }_{\mathrm{SF}}$ (lines). Both datasets are scaled to reference values (see Sec. 4b1 for details) to obtain a better agreement between theory and simulation.

Figure 10

(a) $P\left(\mathrm{\Delta }\right)$ and (b) $P\left({\mathrm{\Delta }}_{AB}\right)$ at different $\phi$ for ${\phi }_0=2.5$.

Figure 11

(a) Finite-size behavior of $P\left({\mathrm{\Delta }}_{AB}\right)$ at $\phi =3.0\approx {\phi }_{\mathrm{G}}$ and (b) density evolution of the rescaled skewness $\mathrm{\Gamma }$ for three different $N$ and for ${\phi }_0=2.5$.

Figure 12

Finite-size behavior of the two terms of the definition of the susceptibility in Eq. (24): (a) $\overline{〈{\mathrm{\Delta }}_{AB}〉}$, and (b) $N(\overline{〈{\mathrm{\Delta }}_{AB}^2〉}-{\overline{〈{\mathrm{\Delta }}_{AB}〉}}^2)$, for ${\phi }_0=2.50$. The dashed lines are fits of a second-order polynomial in $1/N$ to the data. The $1/N\to 0$ extrapolation of these two fits have been used to extract the $N\to \infty$ limit of $\chi$, given in Fig. 13 by a dark solid line.

Figure 13

(a) Inverse caging susceptibility ${\chi }^{-1}$ and (b) skewness $\mathrm{\Gamma }$ as functions of $\phi$ for different ${\phi }_0$. Upon approaching ${\phi }_{\mathrm{G}},\chi$ apparently diverges as $|\phi -{\phi }_{\mathrm{G}}^{\chi }{|}^{-1}$, except for ${\phi }_0=1.8$. Note that the divergence of $\chi$ coincides with the maximum of $\mathrm{\Gamma }$. The $N\to \infty$ extrapolation of the curves in Fig. 12 were used to obtain the solid line.

Figure 14

The numerical susceptibilities (scatters) are compared with theoretical predictions (lines). Both datasets are scaled to reference values (see Sec. 4b3 for details) to obtain a better agreement between theory and simulation. We also report the linear fits to the numerical data used to determine ${\phi }_{\mathrm{G}}^{\chi }$ (dashed lines). This linear fit overestimates the transition point by roughly 2%.

Figure 15

(a) The susceptibility $\chi$ and (b) the skewness $\mathrm{\Gamma }$ averaged over the ensemble of clones are compared with those of four individual samples, for ${\phi }_0=2.50$.

Figure 16

Inverse reduced pressure $1/p$ density $\phi$ phase diagram of the MK model in $d=3$. The liquid EOS follows Eq. (6), while a specific state follows a glass EOS from ${\phi }_0$ up to jamming at ${\phi }_{\mathrm{J}}$. Numerical estimates of the Gardner transition evolve with ${\phi }_0$ similarly as the theoretical predictions [17].