Universal activated aging and weak ergodicity breaking in spin and structural glasses

TL;DR

The paper develops a unified, landscape-based framework for activated aging in glasses by introducing a generalized trap model (GTM) that includes finite-size corrections to barrier-energy distributions. The authors derive aging dynamics through the arcsin law, predicting a WEB phase at $T_{ m WEB}$ distinct from the SEB transition temperature, and they show a logarithmic decay of correlations due to activation clusters, with a finite-size–dependent logarithmic term. They validate the GTM with random-energy models (Gaussian and exponential), barrier-tree analyses, and structural-glass simulations (WCA and amorphous SiO$_2$), and they establish a barrier-tree–based tree-expansion theory that yields universal barrier statistics and an intrinsic static length $\xi_{\rm ag}$ that can be extracted from non-equilibrium dynamics. The results yield a unified WEB ergodic phase diagram, connect aging dynamics to RFOT scaling, and demonstrate a consistent picture across spin and structural glasses, providing a practical route to infer static lengths from aging data and supporting RFOT predictions. Overall, the work offers a comprehensive, quantitative theory for activated aging with broad implications for understanding glassy dynamics and the emergence of static length scales from non-equilibrium measurements.

Abstract

Glasses possess complex energy landscapes and exhibit non-equilibrium aging dynamics. Here, we propose a generalized trap model for activated aging based on a key static property of the energy landscape: the distribution of energy barriers. Our theory predicts that, upon cooling, weak ergodicity breaking (WEB) in quenching dynamics occurs prior to strong ergodicity breaking in equilibrium dynamics. Furthermore, the theory indicates that the characteristic size of activation clusters can be deduced from the logarithmic decay of the time-correlation function. We rigorously test the model's assumptions and predictions using the simplest spin glass model - the random energy model. The predicted aging behavior is also universally observed in paradigmatic structural glasses, including the Weeks-Chandler-Andersen (WCA) model and amorphous silica. Remarkably, applying our framework to the WCA model allows us to extract a static length from the non equilibrium dynamics, extending its observable growth range from a mere factor of 2-3 to a full order of magnitude and providing supportive evidence for the random first-order transition scenario. Finally, we propose a unified ergodic-WEB phase diagram for aging dynamics in general glassy systems.

Figures (27)

• Figure 1: Basic aging phenomenon in glasses. Data are obtained for the WCA model at $T=0.25$ with $N=8000$ particles, quenched from $T_{\rm eq} = 5$ to $T$. (a) $-dE/dt_{\rm w}$ as a function of $t_{\rm w}$ at a few different $T$. The solid line indicates $-dE/dt_{\rm w} \sim t_{\rm w}^{-1}$, i.e., $E(t_{\rm w}) \sim - \ln(t_{\rm w})$. (b) Non-equilibrium decorrelation time $\tau_{\rm neq}$ as a function of $t_{\rm w}$. The solid line represents $\tau_{\rm neq} \sim t_{\rm w}$. (c) $F(t_{\rm w}, t_{\rm w}+t)$ for $t_{\rm w} = 0.2, 2, 20, 2\times 10^2, 2\times 10^3, 2\times 10^4, 8\times 10^5$ (black curves; from left to right), and $F_w(t_{\rm w})$ with $w=1/2$ and $w=3/2$. The dashed line indicates the nonergodicity parameter $f$. (c) $F(t_{\rm w}, t_{\rm w}+t)$ plotted as a function of $w = t/t_{\rm w}$ for intermediate waiting times $t_{\rm w} = 2, 20, 2\times 10^2, 2\times 10^3$. The blue line is $H$ given by Eq. (\ref{['eq:Pi']}) at the given $\hat{x}=T/T_{\rm WEB}$, and the red line is $\tilde{H} = f H$.
• Figure 2: Aging function $\Pi_w(t_{\rm w})$ for the (a,b) E-REM and (c-d) G-REM. In both models, $w=0.5$. Open points are MC simulation data. (a) E-REM results with $N=20$ (points). Data in the plateau regime are fitted by $\Pi(t_{\rm w}) = H \left( 1 + A \, t_{\rm w}^{-\alpha} \right)$ (lines). (b) E-REM data at $T=0.75$. The purple dashed line represents the short-time behavior $\Pi_w^{\rm s}(t_{\rm w}) = 1-Ct_{\rm w}$ with $C \approx 0.214$. The dotted horizontal line represents the BTM plateau $H(w,x) \approx 0.303$. The thermalization time $\tau_{\rm th} = \exp(N/T)$ is marked by crosses. (c) G-REM results with $N=128$ (points). Data at large times are fitted by $\Pi(t_{\rm w}) = H \left( 1 + A \, t_{\rm w}^{-\alpha} \right) - k \ln t_{\rm w}$ (lines). (d) At $T=0.75$, the G-REM data are dominated by logarithmic decay $\Pi(t_{\rm w}) \sim - \frac{B}{N} \ln t_{\rm w}$ (lines) in small systems, where $B=0.7$ is a fitting parameter. The horizontal dashed lines represent the GTM plateau $H(w,\hat{x}) \approx 0.854$ and the BTM plateau $H(w,x) \approx 0.136$. The fitting parameters $H$ and $\alpha$ for both models are plotted in Fig. \ref{['fig:theory']}(a,b).
• Figure 3: Barrier energy distributions in the G-REM and E-REM. (a) A sub-tree formed by local minima and saddle points for the $N=6$ G-REM obtained by the BT algorithm, with corresponding spin configurations and energies ($E$-axis) indicated. (inset) Illustration of two traps: the blue nodes are non-minimum-non-saddle configurations; the barrier energy $\Delta E$ is the energy difference between the local minimum and saddle point. (b) Distributions $p_{\rm G-REM}^{\rm BT}(\Delta E)$ and $p_{\rm E-REM}^{\rm BT}(\Delta E)$ of the G-REM and E-REM by the BT algorithm ($N=20$), compared to $p_{\rm BTM}(\Delta E)$ (Eq. \ref{['eq:BTM']}) and $p_{\rm GTM}(\Delta E)$ (Eq. \ref{['eq:GTM']} with $a=1.12$ and $b=0.96$). (c) Remainder $\ln p(\Delta E) + \Delta E/T_{\rm c}$ for the G-REM (black) and E-REM (red). Open and filled points are obtained by the BT algorithm and TE theory respectively. Lines represent fitting to a Gaussian function. The variance $\sigma^2$ and mean $\bar{E}$ obtained by the Gaussian fitting are plotted in (d) as functions of $N$. Linear fitting of the data in (d) gives $a=1.12$ and $b=0.96$ ($\sigma^2=aN$ and $\bar{E}=bN$).
• Figure 4: WEB in (a,b) E-REM and (c,d) G-REM. In (a,c), $\Pi_w(t_{\rm w})$ is obtained with $N=20$ and $w=0.5$ by MC simulations at the rescaled time $\tilde{t}_{\rm w} = C t_{\rm w}$ with $C = \frac{w \hat{x}}{1+\hat{x}}$. The solid line is $\Pi^{0}_w(\tilde{t}_{\rm w})=1-\tilde{t}_{\rm w}$ (see Eq. \ref{['eq:scaling_GTM']}). The dotted horizontal line is $\Pi_{\rm thr}=0.2$. The time scales $\tilde{\tau}^0$ (circle) and $\tilde{\tau}$ (cross) are marked. (b,d) $R=\tilde{\tau}/\tilde{\tau}^0$ as a function of $T$ for different $N$. (inset) $R$ as a function of $N$ at $T=1.17 \, T_{\rm c}$.
• Figure 5: Aging in (a-c) WCA and (d-f) amorphous silica models. (a) $F_w(t_{\rm w})$ obtained by MD simulations of the WCA model, with $N=16000$. The data in the intermediate time regime are fitted to $F_w(t_{\rm w}) = \tilde{H} \left( 1 + A \, t_{\rm w}^{-\alpha} \right) - k \ln t_{\rm w}$ (solid lines). At the two lowest $T$, $\tilde{H}$ is estimated by fitting the plateau (dashed lines), and $\alpha$ cannot be determined due to the presence of a dip. (b) $F_w(t_{\rm w})$ at $T=0.26$ with different $N$. Inset is the close-up for the data in $1 < t_{\rm w} < 10^3$, fitted to $F_w(t_{\rm w}) \sim -k \ln t_{\rm w}$ (lines). The parameter $k$ obtained in this way is plotted in (c) as a function of $N$ at different $T$. The crossover size $n$ is determined by the intersect of two linear fitting lines (when $T<0.35$). (d-e) Similar plots for amorphous silica (${\rm SiO}_2$). In (d,f), $N=31944$; in (e), $T=4000$ K. In all panels, $w=1/2$. The parameters $H$ and $\alpha$ are plotted in Fig. \ref{['fig:theory']}(a,b) for both models.