Strong ergodicity breaking in dynamical mean-field equations for mixed p-spin glasses

Abstract

The analytical solution to the out-of-equilibrium dynamics of mean-field spin glasses has profoundly shaped our understanding of glassy dynamics, which take place in many diverse physical systems. In particular, the idea that during the aging dynamics, the evolution becomes slower and slower, but keeps wandering in an unbounded space (a manifold of marginal states), thus forgetting any previously found configuration, has been one of the key hypotheses to achieve an analytical solution. This hypothesis, called weak ergodicity breaking, has recently been questioned by numerical simulations and attempts to solve the dynamical mean-field equations (DMFE). In this work, we introduce a new integration scheme for solving DMFE that allows us to reach very large integration times, $t=O(10^6)$, in the solution of the spherical 3+4-spin model, quenched from close to the mode coupling temperature down to zero temperature. Thanks to this new solution, we can provide solid evidence for \emph{strong ergodicity breaking} in the out-of-equilibrium dynamics on mixed $p$-spin glass models. Our solution to the DMFE shows that the out-of-equilibrium dynamics undergo aging, but in a restricted space: the initial condition is never forgotten, and the dynamics take place closer and closer to configurations reached at later times. During this new restricted aging dynamics, the fluctuation-dissipation relation is richer than expected.

Figures (6)

• Figure 1: The grid in the $(t,t')$ plane used to integrate the DMFE in the present work is a subset of a regular grid of step $\Delta t$. Actually, only the half with $t'\le t$ is evaluated. We choose the integration grid to be dense in the bands close to $t'=0$ and $t'=t$, as there the gradients are larger. The grid becomes sparser and sparser with increasing time, away from the dense bands.
• Figure 2: Decay of the correlation with the initial condition $C(t,0)$, the evolution of the response $R(t,0)$ and energy relaxation $E(t)$ as a function of time. All the plots are obtained for the $(3+4)$--spin model starting at a temperature $T=1.001\,T_\text{\tiny MCT}$ ($T_\text{\tiny MCT}=0.805166$) and quenching to zero. The new integration scheme allows us to reach much longer times than in previous works.
• Figure 3: The correlation $C(t+t_\text{w},t_\text{w})$ as a function of the rescaled time $t/t_\text{w}$ for a mixed $(3+4)$--spin model started at $T=1.001\,T_\text{\tiny MCT}$ and quenched to $T_f=0$. The curves are for different waiting times $t_\text{w}=2^n$. The system ages in a region of the configurational space, which shrinks as time passes.
• Figure 4: Breakdown of the fluctuation-dissipation relation in a mixed $(3+4)$--spin model started at $T=1.001\,T_\text{\tiny MCT}$ and quenched to $T_f=0$. The gray region is inaccessible to the system dynamics as it is below the lowest achievable correlation $C(\infty,0)=0.658$. The curves are for different times $t=2^n$ and are plotted parametrically in $t'\in[0,t]$. Three regimes with different effective temperatures are visible in the plot. Data with the largest times are approaching the asymptotic fluctuation-dissipation curve (see inset).
• Figure 5: Decay of the correlation $C(t,0)$ in a zero temperature relaxation, starting from equilibrium at $T=1.001\,T_\text{\tiny MCT}$ (same data shown in the upper panel of Fig. \ref{['fig:1']}). The data marked with a blue thick line are those available prior to the present work, and a power law fit to the last decade of these data (dashed line) makes the WEB scenario hard to exclude. The new data collected in the present work extend to $t \sim O(10^6)$ and make the SEB scenario sure, without any extrapolation.