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 (DMFEs). In this Letter, we introduce a new integration scheme for solving DMFEs 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 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.

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• Nonequilibrium statistical mechanics
• Disordered systems
• Glasses
• Spin glasses
• Complex Langevin dynamics