The simplest spin glass revisited: finite-size effects of the energy landscape can modify aging dynamics in the thermodynamic limit

Abstract: The random energy model is one of the few glass models whose asymptotic activated aging dynamics are solvable. However, the existing aging theory, i.e., Bouchaud's trap model, does not agree with dynamical simulation results obtained in finite-sized systems. Here we show that this discrepancy originates from non-negligible finite-size corrections in the energy barrier distributions. The finite-size effects add a logarithmic decay term in the time-correlation aging function, which destroys the asymptotic large-time plateau predicted by Bouchaud's trap model in the spin glass phase. Surprisingly, the finite-size effects also give corrections, preserved even in the thermodynamic limit, to the value of the asymptotic plateau. It results in an unexpected dynamical transition where weak ergodicity breaking occurs, at a temperature $T_{\rm d}$ above the thermodynamic spin-glass transition temperature $T_{\rm c}$. Based on the barrier distributions obtained by a numerical barrier-tree method and an expansion theory, we propose a generalized trap model to incorporate such finite-size effects. The theoretically derived aging behavior of the generalized trap model explains the Monte-Carlo dynamical simulation data of random energy models with Gaussian and exponential random energies. Our results suggest that the double limits of large system size and long time are not interchangeable for the activated aging dynamics.