Complexity of energy barriers in mean-field glassy systems

Abstract

We analyze the energy barriers that allow escapes from a given local minimum in a mean-field model of glasses. We perform this study by using the Kac-Rice method and computing the typical number of critical points of the energy function at a given distance from the minimum. We analyze their Hessian in terms of random matrix theory and show that for a certain regime of energies and distances critical points are index-one saddles and are associated to barriers. We find that the lowest barrier, important for activated dynamics at low temperature, is strictly lower than the "threshold" level above which saddles proliferate. We characterize how the quenched complexity of barriers, important for activated process at finite temperature, depends on the energy of the barrier, the energy of the initial minimum, and the distance between them. The overall picture gained from this study is expected to hold generically for mean-field models of the glass transition. arXiv:1809.05440v1 [cond-mat.dis-nn]

Dynamical instantons and activated processes in mean-field glass models

“…Similarly, the analysis of Ref. [22] reveals that for each choice of q, ε 2 , the energy landscape is dominated by one specific type of stationary points that are either local minima or index-1 saddles, see Fig. 2.…”

“…We focus on the energy landscape associated to the p-spin spherical model: (1) has been the subject of an extensive amount of research devoted to understanding its statistical properties, which started with the earlier investigations [26][27][28][29][30][31] and culminated in the most recent results [22,23,32,33]. These works highlighted a peculiar organization of the landscape stationary points in terms of their energy density ε = E/N and of their stability: while at large value of the energy the landscape is dominated by saddles with a huge index (i.e., number of unstable directions), the local minima and low-index saddles concentrate at the bottom of the landscape, below a critical threshold value of the energy density ε th .…”

“…In particular, it is important to scan the landscape in the vicinity of any of its stationary points. This information is accessible via large deviation techniques by computing the complexity of the stationary points constrained to be at fixed, non-zero overlap from the reference stationary point [22][23][24]. In Ref.…”

“…In Ref. [22] we computed the complexity of the typical stationary points that are found in the vicinity of a reference minimum, extracted uniformly from the ensemble of minima having a given energy density larger than the ground state ε gs and smaller than the threshold ε th . Henceforth we denote with s 1 the reference minimum, and with s 2 a stationary point at overlap…”

“…Here we provide the first computation of such dynamical instanton and characterize the properties of the new minima reached after barrier-crossing. In order to achieve this goal we make use of the results we obtained recently on the number of the stationary points constrained to be at fixed overlap q (or distance d) from a given minimum in a prototypical energy landscape [22,23] (see also [24] for a related analysis). These studies showed that, given an arbitrary local minimum s 1 of the energy landscape with energy density ε 1 , the landscape in its vicinity is populated by index-1 saddles, that constitute available escape states when the system is trapped in s 1 .…”