Strong ergodicity breaking in aging of mean-field spin glasses.

Out-of-equilibrium relaxation processes show aging if they become slower as time passes. Aging processes are ubiquitous and play a fundamental role in the physics of glasses and spin glasses and in other applications (e.g., in algorithms minimizing complex cost/loss functions). The theory of aging in the out-of-equilibrium dynamics of mean-field spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However, this solution is based on assumptions (e.g., the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work, we present the results of an extraordinary large set of numerical simulations of the prototypical mean-field spin glass models, namely the Sherrington-Kirkpatrick and the Viana-Bray models. Thanks to a very intensive use of graphics processing units (GPUs), we have been able to run the latter model for more than [Formula: see text] spin updates and thus safely extrapolate the numerical data both in the thermodynamical limit and in the large times limit. The measurements of the two-times correlation functions in isothermal aging after a quench from a random initial configuration to a temperature [Formula: see text] provides clear evidence that, at large times, such correlations do not decay to zero as expected by assuming weak ergodicity breaking. We conclude that strong ergodicity breaking takes place in mean-field spin glasses aging dynamics which, asymptotically, takes place in a confined configurational space. Theoretical models for the aging dynamics need to be revised accordingly.

Magnetic field chaos in the Sherrington-Kirkpatrick model.

We study the Sherrington-Kirkpatrick model, both above and below the de Almeida-Thouless line, by using a modified version of the Parallel Tempering algorithm in which the system is allowed to move between different values of the magnetic field h. The behavior of the probability distribution of the overlap between two replicas at different values of the magnetic field h(0) and h(1) gives clear evidence for the presence of magnetic field chaos already for moderate system sizes, in contrast to the case of temperature chaos, which is not visible on system sizes that can currently be thermalized.

Numerical study of the Sherrington-Kirkpatrick model in a magnetic field.

We study numerically the Sherrington-Kirkpatrick model as a function of the magnetic field h, with fixed temperature T=0.6T(c). We investigate the finite size scaling behavior of several quantities, such as the spin-glass susceptibility, searching for numerical evidences of the transition on the de Almeida-Thouless line. We find strong corrections to scaling which make difficult to locate the transition point. This shows, in a simple case, the extreme difficulties of spin-glass simulations in a nonzero magnetic field. Next, we study various sum rules (consequences of stochastic stability) involving overlaps between three and four replicas, which appear to be numerically well satisfied, and in a nontrivial way. Finally, we present data on P(q) for a large lattice size (N=3200) at low temperature T=0.4T(c), where the shape predicted by the replica symmetry breaking solution of the model for a nonzero magnetic field is visible.

Overlap distribution of the three-dimensional Ising model.

We study the Parisi overlap probability density P(L)(q) for the three-dimensional Ising ferromagnet by means of Monte Carlo (MC) simulations. At the critical point, P(L)(q) is peaked around q=0 in contrast with the double peaked magnetic probability density. We give particular attention to the tails of the overlap distribution at the critical point, which we control over up to 500 orders of magnitude by using the multioverlap MC algorithm. Below the critical temperature, interface tension estimates from the overlap probability density are given and their approach to the infinite volume limit appears to be smoother than for estimates from the magnetization.

Functional form of the Parisi overlap distribution for the three-dimensional Edwards-Anderson Ising spin glass.

Recently, it has been conjectured that the statistics of extremes is of relevance for a large class of correlated systems. For certain probability densities this predicts the characteristic large x falloff behavior f(x) approximately exp(-ae(x)), a>0. Using a multicanonical Monte Carlo technique, we have measured the Parisi overlap distribution P(q) for the three-dimensional Edward-Anderson Ising spin glass at and below the critical temperature We find that a probability distribution related to extreme-order statistics gives an excellent description of P(q) over about 80 orders of magnitude.