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WORLD SCIENTIFIC eBooks

Dynamical Mean-Field Theory and the Aging Dynamics

A. Crisanti, Silvio Franz, Jorge Kurchan B, A. Maiorano

Published Aug 1, 2023• 3 citations
Publisher

Abstract

The study of the equilibrium and the out-of-equilibrium dynamics of spin glasses is crucial to understand some of the characterizing properties of disordered systems, including slow relaxation times and aging. The current chapter focuses on recent developments on this relevant topic. In the first part of the chapter, Andrea Crisanti discusses the relation between the equilibrium dynamics of mean-field spin glasses and RSB, briefly reviewing the pioneering ideas of Sommers and Sompolinsky and the subsequent theory by Crisanti, Horner and Sommers. Jorge Kurchan and Silvio Franz focus on the slow out-of-equilibrium glassy dynamics, in both mean-field models and finite-dimensional models, connecting the RSB scheme of Parisi with the concept of multithermalization. Finally, Andrea Maiorano overviews recent results on the presence (and absence) of weak ergodicity breaking in mean-field spin glasses, a pivotal assumption in many theoretical studies focusing on aging. © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

DOI
10.1142/9789811273926_0009

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The study of the equilibrium and the out-of-equilibrium dynamics of spin glasses is crucial to understand some of the characterizing properties of disordered systems, including slow relaxation times and aging. The current chapter focuses on recent developments on this relevant topic. In the first part of the chapter, Andrea Crisanti discusses the relation between the equilibrium dynamics of mean-field spin glasses and RSB, briefly reviewing the pioneering ideas of Sommers and Sompolinsky and the subsequent theory by Crisanti, Horner and Sommers. Jorge Kurchan and Silvio Franz focus on the slow out-of-equilibrium glassy dynamics, in both mean-field models and finite-dimensional models, connecting the RSB scheme of Parisi with the concept of multithermalization. Finally, Andrea Maiorano overviews recent results on the presence (and absence) of weak ergodicity breaking in mean-field spin glasses, a pivotal assumption in many theoretical studies focusing on aging. © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
Glassy materials are characterized by very slow dynamics, associated with a dramatic slowdown of molecular relaxation in structural glasses, and with a dramatic slowdown of spin relaxation in spin glasses. This slowdown of the dynamics has been captured in great part by the results obtained by dynamical mean field theories. In the case of supercooled liquids, the mean-field mode-coupling approach 1 has had some success in describing various features of the relaxation. In the case of spin glasses, a dynamical mean field theory based on a Langevin dynamics for the spins, examined within a functional integral formulation of the Martin-Siggia-Rose approach 2,3,4,5 , has been used to study the long-time relaxation. The dynamical mean field theory of spin glasses has successfully captured 3,4,5 some unusual properties of the spin dynamics, associated with the lack of equilibration, including the presence of physical aging and the breakdown of the equilibrium fluctuation-dissipation relations. However, mean field theories do not allow direct access to a description of the fluctuations in the dynamics. It turns out that fluctuations in the dynamics of glassy systems can in fact be rather strong, as it has been underlined by the discovery of dynamical heterogeneities 6,7 . Dynamical heterogeneities are nanometer-scale regions of molecules
We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical 2+p2+p2+p spin-glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, H\"orner, and Sommers [Z. Phys. B: Condens. Matter 92, 257 (1993)] dynamical solution of the spherical ppp-spin spin-glass model, reproducing a stable static limit that coincides, in the case of a one step replica symmetry breaking ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off equilibrium. We formally extend our analysis to any number of replica symmetry breakings RRR. In the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, both formulations lead to the Parisi antiparabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The formulation does not contain the additional order parameter $\ensuremath{\Delta}$ of the Sompolinsky theory.
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
We present a numerical study of the nonequilibrium dynamics of the Sherrington-Kirkpatrick model. We analyze the behavior of the system for different system sizes N and time scales t, in order to identify the correct out-of-equilibrium regime. We find evidence of aging behavior in the dynamics of the system, compatible with a weak-ergodicity breaking scenario. In addition, we obtain a pictorial characterization of the out-of-equilibrium phase space that presents some analogies with the hierarchy of states found in equilibrium.