Non-equilibrium dynamics of spin facilitated glass models

We consider the dynamics of spin facilitated models of glasses in the nonequilibrium aging regime following a sudden quench from high to low temperatures. We briefly review known results obtained for the broad class of kinetically constrained models, and then present new results for the behaviour of the one-spin facilitated Fredrickson-Andersen and East models in various spatial dimensions. The time evolution of one-time quantities, such as the energy density, and the detailed properties of two-time correlation and response functions are studied using a combination of theoretical approaches, including exact mappings of master operators and reductions to integrable quantum spin chains, field theory and renormalization group, and independent interval and timescale separation methods. The resulting analytical predictions are confirmed by means of detailed numerical simulations. The models we consider are characterized by trivial static properties, with no finite temperature singularities, but they nevertheless display a surprising variety of dynamic behaviour during aging, which can be directly related to the existence and growth in time of dynamic lengthscales. Well-behaved fluctuation-dissipation ratios can be defined for these models, and we study their properties in detail. We confirm in particular the existence of negative fluctuation-dissipation ratios for a large number of observables. Our results suggest that well-defined violations of fluctuation-dissipation relations, of a purely dynamic origin and unrelated to the thermodynamic concept of effective temperatures, could in general be present in non-equilibrium glassy materials. PACS numbers: 05.70.Ln, 05.40.-a, 64.70.Pf, 75.40.Gb Non-equilibrium dynamics of spin facilitated models 2 1. Why study the aging regime of spin facilitated models? 1.1. A brief survey of kinetically constrained models This paper is concerned with the dynamics of spin facilitated models of glasses in the non-equilibrium aging regime following a sudden quench from high temperature to the very low temperature glassy regime. Spin facilitated models belong to the broader family of kinetically constrained models (KCMs). These are simple statistical mechanics models which display many of the dynamical features observed in real glassy materials, such as supercooled liquids [1], spin glasses [2], or soft disordered materials [3]. KCMs are generically defined from a simple, usually non-interacting, Hamiltonian. The complexity of glasses is encoded in specific local dynamical rules, or kinetic constraints. For an extensive review of early results on KCMs see [4]. In this paper we focus on spin facilitated models, in particular Fredrickson-Andersen (FA) [5] and East models [6], but we expect that similar behaviour to the one we describe here will also be found in other KCMs such as constrained lattice gases. The main insight of FA [5] was to devise models that are simplistic, as compared to realistic interacting molecular systems, but whose macroscopic behaviour was in agreement with the phenomenology of liquids approaching the glass transition [5, 7], displaying a super-Arrhenius increase of relaxation timescales on decreasing the temperature and non-exponential relaxation functions at equilibrium. Early studies also demonstrated that when suddenly quenched to very low temperatures the subsequent non-equilibrium aging dynamics of the models compares well with experimental observations on aging liquids [8, 9]. Initially it was suggested that FA models would display finite temperature dynamic transitions similar to the one predicted by the mode-coupling theory of supercooled liquids [5], but it was soon realized that most KCMs do not display such singularity, and timescales in fact only diverge in the limit of zero temperature [4, 10, 11]. This implies in particular that after a quench to any non-zero temperature these systems are eventually able to reach thermal equilibrium. The time window of the aging regime, however, becomes large at low temperatures, and the detailed study of this far from equilibrium aging time regime will be the main subject of this paper. There has been much interest in KCMs recently. This is partly due to the realization [12, 13, 14, 15, 16, 17, 18] that their dynamics is characterized by dynamic heterogeneity, that is, non-trivial spatio-temporal fluctuations of the local relaxation, which is also a hallmark of supercooled liquids [19]. Many of the studies relating to dynamic heterogeneity in KCMs are very recent indeed, having appeared since the review [4] was compiled. These studies characterize in great detail the heterogeneous dynamics of KCMs. They include: papers defining and quantifying relevant dynamic lengthscales in KCMs in equilibrium and their relation to relaxation timescales [10, 11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]; studies of more qualitative or phenomenological consequences of kinetic constraints, including dynamic heterogeneity and activated dynamics, for the physics of glassy systems [31, 32, 33, 34, 35, 36]; and the definition and analysis of new KCMs where kinetic rules, or lattice geometry, are Non-equilibrium dynamics of spin facilitated models 3 tuned to explore in more detail the range of possible behaviours that can be observed in glass models [37, 38, 39, 40, 41, 42]. These numerous recent studies have in turn been instrumental in encouraging numerical and experimental efforts to measure and characterize in more detail dynamic heterogeneity in supercooled liquids [43, 44, 45, 46, 47, 48, 49, 50], granular materials [51, 52], colloidal systems [50, 53, 54, 55], and soft glassy materials [56, 57, 58, 59]. 1.2. Aging dynamics in glassy systems: early studies and open questions The previous section suggests that dynamic heterogeneity and activated dynamics, which have been well-studied and characterized at thermal equilibrium in various models and systems approaching the glass transition, play central roles in glassy dynamics. However, when moving deeper into the glass phase, glassy materials cannot be equilibrated anymore on experimental or numerical timescales. In this non-equilibrium state, physical properties are not stationary, and the system displays aging behaviour [2]. Experimentally, aging has been well studied at the macroscopic level, in systems as diverse as polymers [60], structural and spin glasses [2], and soft materials [3]. A full understanding of the non-equilibrium glassy state remains a central theoretical challenge [2]. Theoretical studies of mean-field models have provided important insights into the aging dynamics of both structural and spin glasses [2, 61, 62]. In mean-field models, thermal equilibrium is never reached, and aging proceeds by downhill motion in an increasingly flat free energy landscape [63]. Time translational invariance is broken, and two-time correlation and response functions depend on both their arguments. The fluctuation-dissipation theorem (FDT), which relates equilibrium correlation and response functions, does not apply in the aging regime, but a generalized form is shown to hold [64]. This is defined in terms of the two-time connected correlation function for some generic observable A(t), C(t, tw) = 〈A(t)A(tw)〉 − 〈A(t)〉〈A(tw)〉, (1) with t ≥ tw, and the corresponding two-time (impulse) response function R(t, tw) = T δ〈A(t)〉 δh(tw) ∣∣∣∣ h=0 . (2) Here h denotes the thermodynamically conjugate field to the observable A so that the perturbation to the Hamiltonian (or energy function) is δE = −hA, and angled brackets indicate an average over initial conditions and any stochasticity in the dynamics. Note that we have absorbed the temperature T in the definition of the response. The associated generalized FDT is then R(t, tw) = X(t, tw) ∂ ∂tw C(t, tw), (3) with X(t, tw) the so-called fluctuation-dissipation ratio (FDR). At equilibrium, correlation and response functions are time translation invariant, depending only on Non-equilibrium dynamics of spin facilitated models 4 τ = t− tw, and equilibrium FDT imposes that X(t, tw) = 1 at all times. A parametric fluctuation-dissipation (FD) plot of the step response or susceptibility