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arXiv:cond-mat/0109150 · Physical Review Letters

Separation of Time Scales and Reparametrization Invariance for Aging Systems

Claudio Chamon, Malcolm P. Kennett, Horacio E. Castillo, Leticia F. Cugliandolo

Published Oct 31, 2002• 72 citations
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Abstract

We show that the generating functional describing the slow dynamics of spin-glass systems is invariant under reparametrizations of the time. This result is general and applies for both infinite and short-range models. It follows simply from the assumption that a separation between short time scales and long time scales exists in the system, and from the constraints of causality and unitarity. Global-time reparametrization invariance suggests that the low action excitations in a spin-glass may be smoothly spatially varying time reparametrizations. These Goldstone modes may provide the basis for an analytic dynamical theory of short-range spin glasses.

DOI
10.1103/physrevlett.89.217201

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We show that the generating functional describing the slow dynamics of spin-glass systems is invariant under reparametrizations of the time. This result is general and applies for both infinite and short-range models. It follows simply from the assumption that a separation between short time scales and long time scales exists in the system, and from the constraints of causality and unitarity. Global-time reparametrization invariance suggests that the low action excitations in a spin-glass may be smoothly spatially varying time reparametrizations. These Goldstone modes may provide the basis for an analytic dynamical theory of short-range spin glasses.
We study the long-time aging dynamics of spin-glass models with two-spin interactions by performing a renormalization group (RG) transformation on the time variable in the nonequilibrium dynamical generating functional. We obtain the RG equations and find that the flow converges to an exact fixed point. We show that this fixed point is invariant under reparametrizations of the time variable. This continuous symmetry is broken, as evidenced by the fact that the observed correlations and responses are not invariant under it. We argue that this gives rise to the presence of Goldstone modes, and that those Goldstone modes shape the behavior of fluctuations in the nonequilibrium dynamics.
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
Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations. Spin glasses (SG) exhibit characteristic slow dynamics below the SG transition temperature. 1)-3) Recently such slow dynamics, in particular aging phenomena, has been much attractive both in experimental and theoretical studies. Several attempts have been made so far to explain the slow dynamics. There are mainly two distinct
The deviations from full or pure aging behavior, i.e. perfect t/twt/t_wt/tw​ scaling of the correlation and response functions of aging glassy systems, are not well understood theoretically. Recent experiments of Rodriguez et al. (Phys. Rev. Lett. 91, 037203 (2003)) have shown that full aging applies to the thermoremanent magnetization in the limit of infinite cooling rate during the initial quench. In numerical models, instantaneous thermal quenches can be--and usually are--applied. This has motivated the present numerical investigations of the aging behavior of Edwards-Anderson Ising spin glass models with both short- and long-range Gaussian interactions, respectively. We sample the distribution of residence time ttt in suitably defined metastable valleys entered at age twt_wtw​, finding that the deviations from t/twt/t_wt/tw​ scaling are small and decrease systematically as the system size grows and/or the temperature decreases. Finally, the connection between this behavior and the scaling of the correlation function itself is discussed.
We study the long time dynamics of quantum spin glasses of rotors using the nonequilibrium Schwinger-Keldysh formalism. These models are known to have a quantum phase transition from a paramagnetic to a spin-glass phase, which we approach by looking at the divergence of the spin-relaxation rate at the transition point. In the aging regime, we determine the dynamical equations governing the time evolution of the spin response and correlation functions, and show that all terms in the equations that arise solely from quantum effects are irrelevant at long times under time reparametrization group (RPG) transformations. At long times, quantum effects enter only through the renormalization of the parameters in the dynamical equations for the classical counterpart of the rotor model. Consequently, quantum effects only modify the out-of-equilibrium fluctuation-dissipation relation (OEFDR), i.e. the ratio X between the temperature and the effective temperature, but not the form of the classical OEFDR.
We show that the algebra of Parisi ultrametric matrices is recovered by the real-time, replica-free, Dyson-Keldysh equations of infinite-range quantum spin glasses in the late time glassy limit. This connects to earlier results on classical and quantum systems showing how ultrametricity emerges from the persistent slow aging dynamics of the glass phase. The stationary spin glass state thereby spontaneously breaks thermal symmetry, or the Kubo-Martin-Schwinger relation of a state in global thermal equilibrium. We describe the Keldysh path integral of the infinite-range Ising model in transverse and longitudinal fields, and in the context of the Landau expansion of the action functional, show how the long-time limit connects to the full replica symmetry breaking obtained in the equilibrium formalism. We also illustrate our formalism by applying it to the spherical quantum p p -spin model, which only exhibits one-step replica symmetry breaking.