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arXiv:cond-mat/0511486 · Physical Review Letters

Dynamic Compressibility and Aging in Wigner Crystals and Quantum Glasses

Leticia F. Cugliandolo, Thierry Giamarchi, Pierre Le Doussal

Published Jun 1, 2006• 17 citations
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Abstract

We study the nonequilibrium linear response of quantum elastic systems pinned by quenched disorder with Schwinger-Keldysh real-time techniques complemented by a mean-field variational approach. We find (i) a quasiequilibrium regime in which the analytic continuation from the imaginary-time replica results holds provided the marginality condition is enforced, and (ii) an aging regime. The conductivity and compressibility are computed. The latter is found to cross over from its dynamic to static value on a scale set by the waiting time after a quench, an effect which can be probed in experiments in, e.g., Wigner glasses.

DOI
10.1103/physrevlett.96.217203

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We study the nonequilibrium linear response of quantum elastic systems pinned by quenched disorder with Schwinger-Keldysh real-time techniques complemented by a mean-field variational approach. We find (i) a quasiequilibrium regime in which the analytic continuation from the imaginary-time replica results holds provided the marginality condition is enforced, and (ii) an aging regime. The conductivity and compressibility are computed. The latter is found to cross over from its dynamic to static value on a scale set by the waiting time after a quench, an effect which can be probed in experiments in, e.g., Wigner glasses.
We apply the Gaussian variational method (GVM) to study the equilibrium statistical mechanics of two related systems: (i) classical elastic manifolds, such as flux lattices, in the presence of columnar disorder correlated along the \ensuremath{\tau} direction, and (ii) interacting quantum particles in a static random potential. We find localization by disorder, the localized phase being described by a replica-symmetry-broken solution confined to the mode \ensuremath{\omega}=0. For classical systems we compute the correlation function of relative displacements. In d=2+1, in the absence of dislocations, the GVM allows one to describe the Bose glass phase. Along the columns the displacements saturate at a length ${\mathit{l}}_{\mathrm{\ensuremath{\perp}}}$, indicating flux-line localization. Perpendicularly to the columns long-range order is destroyed. We find a divergent tilt modulus c44{\mathit{c}}_{44}c44​=\ensuremath{\infty} and a x\ensuremath{\sim}${\mathrm{\ensuremath{\tau}}}^{1/2}$ scaling. Quantum systems are studied using the analytic continuation from imaginary to real time \ensuremath{\tau}\ensuremath{\rightarrow}it. We compute the conductivity and find that it behaves at small frequency as \ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}^{2}$ in all dimensions (d4) for which disorder is relevant. We compute the quantum localization length \ensuremath{\xi}. In d=1, where the model also describes interacting fermions in a static random potential, we find a delocalization transition and obtain analytically both the low- and high-frequency behavior of the conductivity for any value of the interaction. We show that the marginality condition appears as the condition to obtain the correct physical behavior. Agreement with renormalization group results is found whenever it can be compared. \textcopyright{} 1996 The American Physical Society.
We set the formalism to study the way in which the choice of canonical equilibrium initial conditions affect the real-time dynamics of quantum disordered models. We use a path integral formulation on a time contour with real and imaginary time branches. The factorisation of the time-integration paths usually assumed in field-theoretical studies breaks down due to the averaging over quenched randomness. We derive the set of Schwinger-Dyson dynamical equations that govern the evolution of linear response and correlation functions. The solution of these equations is not straightforward as it needs, as an input, the full imaginary-time (or Matsubara frequency) dependence of the correlation in equilibrium. We check some limiting cases (equilibrium dynamics, classical limit) and we set the stage for the analytic and numerical analysis of quenches in random quantum systems.
Abstract We set the formalism to study the way in which the choice of canonical equilibrium initial conditions affect the real-time dynamics of quantum disordered models. We use a path integral formulation on a time contour with real and imaginary time branches. The factorisation of the time-integration paths usually assumed in field-theoretical studies breaks down due to the averaging over quenched randomness. We derive the set of Schwinger–Dyson dynamical equations that govern the evolution of linear response and correlation functions. The solution of these equations is not straightforward as it needs, as an input, the full imaginary-time (or Matsubara frequency) dependence of the correlation in equilibrium. We check some limiting cases (equilibrium dynamics, classical limit) and we set the stage for the analytic and numerical analysis of quenches in random quantum systems.
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 develop a systematic analytic approach to aging effects in quantum disordered systems in contact with an environment. Within the closed-time path-integral formalism we include dissipation by coupling the system to a set of independent harmonic oscillators that mimic a quantum thermal bath. After integrating over the bath variables and averaging over disorder we obtain an effective action that determines the real-time dynamics of the system. The classical limit yields the Martin-Siggia-Rose generating functional associated to a colored noise. We apply this general formalism to a prototype model related to the p spin glass. We show that the model has a dynamic phase transition separating the paramagnetic from the spin-glass phase and that quantum fluctuations depress the transition temperature until a quantum critical point is reached. We show that the dynamics in the paramagnetic phase is stationary but presents an interesting crossover from a region controlled by the classical critical point to another one controlled by the quantum critical point. The most characteristic property of the dynamics in a glassy phase, namely, aging, survives the quantum fluctuations. In the subcritical region the quantum fluctuation-dissipation theorem is modified in a way that is consistent with the notion of effective temperatures introduced for the classical case. We discuss these results in connection with recent experiments in dipolar quantum spin glasses and the relevance of the effective temperatures with respect to the understanding of the low-temperature dynamics.
Using the renormalization group method we investigate the nonequilibrium relaxation of the (Cardy-Ostlund) 2D random sine-Gordon model, which describes pinned arrays of lines. Its statics exhibit a marginal (theta = 0) glass phase for T < Tg described by a line of fixed points. We obtain the universal scaling functions for two-time dynamical response and correlations near Tg for various initial conditions, as well as the autocorrelation exponent. The fluctuation dissipation ratio is found to be nontrivial and continuously dependent on T.