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.
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.
Using a functional renormalization group (RG) method, we reexamine the glass phase of the two-dimensional random-field sine Gordon model. It is described by a line of fixed points with a super-roughening amplitude $\overline{[u(0)\ensuremath{-}u(r){]}^{2}}\ensuremath{\sim}A(T){\mathrm{ln}}^{2}r$ as temperature T is varied. By speculating that this line is identical for all T to the one found in disordered free-fermion models via exact results from ``nearly conformal'' field theory, one would predict A(T=0)=0, contradicting numerics. We point out that this result may be related to failure of dimensional reduction, and that a functional RG method incorporating higher harmonics and nonanalytic operators predicts a nonzero A(T=0) which compares reasonably well with numerics.
We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.
We study two-dimensional triangular elastic lattices in a background of point disorder, excluding dislocations (tethered network). Using both (replica symmetric) static and (equilibrium) dynamic renormalization group (RG) for the corresponding N=2 component model, we find a transition to a glass phase for TTg, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with uT2\ensuremath{\sim}uL2\ensuremath{\sim}A1ln 2r, with universal subdominant anisotropy uT2-uL2\ensuremath{\sim}A2ln r, where A1 and A2 depend continuously on temperature and the Poisson ratio \ensuremath{\sigma}. We also obtain the continuously varying dynamical exponent z. For the Cardy-Ostlund N=1 model, a particular case of the above model, we point out a discrepancy in the value of A1 with other published results in the literature. We find that our result reconciles the order of magnitude of the RG predictions with the most recent numerical simulations.
We study thermally activated, low-temperature equilibrium dynamics of elastic systems pinned by disorder using one loop functional renormalization group (FRG). Through a series of increasingly complete approximations, we investigate how the field theory reveals the glassy nature of the dynamics, in particular divergent barriers and barrier distributions controling the spectrum of relaxation times. First, we naively assume a single relaxation time tau(k) for each wave vector k, leading to analytical expressions for equilibrium dynamical response and correlations. These exhibit two distinct scaling regimes (scaling variables T k(theta) ln t and t/ tau(k), respectively, with T the temperature, theta the energy fluctuation exponent, and tau(k) approximately e(c k(-theta) /T) ) and are easily extended to quasiequilibrium and aging regimes. A careful study of the dynamical operators encoding for fluctuations of the relaxation times shows that this first approach is unsatisfactory. A second stage of approximation including these fluctuations, based on a truncation of the dynamical effective action to a random friction model, yields a size (L) dependent log-normal distribution of relaxation times (effective barriers centered around Ltheta and of fluctuations approximately L(theta/2) ) and some procedure to estimate dynamical scaling functions. Finally, we study the full structure of the running dynamical effective action within the field theory. We find that relaxation time distributions are nontrivial (broad but not log normal) and encoded in a closed hierarchy of FRG equations divided into levels p=0,1, em leader, corresponding to vertices proportional to the pth power of frequency omega(p). We show how each level p can be solved independently of higher ones, the lowest one (p=0) comprising the statics. A thermal boundary layer ansatz (TBLA) appears as a consistent solution. It extends the one discovered in the statics which was shown to embody droplet thermal fluctuations. Although perturbative control remains a challenge, the structure of the dynamical TBLA which encodes barrier distributions opens the way for deeper understanding of the field theory approach to glasses.
We study the nonequilibrium aging behavior of the gauge glass model in three dimensions at the critical temperature. We perform Monte Carlo simulations with a Metropolis update, and correlation and response functions are calculated for different waiting times. We obtain a multiplicative aging scaling of the correlation and response functions, calculating the aging exponent b and the nonequilibrium autocorrelation decay exponent ${\ensuremath{\lambda}}_{c}/{z}_{c}$. We also analyze the fluctuation-dissipation relationship at the critical temperature, obtaining the critical fluctuation-dissipation ratio ${X}_{\ensuremath{\infty}}$. By comparing our results with the aging scaling reported previously for a model of interacting flux lines in the vortex-glass regime, we found that the exponents for both models are very different.
We investigate, analytically near the dimension d(uc) =4 and numerically in d=3 , the nonequilibrium relaxational dynamics of the randomly diluted Ising model at criticality. Using the exact renormalization-group method to one loop, we compute the two times t, t(w) correlation function and fluctuation dissipation ratio (FDR) for any Fourier mode of the order parameter, of finite wave vector q . In the large time separation limit, the FDR is found to reach a nontrivial value X(infinity) independently of (small) q and coincide with the FDR associated to the total magnetization obtained previously. Explicit calculations in real space show that the FDR associated to the local magnetization converges, in the asymptotic limit, to this same value X(infinity). Through a Monte Carlo simulation, we compute the autocorrelation function in three dimensions, for different values of the dilution fraction p at T(c) (p) . Taking properly into account the corrections to scaling, we find, according to the renormalization-group predictions, that the autocorrelation exponent lambda(c) is independent of p . The analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state.