Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems

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 τ_k for each wave vector k, leading to analytical expressions for equilibrium dynamical response and correlations. These exhibit two distinct scaling regimes (scaling variables Tk^θ ln t and t∕τ_k, respectively, with T the temperature, θ the energy fluctuation exponent, and τ_k∼e^ck−θ∕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 L^θ and of fluctuations ∼L^θ∕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,…, corresponding to vertices proportional to the pth power of frequency ω^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.