Growing correlations and aging of an elastic line in a random potential

Abstract

We study the thermally assisted relaxation of a directed elastic line in a two-dimensional quenched random potential by solving numerically the Edwards-Wilkinson equation and the Monte Carlo dynamics of a solid-on-solid lattice model. We show that the aging dynamics is governed by a growing correlation length displaying two regimes: an initial thermally dominated power-law growth which crosses over, at a static temperature-dependent correlation length ${L}_{T}\ensuremath{\sim}{T}^{3}$, to a logarithmic growth consistent with an algebraic growth of barriers. We present scaling arguments to deal with the crossover-induced geometrical and dynamical effects. This analysis allows us to explain why the results of most numerical studies so far have been described with effective power laws and also permits us to determine the observed anomalous temperature dependence of the characteristic growth exponents. We argue that a similar mechanism should be at work in other disordered systems. We generalize the Family-Vicsek stationary scaling law to describe the roughness by incorporating the waiting-time dependence or age of the initial configuration. The analysis of the two-time linear response and correlation functions shows that a well-defined effective temperature exists in the power-law regime. Finally, we discuss the relevance of our results for the slow dynamics of vortex glasses in high-${T}_{c}$ superconductors.