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

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 $L_T \sim T^3$, to a logarithmic growth consistent with an algebraic growth of barriers. We present a scaling arguments to deal with the crossover-induced geometrical and dynamical effects. This analysis allows to explain why the results of most numerical studies so far have been described with effective power-laws and also permits 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-Tc superconductors.