Spatiotemporal scaling for out-of-equilibrium relaxation dynamics of an elastic manifold in random media: Crossover between diffusive and glassy regimes

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We study the relaxation dynamics of a three-dimensional elastic manifold in a random potential from a uniform initial condition by numerically solving the Langevin equation. We observe the growth of the roughness of the system up to larger wavelengths with time. We analyze the structure factor in detail and find a compact scaling ansatz describing two distinct time regimes and the crossover between them. We find that a short-time regime corresponding to a length scale smaller than the Larkin length ${L}_{c}$ is well described by the Larkin model, which predicts a power-law growth of the domain size $L(t)$. Longer-time behavior exhibits a glassy regime with slower growth of $L(t)$.