Creep and depinning in disordered media

Elastic systems driven in a disordered medium exhibit a depinning transition at zero temperature and a creep regime at finite temperature and slow drive f. We derive functional renormalization-group equations which allow us to describe in detail the properties of the slowly moving states in both cases. Since they hold at finite velocity v, they allow us to remedy some shortcomings of the previous approaches to zero-temperature depinning. In particular, they enable us to derive the depinning law directly from the equation of motion, with no artificial prescription or additional physical assumptions, such as a scaling relation among the exponents. Our approach provides a controlled framework to establish under which conditions the depinning regime is universal. It explicitly demonstrates that the random potential seen by a moving extended system evolves at large scale to a random field and yields a self-contained picture for the size of the avalanches associated with the deterministic motion. At finite temperature $T>\mathrm{}0\mathrm{}$ we find that the effective barriers grow with length scale as the energy differences between neighboring metastable states, and demonstrate the resulting activated creep law $v\sim \exp (-{\mathrm{Cf}}^{-\mu }/T)$ where the exponent μ is obtained in a $\epsilon =4\mathrm{}-D$ expansion $(D$ is the internal dimension of the interface). Our approach also provides quantitatively an interesting scenario for creep motion as it allows us to identify several intermediate length scales. In particular, we unveil a “depinninglike” regime at scales larger than the activation scale, with avalanches spreading from the thermal nucleus scale up to the much larger correlation length $R_V.$ We predict that $R_V\sim T^{-\sigma }{f}^{-\lambda }$ diverges at small drive and temperature with exponents $\sigma ,\lambda$ that we determine.