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 to describe in details the properties of the slowly moving states in both cases. Since they hold at finite velocity v, they allow 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 > 0 we find that the effective barriers grow with lenghtscale as the energy differences between neighboring metastable states, and demonstrate the resulting activated creep law v ∼ exp −C f −µ /T where the exponent µ is obtained in a ǫ = 4 − D expansion (D is the internal dimension of the interface). Our approach also provides quantitatively a new scenario for creep motion as it allows to identify several intermediate lengthscales. In particular, we unveil a novel "depinning-like" regime at scales larger than the activation scale, with avalanches spreading from the thermal nucleus scale up to the much larger correlation length RV . We predict that RV ∼ T −σ f −λ diverges at small drive and temperature with exponents σ, λ that we determine.

We construct the field theory which describes the universal properties of the quasi-static isotropic depinning transition for interfaces and elastic periodic systems at zero temperature, taking properly into account the nonanalytic form of the dynamical action. This cures the inability of the 1-loop flow-equations to distinguish between statics and quasi-static depinning, and thus to account for the irreversibility of the latter. We prove two-loop renormalizability, obtain the 2-loop β-function and show the generation of "irreversible" anomalous terms, originating from the non-analytic nature of the theory, which cause the statics and driven dynamics to differ at 2-loop order. We obtain the roughness exponent ζ and dynamical exponent z to order ǫ 2 . This allows to test several previous conjectures made on the basis of the 1-loop result. First it demonstrates that randomfield disorder does indeed attract all disorder of shorter range. It also shows that the conjecture ζ = ǫ/3 is incorrect, and allows to compute the violations, as ζ = ǫ 3(1 + 0.14331ǫ), ǫ = 4 − d. This solves a longstanding discrepancy with simulations. For long-range elasticity it yields ζ = ǫ 3(1 + 0.39735ǫ), ǫ = 2 − d (vs. the standard prediction ζ = 1/3 for d = 1), in reasonable agreement with the most recent simulations. The high value of ζ ≈ 0.5 found in experiments both on the contact line depinning of liquid Helium and on slow crack fronts is discussed.