Two-loop functional renormalization group theory of the depinning transition

We construct the field theory of quasistatic isotropic depinning 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 one-loop flow equations to distinguish between statics and quasistatic depinning, and thus to account for the irreversibility of the latter. We prove two-loop renormalizability, obtain the two-loop beta-function and show the generation of "irreversible" anomalous terms, resulting from the nonanalyticity of the theory, which cause statics and driven dynamics to differ at two loops. We give the exponents zeta (roughness) and z (dynamics) to order epsilon(2). This tests previous conjectures based on the one-loop result: It shows that random-field disorder indeed attracts all shorter range disorder. The conjecture zeta=epsilon/3 is incorrect, with violation zeta=(epsilon/3)(1+0.14331epsilon), epsilon=4-d. This solves a longstanding discrepancy with simulations. For long-range elasticity zeta=(epsilon/3)(1+0.39735epsilon), epsilon=2-d (vs the standard prediction zeta=1/3 for d=1), in reasonable agreement with simulations. The high value of zetaapproximate to0.5 in experiments both on Helium contact line depinning and on slow crack fronts is discussed.