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 β-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 ζ (roughness) and z (dynamics) to order ε². This tests previous conjectures based on the one-loop result: It shows that random-field disorder indeed attracts all shorter range disorder. The conjecture ζ=ε/3 is incorrect, with violation ζ=(ε/3)(1+0.14331ε), ε=4-d. This solves a longstanding discrepancy with simulations. For long-range elasticity ζ=(ε/3)(1+0.39735ε), ε=2-d (vs the standard prediction ζ=1/3 for $d=1),$ in reasonable agreement with simulations. The high value of ζ≈0.5 in experiments both on Helium contact line depinning and on slow crack fronts is discussed.