Two-loop functional renormalization group theory of the depinning transition

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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 non-analytic 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 beta-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 zeta and dynamical exponent z to order epsilon^2. This allows to test several previous conjectures made on the basis of the 1-loop result. First it demonstrates that random-field disorder does indeed attract all disorder of shorter range. It also shows that the conjecture zeta=epsilon/3 is incorrect, and allows to compute the violations, as zeta=epsilon/3 (1 + 0.14331 epsilon), epsilon=4-d. This solves a longstanding discrepancy with simulations. For long-range elasticity it yields zeta=epsilon/3 (1 + 0.39735 epsilon), epsilon=2-d (vs. the standard prediction zeta=1/3 for d=1), in reasonable agreement with the most recent simulations. The high value of zeta approximately 0.5 found in experiments both on the contact line depinning of liquid Helium and on slow crack fronts is discussed.