Two-loop functional renormalization group theory of the depinning transition

Abstract

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.

Deblocking of interacting particle assemblies: from pinning to jamming

“…In most situations, however, interactions between particles should be explicitly taken into account. In this case we will still observe a depinning transition, but its quantitative and qualitative features will change [52][53][54][55][56][57][58]. It is convenient to first study an interacting particle system in the elastic approximation, in which the pinning forces are not strong enough to break the topological properties of the particle system.…”

“…These elastic equations are still impossible to solve exactly, but several results have been obtained using scaling theories and renormalization group calculations [52][53][54][55][56][57][58]. In the case of long-range interactions the gradients are replaced by a non-local interaction kernel and the equation becomes…”

Universal distribution of threshold forces at the depinning transition

“…The splitted diagrammatics for the perturbation theory in disorder ∆ was developed in Refs. [16,17]. It is known that naive perturbation theory, obtained by taking for ∆ an analytic function exhibits the property of dimensional reduction and fails to describe the physics, giving for example an incorrect roughness exponent.…”

“…[14,15] that the corresponding functional renormalization group (FRG) provides an adequate description of the depinning transition if one considers the non-analytic renormalized function ∆(u). It was shown only recently that the FRG can be unambiguously extended to higher loop order and that the underlying non-analytic field theory is renormalizable [16,17]. The FRG equation for ∆(u) has two main non-trivial stable fixed points which describe periodic and interface universality classes.…”

“…This cusp accounts for the existence of the critical threshold force f c ∼ ∆ * ′ (0 + ). The corresponding critical exponents have been computed to second order in ε = 4 − d [16,17].…”