2-loop functional renormalization group theory of the depinning transition

Functional renormalization group for anisotropic depinning and relation to branching processes

Pierre Le Doussal¹, Kay Jörg Wiese²
¹CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
²KITP, University of California at Santa Barbara, Santa Barbara, CA 93106-4030, USA

Abstract

Using the functional renormalization group, we study the depinning of elastic objects in presence of anisotropy. We explicitly demonstrate how the KPZ-term is always generated, even in the limit of vanishing velocity, except where excluded by symmetry. This mechanism has two steps: First a non-analytic disorder-distribution is generated under renormalization beyond the Larkin-length. This non-analyticity then generates the KPZ-term. We compute the β-function to one loop taking properly into account the non-analyticity. This gives rise to additional terms, missed in earlier studies. A crucial question is whether the non-renormalization of the KPZ-coupling found at 1-loop order extends beyond the leading one. Using a Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all orders. The resulting flow-equations describe a variety of physical situations: We study manifolds in periodic disorder, relevant for charge density waves, as well as in non-periodic disorder. Further the elasticity of the manifold can either be short-range (SR) or long-range (LR). A careful analysis of the flow yields several non-trivial fixed points. All these fixed points are transient since they possess one unstable direction towards a runaway flow, which leaves open the question of the upper critical dimension. The runaway flow is dominated by a Landau-ghost-mode. For LR elasticity, relevant for contact line depinning, we show that there are two phases depending on the strength of the KPZ coupling. For SR elasticity, using the Cole-Hopf transformed theory we identify a non-trivial 3-dimensional subspace which is invariant to all orders and contains all above fixed points as well as the Landau-mode. It belongs to a class of theories which describe branching and reaction-diffusion processes, of which some have been mapped onto directed percolation.

cond-mat/0208204 [pdf]
Phys. Rev. E 67, 016121 (2003) [pdf]

Referee Report (Phys. Rev. E)

This manuscript presents a detailed functional renormalization group analysis of depinning transitions in anisotropic media. This is an important problem that was studied several years ago by a variety of techniques, with limited success. There was a hiatus due to the difficulty of carrying out the calculations (and simulations) further. In the last year there has been renewed interest in this problem due to improved simulations, and potential relevance to contact lines and earthquakes. Also the authors succeeded in performaing functional RG (for the isotropic version of the problem) to second order, indicating shortcomings of previous analysis, and pointing to further work. In the current manuscript the authors apply their functional RG expertize to the anisotropic depinning problem. Along the way they quantify some earlier conjectures, and also provide intriguing glimpses of posible connections to other problems (branching process). This is by no means the final word on this topic (the identified fixed points correspond to phase transitions, rather than stable phases), but a very important breakthrough. Despite its detailed nature, the paper is well written and nicely presented. As it includes much important new information, I recommend its publication in Physcial Review E, in its present form.