Functional renormalization group for anisotropic depinning and relation to branching processes

TL;DR

This paper uses the functional renormalization group to analyze anisotropic depinning of elastic objects, revealing the generation of KPZ terms, fixed points, and connections to branching processes and directed percolation.

Contribution

It demonstrates the all-order non-renormalization of the KPZ coupling and links depinning phenomena to branching and reaction-diffusion processes.

Findings

KPZ term is generated in anisotropic depinning.

Fixed points are identified but are transient with runaway flows.

A subspace invariant under all orders relates to branching and reaction-diffusion processes.

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. We compute the beta-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. 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 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.