Numerical Calculation of the Functional renormalization group fixed-point functions at the depinning transition

TL;DR

This paper numerically computes the functional renormalization group functions at the depinning transition of an elastic line in disordered potentials, revealing universal cusp behaviors linked to avalanches, consistent with FRG predictions.

Contribution

It introduces a numerical method to determine FRG functions at depinning, confirming theoretical predictions about cusp forms and avalanche effects.

Findings

Universal cusp in second cumulant Delta(u) matches 2-loop FRG

Cusp in third cumulant also observed and characterized

Results are consistent for both RB and RF disorder types

Abstract

We compute numerically the sequence of successive pinned configurations of an elastic line pulled quasi-statically by a spring in a random bond (RB) and random field (RF) potential. Measuring the fluctuations of the center of mass of the line allows to obtain the functional renormalization group (FRG) functions at the depinning transition. The universal form of the second cumulant Delta(u) is found to have a linear cusp at the origin, to be identical for RB and RF, different from the statics, and in good agreement with 2-loop FRG. The cusp is due to avalanches, which we visualize. Avalanches also produce a cusp in the third cumulant, whose universal form is obtained, as predicted by FRG.