Perturbation theory for ac-driven interfaces in random media

TL;DR

This paper investigates the behavior of driven elastic manifolds in disordered media using perturbation theory and mean-field approximation, revealing dimensional thresholds for regularity and providing analytical and numerical insights.

Contribution

It demonstrates that perturbation theory becomes regular for dimensions greater than four and introduces a resummation method for mean-field diagrams to handle divergences.

Findings

Perturbation theory diverges for D ≤ 4 due to non-regular terms.

For D > 4, perturbation expansion remains regular at all orders.

Resummation of diagrams in mean-field theory cancels unbounded contributions.

Abstract

We study $D$-dimensional elastic manifolds driven by ac-forces in a disordered environment using a perturbation expansion in the disorder strength and the mean-field approximation. We find, that for $D\le 4$ perturbation theory produces non-regular terms that grow unboundedly in time. The origin of these non-regular terms is explained. By using a graphical representation we argue that the perturbation expansion is regular to all orders for $D>4$. Moreover, for the corresponding mean-field problem we prove that ill-behaved diagrams can be resummed in a way, that their unbounded parts mutually cancel. Our analytical results are supported by numerical investigations. Furthermore, we conjecture the scaling of the Fourier coefficients of the mean velocity with the amplitude of the driving force $h$.