The Functional Renormalization Group Treatment of Disordered Systems: a Review

TL;DR

This review discusses the application of the functional renormalization group to disordered systems, highlighting the necessity of a functional approach, non-analytic disorder distributions, and exact solutions for elastic manifolds.

Contribution

It provides a comprehensive overview of the functional renormalization group method for disordered systems, including non-perturbative solutions and comparisons with replica methods.

Findings

Disorder distributions become non-analytic after renormalization.

Exact solution for elastic manifolds in the limit of infinite dimensions.

Analysis of depinning phenomena and interface width distributions.

Abstract

We review current progress in the functional renormalization group treatment of disordered systems. After an elementary introduction into the phenomenology, we show why in the context of disordered systems a functional renormalization group treatment is necessary, contrary to pure systems, where renormalization of a single coupling constant is sufficient. This leads to a disorder distribution, which after a finite renomalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how a renormalizable field theory can be constructed, even beyond 1-loop order. We then discuss an elastic manifold imbedded in N dimensions, and give the exact solution for N -> oo. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. We finally discuss depinning, both isotropic and anisotropic, and the scaling function for the width distribution of an interface.