Functional Renormalization for Disordered Systems, Basic Recipes and Gourmet Dishes

TL;DR

This paper introduces the functional renormalization group approach to disordered systems, highlighting its necessity, non-analytic disorder distributions, and applications to elastic manifolds, random field magnets, and depinning phenomena.

Contribution

It provides a pedagogical overview, constructs a renormalizable field theory beyond leading order, and compares exact solutions with replica symmetry breaking predictions.

Findings

Disorder distribution becomes non-analytic after finite renormalization.

Exact solution for elastic manifold in infinite dimensions.

Comparison of solutions with Gaussian replica variational ansatz.

Abstract

We give a pedagogical introduction into the functional renormalization group treatment of disordered systems. After a review of its 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 renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how the non-analyticity can be measured in a simulation or experiment. We then construct a renormalizable field theory beyond leading order. We discuss an elastic manifold embedded in N dimensions, and give the exact solution for N to infinity. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. We further consider random field magnets, and supersymmetry. We finally discuss depinning, both isotropic and anisotropic, and universal scaling function.