Disordered Systems and the Functional Renormalization Group: A Pedagogical Introduction

TL;DR

This paper provides a pedagogical overview of disordered systems, emphasizing the functional renormalization group (FRG) method, its challenges, and recent advances in understanding elastic manifolds within disordered environments.

Contribution

It introduces the FRG approach to disordered systems, explaining its principles, challenges, and recent progress, complementing existing methods like Gaussian variational and RSB techniques.

Findings

FRG is a powerful method for strongly disordered systems

Current progress includes understanding elastic manifolds in disorder

Challenges remain in implementing FRG effectively

Abstract

In this article, we review basic facts about disordered systems, especially the existence of many metastable states and and the resulting failure of dimensional reduction. Besides techniques based on the Gaussian variational method and replica-symmetry breaking (RSB), the functional renormalization group (FRG) is the only general method capable of attacking strongly disordered systems. We explain the basic ideas of the latter method and why it is difficult to implement. We finally review current progress for elastic manifolds in disorder.