(Perturbed) Conformal Field Theory Applied To 2D Disordered Systems: An Introduction

TL;DR

The paper addresses how perturbed conformal field theory can describe two-dimensional disordered systems, with motivation from localization transitions and quantum Hall physics. It presents four complementary frameworks—the direct method, the supersymmetric method, the replica method, and the variational method—for analyzing disorder within CFT. A key result is that disordered WZW models at large impurity density factorize into a non-compact coset WZW sector on ${\cal G}^C/{\cal G}$ with level $-2h^v$ together with copies of the original WZW model; in the supersymmetric approach, the relevant algebra is the affine $OSp(2N|2N)$ with zero superdimension, and the replica method yields insight into the RG flow of the random phase sine-Gordon model. The work provides a framework for connecting disordered 2D physics to conformal theory and suggests concrete critical theories for gaussian disordered systems.

Abstract

We describe applications of (perturbed) conformal field theories to two-dimensional disordered systems. We present various methods of study~: (i) {\it A direct method} in which we compute the explicit disorder dependence of the correlation functions for any sample of the disorder. This method seems to be specific to two dimensions. The examples we use are disordered versions of the Abelian and non-Abelian WZW models. We show that the disordered WZW model over the Lie group $\CG$ at level $k$ is equivalent at large impurity density to the product of the WZW model over the coset space $\CG^C/\CG$ at level $(-2h^v)$ times an arbitrary number of copies of the original WZW model. (ii) {\it The supersymmetric method} is introduced using the random bond Ising model and the random Dirac theory as examples. In particular, we show that the relevent algebra is the affine $OSp(2N|2N)$ Lie superalgebra, an algebra with zero superdimension. (iii) {\it The replica method} is introduced using the random phase sine-Gordon model as example. We describe particularities of its renormalization group flow. (iv) {\it A variationnal approach} is also presented using the random phase sine-Gordon model as example. Lectures presented at the '95 Cargese Summer School on "Low dimensional application of quantum field theory".