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

TL;DR

This paper introduces various methods for applying (perturbed) conformal field theories to analyze two-dimensional disordered systems, providing explicit calculations and theoretical insights into models like disordered WZW, random Ising, and sine-Gordon.

Contribution

It presents a comprehensive overview of methods including direct, supersymmetric, replica, and variational approaches for studying 2D disordered systems using conformal field theory.

Findings

Disordered WZW model at large impurity density relates to coset models and original WZW models.

Supersymmetric method reveals the relevance of the affine OSp(2N|2N) Lie superalgebra.

Renormalization group flow characteristics of the random phase sine-Gordon model.

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".