Non-unitary Conformal Field Theory and Logarithmic Operators for Disordered Systems

TL;DR

This paper explores non-unitary conformal field theories with logarithmic operators in disordered systems, revealing a supersymmetric structure and providing exact four-point functions relevant to critical phenomena like the quantum Hall transition.

Contribution

It introduces a superalgebra framework for analyzing disordered systems, determining conformal weights, and deriving logarithmic four-point functions using conformal field theory methods.

Findings

Identification of osp(2/2)_1 symmetry at critical points

Derivation of conformal weights including negative values

Explicit four-point functions with logarithmic dependence

Abstract

We consider the supersymmetric approach to gaussian disordered systems like the random bond Ising model and Dirac model with random mass and random potential. These models appeared in particular in the study of the integer quantum Hall transition. The supersymmetric approach reveals an osp(2/2)_1 affine symmetry at the pure critical point. A similar symmetry should hold at other fixed points. We apply methods of conformal field theory to determine the conformal weights at all levels. These weights can generically be negative because of non-unitarity. Constraints such as locality allow us to quantize the level k and the conformal dimensions. This provides a class of (possibly disordered) critical points in two spatial dimensions. Solving the Knizhnik-Zamolodchikov equations we obtain a set of four-point functions which exhibit a logarithmic dependence. These functions are related to logarithmic operators. We show how all such features have a natural setting in the superalgebra approach as long as gaussian disorder is concerned.