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

TL;DR

The paper develops a unified, non-unitary conformal framework for Gaussian disordered systems in two dimensions by exploiting an $osp(2/2)_k$ current algebra arising from Efetov’s supersymmetric method. Using the Sugawara construction, it obtains a vanishing central charge, fixed conformal data for primary fields, and a locality condition that quantizes the level to $k=2-rac{1}{l}$. The analysis reveals that logarithmic operators emerge from indecomposable representations, and four-point functions are governed by Knizhnik–Zamolodchikov equations, with logarithmic dependence appearing in generic cases and in special integer-related CFT scales. These results provide a natural setting for logarithmic CFT in disordered systems and connect to multifractality and density-of-states phenomena in the quantum Hall context. Overall, the work offers a concrete, algebraic route to understanding non-unitary critical points and their rich operator content in 2D disordered quantum field theories.

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.