Orbifold Duality Symmetries and Quantum Hall systems

TL;DR

This paper explores how chiral orbifold conformal field theories can describe edge states in quantum Hall systems, extending known duality symmetries from toroidal models to orbifolds, revealing new fractional quantum Hall states.

Contribution

It generalizes the duality framework for quantum Hall edge states from toroidal to orbifold conformal field theories, introducing a formalism for Hall conductance transformations and analyzing a Z_3 orbifold model.

Findings

New classes of filling fractions with fractional electric charge

Orbifold duality groups relate different quantum Hall states

Connection between orbifold edge theories and Luttinger liquids

Abstract

We consider the possible role that chiral orbifold conformal field theories may play in describing the edge state theories of quantum Hall systems. This is a generalization of work that already exists in the literature, where it has been shown that 1+1 chiral bosons living on a n-dimensional torus, and which couple to a U_1 gauge field, give rise to anomalous electric currents, the anomaly being related to the Hall conductivity. The well known $O(n,n;Z)$ duality group associated with such toroidal conformal field theories transforms the edge states and Hall conductivities in a way which makes interesting connections between different theories, e.g. between systems exhibiting the integer and fractional quantum Hall effect. In this paper we try to explore the extension of these constructions to the case where such bosons live on a n-dimensional orbifold. We give a general formalism for discussing the relevant quantities like the Hall conductance and their transformation under the duality groups present in orbifold compactifications. We illustrate these ideas by presenting a detailed analysis of a toy model based on the two-dimensional Z_3 orbifold. In this model we obtain new classes of filling fractions, which generally the correspond to fermionic edge states carrying fractional electric charge. We also consider the relation between orbifold edge theories and Luttinger liquids (LL's), which in the past have provided important insights into the physics of quantum Hall systems.