A Unified Conformal Field Theory Description of Paired Quantum Hall States

TL;DR

This paper constructs a consistent unitary conformal field theory description of the Haldane-Rezayi paired quantum Hall state, linking it to the 331 state and clarifying its edge excitations and relation to other states.

Contribution

It develops a complete unitary CFT framework for the Haldane-Rezayi state, connecting non-unitary and unitary theories and relating it to the 331 and Pfaffian states.

Findings

Unitary c=1 theory describes Haldane-Rezayi edge excitations.

The unitary theory matches that of the 331 state.

No alternative rational CFT descriptions exist for the Haldane-Rezayi state.

Abstract

The wave functions of the Haldane-Rezayi paired Hall state have been previously described by a non-unitary conformal field theory with central charge c=-2. Moreover, a relation with the c=1 unitary Weyl fermion has been suggested. We construct the complete unitary theory and show that it consistently describes the edge excitations of the Haldane-Rezayi state. Actually, we show that the unitary (c=1) and non-unitary (c=-2) theories are related by a local map between the two sets of fields and by a suitable change of conjugation. The unitary theory of the Haldane-Rezayi state is found to be the same as that of the 331 paired Hall state. Furthermore, the analysis of modular invariant partition functions shows that no alternative unitary descriptions are possible for the Haldane-Rezayi state within the class of rational conformal field theories with abelian current algebra. Finally, the known c=3/2 conformal theory of the Pfaffian state is also obtained from the 331 theory by a reduction of degrees of freedom which can be physically realized in the double-layer Hall systems.