Effective Chern-Simons Theories of Pfaffian and Parafermionic Quantum Hall States, and Orbifold Conformal Field Theories

Abstract

We present a pure Chern-Simons formulation of families of interesting Conformal Field Theories describing edge states of non-Abelian Quantum Hall states. These theories contain two Abelian Chern-Simons fields describing the electromagnetically charged and neutral sectors of these models, respectively. The charged sector is the usual Abelian Chern-Simons theory that successfully describes Laughlin-type incompressible fluids. The neutral sector is a 2+1-dimensional theory analogous to the 1+1-dimensional orbifold conformal field theories. It is based on the gauge group O(2) which contains a ${\mathbf{Z}}_2$ disconnected group manifold, which is the salient feature of this theory. At level q, the Abelian theory of the neutral sector gives rise to a ${\mathbf{Z}}_{2q}$ symmetry, which is further reduced by imposing the ${\mathbf{Z}}_2$ symmetry of charge-conjugation invariance. The remaining ${\mathbf{Z}}_q$ symmetry of the neutral sector is the origin of the non-Abelian statistics of the (fermionic) q-Pfaffian states.

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• The nu=5/2 quantum Hall state revisited: spontaneous breaking of the chiral fermion parity and phase transition between abelian and non-abelian statistics
• Effective theory of fractional topological insulators in two spatial dimensions
• Charge and spin fractionalization in strongly correlated topological insulators
• Topological Superconductor from the Quantum Hall Phase: Effective Field Theory Description