Degeneracy of non-abelian quantum Hall states on the torus: domain walls and conformal field theory

Abstract

We analyze the non-abelian Read-Rezayi quantum Hall states on the torus, where it is natural to employ a mapping of the many-body problem onto a one-dimensional lattice model. On the thin torus--the Tao-Thouless (TT) limit--the interacting many-body problem is exactly solvable. The Read-Rezayi states at filling $\nu =\frac{k}{kM+2}$ are known to be exact ground states of a local repulsive $k+1$-body interaction, and in the TT limit this is manifested in that all states in the ground state manifold have exactly $k$ particles on any $kM+2$ consecutive sites. For $M\ne 0$ the two-body correlations of these states also imply that there is no more than one particle on $M$ adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domain walls between the ground states, and we show that the number of distinct domain wall patterns gives rise to the nontrivial degeneracies, required by the non-abelian statistics of these states. In the second part of the paper we consider the quasihole degeneracies from a conformal field theory (CFT) perspective, and show that the counting of the domain wall patterns maps one to one on the CFT counting via the fusion rules. Moreover we extend the CFT analysis to topologies of higher genus.

15 pages

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• From Irrational to Non-Unitary: on the Haffnian and Haldane-Rezayi wave functions
• Domain walls, fusion rules and conformal field theory in the quantum Hall regime
• Zero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization
• Solvable models for unitary and non-unitary topological phases
• Algebraic approach to the study of zero modes of Haldane pseudopotentials
• Effective spin chains for fractional quantum Hall states
• Spin chain description of rotating bosons at $\nu =1$
• Conformal field theory construction for nonabelian hierarchy wave functions