Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformalfield theory

Abstract

We analyze the non-Abelian Read–Rezayi quantum Hall states on the torus, where it isnatural to employ a mapping of the many-body problem onto a one-dimensionallattice model. On the thin torus—the Tao–Thouless (TT) limit—the interactingmany-body problem is exactly solvable. The Read–Rezayi states at fillingν = k/(kM+2) are known to be exact ground states of a local repulsivek+1-bodyinteraction, and in the TT limit this is manifested in that all states in the ground state manifold have exactlyk particleson any kM+2 consecutive sites. For the two-body correlations of these states also imply that there is no more than one particle onM adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domainwalls between the ground states, and we show that the number of distinct domain wallpatterns gives rise to the nontrivial degeneracies, required by the non-Abelianstatistics of these states. In the second part of the paper we consider the quasiholedegeneracies from a conformal field theory (CFT) perspective, and show that thecounting of the domain wall patterns maps one to one on the CFT counting via thefusion rules. Moreover we extend the CFT analysis to topologies of higher genus.