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

TL;DR

This paper studies non-abelian Read-Rezayi quantum Hall states on a torus, using a lattice model in the thin torus limit, revealing domain wall patterns and conformal field theory connections that explain their degeneracies and non-abelian statistics.

Contribution

It provides an exact analysis of the ground state degeneracies and domain wall structures of Read-Rezayi states on the torus, linking lattice models to conformal field theory.

Findings

Exact ground state degeneracies are characterized by domain wall patterns.

Domain walls correspond to quasiparticles and quasiholes with non-abelian statistics.

CFT fusion rules match the counting of domain wall configurations.

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 $M\neq 0$ 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.