Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus

Abstract

We analyze the Moore-Read Pfaffian state on a thin torus. The known sixfold degeneracy is realized by two inequivalent crystalline states with a four- and twofold degeneracy, respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension ${2}^{n\ensuremath{-}1}$ for $2n\ensuremath{-}k$ holes and $k$ particles at fixed positions and assign each a charge $\ifmmode\pm\else\textpm\fi{}e∕4$. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model Hamiltonian and a small number of particles support the presence of a stable phase with degenerate vacua and quarter-charged domain walls also away from the thin-torus limit. A spin-chain Hamiltonian encodes the degenerate vacua and the various domain walls.