The Pfaffian quantum Hall state made simple--multiple vacua and domain walls on a thin torus

TL;DR

This paper simplifies the understanding of the Moore-Read Pfaffian state on a thin torus, revealing multiple vacua and domain walls, and provides a counting argument for quasihole excitations, supported by numerical and spin chain models.

Contribution

It introduces a simple framework for the Pfaffian state on a thin torus, identifying multiple vacua and domain walls, and extends quasihole excitation properties beyond conformal field theory methods.

Findings

Degeneracy realized by two inequivalent crystalline states

Domain walls correspond to quasihole and quasiparticle excitations

Numerical evidence supports stable phases with degenerate vacua and quarter-charged domain walls

Abstract

We analyze the Moore-Read Pfaffian state on a thin torus. The known six-fold degeneracy is realized by two inequivalent crystalline states with a four- and two-fold degeneracy respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension $2^{n-1}$ for $2n-k$ holes and $k$ particles at fixed positions and assign each a charge $\pm e/4$. 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 supports 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.