Tracing non-Abelian anyons via impurity particles

Abstract

Non-Abelian excitations are an interesting feature of many fractional quantum Hall phases, including those phases described by the Moore-Read (or Pfaffian) wave function. However, the detection of the non-Abelian quasiparticles is challenging. Here, we consider a system described by the Moore-Read wave function, and assume that impurity particles bind to its quasiholes. Then, the angular momentum of the impurities, reflected also by the impurity density, provides a useful witness of the physics of the non-Abelian excitations. By demanding that the impurities are constrained to the lowest Landau level, we are able to write down the corresponding many-body wave function describing both the Moore-Read liquid and the impurities. Through Monte Carlo sampling we determine the impurity angular momentum, and we show that it suggests a quantum-statistical parameter $\alpha = a\nu -b +P/2$ for the quasiholes, where $\alpha$ ranges from $0$ for bosons to $1$ for fermions. A reasonable agreement with the Monte Carlo results is obtained for $a=1/4$, $b=1/8$ and $P=0,1$ depending on the parity of the particle number in the Moore-Read liquid. This parity-dependence of the angular momentum serves as an unambiguous demonstration of the non-Abelian nature of the excitations. In addition to the studies of excitations in the Moore-Read liquid, we also apply our scheme to Laughlin liquids, for which we focus on interacting bosonic impurities. With this, the impurities themselves form Laughlin states, which allows for a study of hierarchical fractional quantum Hall states.