A bosonic matrix product state description of Read-Rezayi states and its application to quasi-hole spins

Abstract

We study the $k=3$ Read-Rezayi quantum Hall state by means of a purely bosonic matrix product state formulation, which is described in detail. We calculate the density profiles in the presence of bulk quasi-holes of six different types: one for each $\mathbb{Z}_3$ parafermion sector. From the density profiles, we calculate the (local) spins of these quasi-holes. By employing a spin-statistics relation, we obtain the exchange statistics parameters. Our results, which are entirely based on local properties of the quasi-holes, corroborate previous results obtained by explicitly braiding quasi-holes, showing that the exchange statistics can be read off from the monodromy properties of the wave functions, i.e., that the associated Berry phase vanishes. We also discuss the entanglement spectrum, to show that our bosonic matrix product state formulation correctly captures the $\mathbb{Z}_3$ parafermionic structure of the $k=3$ Read-Rezayi states.