Projective construction of the $\mathbb{Z}_k$ Read-Rezayi fractional quantum Hall states and their excitations on the torus geometry

TL;DR

This paper introduces a new projective construction method for $ ext{Z}_k$ Read-Rezayi fractional quantum Hall states on the torus, improving the completeness of ground state representations and analyzing excitations.

Contribution

It develops an alternative symmetrization-based construction that recovers the full ground state manifold and quasihole states on the torus, extending to plane and sphere geometries.

Findings

Complete ground state and quasihole manifolds recovered numerically

Accurate approximations of neutral excitation modes achieved

Construction extended to multiple geometries

Abstract

Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the $\mathbb{Z}_k$ Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of $\mathbb{Z}_k$ ground states on the torus. We develop an alternative projective construction of $\mathbb{Z}_k$ Read-Rezayi states that complements the existing one. On the multi-layer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend the new symmetrization scheme to the plane and sphere geometries.