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Multilayer fractional quantum Hall wave functions can be used to constructthe non-Abelian states of the $\mathbb{Z}_k$ Read-Rezayi series uponsymmetrization over the layer index. Unfortunately, this construction does notyield the complete set of $\mathbb{Z}_k$ ground states on the torus. We developan alternative projective construction of $\mathbb{Z}_k$ Read-Rezayi statesthat complements the existing one. On the multi-layer torus geometry, ourconstruction consists of introducing twisted boundary conditions connecting thelayers before performing the symmetrization. We give a comprehensive account ofthis construction for bosonic states, and numerically show that the full groundstate and quasihole manifolds are recovered for all computationally accessiblesystem sizes. Furthermore, we analyze the neutral excitation modes above theMoore-Read on the torus through an extensive exact diagonalization study. Weshow numerically that our construction can be used to obtain excellentapproximations to these modes. Finally, we extend the new symmetrization schemeto the plane and sphere geometries.

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