Effective field theory and projective construction for $Z_k$ parafermion fractional quantum Hall states

The projective construction is a powerful approach to deriving the bulk and edge field theories of non-Abelian fractional quantum Hall (FQH) states and yields an understanding of non-Abelian FQH states in terms of the simpler integer quantum Hall states. Here we show how to apply the projective construction to the $Z_k$ parafermion (Laughlin/Moore-Read/Read-Rezayi) FQH states, which occur at filling fraction $\nu =k/\left(kM+2\right)$. This allows us to derive the bulk low-energy effective field theory for these topological phases, which is found to be a Chern-Simons theory at level 1 with a $U\left(M\right)\times Sp\left(2k\right)$ gauge field. This approach also helps us understand the non-Abelian quasiholes in terms of holes of the integer quantum Hall states.