U(1) X U(1) XI Z2 Chern-Simons theory and Z4 parafermion fractional quantum Hall states

We study U(1) X U(1) XI Z(2) Chern-Simons theory with integral coupling constants (k, l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the U(1) X U(1) XI Z(2) Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z(2) vortices in the U(1) X U(1) XI Z(2) Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z(2) vortices on a sphere. These results allow us to show that l = 3 U(1) X U(1) XI Z(2) Chern-Simons theory is the low-energy effective theory for the Z(4) parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction nu = 2/2k-3. The U(1) X U(1) XI Z(2) theory is more useful than an alternative SU(2)(4) X U(1)/U(1) Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the nu = 2/3 phase diagram.