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$U(1) \times U(1) \rtimes Z_2$ Chern-Simons Theory and Z_4 Parafermion Fractional Quantum Hall States

We study $U(1) \times U(1) \rtimes Z_2$ Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the $U(1) \times U(1) \rtimes Z_2$ 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 $U(1) \times U(1)\rtimes Z_2$ 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 $U(1)\times U(1) \rtimes Z_2$ 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 $ν= 2/(2k - 3)$. The $U(1)\times U(1) \rtimes Z_2$ theory is more useful than an alternative $SU(2)_4\times U(1)/ U(1)$ 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 $ν= 2/3$ phase diagram.