U(1)×U(1)⋊Z2Chern-Simons theory andZ4parafermion fractional quantum Hall states

Abstract

We study $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory with integral coupling constants $(k,l)$ and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ 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 $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ 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 $l=3$ $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ 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 $\ensuremath{\nu}=\frac{2}{2k\ensuremath{-}3}$. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ theory is more useful than an alternative $SU{(2)}_{4}\ifmmode\times\else\texttimes\fi{}U(1)∕U(1)$ 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 $\ensuremath{\nu}=2∕3$ phase diagram.