$U\left(1\right)\times U\left(1\right)⋊Z_2$ Chern-Simons theory and $Z_4$ parafermion fractional quantum Hall states

We study $U\left(1\right)\times U\left(1\right)⋊Z_2$ Chern-Simons theory with integral coupling constants $(k,l)$ and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the $U\left(1\right)\times U\left(1\right)⋊Z_2$ 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 $U\left(1\right)\times U\left(1\right)⋊Z_2$ 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 $l=3$ $U\left(1\right)\times U\left(1\right)⋊Z_2$ 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 $\nu =\frac{2}{2k-3}$. The $U\left(1\right)\times U\left(1\right)⋊Z_2$ theory is more useful than an alternative $SU{\left(2\right)}_4\times U\left(1\right)∕U\left(1\right)$ 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.

Figure 1

Canonical homology basis for ${\Sigma }_g$.

Figure 2

(a) A single twist along the $a_g$ direction. (b) Take two copies of ${\Sigma }_g$, cut them along the $a_g$ cycle, and glue them together as shown. This yields a genus $2g-1$ surface. For the case $g=3$, we see explicitly that a genus 5 surface is obtained.

Figure 3

Canonical homology basis for ${\Sigma }_g$.

Figure 4

Consider the diffeomorphism $f$, which takes the neighborhood of a pair of $Z_2$ vortices to the end of a cylinder. We can imagine $f$ as a composition of two maps, the first which expands the cut $\gamma$ to a hole, and a second one which maps the result to the end of a cylinder.

Figure 5

Two pairs of $Z_2$ vortices on a sphere. This sequence of diffeomorphisms illustrates that this situation is equivalent to $M$ being a cylinder.

Figure 6

A single pair of $Z_2$ vortices on a sphere. This sequence of diffemorphisms illustrates that this situation is equivalent to $M$ being a hemisphere, but with a different set of boundary conditions on $A_{\mu }$.

Figure 7

Four pairs of $Z_2$ vortices on a sphere. (a) This sequence of diffeomorphisms shows that we can think of the situation with this many vortices as a gauge field defined on the surface shown in the lower left figure, which looks like half of a genus $g=3$ surface. (b) The figure in the lower left of (a) can be cut open as shown here. The arrows on the figure indicate how the points on the boundaries should be identified. (c) Two copies of the figure in (b). (d) Gluing together two copies along their boundaries gives a genus $g=3$ surface. For general $n$, this procedure gives a surface of genus $g=n-1$.