Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states

We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the $\left(ppq\right)$ Abelian bilayer states to a set of non-Abelian FQH states, which we dub orbifold FQH states, of which the $Z_4$ parafermion (Read-Rezayi) state is a special case. This provides an example in which $Z_2$ electron fractionalization leads to non-Abelian topological phases. The naive “ideal” wave functions and ideal Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases but, instead, lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern-of-zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the $\left(ppq\right)$ bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fractions $8/3$ and $12/5$ and as a way to tune to non-Abelian states in double-layer or wide quantum wells.

Figure 1

How to see that $Z_2$ vortices at fixed locations come with a degeneracy of states and should therefore be non-Abelian. (a) The space is deformed arbitrarily; (b) we consider a doubled space with genus $g=n-1$, where $n$ is the number of pairs of $Z_2$ vortices. Because of the boundary conditions of the two gauge fields $a$ and $\tilde{a}$, we can define a single continuous gauge field on the doubled space, which leaves us with $U\left(1\right)$ CS theory on a genus $g=n-1$ surface. Thus the number of states in the presence of $n$ pairs of $Z_2$ vortices grows exponentially in $n$. In this sense, $Z_2$ vortices are like “genons:” they effectively change the genus of the manifold.

Figure 2

Here we try to illustrate the different ways of describing the topological order of the orbifold FQH states and how they are related. The $U\left(1\right)\times U\left(1\right)\times Z_2$ CS theory is a bulk topological field theory, but we do not know how to compute all of the topological properties from this theory. Closely related is the slave Ising theory gauge theory presented in Sec. 3, which we believe provides a lattice regularization of the $U\left(1\right)\times U\left(1\right)⋊Z_2$ CS theory. The spectrum of the edge theory for these bulk topological theories is conjectured to be generated by the electron operator ${\Psi }_e=\cos \left(\sqrt{N/2}\varphi \right)e^{i\sqrt{(p+q)/2}{\phi }_c}$. We also conjecture that this edge theory is equivalent to what one would obtain for the edge theory by taking the electron operator to be ${\Psi }_e={\phi }_N^1{e}^{i\sqrt{(p+q)/2}{\phi }_c}$ and embedding it into the holomorphic half of the $Z_2$ orbifold $\times U{\left(1\right)}_{\mathrm{charge}}$ CFT.