Fibonacci Anyons From Abelian Bilayer Quantum Hall States

The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this Letter, we consider a simple bilayer fractional quantum Hall system with the $1/3$ Laughlin state in each layer. We show that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous “Fibonacci” anyon, whose non-Abelian statistics is powerful enough for universal TQC. Our analysis rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions. We provide evidence that the transition can be continuous, at which point the charge gap remains open while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at $\nu =2/3$ in bilayers, as past experiments may not have definitively ruled them out.

Figure 1

$\nu =2/3$ proposed global phase diagram, for interlayer tunneling on the order of interaction strengths. We find possible continuous transitions between four different states, as described in the main text. $m_1$, $m_2$ are phenomenological parameters in the effective theory and drive the two types of Higgs transitions. They depend on interlayer tunneling and inter- or intralayer interactions in a way which requires further study. The minimal quasiparticle charge $e^{*}$ distinguishes (b) from the others. The others must be distinguished in principle through detecting phase transitions in the neutral sector, or through tunneling or interferometry measurements.