Composite particle construction of the Fibonacci fractional quantum Hall state

The Fibonacci topological order is the simplest platform for a universal topological quantum computer. While the $\nu =12/5$ fractional quantum Hall (QH) state has been proposed to support a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a QH system has been lacking. We use non-Abelian dualities to construct a Fibonacci state of bosons at filling $\nu =2$ starting from a trilayer of integer QH states. Our parent theory consists of bosonic composite vortices coupled to fluctuating $U\left(2\right)$ gauge fields, which is dual to the theory of Laughlin quasiparticles. The Fibonacci state is obtained by interlayer clustering of the composite vortices, along with flux attachment. We use this framework to motivate a wave function for the Fibonacci state.

Figure 1

A schematic of our construction of the (a) $U{\left(2\right)}_3$ and (b) $U{\left(2\right)}_{3,1}$ Fibonacci states. These are FQH states of fermions and bosons, respectively. Here $\leftrightarrow$ denotes duality between the theories of Laughlin quasiparticles and composite vortices. The non-Abelian state is obtained by clustering of the dual composite vortices between the layers.

Figure 2

Root system of $G_2$ labeled by the corresponding ${\left(G_2\right)}_1$ current generators. The green circles indicate the operators we identify as the boson operators.