Composite particle construction of the Fibonacci fractional quantum Hall state

The Fibonacci topological order is the simplest platform for a universal quantum computer, consisting of single type non-Abelian anyon, $\tau$, with fusion rule $\tau\times\tau=1+\tau$. While it has been proposed that anyon spectrum $\nu=12/5$ fractional Hall state includes sector, dynamical picture how pure may emerge in system lacking. Here we use recently dualities to construct bosons at filling $\nu=2$ starting from trilayer integer states. Our parent theory consists bosonic "composite vortices" coupled fluctuating $U(2)$ gauge fields, which related standard Laughlin quasiparticles by duality. obtained clustering composite vortices between layers, along flux attachment, procedure reminiscent Read-Rezayi We further this framework motivate wave function state.