A composite particle construction of the Fibonacci fractional quantum Hall state

TL;DR

This paper constructs a Fibonacci fractional quantum Hall state using non-Abelian dualities and a trilayer bosonic system, providing a dynamical picture and a wave function proposal for this topologically ordered phase.

Contribution

It introduces a new construction of the Fibonacci quantum Hall state from a trilayer bosonic system via non-Abelian dualities, advancing understanding of its emergence.

Findings

Fibonacci state constructed from trilayer bosons using dualities

Clustering of composite vortices leads to Fibonacci order

Proposed wave function for the Fibonacci fractional quantum Hall state

Abstract

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