Exact diagonalization study of $\nu=1/3+1/3$ bilayer quantum Hall in search of Fibonnacci anyons

Non-abelian states with Fibonnacci anyons that can support universal topological quantum computation have been elusive. Recently it has been proposed that a remarkably simple setting of Abelian quantum Hall bilayer could support an exotic state with Fibonnacci anyons (Vaezi and Barkeshli, arXiv:1403.3383). Here we explore $\nu=1/3+1/3$ bilayer quantum Hall system considering different possibilities for interaction between bilayers using exact diagonalization. We find a sizable region in phase space potentially exhibiting topological degeneracy expected of Fibonnacci anyon states.