Fibonacci anyons and charge density order in the 12/5 and 13/5 quantum Hall plateaus

The ν=12/5 fractional quantum Hall plateau observed in GaAs semiconductor wells is a suspect in the search for non-Abelian Fibonacci anyons. Using the infinite density matrix renormalization group, we find clear evidence that fillings ν=12/5 and ν=13/5 are in the k=3 Read-Rezayi phase in the absence of particle-hole symmetry breaking effects. The lowest energy charged excitation is identified as a non-Abelian Fibonacci anyon, distinguished from its Abelian counterpart by its local quadrupole moment. However, several experiments at ν=13/5 observe a reentrant integer quantum Hall effect, implying particle-hole symmetry is broken. We rule out spin polarization as the origin of the asymmetry. Further, we point out extremely close energetic competition between the Read-Rezayi phase and a reentrant integer quantum Hall phase. This competition suggests that even small particle-hole symmetry breaking perturbations can explain the experimentally observed asymmetry between ν=12/5 and 13/5. We find that at ν=12/5 Landau level mixing favors the Read-Rezayi phase over the reentrant phase.