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

Abstract

The $\nu =12/5$ fractional quantum Hall plateau observed in $\mathrm{GaAs}$ 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 $\nu =12/5$ and $\nu =13/5$ 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 $\nu =13/5$ observe a re-entrant 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 re-entrant integer quantum Hall phase. This competition suggests that even small particle-hole symmetry-breaking perturbations can explain the experimentally observed asymmetry between $\nu =12/5$ and $13/5$. We find that at $\nu =12/5$ Landau level mixing favors the Read-Rezayi phase over the re-entrant phase.

8 pages + appendices, 14 figures. v2 added data for Landau level mixing

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