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

TL;DR

This paper provides evidence that the fractional quantum Hall states at filling factors 12/5 and 13/5 are in the non-Abelian Fibonacci anyon phase, with implications for understanding particle-hole symmetry breaking and phase competition.

Contribution

The study uses infinite density matrix renormalization group to identify the Read-Rezayi phase at 12/5 and 13/5, revealing the nature of charged excitations as Fibonacci anyons and analyzing symmetry-breaking effects.

Findings

12/5 and 13/5 states are in the k=3 Read-Rezayi phase

Fibonacci anyons identified as lowest energy charged excitations

Particle-hole symmetry breaking explains experimental asymmetry

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.