Abelian and Non-Abelian States in $\nu=2/3$ Bilayer Fractional Quantum Hall Systems

TL;DR

This study investigates the phase diagram of fractional quantum Hall systems at various filling fractions, finding that certain non-Abelian states are more likely in single-component systems, with implications for topological quantum computation.

Contribution

It provides the first systematic numerical analysis of non-Abelian states at $ u=2/3$, especially the $Z_4$ parafermion state, in bilayer and single-component FQH systems.

Findings

No $Z_4$ state in $ u=2/3$ bilayers.

$Z_4$ state has higher overlap at $ u=8/3$, indicating potential non-Abelian nature.

Results from exact diagonalization and variational Monte Carlo agree qualitatively.

Abstract

There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two- component FQH systems at total filling fraction $\nu = n+ 2/3$, for integer $n$. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here, we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\nu = n+2/3$, including in particular the possibility of the non-Abelian $Z_4$ parafermion state. In $\nu = 2/3$ bilayers, we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the $Z_4$ state. On the other hand, in single-component systems at $\nu = 8/3$, we find that the $Z_4$ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\nu = 8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.