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

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.

5 pages, 4 figures; v2 is the expanded published version with 7 pages, 6 figures, and additional references

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