  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.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics