Abelian and non-Abelian states in $\nu =2/3$ bilayer fractional quantum Hall systems

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.

Figure 1

(a)–(f) Shifts of $\mathcal{S}=3$, 0, and 1 from left to right for $N=8$ electrons. (a)–(c) The overlap with ${\mathrm{\Psi }}_{330},2/3$ Laughlin, and the pseudospin singlet (112), respectively. (d)–(f) The energy gap. (g) The overlap with $Z_4$ state for $N=12$ at $\mathcal{S}=3$. (h) The gap for $N=12$ and shift $\mathcal{S}=3$. (i) Displays the resulting quantum phase diagram.

Figure 2

(a)–(f) Shifts of $\mathcal{S}=3$, 0, and 1 from left to right for $N=6$ electrons in the lowest Landau level. (a)–(c) The overlap with ${\mathrm{\Psi }}_{330},2/3$ Laughlin, and the pseudospin singlet (112), respectively. (d)–(f) The energy gap.

Figure 3

(a)–(f) Shifts of $\mathcal{S}=3$, 0, and 1 from left to right for $N=10$ electrons in the lowest Landau level. (a)–(c) The overlap with ${\mathrm{\Psi }}_{330},2/3$ Laughlin, and the pseudospin singlet (112), respectively. (d)–(f) The energy gap. The white space beyond $d/l>5.5$ for $S=0$ was not calculated and therefore left blank.

Figure 4

Overlap and gap calculations for $N=16$ electrons in the LLL in the single-component limit. Top left (a) shows overlaps with the $Z_4$ state, at $\mathcal{S}=3$; top right (b) shows overlaps with the Laughlin state, at $\mathcal{S}=0$. Lower panels (c) and (d) show the gaps at $S=3$ and $S=0$.

Figure 5

Overlaps (a) and (b) and energy gaps (c) and (d) for $N=16$ electrons in the SLL in the single-component limit. See caption of Fig. 4.

Figure 6

The quantum phase diagram determined by variational energies at thickness $w=0$. The contour lines depict the energy advantage $\delta E$ of each dominant phase. In the SLL, the Laughlin state does not have a clear energy advantage over the $Z_4$ state.