The enigma of the $\nu =2+3/8$ fractional quantum Hall effect

The fractional quantum Hall effect at $\nu =2+3/8$, which has been definitively observed, is one of the last fractions for which no viable explanation has so far been demonstrated. Our detailed study suggests that it belongs to a new class of exotic states described by the Bonderson-Slingerland wave function. Its excitations are non-Abelian anyons similar to those of the well studied Pfaffian state at 5/2, but its wave function has a more complex structure. Using the effective edge theory, we make predictions for various measurable quantities that should enable a confirmation of the underlying topological order of this state.

Figure 1

Pair correlation function $g\left(r\right)$ for the exact second Landau level Coulomb state (red), and the Bonderson-Slingerland trial state of Eq. (3) (black) at filling factor $\nu =2+3/8$ for $N=12$ electrons. Inset: spectrum for the fully polarized $\nu =3/8$ state in the second LL obtained by exact diagonalization at $N=12$ and $2Q=31$ for the exact second Landau level Coulomb interaction.

Figure 2

Energy spectra of the $N=12,N_{\varphi }=32$ system on torus with aspect ratio (a) $R=1.0$ and (b) $R=1.5$.

Figure 3

Entanglement spectrum of the $N=12,2Q=31$ system on sphere. $S$ $\left(R\right)$ is the southern (northern) hemisphere. The number of electrons in the southern hemisphere is $N_A=6$. The entanglement levels are labeled by the $z$-component angular momentum of the southern hemisphere $L_z^s$.