Prediction of a Non-Abelian Fractional Quantum Hall State with $f$-Wave Pairing of Composite Fermions in Wide Quantum Wells

We theoretically investigate the nature of the state at the quarter filled lowest Landau level and predict that, as the quantum well width is increased, a transition occurs from the composite fermion Fermi sea into a novel non-Abelian fractional quantum Hall state that is topologically equivalent to $f$-wave pairing of composite fermions. This state is topologically distinct from the familiar $p$-wave paired Pfaffian state. We compare our calculated phase diagram with experiments and make predictions for many observable quantities.

Figure 1

(a) Energies of several candidate states at $\nu =1/4$ as a function of density $\rho$ for a quantum well of width 60 nm. The different states are labeled as shown on the figure. All energies are thermodynamic values, measured relative to the energy of the Pfaffian state. Only the CFFS and 22 111 states become ground states for the parameters studied. (Inset) The electron density as a function of transverse position for densities $1.5\times {10}^{11}$ and $2.0\times {10}^{11}{\mathrm{cm}}^{-2}$ as given by LDA for a quantum well of width 60 nm. (b) The calculated phase diagram at $\nu =1/4$ as a function of the quantum well width and density. In the region of parameter space shown in the figure only the CFFS and 22 111 states are realized. We also include experimental results, shown by black squares, taken from Refs. [13, 14]. (c) Energies of several bilayer states as a function of the layer separation $d/l$. We have studied 11 liquid states (Table 2) and 24 crystal states (Supplemental Material [33]). Here we omit the high energy states (see Supplemental Material for more complete results) and show the energies of the $(1/5,1/5|3)$ state, the pseudospin singlet CFFS, the pseudospin polarized CFFS, and several crystal states (notation explained in Supplemental Material [33]). All energies are measured relative to the $(1/5,1/5|3)$ state. No FQHE is stabilized.