Neutral Fermion Excitations in the Moore-Read State at Filling Factor $\nu =5/2$

We present evidence supporting the weakly paired Moore-Read phase in the half-filled second Landau level, focusing on some of the qualitative features of its excitations. Based on numerical studies, we show that systems with odd particle number at the flux $N_{\varphi }=2N-3$ can be interpreted as a neutral fermion mode of one unpaired fermion, which is gapped. The mode is found to have two distinct minima, providing a signature that could be observed by photoluminescence. In the presence of two quasiparticles the same neutral fermion excitation is shown to be gapless, confirming expectations for non-Abelian statistics of the Ising model with degenerate fusion channels 1 and $\psi$.

Figure 1

Energy spectra (bare interaction energy $E$ versus angular momentum $L$) of $N=15$ electrons in a half-filled second LL, interpreted in terms of the neutral fermion (NF) excitation in the Pfaffian (Pf) ground state. Different values of the magnetic flux are $N_{\varphi }=26$ (top), 27 (center), and 28 (bottom), corresponding to the NF with additional pair of charged quasielectrons (QEs), NF alone, and NF with additional pair of charged quasiholes (QHs), respectively. Different interactions are ${\mathcal{H}}_C=\mathrm{\text{pure}}$ Coulomb (left), ${\mathcal{H}}_1=\mathrm{\text{Coulomb}}$ with an additional enhancement of the $m=1$ pair pseudopotential (center), and ${\mathcal{H}}_{\mathrm{Pf}}=\mathrm{\text{three-body}}$ Pfaffian Hamiltonian with the only triplet pseudopotential at $m=3$ (right). Labels indicate squared overlaps $|⟨{\mathcal{H}}_{C/1}|{\mathcal{H}}_{\mathrm{Pf}}⟩{|}^2$ for the low-lying bands.

Figure 2

Dispersions (energy $E$ versus wave vector $k$) of the neutral fermion (NF) collective modes of the half-filled second LL, estimated from the systems of $N\le 19$ electrons at the magnetic flux $N_{\varphi }=2N-3$, for the different Hamiltonians of Fig. 1. Gray dashed lines in (b) show the evolution of ${\Delta }_{\mathrm{NF}}$ with $\delta V_1$. For comparison, (c) also shows the magnetoroton mode for ${\mathcal{H}}_{\mathrm{Pf}}$.

Figure 3

Comparison of the total energies of a pair of QEs or QHs, with and without an additional NF (see text for the precise definition), for systems of different size $N$ and for the different Hamiltonians of Fig. 1. Top: energies $⟨E_{\alpha }⟩$ averaged over all $2\mathrm{QE}/2\mathrm{QH}$ ($+\mathrm{NF}$) states; bottom: energies $E_{\mathrm{prox}}$ taken for the smallest average QE-QE or QH-QH distance. While $⟨E_{\alpha }⟩$ extrapolate to similar values with or without NF, the differences are significant for $E_{\mathrm{prox}}$.