Nature of Excitations of the $\frac{5}{2}$ Fractional Quantum Hall Effect

It is shown, with the help of exact diagonalization studies on systems with up to 16 electrons, in the presence of up to two delta function impurities, that the Pfaffian model is not accurate for the actual quasiholes and quasiparticles of the $\frac{5}{2}$ fractional quantum Hall effect. Implications for non-Abelian statistics are discussed.

Figure 1

Left panel: Charge densities for the ground state for ${\Psi }_{2\mathrm{\text{−}}\mathrm{qh}}^{\mathrm{Pf}}$ (a), ${\Psi }_{2\mathrm{\text{−}}\mathrm{qh}}^{(3)}$ (b), and ${\Psi }_{2\mathrm{\text{−}}\mathrm{qh}}^{\mathrm{C}}$ (d) in the presence of two delta function impurities at the two poles, and for ${\Psi }_{2\mathrm{\text{−}}\mathrm{qh}}^{\mathrm{C}}$ (c) in the presence of one delta function impurity at the north pole. The results are shown for $N=12$ (dashed lines), $N=14$ (dotted lines) and $N=16$ (solid lines) electrons at $2Q=2N-2$. The density for the Pfaffian wave function ${\Psi }_{2\mathrm{\text{−}}\mathrm{qh}}^{\mathrm{Pf}}$ (obtained by Monte Carlo) is also shown for comparison, with the quasiholes placed on the two poles. When the ground state has $L_z\ne 0$, there are two degenerate states at $\pm L_z$; we have shown only one of them for simplicity. The normalization is chosen to ensure that the integral of the entire sphere gives $N$. Right panel: the integrated excess charge for each density, normalized so that the total charge excess is $\frac{1}{2}$. Two spatially separated quasiholes will exhibit a step at charge $\frac{1}{4}$.

Figure 2

Same as in Fig. 1, but for quasiparticles. There are only three panels, because a Pfaffian wave function analogous to ${\Psi }_{2\mathrm{\text{−}}\mathrm{qh}}$ is not known.