Transition from Abelian to non-Abelian quantum liquids in the second Landau level

The search for non-Abelian quantum Hall states in the second Landau level is narrowed to the range of filling factors $7/3<{\nu }_e<8/3$. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau-level mixing, supports the prominent non-Abelian candidates at ${\nu }_e=5/2$ (Moore-Read) and 12/5 (Read-Rezayi). Outside of it, the four-flux noninteracting composite fermion model is validated. The borderline ${\nu }_e=7/3$ state is adiabatically connected to the Laughlin liquid, but its short-range correlations and charge excitations are different.

Figure 1

(Color online) Ternary contour plots of the neutral excitation gaps $\Delta$ for $N$ fermions in a partially filled Landau level with shell angular momentum $\ell$, interacting via model pseudopotential $U_m$. In each plot, three corners of the big triangles correspond to $U_m={\delta }_{m,\mu }$ with $\mu =1,3,5$ marked in each corner. Points relevant for the actual interactions in different electron or CF Landau levels are marked on a triangle in frame (i). Different candidate incompressible states are indicated in frames (a)–(h), labeled by filling factors $\nu$ and shifts $\gamma$ assigned to each finite system $\left(N,2\ell \right)$. For example, the Moore-Read state is marked as (b) in frames (a) and (b).

Figure 2

(Color online) Symmetric second-order differences ${\delta }^{2}E$ of the ground-state energy per particle $E$ for $N=10$ and 12 electrons in ${\text{LL}}_0$ and ${\text{LL}}_1$, as a function of shell angular momentum $\ell$, for layer widths $w=0$ and $3\lambda$. Candidate incompressible states are marked as “$\Phi \nu$,” where $\Phi =P$, $\overline{P}$, $L$, $J$, and $B$ denote Moore-Read pfaffian, antipfaffian, Laughlin and Jain states, and the state of Ref. 9. Squared overlaps ${\left|\chi \right|}^2$ between the states repeating in both LLs at $\nu \le \frac{1}{3}$ are indicated. $\lambda$ is the magnetic length.

Figure 3

(Color online) (a) Size extrapolation of the ground-state energies per particle $E$ calculated for $N$ electrons in the second Landau level. Competing series of candidate incompressible states with filling factors $\nu =\frac{1}{3}$ and $\frac{2}{5}$ are distinguished by their shifts $\gamma$. Energy corrections $dE$ due to Landau-level mixing were obtained for $N=10$ (except for $\nu =\frac{2}{5}$ and $\gamma =2$, where we used $N=8$), assuming that $B=2.6\text{T}$. (b) Dependence of the extrapolated neutral gaps $\Delta$ on the layer width $w$ for the $\nu =\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{5}$ nondegenerate ground states in the second Landau level. For $\nu =\frac{1}{2}$, circles (1) and (2) show gap reduction due to Landau-level mixing relevant for experimental Refs. 14, 12, respectively.

Figure 4

(Color online) Quasiparticle energies $\epsilon$ at the filling factors $\nu =\frac{1}{3}$ and $\frac{1}{2}$ in the second Landau level: (a) and (c) Size extrapolation of the $N$-electron results for the layer widths $w=0$ and $3\lambda$. (b) and (d) Extrapolated quasielectron and quasihole energies, and their sum $\tilde{\Delta }$, as a function of width $w$. In (b), curves for the lowest Landau level are added for comparison. In (d), dotted line includes Landau-level mixing $\left(B=2.6\text{T}\right)$.