Non-Abelian quantized Hall states of electrons at filling factors 12/5 and 13/5 in the first excited Landau level

We present results of extensive numerical calculations on the ground state of electrons in the first excited $\left(n=1\right)$ Landau level with Coulomb interactions, and including nonzero thickness effects, for filling factors 12/5 and 13/5 in the torus geometry. In a region that includes these experimentally relevant values, we find that the energy spectrum and the overlaps with the trial states support the previous hypothesis that the system is in the non-Abelian $k=3$ liquid phase we introduced in a previous paper.

Figure 1

(Color online) Low-lying spectrum for 18 electrons on the torus on the torus vs $K=\left|\mathbf{K}\right|$ for hexagonal unit cell. Energies are in units of $e^2/{\ell }_B$. The interaction Hamiltonian is the $n=1$ Coulomb interaction, including nonzero thickness with parameter $w=2{\ell }_B$ and $\delta V_1=-0.0035$.

Figure 2

(Color online) The sum of the squared overlap of each member of the ground-state doublet for $n=1$ Coulomb interaction (for $w=2$ and $w$) with the trial two-dimensional subspace, as a function of $\delta V_1$ on the torus, for 15 electrons. (Also included is the same for a quantum well wave function in $x_3$, labeled QW, with $w=2.75$.) Also, the squared overlap of one of the ground-state doublets for $w=2$ with the H-CF state, for both 15 and 18 electrons, as a function of $\delta V_1$ in the same system: solid line—lowest in doublet; dashed line—next lowest.

Figure 3

(Color online) The low-lying energy spectrum as a function of $\delta V_1$ on the torus for 15 particles. The $\mathbf{K}=\mathbf{0}$ and $\mathbf{K}\ne \mathbf{0}$ levels are shown as distinct symbols.

Figure 4

(Color online) A three-dimensional plot of the total squared overlap with the RR trial subspace for $N=15$ particles on the torus, with $w=2$, as a function of both $\delta V_1$ and $\delta V_3$.

Figure 5

(Color online) The same as Fig. 4 but shown as a contour plot.

Figure 6

(Color online) A contour plot of the level splitting between the two lowest $\mathbf{K}=\mathbf{0}$ states for 15 particles as a function of $\delta V_1$ and $\delta V_3$.

Figure 7

(Color online) Low-lying energy spectrum, shown as the difference from the ground-state energy, for 18 particles as a function of $\delta V_1$. Again, states at $\mathbf{K}=\mathbf{0}$ and $\mathbf{K}\ne \mathbf{0}$ are shown with distinct symbols.