Pfaffian and fragmented states at $\nu =\frac{5}{2}$ in quantum Hall droplets

When a gas of electrons is confined to two dimensions, application of a strong magnetic field may lead to startling phenomena such as emergence of electron pairing. According to a theory this manifests itself as appearance of the fractional quantum Hall effect with a quantized conductivity at an unusual half-integer $\nu =\frac{5}{2}$ Landau level filling. Here we show that similar electron pairing may occur in quantum dots where the gas of electrons is trapped by external electric potentials into small quantum Hall droplets. However, we also find theoretical and experimental evidence that, depending on the shape of the external potential, the paired electron state can break down, which leads to a fragmentation of charge and spin densities into incompressible domains. The fragmentation of the quantum Hall states could be an issue in the proposed experiments that aim to probe for non-Abelian quasiparticle characteristics of the $\nu =\frac{5}{2}$ quantum Hall state.

Figure 1

(Color online) The overlaps of the Pfaffian wave function with the corresponding exact state at the lowest (LLL), second (SLL), and third (TLL) Landau level in the case of a Coulombic electron-electron interaction. $N^{\prime }$ denotes the number of electrons in the half-filled Landau level.

Figure 2

(Color online) Overlap of the Pfaffian wave function with the corresponding exact state for $N^{\prime }=6$ and $N^{\prime }=8$ electrons in the half-filled Landau level, respectively, using (a)–(b) screened electron-electron potential with screening length $d_S$ and (c)–(d) softened potential due to finite sample thickness $d_T$ for electrons at the LLL corresponding to $\nu =\frac{1}{2}$, and the SLL corresponding to $\nu =\frac{5}{2}$.

Figure 3

(Color online) (a) Kohn-Sham energy spectrum of a 60-electron quantum dot as a function of single-particle angular momentum $L$ calculated from the density-functional theory with spin-compensated orbitals. The density of states of the SLL is high near the Fermi energy $E_F$. The magnetic field is 2.125 T which corresponds to a filling factor of $\nu =\frac{5}{2}$. (b) The corresponding energy spectrum from spin-density-functional theory shows lifting of the degeneracy near the Fermi level via complete spin polarization of the SLL. The LLL remains spin compensated. Spin $\uparrow$ $(\downarrow )$ corresponds to spin orientation parallel (antiparallel) to the magnetic field. The spin splitting due to many-body effects is about 1.5 meV at $L=0$. In comparison, the Zeeman splitting is about 0.05 meV.

Figure 4

(Color online) Total electron density ${\rho }_{\uparrow }+{\rho }_{\downarrow }$ (full region) and the net spin density ${\rho }_{\uparrow }-{\rho }_{\downarrow }$ (transparent blue region) of quantum Hall states in a quantum dot at (a) $\nu =2$, at (b) an intermediate state between $\nu =2$ and $\nu =\frac{5}{2}$, and at (c) $\nu =\frac{5}{2}$. The latter two show fragmented charge and spin densities with spin-compensated $\nu =2$ region at the edges and spin-polarized $\nu =3$ at the center. The densities were calculated with the spin-density-functional theory for a 60-electron quantum dot. The net spin-up density is due to spin polarization of the second-lowest Landau level.

Figure 5

(Color online) Ground-state energy of the $N=47$ quantum dot calculated with the density-functional theory for the rotationally symmetric eigenstates of the angular momentum $L$ (blue curve) and for the ground states in the symmetry-unrestricted approach (magenta curve). The corresponding chemical potentials $\mu \left(N=47\to 48\right)$ calculated from the theory are shown in the lower panel together with experimental data from a lateral quantum-dot device (sample B). Dashed lines correspond to the boundaries of the spin-droplet regime. The insets show the fragmented spin and charge densities of three of the corresponding states (cf. Fig. 4). The strength of the parabolic confining potential of the quantum dot is ${\omega }_0=2\text{meV}$ in the calculations.

Figure 6

(Color online) Chemical potentials calculated with (a) the density-functional theory and measured from (b) vertical and (c) lateral quantum-dot devices for various electron numbers. Both experiments show the signal associated with the polarization of the second-lowest Landau level at $\nu =5/2$ in the peak position data when $N\gtrsim 30$ in agreement with the theoretical result. The confinement strength ranged from 2 to 4 meV depending on the electron number. The data for the vertical device in (b) are courtesy of Kouwenhoven (Ref. 34) and the data for the lateral device in (c) are courtesy of Sachrajda (Ref. 38).

Figure 7

(Color online) Checkerboard pattern of transport current in density-functional theory (left panel) and spin-blockade experiments (right panel). The lowest current densities correspond to electron transport via states in the second-lowest Landau level, near the core of the quantum dot. The current density in experiments has been amplified in high magnetic fields with a linear function to compensate for the general attenuation of the signal.

Figure 8

(Color online) Comparison of the ground-state energies for given total spin $S$ in the DFT and the QMC method. The number of electrons $N=48$. The linewidths in the QMC denote the statistical error in the results. Only the ground state and the $S=0$ state are shown in the DFT result. The strength of the parabolic confining potential of the quantum dot is ${\omega }_0=2\text{meV}$ in the calculations.