Quantum Hall droplet laterally coupled to a quantum ring

We study a two-dimensional cylindrically symmetric electron droplet separated from a surrounding electron ring by a tunable barrier using the exact diagonalization method. The magnetic field is assumed strong so that the electrons become spin polarized and reside on the lowest Fock-Darwin band. We calculate the ground state phase diagram for six electrons. At weak coupling, the phase diagram exhibits a clear diamond structure due to the blockade caused by the angular momentum difference between the two systems. We find separate excitations of the droplet and the ring as well as the transfer of charge between the two parts of the system. At strong coupling, interactions destroy the coherent structure of the phase diagram, while individual phases are still heavily affected by the potential barrier.

Figure 1

Scaling of the potential barrier parameter $C=\left(l^3/a_{\ast }\right)\hslash \omega \beta$ with the magnetic field for $\beta =1$.

Figure 2

(Color online) (a) The ground-state angular-momentum phase diagram for a strong barrier $\left(\beta =1\right)$ six-electron system as a function of the position of the potential barrier $r_0$ indicated by black lines, and the magnetic field $B$. The curvilinearity of the coordinate axis is due to the reason that when we move in the vertical direction in the phase diagram, $r_0$ in units of the oscillator length $l$ remains dimensionless constant as the magnetic field changes. The diagram separates into seven branches according to the number of electrons in the inner droplet $N_I$ indicated in the figure. (b) The same for a weak barrier $\left(\beta =0.2\right)$. The color (lightness) indicates the angular momentum. For example, yellow (the lightest gray) corresponds to $M=45$, the angular momentum for filling fraction $\nu =1/3$ state found at large $r_0$.

Figure 3

The ground state angular momentum along lines of constant $r_0$ in the $N_I=6$ branch of Fig. 2a.

Figure 4

(Color online) The total ground state angular momentum (solid line) and the angular momentum of the ring (black dots) along lines of constant $r_0$ in the $N_I=0,1,2,3,4$ and 5 branches of Fig. 2a. The vertical dashed red lines mark the transitions of the inner system. Occupation numbers for the ground states in panels (a)–(f) evaluated at the first magnetic field value at each plateau, the uppermost corresponding to $B=0\text{T}$.

Figure 5

(Color online) The same as Fig. 5 for $\beta =0.2$.

Figure 6

The edge Green’s functions at three radial distances $r$ for the two states in the $\nu =1/3$ topological phase marked by the squares in Fig. 2b. The black square corresponds to the black line.

Figure 7

The edge Green’s function at three radial distances $r$ for at the point $r_0=30\text{nm}$, $B=5.7\text{T}$, and $\beta =0.2$ marked in the strong coupling phase diagram of Fig. 2b.