Currents in a many-particle parabolic quantum dot under a strong magnetic field

Currents in a few-electron parabolic quantum dot placed into a perpendicular magnetic field are considered. We show that traditional ways of investigating the Wigner crystallization by studying the charge density correlation function can be supplemented by the examination of the density-current correlator. However, care must be exercised when constructing the correct projection of the multidimensional wave function space. The interplay between the magnetic field and Euler-liquid-like behavior of the electron liquid gives rise to persistent and local currents in quantum dots. We demonstrate these phenomena by collating a quasiclassical theory valid in high magnetic fields and an exact numerical solution of the many-body problem.

Figure 1

Local coordinates for three electrons on a ring.

Figure 2

The radial distribution of the density $\left(\rho \right)$ and azimuthal current density $\left(j_{\phi }\right)$ for three electrons in a quantum dot with $\lambda =4$ for angular momentum $M=-18$. The formation of a ring structure is apparent. Note that the currents flow in opposite directions on the inner and outer edges of the ring. The peak of $\rho$ and zero of $j_{\phi }$ are close to the classical electron ring radius $a_0\approx 1.322$ marked by the dot.

Figure 3

The net current crossing the radius of quantum dots containing two, three, and four electrons calculated by exact diagonalization. The angular momenta up to $\mid M\mid =20$ are included. We observe a sawtoothlike behavior reminiscent of persistent currents in quantum rings.

Figure 4

Double-logarithm plot of the current jumps $\Delta I$ versus the dimensionless magnetic field $B$. The symbols (solid lines) denote the numerical exact diagonalization (quasiclassical) results.

Figure 5

The divergence of the density-current correlator in a two-electron quantum dot. The angular momentum is $M=-18$. Panel (a) corresponds to the the lower end (i.e., $B=17.75$) of the magnetic field interval where this state is the ground state, panel (b) is plotted for the medium value $(B=18.25)$, and panel (c) is obtained at the higher end $(B=18.75)$ of this interval. Uncompensated global currents are visible. Dark (light) areas correspond to generation (extinction) of the current. The dotted curve denotes the classical radius of the quantum dot.

Figure 6

The curl of the density-current correlation function for quantum dots containing two, three, and four electrons. The ring of the classical radius is marked by the dotted line, and one electron is pinned at the intersection of this ring with the negative part of the $x$ axis (indicated by the dot). The ground state of $M=-18$ is shown. Dark (light) areas correspond to positive (negative) vorticity, and the solid black line denotes the separating zero-vorticity contour. The quasiclassical prediction for this contour is marked by the white line.

Figure 7

Many-electron density.