Electron-Hole Duality and Vortex Rings in Quantum Dots

In a quantum-mechanical system, particle-hole duality implies that instead of studying particles, we can get equivalent information by studying the missing particles, the so-called holes. Using this duality picture for fermions in a rotating trap the vortices appear as holes in the Fermi sea. Here we predict that the formation of vortices in quantum dots at high magnetic fields causes oscillations in the energy spectrum which can be experimentally observed using accurate tunneling spectroscopy. We use the duality picture to show that these oscillations are caused by the localization of vortices in rings.

Figure 1

Energy spectrum of 20 electrons in a quantum dot as a function of the angular momentum. The solid line connects the lowest-energy states for each angular momentum and shows the characteristic oscillation caused by the vortex localization. The vortex textures are shown schematically below the regions for 1 to 4 vortices. (A smooth function of the angular momentum, $f(L)=A+BL+C{L}^2$, was subtracted from the energies to mimic the effect of the magnetic field on the total energy.) The energy is given in effective atomic units or ${\mathrm{Ha}}^{*}=m^{*}{e}^4/{\hslash }^2(4\pi ϵ{ϵ}_0{)}^2$, where $m^{*}$ is the effective mass and $ϵ$ the dielectric constant of the semiconductor in question.

Figure 2

Vortex-vortex pair correlation function for the four-vortex state in a quantum dot with 20 electrons. The reference point is marked with a black dot. Part (a) shows the result for the calculation of 20 electrons with angular momentum 246 and part (b) shows the result for the calculation of the conjugate system with 4 holes.

Figure 3

Vortex-vortex correlations of the two lowest-energy states for 30 electron quantum dots at angular momentum 555. The ground state (a) shows 6 vortices localized in a ring, while the first excited state (b) shows a five vortex ring with one vortex in the center.