When a system consisting of many interacting particles is set rotating, it may form vortices. This is familiar to us from every-day life: you can observe vortices while stirring your coffee or watching a hurricane. In the world of quantum mechanics, famous examples of vortices are superconducting films 1 and rotating bosonic 4 He or fermionic 3 He liquids 2,3 . Vortices are also observed in rotating Bose-Einstein condensates in atomic traps 4,5,6 and are predicted to exist 7 for paired fermionic atoms 8,9 . Here we show that the rotation of trapped particles with a repulsive interaction leads to a similar vortex formation, regardless of whether the particles are bosons or (unpaired) fermions. The exact, quantum mechanical many-particle wave function provides evidence that in fact, the mechanism of this vortex formation is the same for boson and fermion systems.Let us now consider a number of identical particles with repulsive interparticle interactions confined in a harmonic trap under rotation. These particles could be electrons in a quantum dot 10 , positive or negative ions, or neutral atoms in boson or fermion condensates 11 . Though simple to describe, this quantum mechanical many-body problem is extremely complex and in general not solvable exactly. Consequently, in rotating systems the formation of vortices and their mutual interaction is usually described using a mean field approximation. In superconductors this is the Ginzburg-Landau method 1 . For Bose-Einstein condensates, one often applies the Gross-Pitaevskii equation 11,12 . In this way, Butts and Rokhsar 13 found successive transitions between stable patterns of singly-quantised vortices, as the angular momentum was increased. A single vortex appears when the angular momentum L is equal to the number of particles N , two vortices appear at L ∼ 1.75N and three vortices at L ∼ 2.1N (see Refs. 13,14,15,16 ). For quantum dots in strong magnetic fields, the occurrence of vortices was very recently discussed by Saarikoski et al.17 .Based on the rigorous solution of the many-particle Hamiltonian, we show that striking similarities between the boson and fermion systems exist: the vortex formation is indeed universal for both kinds of particles, and the many-particle configurations generating these vortices are the same. For a small number of particles, the many-body Hamilton operator can be diagonalised numerically. We use a single particle basis of Gaussian functions to span the Hilbert space. These Gaussians are eigenstates of the trap for radial quantum number n = 0 and different single-particle angular momenta. These states dominate for large total angular momenta L. We only consider one spin state, i.e. bosons with zero spin or spin-polarised fermions. Numerical feasibility limits calculations to small particle numbers N . However, the advantages of our approach as compared to mean field methods are that our solutions (i) are exact (up to numerical accuracy), (ii) maintain the circular symmetry and thus have a good angular momentum, and (iii) allow ...

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.

ABSTRACT:Using quantum dot artificial atoms as a simple toy model, we reflect on the question of whether spin density functional theory (SDFT) can accurately describe correlation effects in low-dimensional fermion systems. Different expressions for the local density approximation of the exchange-correlation energy for the twodimensional electron gas, such as the much-used functional of Tanatar and Ceperley, and the recent suggestion by Attaccalite et al., are compared with the results of a numerical diagonalization of the many-body Hamiltonian matrix in the limit of small electron numbers. For systems with degeneracies, as shown in the present work for the example of a spin triplet with S ϭ 1, the direct comparison with configuration interaction (CI) methods demonstrates that the spin representation of SDFT may, under certain circumstances, produce artificial energy splittings between states that belong to the same spin multiplet. For a singlet ground state with S ϭ S z ϭ 0, however, the correlation functions of the CI solutions confirm the spin-density wave states found earlier within the SDFT method.

Vortices can form when finite quantal systems are set rotating. In the limit of small particle numbers, the vortex formation in a harmonically trapped fermion system, with repulsively interacting particles, shows similarities to the corresponding boson system, with vortices entering the rotating cloud for increasing rotation. For a larger number of fermions, N ≳ 15, the fermion vortices compete and co-exist with (Chamon–Wen) edge-reconstructed ground states, forcing some ground states, as for example the central single vortex, into the spectrum of excited states. Experimentally, the fermion system could, for instance, be electrons in a semiconductor heterostructure, a quantum dot, and the corresponding boson system, a Bose–Einstein condensate in a magneto optical trap.

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