Electrons in quantum dots: A comparison of interaction energies

Recent studies of the effects of electron-electron interactions in quantum dots have utilized differing forms for the particle pair-interaction energy: the Coulomb energy and an energy that varies quadratically with particle separation (harmonic interaction). These two models have fundamentally different ground states for quantum dots in high magnetic fields. The ground state for the Coulomb case can have large total angular momentum, while the harmonic ground state is always at the minimum angular momentum. We bridge these two models, and show that the harmonic interaction is valid, i.e., is a good approximation to the Coulomb interaction, in systems where the electrons are strongly confined in the dot. We also demonstrate that while the Laughlin wave function is an exact eigenstate of the harmonic interaction, the harmonic interaction does not exhibit the fractional quantum Hall ground state because the relative angular momentum of the particles is not constrained to be the same for all particle pairs as for the Laughlin picture. This fact limits the validity of the harmonic interaction to systems with strong confinement, where the fractional quantum Hall effect is quenched.