Electron correlations in charge coupled vertically stacked quantum rings

We study spatial correlations for two electrostatically coupled electrons in a pair of vertically stacked quantum rings with configuration-interaction approach. We demonstrate that for distant quantum rings, the correlations undergo abrupt oscillations with the external magnetic field, which are strong for odd multiples of the flux quantum for which the electrons distinctly avoid being localized one above the other and are negligible for other values of the flux. The oscillations of the correlation strength vanish in the single-ring limit for which the electrons form a Wigner molecule. The wave function of the Hartree approximation is frustrated between reproducing the exact angular momentum eigenstate or the correlation between the electrons. We show that for distant rings the mean-field solution breaks the rotational symmetry of the system in a reentrant manner near odd multiples of half of the flux quantum to account for the correlation appearing in the exact solution.