  In this work we investigate Wigner localization at very low densities by
means of the exact diagonalization of the Hamiltonian. This yields numerically
exact results. In particular, we study a quasi-one-dimensional system of two
electrons that are confined to a ring by three-dimensional gaussians placed
along the ring perimeter. To characterize the Wigner localization we study
several appropriate observables, namely the two-body reduced density matrix,
the localization tensor and the particle-hole entropy. We show that the
localization tensor is the most promising quantity to study Wigner localization
since it accurately captures the transition from the delocalized to the
localized state and it can be applied to systems of all sizes.
