  Strong repelling interactions between a few fermions or bosons confined in
two-dimensional circular traps lead to particle localization and formation of
quantum Wigner molecules (QWMs) possessing definite point-group space
symmetries. These point-group symmetries are "hidden" (or emergent), namely
they cannot be traced in the circular single-particle densities (SPDs)
associated with the exact many-body wave functions, but they are manifested as
characteristic signatures in the ro-vibrational spectra. An example, among
many, are the few-body QWM states under a high magnetic field or at fast
rotation, which are precursor states for the fractional quantum Hall effect.
The hidden geometric symmetries can be directly revealed by using spin-resolved
conditional probability distributions, which are extracted from
configuration-interaction (CI), exact-diagonalization wave functions. The
hidden symmetries can also be revealed in the CI SPDs by reducing the symmetry
of the trap (from circular to elliptic to quasi-linear). In addition the hidden
symmetries are directly connected to the explicitly broken-symmetry (BS)
solutions of mean-field approaches, such as unrestricted Hartree-Fock (UHF). A
companion step of restoration of the broken symmetries via projection operators
applied on the BS-UHF solutions produces wave functions directly comparable to
the CI ones, and sheds further light into the role played by the emergence of
hidden symmetries in the exact many-body wave functions. Illustrative examples
of the importance of hidden symmetries in the many-body problem of few
electrons in semiconductor quantum dots and of few ultracold atoms in optical
traps (where unprecedented control of the interparticle interaction has been
experimentally achieved recently) will be presented.
