  We consider quantum phases of tightly-confined spin-2 bosons in an external
field under the presence of rotationally-invariant interactions. Generalizing
previous treatments, we show how this system can be mapped onto a quantum rotor
model. Within the rotor framework, low-energy excitations about fragmented
states, which cannot be accessed within standard Bogoliubov theory, can be
obtained. In the spatially extended system in the thermodynamic limit there
exists a mean-field ground state degeneracy between a family of nematic states
for appropriate interaction parameters. It has been established that quantum
fluctuations lift this degeneracy through the mechanism of order-by-disorder
and select either a uniaxial or square-biaxial ground state. On the other hand,
in the full quantum treatment of the analogous single-spatial mode problem with
finite particle number it is known that, due to symmetry restoring
fluctuations, there is a unique ground state across the entire nematic region
of the phase diagram. Within the established rotor framework we investigate the
possible quantum phases under the presence of a quadratic Zeeman field, a
problem which has previously received little attention. By investigating wave
function overlaps we do not find any signatures of the order-by-disorder
phenomenon which is present in the continuum case. Motivated by this we
consider an alternative external potential which breaks less symmetry than the
quadratic Zeeman field. For this case we do find the phenomenon of
order-by-disorder in the fully quantum system. This is established within the
rotor framework and with exact diagonalization.
