  d- and f-shells support a large number of local degrees of freedom: dipoles,
quadrupoles, octupoles, hexadecapoles, etc. Usually, the ordering of any
multipole component leaves the system sufficiently symmetrical to allow a
second symmetry breaking transition. Assuming that a second continuous phase
transition occurs, we classify the possibilities. We construct the symmetry
group of the first ordered phase, and then re-classify the order parameters in
the new symmetry. While this is straightforward for dipole or quadrupole order,
it is less familiar for octupole order. We give a group theoretical analysis,
and some illustrative mean field calculations, for the hypothetical case when a
second ordering transition modifies the primary T(xyz) octupolar ordering in a
tetragonal system like URu2Si2. If quadrupoles appear in the second phase
transition, they must be accompanied by a time-reversal-odd multipole as an
induced order parameter. For O(xy), O(zx), or O(yz) quadrupoles, this would be
one of the components of J, which should be easy either to check or to rule
out. However, a pre-existing octupolar symmetry can also be broken by a
transition to a new octupole--hexadecapole order, or by a combination of O(22)
quadrupole and triakontadipole order. It is interesting to notice that if
recent NQR results (Saitoh et al, 2005) on URu2Si2 are interpreted as a hint
that the onset of octupolar hidden order at T0=17K is followed by quadrupolar
ordering at T* = 13.5K, this sequence of events may fit several of the
scenarios found in our general classification scheme. However, we have to await
further evidence showing that the NQR anomalies at T* = 13.5K are associated
with an equilibrium phase transition.
