  The five-band Hubbard model for a $d$ band with one electron per site is a
model which has very interesting properties when the relevant ions are located
at sites with high (e. g. cubic) symmetry. In that case, if the crystal field
splitting is large one may consider excitations confined to the lowest
threefold degenerate $t_{2g}$ orbital states. When the electron hopping matrix
element ($t$) is much smaller than the on-site Coulomb interaction energy
($U$), the Hubbard model can be mapped onto the well-known effective
Hamiltonian (at order $t^{2}/U$) derived by Kugel and Khomskii (KK). Recently
we have shown that the KK Hamiltonian does not support long range spin order at
any nonzero temperature due to several novel hidden symmetries that it
possesses. Here we extend our theory to show that these symmetries also apply
to the underlying three-band Hubbard model. Using these symmetries we develop a
rigorous Mermin-Wagner construction, which shows that the three-band Hubbard
model does not support spontaneous long-range spin order at any nonzero
temperature and at any order in $t/U$ -- despite the three-dimensional lattice
structure. Introduction of spin-orbit coupling does allow spin ordering, but
even then the excitation spectrum is gapless due to a subtle continuous
symmetry. Finally we showed that these hidden symmetries dramatically simplify
the numerical exact diagonalization studies of finite clusters.
