  The Kugel-Khomskii Hamiltonian for cubic titanates describes spin and orbital
superexchange interactions between $d^1$ ions having three-fold degenerate
$t_{2g}$ orbitals. Since orbitals do not couple along "inactive" axes,
perpendicular to the orbital planes, the total number of electrons in $|\alpha
>$ orbitals in any such plane and the corresponding total spin are both
conserved. A Mermin-Wagner construction shows that there is no long-range spin
ordering at nonzero temperatures. Inclusion of spin-orbit coupling allows such
ordering, but even then the excitation spectrum is gapless due to a continuous
symmetry. Thus, the observed order and gap require more symmetry breaking
terms.
