Unusual Symmetries in the Kugel-Khomskii Hamiltonian

The Kugel-Khomskii Hamiltonian for cubic titanates describes spin and orbital superexchange interactions between d¹ ions having threefold degenerate t_2g orbitals. Since orbitals do not couple along “inactive” axes, perpendicular to the orbital planes, the total number of electrons in |α⟩ 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.