Hidden symmetries and their consequences in $t_{2g}$ cubic perovskites

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 $\mathrm{}\left(U\right),$ 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. The 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 show that these hidden symmetries dramatically simplify the numerical exact diagonalization studies of finite clusters.