Spin-orbital interaction for face-sharing octahedra: Realization of a highly symmetric SU(4) model

Specific features of orbital and spin structure of transition-metal compounds in the case of the face-sharing MO₆ octahedra are analyzed. In this geometry, we consider the form of the spin-orbital Hamiltonian for transition-metal ions with double (eg^σ) or triple (t2g) orbital degeneracy. Trigonal distortions typical of the structures with face-sharing octahedra lead to splitting of t2g orbitals into an a1g singlet and eg^π doublet. For both doublets (eg^σ and eg^π), in the case of one electron or hole per site, we arrive at a symmetric model with the orbital and spin interaction of the Heisenberg type and the Hamiltonian of unexpectedly high symmetry: SU(4). Thus, many real materials with this geometry can serve as a testing ground for checking the prediction of this interesting theoretical model. We also compare general trends in the spin-orbital (``Kugel-Khomskii'') exchange interaction for three typical situations: those of MO₆ octahedra with common corner, common edge, and the present case of common face, which has not been considered yet.