The cubic Kugel–Khomskii model for triply degenerate t_2g electrons

The cubic Kugel–Khomskii Hamiltonian for titanates describes spin and orbital superexchange interactions between d¹ ions in an ideal cubic perovskite structure in which the three t_2g orbitals are degenerate in energy, and electron hopping in the presence of large Coulomb interactions is constrained by cubic site symmetry. We review results for the unusual symmetry of this model and give a simple physical argument that explains why this symmetry prevents long-range spin order at non-zero temperatures. We also review the Landau theory of the disordered phase of this model, which gives rise to susceptibilities that are dispersionless along one wavevector axis. We present new results for the mean-field equations, which describe possible long-range order (in the presence of suitable stabilizing perturbations). We also analyse the role of thermal and quantum fluctuations and for the first time give a renormalization group analysis of this model in d spatial dimensions. Finally, we briefly review extensions of this model which are needed to describe real systems, such as LaTiO₃.