Landau expansion for the Kugel-Khomskii $t_{2g}$ Hamiltonian

The Kugel-Khomskii (KK) Hamiltonian describes spin and orbital superexchange interactions between $d^1$ ions in an ideal cubic perovskite structure, in which the three $t_{2g}$ orbitals are degenerate in energy and electron hopping is constrained by cubic site symmetry. In this paper we implement a variational approach to mean-field theory in which each site i has its own $n\times n$ single-site density matrix $\mathit{\rho }\mathrm{}\left(i\right),$ where n, the number of allowed single-particle states, is 6 (3 orbital times 2 spin states). The variational free energy from this 35 parameter density matrix is shown to exhibit the unusual symmetries noted previously, which lead to a wave-vector-dependent susceptibility for spins in $\alpha$ orbitals which is dispersionless in the $q_{\alpha }$ direction. Thus, for the cubic KK model itself, mean-field theory does not provide wavevector “selection,” in agreement with rigorous symmetry arguments. We consider the effect of including various perturbations. When spin-orbit interactions are introduced, the susceptibility has dispersion in all directions in $\mathbf{q}$ space, but the resulting antiferromagnetic mean-field state is degenerate with respect to global rotation of the staggered spin, implying that the spin-wave spectrum is gapless. This possibly surprising conclusion is also consistent with rigorous symmetry arguments. When next-nearest-neighbor hopping is included, staggered moments of all orbitals appear, but the sum of these moments is zero, yielding an exotic state with long-range order without long-range spin order. The effect of a Hund’s rule coupling of sufficient strength is to produce a state with orbital order.