The rather complex interplay of orbital and spin degrees of freedom can be formulated in terms of spin-orbital models, which are derived from more general multi-band Hubbard models. Central for the theoretical discussion of spin-orbital physics is the superexchange interaction, which for orbital degenerate systems lead to quite complex spin-orbital (or Kugel-Khomskii type) models
H = ∑<i,j>[Jij(Ti ,Tj)Si · Sj + Kij(Ti ,Tj)],
Such systems can also be viewed as Heisenberg spin models H = J∑ Si · Sj, where the exchange parameters Jij depend on the bond direction and the orbital occupation.
Exchange parameters Jij and Kij are now no longer constants but operators. These are expressed by pseudospin operators Ti which represent the orbital degrees of freedom. Thus the size and even the sign of Jij in a spin-orbital state may fluctuate, i.e. determined by the quantum dynamics of spins and orbitals. Candidates for such strongly entangled spin and orbital degrees of freedom are t2g electron systems, i.e., like the vanadate perovskites, which form a central challenge in our research.
In more conventional cases, however, the orbital state is frozen due to strong Jahn-Teller interactions with the lattice, i.e., leading to a quenching of the orbital dynamics. In general the balance of this interplay depends not only on the strength of the electron-lattice interaction but also on the particular state and its spin-orbital correlations. Frequently also the relativistic spin-orbit coupling HLS = Λ∑i(Si · Ii) contributes. Then the spin part of the Hamiltonian will also turn out anisotropic.
(The logo indicates the orbital order in the a-b plane of LaMnO3.)