Spin-1 effective Hamiltonian with three degenerate orbitals: An application to the case of V2O3

Motivated by recent neutron and x-ray observations in ${\mathrm{V}}_{2}{\mathrm{O}}_{3},$ we derive the effective Hamiltonian in the strong coupling limit of an Hubbard model with three degenerate ${t}_{2g}$ states containing two electrons coupled to spin $S=1,$ and use it to reexamine the low-temperature ground-state properties of this compound. An axial trigonal distortion of the cubic states is also taken into account. Since there are no assumptions about the symmetry properties of the hopping integrals involved, the resulting spin-orbital Hamiltonian can be generally applied to any crystallographic configuration of the transition metal ion giving rise to degenerate ${t}_{2g}$ orbitals. Specializing to the case of ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$ we consider the low-temperature antiferromagnetic insulating phase. We find two variational regimes, depending on the relative size of the correlation energy of the vertical pairs and the in-plane interaction energy. The former favors the formation of stable molecules throughout the crystal, while the latter tends to break this correlated state. Using the appropriate variational wave functions we determine in both cases the minimizing orbital solutions for various spin configurations, compare their energies and draw the corresponding phase diagrams in the space of the relevant parameters of the problem. We find that none of the symmetry-breaking stable phases with the real spin structure presents an orbital ordering compatible with the magnetic space group indicated by very recent observations of nonreciprocal x-ray gyrotropy in ${\mathrm{V}}_{2}{\mathrm{O}}_{3}.$ We do, however, find a compatible solution with very small excitation energy in two distinct regions of the phase space, which might turn into the true ground state of ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$ due to the favorable coupling with the lattice. We illustrate merits and drawbacks of the various solutions and discuss them in relation to the present experimental evidence.