Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 4 / a / Solution

The microscopic electric interaction is spin independent to leading order, but the Pauli exclusion principle ties the spatial symmetry of an electronic state to its spin symmetry. Different spin combinations therefore have different Coulomb and kinetic energies. Projecting those energies onto localized low-energy moments often gives an approximately rotationally invariant bilinear exchange Hamiltonian operator $-J{\mathbf{S}}_1\cdot {\mathbf{S}}_2$. For spin one half, the spin singlet state and triplet state energies are distinguished by ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=-3/4$ and $1/4$, so any isotropic two-level spin singlet state–triplet state splitting can be written in this form plus a constant. More general spins or orbital degeneracy can require extra interactions.

An antiferromagnetic oxide commonly has superexchange through a ligand such as oxygen. Virtual hopping between half-filled metal orbitals lowers the energy of antiparallel spins more strongly because the corresponding intermediate doubly occupied configurations are allowed by the Pauli exclusion principle. The simple one-orbital Hubbard model limit gives $+4t^2{\mathbf{S}}_1\cdot {\mathbf{S}}_2/U$, equivalent to negative $J$ in the question's minus-sign convention. Bond angles and orbital occupancy can change the sign in more complicated oxides, so superexchange is not universally antiferromagnetic irrespective of geometry.