Exchange interaction

The exchange interaction is an effective interaction between spins originating in the quantum mechanics of Electrons. Its sign convention must accompany any formula. For the illustrative classical fixed-length spin pair used here, writeWith this plus-sign convention, $J>0$ favors antiparallel spins: $E(\pi )=-J{S}^2$, compared with $E(0)=+J{S}^2$. The energy difference is $2J{S}^2$. If instead the Hamiltonian is written with an overall minus sign, the same physical preference has a negative coupling. This prevents a sign convention in the Ising model from being mistaken for a disagreement about the ordering.

Let $a,b$ be distinct orthonormal spatial orbitals occupied by two identical spin one-half particles, with a weak repulsive Coulomb interaction. The symmetric and antisymmetric spatial combinations have first-order energy shifts $J+K$ and $J-K$, where $J$ is the direct Coulomb integral andThe Coulomb kernel has positive Fourier transform $4\pi /|\mathbf{k}{|}^2$, so $K\ge 0$. Fermionic exchange symmetry pairs the symmetric spatial function with the spin-one-half singlet state and the antisymmetric one with the spin-one-half triplet state. For these fixed orbitals the triplet lies lower by $2K$ when $K>0$. This does not compare arbitrary singlets and triplets with different orbital occupations; for negligible orbital overlap the splitting can vanish.

Superexchange couples localized moments through virtual hopping via intermediate, often nonmagnetic, ligand orbitals. Half-filled orbitals sharing an effective hopping amplitude $t$ give an antiferromagnetic scale $J\simeq 4|t{|}^2/U$ in the repulsive one-band limit. Ligand geometry, orbital occupation and exchange within intermediate states can change the balance and allow ferromagnetic coupling. Real charge motion can remain gapped while these virtual processes couple an insulating magnet's spins.

Direct exchange comes from Coulomb matrix elements between overlapping localized electronic orbitals together with the antisymmetry required by the Pauli exclusion principle. It produces a spin-dependent energy without being a classical dipole interaction. Its competition with kinetic processes and orbital occupation helps determine whether neighbouring moments prefer parallel or antiparallel alignment.