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.

Magnetic moments can interact through intermediate orbitals or itinerant carriers rather than direct overlap alone. Superexchange uses virtual charge transfer, the RKKY interaction uses conduction-electron spin response, and double exchange gains carrier kinetic energy through spin-dependent hopping. More than one mechanism can operate in the same solid, with different signs and ranges.

A local moment induces conduction-electron spin polarization, which mediates a second-order coupling to another moment. The static electronic spin susceptibility fixes the spatial interaction. For a three-dimensional spherical Fermi surface its leading large-distance form oscillates as $\cos (2k_{F}R)/R^3$, up to the overall sign convention and prefactor. The sign alternates with separation. Other band structures and dimensions change this form. ^[ref]

For localized orbitals with an exponential tail $e^{-\kappa r}$, overlap-mediated hopping over separation $d$ inherits an exponentially small factor $e^{-\kappa d}$, multiplied by geometry-dependent polynomial factors. A kinetic-exchange scale then falls approximately as $e^{-2\kappa d}$. Comparisons must use actual nearest-neighbour distances, not interchange lattice constants between different crystal structures. The estimate is not a replacement for a screened solid-state orbital calculation.

A localized magnetic moment can polarize a metal's conduction electrons, whose response changes the energy of a second moment. This indirect exchange interaction can be long ranged and oscillate in sign with separation because of the Fermi surface. In the three-dimensional isotropic free-electron asymptotic limit its range dependence is proportional to $\cos (2k_{F}r)/r^3$, with coupling-dependent prefactor and sign convention. The original calculation for nuclear moments used their contact hyperfine interaction with conduction electrons; it was motivated by nuclear magnetic resonance line widths in metallic silver.^[ref]

Double exchange couples localized magnetic moments through actual carrier hopping between sites of different valence. Strong local Hund's coupling makes hopping energetically easier when neighboring core spins align, lowering the carrier kinetic energy. In the classical strong-coupling limit, a common effective hopping is $t_{\mathrm{eff}}=t\cos (\theta /2)$ for relative spin angle $\theta$; aligned cores maximize its magnitude. Unlike insulating superexchange, this mechanism directly connects carrier motion and magnetic alignment.^[ref]

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.

In a minimal hole description of a metal–ligand–metal path, two holes initially occupy separate metal orbitals and the ligand has no hole. Creating a virtual ligand hole costs a charge-transfer energy $\Delta$. Two hops generate an effective metal–metal hopping of order $t_{pd}^2/\Delta$; two more hops through metal double occupancy generate the antiferromagnetic scale in the title. Ligand double occupancy adds a channel with denominator $2\Delta +U_p$. Orbital geometry and intra-atomic exchange determine when the antiferromagnetic channel dominates.

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.

A direct intersite Coulomb exchange integral acts without an intermediate doubly occupied state. In a repulsive one-band Hubbard model, by contrast, virtual hopping into a doubly occupied state lowers the singlet energy by $4t^2/U$ relative to the triplet. The latter is antiferromagnetic kinetic exchange, even when hopping is directly between neighbouring magnetic orbitals. Ligand-mediated superexchange introduces an intermediate ligand orbital as well.