Exchange interactions in magnetic insulators

Abstract

An account is given of the theory of the exchange interactions between localized moments in an insulating crystal, using recently developed techniques. The Hamiltonian, H , can be a quite general one, and it is treated by degenerate perturbation theory in which the unperturbed Hamiltonian is a defined quantity, and chosen to have the same symmetry properties as H . In particular it is invariant with respect to interchanges of electrons. The perturbation expansion leads to an expression in second quantized operators, which is recast into an equivalent operator in angular momentum operators, defined for the sites. The result is the ‘spin-Hamiltonian’. The first order perturbation terms give large single-site terms and reproduce crystal field theory. Corrections to this, and the exchange and other interactions, appear mainly in second order, and they renormalize spin-Hamiltonian parameters which are already present in first order. Crystal field theory and the theory of exchange interactions, which have usually been treated separately in the past, are nicely unified, and a clear route is provided whereby one can pass from a general Hamiltonian to a spin-Hamiltonian.