Exchange interactions and magnetic force theorem

We critically reexamine the problem of interatomic exchange interactions, which describe total energy change caused by infinitesimal rotations spins near some equilibrium state. For small variations, such interactions can be always related to response function. However, form this relation depend on additional approximations. Particularly, commonly used magnetic force theorem (MFT) prescribes linear between and function, while exact theory requires dependence inverse. explore origin consequences these differences in definition for wide class materials: ferromagnetic Ni, antiferromagnetic NiO, half-metallic CrO2, multiferroic HoMnO3, layered magnets CrCl3 CrI3. While most cases, MFT produces quite reasonable results rigorously justifies long wavelength strong-coupling limits, formulation appears more consistent, especially dealing with two important issues, typically arise interactions: (i) treatment ligand states, (ii) choice suitable variable description spins. Both issues efficiently resolved employing ideas adiabatic spin dynamics supplemented expression interactions. we propose a simple "downfolding" procedure elimination transferring their effect interaction parameters localized Furthermore, argue that moments are low-energy excitations, whole magnetization matrix cause much stronger perturbation system