Exchange interactions and magnetic force theorem

We critically reexamine the problem of interatomic exchange interactions, which describe the total energy change caused by infinitesimal rotations of spins near some equilibrium state in the framework of constrained spin-density functional theory (cSDFT). For small variations of the spin magnetization, such interactions can always be related to the response function (or transverse spin susceptibility). However, the form of this relation can depend on additional approximations supplementing the practical calculations. Particularly, the commonly used magnetic force theorem prescribes the linear relation between the exchange interactions and the response function, while the exact theory requires this dependence to be inverse, as it can be rigorously derived from cSDFT. We explore the origin and consequences of these differences in the definition for a wide class of materials, including ferromagnetic Ni, antiferromagnetic NiO, half-metallic ferromagnetic CrO2 , multiferroic HoMnO3 , and layered van der Waals magnets CrCl3 and CrI3 . While in most of these cases, the magnetic force theorem produces quite reasonable results and can be rigorously justified in the long wavelength and strong-coupling limits, the exact formulation appears to be more consistent, especially in dealing with two important issues, which typically arise in the theory of exchange interactions: (i) the treatment of the ligand spins and (ii) the choice of the suitable variable for the description of infinitesimal rotations in the system of spins within cSDFT. Both issues can be efficiently resolved by employing the ideas of adiabatic spin dynamics supplemented with the exact expression for the exchange interactions. Particularly, the ligand spins can produce quite sizable contributions to the total energy change. For this case, we propose a simple downfolding procedure of elimination of the ligand spins from the model by transferring their effects to the interaction parameters between the localized 3 d spins. Furthermore, the exchange interactions appear to be sensitive to the definition of the variable, which is used to describe the rotations of spins in cSDFT: Generally, the rotations of spin moments and spin magnetization matrix lead to different results. In this respect, we argue that the rotations of spin moments are more suitable for the description of low-energy excitations, while the rotations of the whole magnetization matrix cause much stronger perturbation in the system of spins.