Linear response theories for interatomic exchange interactions

In 1987, Liechtenstein et al. came up with the idea to formulate the problem of interatomic exchange interactions, which would describe the energy change caused by the infinitesimal rotations of spins, in terms of the magnetic susceptibility. The formulation appears to be very generic and, for isotropic systems, expresses the energy change in the form of the Heisenberg model, irrespectively on which microscopic mechanism stands behind the interaction parameters. Moreover, this approach establishes the relationship between the exchange interactions and the electronic structure obtained, for instance, in the first-principles calculations based on the density functional theory. The purpose of this review is to elaborate basic ideas of the linear response theories for the exchange interactions as well as more recent developments. The special attention is paid to the approximations underlying the original method of Liechtenstein et al. in comparison with its more recent and more rigorous extensions, the roles of the on-site Coulomb interactions and the ligand states, and calculations of antisymmetric Dzyaloshinskii-Moriya interactions, which can be performed alongside with the isotropic exchange, within one computational scheme. The abilities of the linear response theories as well as many theoretical nuances, which may arise in the analysis of interatomic exchange interactions, are illustrated on magnetic van der Walls materials Cr$X_3$ ($X$$=$ Cl, I), half-metallic ferromagnet CrO$_2$, ferromagnetic Weyl semimetal Co$_3$Sn$_2$S$_2$, and orthorhombic manganites $A$MnO$_3$ ($A$$=$ La, Ho), known for the peculiar interplay of the lattice distortion, spin, and orbital ordering.