Quantitative theory of magnetic interactions in solids

In this report we review the method of explicit calculations of interatomic exchange interactions of magnetic materials. This involves exchange mechanisms normally referred to as Heisenberg exchange, Dzyaloshinskii-Moriya interaction and anisotropic symmetric exchange. The connection between microscopic theories of the electronic structure, such as density functional theory or dynamical mean field theory, and interatomic exchange, is given in detail. The different aspects of extracting information for an effective spin Hamiltonian that involves thousands of atoms, from electronic structure calculations considering significantly fewer atoms (1-50) is highlighted. Examples of exchange interactions of a large group of materials is presented, which involves heavy elements of the 3d period, alloys between transition metals, Heusler compounds, multilayer systems as well as overlayers and adatoms on a substrate, transition metal oxides, 4f elements, magnetic materials in two dimensions and molecular magnets. Where possible, a comparison to experimental data is made, that naturally becomes focused on the magnon dispersion. The influence of relativity is reviewed for a few cases, as is the importance of dynamical correlations. Development to theories that handle out of equilibrium conditions is also described here. The review ends with a short description of extensions of the theories behind explicit calculations of interatomic exchange, to non-magnetic situations, e.g. that describe chemical (charge) order and superconductivity.