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Self-consistent linear response for the spin-orbit interaction related properties

In many cases, the relativistic spin-orbit (SO) interaction is regarded to be small and can be treated using perturbation theory. The major obstacle on this route comes from the fact that the SO interaction can also polarize the electron system and produce additional contributions to the perturbation theory, arising from the electron-electron interactions. In electronic structure calculations, it may even lead to necessity to abandon the perturbation theory and return to the self-consistently solution of Kohn-Sham-like equations with the effective potential $\hat{v}$, incorporating the effects of the electron-electron interactions and the SO coupling, even though the latter is small. In this work, we present the theory of self-consistent linear response (SCLR), which allows us to get rid of numerical self-consistency and formulate it analytically in the first order of the SO coupling. This strategy is applied to the Hartree-Fock solution of the effective Hubbard model, derived from electronic structure calculations in the Wannier basis. By using $\hat{v}$, obtained from SCLR, one can successfully reproduce results of ordinary calculations for the orbital magnetization and other properties in the first order of the SO coupling. Particularly, SCLR appears to be extremely useful for calculations of antisymmetric Dzyaloshinskii-Moriya (DM) interactions based on the magnetic force theorem. Furthermore, due to the powerful 2n+1 theorem, the SCLR theory allows us to calculate the magnetic anisotropy energy up to the third order of the SO coupling. The fruitfulness of this approach is illustrated on a number of example, including the spin canting in YTiO$_3$ and LaMnO$_3$, formation of spiral magnetic order in BiFeO$_3$, and the magnetic inversion symmetry breaking in BiMnO$_3$, which gives rise to both ferroelectric activity and DM interactions, responsible for the ferromagnetism.