The Dzyaloshinskii-Moriya (DM) interaction, as well as symmetric anisotropic exchange, are important ingredients for stabilizing topologically non-trivial magnetic textures, such as, e.g., skyrmions, merons and hopfions. These types of textures are currently in focus from a fundamental science perspective and they are also discussed in the context of future spintronics information technology. While the theoretical understanding of the Heisenberg exchange interactions is well developed, it is still a challenge to access, from first principles theory, the DM interaction as well as the symmetric anisotropic exchange, which both require a fully-relativistic treatment of the electronic structure, in magnetic systems where substantial electron-electron correlations are present. Here, we present results of a theoretical framework which allows to compute these interactions in any given system and demonstrate its performance for several selected cases, for both bulk and low-dimensional systems. We address several representative cases, including the bulk systems CoPt and FePt, the B20 compounds MnSi and FeGe as well as the low-dimensional transition metal bilayers Co/Pt(111) and Mn/W(001). The effect of electron-electron correlations is analyzed using dynamical mean-field theory on the level of the spin-polarized $T$-matrix + fluctuating exchange (SPTF) approximation, as regards the strength and character of the isotropic (Heisenberg) and anisotropic (DM) interactions in relation to the underlying electronic structure. Our method can be combined with more advanced techniques for treating correlations, e.g., quantum Monte Carlo and exact diagonalization methods for the impurity solver of dynamical mean-field theory. We find that correlation-induced changes of the DM interaction can be rather significant, with up to five-fold modifications in the most distinctive case.
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