Topological properties of complex magnets from an advanced $\textit{ab-initio}$ Wannier description

Berry phases impart an elegant interpretation of fundamental condensed-matter phenomena as a direct consequence of the electrons' adiabatic evolution under the variation of control parameters. This thesis develops advanced $\textit{ab initio}$ methods based ondensity functional theory and applies them to investigate Berry phase effects in complex magnets, rooting in the global properties of two distinct types of phase spaces. The non-trivial geometry of momentum space manifests in intrinsic contributions to the anomalous Hall effect as well as orbital magnetism in solids. While the former has been subject to intensive research in the past decades, our understanding of orbital magnetism in periodic systems is still at a rather premature stage. Even its quantum-mechanical description was elusive until the recent advent of a rigorous but involved Berry phase theory, the overall importance of which is unclear. To resolve this open question, we implement the modern theory of orbital magnetization within the full-potential linearized augmented-plane-wave method that is known for its high precision. By comparing to a commonly applied but simple local approximation, we uncover in this thesis that the Berry phase theory is crucial to predict reliably orbital magnetism in systems studied extensively in spintronics, including thin magnetic heterostructures and topological magnets. Remarkably, we demonstrate that the emergent magnetic field due to the chiral spin structure of non-coplanar antiferromagnets constitutes an effcient mechanism to lift the orbital degeneracy even in the absence of spin-orbit coupling. In a new class of materials to which we refer as topological orbital ferromagnets, the macroscopic magnetization originates solely from pronounced orbital magnetism due to non-local charge currents. We identify promising candidates of film and bulk systems that realize the predicted topological orbital magnetization, without any reference to correlation or spin-orbit effects. Paving the road towards innovative device architectures, the burgeoning research field of spin-orbitronics exploits relativistic phenomena to control electrically magnetism by means of spin-orbit torques. Only recently, these torques and the related Dzyaloshinskii-Moriya interaction were recognized as innately geometrical effects that originate from the global properties of a $\textit{mixed}$ phase space entangling the crystal momentum with the magnetization direction. However, the effcient treatment of such complex higher-dimensional phase spaces sets a central challenge for $\textit{ab initio}$ theory, calling for advanced computational methods. This demand is met by a generalized Wannier interpolation that we develop here in order to describe Berry phase effects in generic parameter spaces precisely. Using the scheme for spin torques and chiral interactions in magnetic heterostructures, we correlate their microscopic origin with the electronic structure, and elucidate the role of chemical composition and disorder. In addition, the developed formalism enables us to evaluate effciently the dependence of these phenomena on the magnetization direction, revealing large anisotropies in the studied systems. Considering the interplay of magnetism and topology, we uncover that magnetically induced band crossings manifest in prominent magneto-electric responses in magnetic insulators. We introduce the concept of mixed Weyl semimetals to establish novel guiding principles for engineering large spin-orbit torques in topologically complex ferromagnets. Moreover, we show that topological phase transitions in these materials are accompanied by drastic changes of the local orbital chemistry