We address the importance of the modern theory of orbital magnetization for spintronics. Based on an allelectron first-principles approach, we demonstrate that the predictive power of the routinely employed "atomcentered" approximation is limited to materials like elemental bulk ferromagnets, while the application of the modern theory of orbital magnetization is crucial in chemically or structurally inhomogeneous systems such as magnetic thin films, and materials exhibiting nontrivial topology in reciprocal and real space, e.g., Chern insulators or noncollinear systems. We find that the modern theory is particularly crucial for describing magnetism in a class of materials that we suggest here-topological orbital ferromagnets. 75.70.Ak, Magnetism is an elementary property of materials, and it is composed of spin and orbital contributions. In contrast to the concept of spin magnetization, which has been relatively well understood and extensively researched in the course of the past decades, our understanding of orbital magnetism in solids has been poor so far, and an ability to describe it reliably has been missing until recently. Both spin and orbital magnetization (OM) are accessible separately, e.g., by means of magnetomechanical [1] or magnetic circular dichroism measurements [2-4], but the orbital contribution to the magnetization of solids is usually overshadowed by the spin counterpart, owing to the orbital moment quenching. However, in certain systems the OM yields an equally important contribution, which can even result in a spin-orbital compensation of magnetization [5][6][7]. Its influence on spin-dependent transport [8][9][10][11], magnetic susceptibility [11], orbital magnetoelectric response [12][13][14], magnetic anisotropy [15], and Dzyaloshinskii-Moriya interaction [16] renders the OM crucial for understanding basic properties of magnets. A spontaneous OM in ferromagnets is a key manifestation of the spinorbit interaction (SOI), lifting in part the quenching mechanism. This interpretation applies to most materials but it fails to explain orbital magnetism in systems where a finite topological OM emerges even without SOI as a result of a nontrivial real-space distribution of spins [17].Addressing the OM in solids is a subtle point as the position operator r is ill-defined in the basis of extended Bloch states. To circumvent this problem in ab initio calculations, the evaluation of the angular momentum operator L is typically restricted locally in space to atom-centered spheres. This atom-centered approximation (ACA) is widely used to study orbital magnetism in solids even nowadays. Rather recently, a rigorous theory of OM was established through three independent approaches [18][19][20][21]. In this so-called modern theory [22-24] the OM is expressed as a genuine bulk property evaluated from the ground-state wave functions:where k is the crystal momentum, [dk] stands for occ n dk/(2π) 3 , |u kn is an eigenstate of the lattice-periodic Hamiltonian H k = e −ik·r He ik·r to the band energy E kn , E F ...
