Quantum Theory of Orbital Magnetization and Its Generalization to Interacting Systems

Abstract

Based on standard perturbation theory, we present a full quantum derivation of the formula for the orbital magnetization in periodic systems. The derivation is generally valid for insulators with or without a Chern number, for metals at zero or finite temperatures, and at weak as well as strong magnetic fields. The formula is shown to be valid in the presence of electron-electron interaction, provided the one-electron energies and wave functions are calculated self-consistently within the framework of the exact current and spin-density functional theory. DOI: 10.1103/PhysRevLett.99.197202 PACS numbers: 75.10.Lp, 73.20.At Magnetism is one of the most important properties of materials. Both spin and orbital motion of electrons can contribute to the total magnetization. While the spin magnetization can already be calculated from first principles with high accuracy by state-of-art methods such as the spin-density functional theory (SDFT), the study of orbital magnetization is still in a comparatively primitive stage.A first difficulty arises from the fact that there is still no theoretically well-established formula for calculating the orbital magnetization of a crystalline solid. The nonlocality of the orbital magnetization operatorM ÿe=2r is the major obstacle to obtaining a closed formula for an extended periodic system. Recently, Xiao et al. [1] and, independently, Thonhauser et al. [2,3], obtained an orbital magnetization formula which avoids the nonlocality problem and looks very promising for applications. However, up to date, there exists no general quantum mechanical derivation of this formula. The derivation presented in Ref.[1] relies on the semiclassical wave-packet dynamics of Bloch electrons [1,4,5], and its validity in the quantum context is not completely clear. On the other hand, the derivation presented in Ref.[2] is quantum mechanical but relies on the existence of localized Wannier functions, and it cannot be easily generalized to metals or insulators with nonzero Chern number. In addition, both derivations are limited to noninteracting systems. The shortcomings of these approaches call for a full quantum mechanical and many-body theory of the orbital magnetization.A second difficulty is that a first principle calculation of the orbital magnetization (taking into account many-body effects) should be based on the spin current density functional theory (SCDFT) [6] rather than the conventional SDFT. Unfortunately, SCDFT has been hindered so far by the lack of reliable expressions for the magnetizationdependent effective potentials. This may partly explain why the orbital moments of ferromagnetic transition metals such as Fe, Co, and Ni calculated in SCDFT were found to be significantly smaller than the experimentally determined values [7]. How problematic these calculations are is well explained in the review article by Richter [8]. The situation, however, has been rapidly changing in recent years. The advent of optimized effective potentials [9,10] which treat exchange exactly and may systemati...

Berry phase theory of Dzyaloshinskii–Moriya interaction and spin–orbit torques

“…However, in the absence of an applied electric field the rightmost terms in Eq. (20) and in Eq. (21) vanish.…”

“…(8) with the quantum theory of OM [20,21] one finds strong formal analogies, where B knij corresponds to the Berry curvature i ∇ k u kn | × |∇ k u kn and A knij corresponds to the orbital moment…”

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