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Understanding the origin of damping mechanisms in magnetization dynamics ofmetallic ferromagnets is a fundamental problem for nonequilibrium many-bodyphysics of systems where quantum conduction electrons interact with localizedspins assumed to be governed by the classical Landau-Lifshitz-Gilbert (LLG)equation. It is also of critical importance for applications, as dampingaffects energy consumption and speed of spintronic and magnonic devices. Sincethe 1970s, a variety of linear-response and scattering theory approaches havebeen developed to produce widely used formulas for computation ofspatially-independent Gilbert scalar parameter as the magnitude of the Gilbertdamping term in the LLG equation. The largely unexploited for this purposeSchwinger-Keldysh field theory (SKFT) offers additional possibilities, such asto rigorously derive an extended LLG equation by integrating quantum electronsout. Here we derive such equation whose Gilbert damping for metallicferromagnets is nonlocal, i.e., dependent on all localized spins at a giventime, and nonuniform, even if all localized spins are collinear and spin-orbitcoupling (SOC) is absent. This is in sharp contrast to standard lore, wherenonlocal damping is considered to emerge only if localized spins arenoncollinear; for such situations, direct comparison on the example of magneticdomain wall shows that SKFT-derived nonlocal damping is an order of magnitudelarger than the previously considered one. Switching on SOC makes such nonlocaldamping anisotropic, in contrast to standard lore where SOC is usuallynecessary to obtain nonzero Gilbert damping scalar parameter. Our analyticalformulas, with their nonlocality being more prominent in low spatialdimensions, are fully corroborated by numerically exact quantum-classicalsimulations.

physical review. b./physical review. b

Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field  theory: Intrinsically nonlocal and nonuniform, and made anisotropic by  spin-orbit coupling