  Understanding the origin of damping mechanisms in magnetization dynamics of
metallic ferromagnets is a fundamental problem for nonequilibrium many-body
physics of systems where quantum conduction electrons interact with localized
spins assumed to be governed by the classical Landau-Lifshitz-Gilbert (LLG)
equation. It is also of critical importance for applications, as damping
affects energy consumption and speed of spintronic and magnonic devices. Since
the 1970s, a variety of linear-response and scattering theory approaches have
been developed to produce widely used formulas for computation of
spatially-independent Gilbert scalar parameter as the magnitude of the Gilbert
damping term in the LLG equation. The largely unexploited for this purpose
Schwinger-Keldysh field theory (SKFT) offers additional possibilities, such as
to rigorously derive an extended LLG equation by integrating quantum electrons
out. Here we derive such equation whose Gilbert damping for metallic
ferromagnets is nonlocal, i.e., dependent on all localized spins at a given
time, and nonuniform, even if all localized spins are collinear and spin-orbit
coupling (SOC) is absent. This is in sharp contrast to standard lore, where
nonlocal damping is considered to emerge only if localized spins are
noncollinear; for such situations, direct comparison on the example of magnetic
domain wall shows that SKFT-derived nonlocal damping is an order of magnitude
larger than the previously considered one. Switching on SOC makes such nonlocal
damping anisotropic, in contrast to standard lore where SOC is usually
necessary to obtain nonzero Gilbert damping scalar parameter. Our analytical
formulas, with their nonlocality being more prominent in low spatial
dimensions, are fully corroborated by numerically exact quantum-classical
simulations.
