Nonlocal damping in metallic ferromagnets from Schwinger-Keldysh field theory

An accurate description of magnetization dynamics 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 phenomenological Landau-Lifshitz-Gilbert (LLG) equation. However, the LLG equation is only exact for the expected value of magnetization of a single isolated localized quantum spin. As such, rigorous justification of the LLG equation has been of great interest for the fields of spintronics and magnonics, where experimental evidence has revealed the need for extensions to it. In this talk, we use the largely unexploited for this purpose Schwinger-Keldysh field theory (SKFT) to rigorously derive an extended LLG equation for localized spins in a conducting ferromagnet by integrating quantum electrons out. Our results contain a damping term that is nonlocal and nonuniform, even if all localized spins are collinear and spin-orbit coupling (SOC) is absent, in sharp contrast to conventional understanding. The ensuing spin dynamics are corroborated by numerically exact quantum-classical simulations in one dimension, and the wavevector dependent damping that emerges for two dimensional spin waves is explored.