Benchmarking the generalized Kadanoff-Baym ansatz and second-order adiabatic expansion using time-dependent spintronic effects: Spin pumping, torque, and inertia

Abstract: The generalized Kadanoff-Baym ansatz (GKBA) [P. Lipavský {\em et al.}, Phys. Rev. B {\bf 34}, 6933 (1986)] has emerged as a popular and numerically efficient algorithm for simplification of nonequilibrium Green's function (NEGF)-based calculations of time-dependent quantum transport. For systems that can be split into classical and quantum degrees of freedom, another popular simplifying strategy is adiabatic expansion (AE) of NEGF [N. Bode {\em et al.}, Phys. Rev. Lett. {\bf 107}, 036804 (2011); S. Deghi {\em et al.}, Phys. Rev. B {\bf 110}, 115409 (2024)] in terms of the velocity of classical degrees of freedom, such as localized magnetic moments (LMMs) in spintronics or coordinates of nuclei in nanoelectronics. Here we compare GKBA and second-order AE with numerically exact benchmarks for two-terminal junctions whose central region hosting quantum electrons and classical LMMs is attached to two semi-infinite normal metal leads. Three simple models are employed to exhibit cornerstone time-dependent effects in spintronics---spin pumping and spin-transfer torque (STT), as well as magnetic inertia as a recently explored phenomenon. We find that GKBA fails to describe pumping of spin current by precessing LMMs, or STT vectors, and thereby induced LMM dynamics. Conversely, the second-order AE matches numerically exact benchmarks for both effects remarkably well, thereby also revealing the essentially {\em nonadiabatic} nature of spin pumping. Thus, AE opens a path toward an accurate description of STT-driven magnetization dynamics, including combination with first-principles Hamiltonians, while incurring a fraction of the cost of time evolution of full NEGF. However, despite including terms up to the second time derivatives of LMMs into AE, this approach fails to capture fast nutational oscillations on top of the precessional motion of LMMs as the hallmark of magnetic inertia.