  The Landau-Lifshitz-Gilbert (LLG) equation is widely used to describe
magnetization dynamics. We develop a unified framework of the microscopic LLG
equation based on the nonequilibrium Green's function formalism. We present a
unified treatment for expressing the microscopic LLG equation in several
limiting cases, including the adiabatic, inertial, and nonadiabatic limits with
respect to the precession frequency for a magnetization with fixed magnitude,
as well as the spatial adiabatic limit for the magnetization with slow
variation in both its magnitude and direction. The coefficients of those terms
in the microscopic LLG equation are explicitly expressed in terms of
nonequilibrium Green's functions. As a concrete example, this microscopic
theory is applied to simulate the dynamics of a magnetic Skyrmion driven by
quantum parametric pumping. Our work provides a practical formalism of the
microscopic LLG equation for exploring magnetization dynamics.

Mesoscale and Nanoscale Physics