Theory of spin magnetization driven by chiral phonons

We construct a general theory of spin magnetization driven by chiral phonons under an adiabatic process, in which atoms rotate around their equilibrium positions with a low phonon frequency. Here the spin magnetization originates from the modulated electronic states with spin-orbital coupling by atomic rotations. Under the adiabatic approximation, the time-dependent spin magnetization can be calculated by a Berry-phase method. In this paper, we focus on its time average, which is evaluated by assuming that the phonon displacement is small. As a result, the time average of the spin magnetization is concisely formulated in the form of the Berry curvature defined in the phonon-displacement space as an intrinsic property of atomic rotations. Our formula for spin magnetization reflects the chiral nature of phonons, and is convenient for $ab$ $initio$ calculations.