A semiclassical theory for the orbital magnetization due to adiabatic evolutions of Bloch electronic states is proposed. It renders a unified theory for the periodic-evolution pumped orbital magnetization and the orbital magnetoelectric response in insulators by revealing that these two phenomena are the only instances where the induced magnetization is gauge invariant. This theory also accounts for the electric-field induced intrinsic orbital magnetization in two-dimensional metals and Chern insulators. We illustrate the orbital magnetization pumped by microscopic local rotations of atoms, which correspond to phonon modes with angular momentum, in toy models based on honeycomb lattice, and the results are comparable to the pumped spin magnetization via strong Rashba spin orbit coupling. We also show the vital role of the orbital magnetoelectricity in validating the Mott relation between the intrinsic nonlinear anomalous Hall and Ettingshausen effects.
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