Berry curvature and orbital moment of the Bloch state are two basic ingredients, in addition to the band energy, that must be included in the formulation of semiclassical dynamics of electrons in crystals, in order to give proper account of thermodynamic and transport properties to first order in the electromagnetic field. These quantities are gauge invariant and have direct physical significance as demonstrated by numerous applications in recent years. Generalization to the case of degenerate bands has also been achieved, with important applications in spin-dependent transport. We have now extended the theory to second order in electromagnetic fields, keeping the structure of the first order theory but greatly expanding the scope of applications. I will illustrate two applications of the extended theory, magneto polarization and nonlinear anomalous Hall effects.
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