Quantum geometry and semiclassical dynamics in inhomogeneous fields

We revisit the problem of nonequilibrium semiclassical electron transport in the presence of inhomogeneous external perturbations such an electric and magnetic fields. By considering the quantum geometry of a Bloch band structure beyond the Berry connection contribution, we develop a systematic approach for computing the geometric contributions to the equations of motion of the position and momentum operators in an N-band system. For this purpose, we extend the notions of Berry curvature and quantum geometric tensor to higher orders in the inhomogeneity of the applied fields by introducing operator-valued geometric quantities at each order. We also demonstrate how to derive the dynamics from a generic coupling between Bloch momentum and an inhomogeneous external field, thus generalizing previous studies.