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Dynamical extension of Hellmann-Feynman theorem and application to nonadiabatic quantum processes in Topological and Correlated Matter

An extension of the Hellmann-Feynman theorem to one employing dynamical parameters that vary with time according to quantum dynamics is rigorously derived, avoiding any linear response or other approximations. The resulting theorem for the dynamics of observables, valid to all orders in external fields, is found to contain generalized Berry curvature type of quantities that incorporate the dynamics through explicit and nontrivial time-dependence; these Berry quantities resemble the so called anomalous terms (in semiclassical equations of motion in solids) but are both of magnetic and electric type, the former being associated with Quantum Hall Effect type of behaviors and the latter with Polarization type of behaviors. By way of application of the new theorem, the quantum equations of motion of a spinless and a spinfull electron in a solid are derived without any adiabatic or semiclassical approximation. The charge current formula for a many-body and interacting spinfull system is also derived and is found to consist of a longitudinal and a transverse part; phenomenological interpretations with respect to polarization and magnetization currents respectively then emerge in a natural way. In addition, a formula for the topological magnetoelectric effect for an interacting spinfull electron system is also provided. By carefully defining singlevaluedness in parameter space, in a nonstandard fashion and in higher rigor than usual, we are able to discuss in clarity the issue of possible obstruction of this singlevaluedness, the associated creation of Berry monopoles in parameter space and the quantization of the flux of Berry curvature (but with nontrivial dynamics included in its definition).