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 parameters that vary with time is derived. The resulting formula for the dynamics of observables is found to have a profound connection to Berry curvature type of quantities that however, incorporate the dynamics. 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 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 single-valuedness in parameter space, in a nonstandard fashion and in higher rigor that usual, we are able to discuss in clarity the issue of possible obstruction of this single-valuedness, the associated creation of "Berry monopoles" in parameter space and the quantization of the flux of Berry curvature (but with dynamics included).