  The phenomenon of charge pumping and the modern theory of electric
polarization are reconsidered by analytically taking into account emergent
non-Hermitian contributions. These are accounted for through the use of an
extended definition of the velocity operator and are determined by means of a
dynamic Hellmann-Feynman theorem (DHFT) that we derive here for the first time.
The DHFT introduces generalized Berry curvatures and it is valid for
calculating observables nonperturbatively, hence with results valid to all
orders of the external fields. By using the extended velocity operator we
rigorously show how the charge pumping is linked up with the boundaries of the
material (with the non-Hermiticity being essential for this connection), and by
means of the DHFT we show that the well-known topological quantization of the
pumped charge breaks down due to a nonintegrable Aharonov-Anandan phase in
driven non-equilibrium processes whenever the periodic gauge cannot be applied
to the Floquet-Bloch states. Likewise, we show that the electronic polarization
change has an additional non-Hermitian contribution, which is overlooked in the
modern theory of electric polarization. The non-Hermitian contribution is by
definition a bulk quantity that may equally be evaluated as a boundary quantity
due to a symmetric structure that allows the bulk integration to be transformed
into a boundary one. This non-Hermitian contribution is very sensitive to the
realistic boundary conditions imposed on the wavefunctions and it is therefore
expected to be significant in biased insulators where charge accumulation over
their boundaries is present during the process that causes the polarization
change. Finally, we show how a well-defined surface-charge theorem can be
formulated in terms of the boundary non-Hermitian contribution.
