Topological nature of polarization and charge pumping in ferroelectrics

Electric polarization or transferred charge due to an adiabatic change of external parameters $\vec{Q}$ is expressed in terms of a vector field defined in the $\vec{Q}$ space. This vector field is characterized by strings, i.e., trajectories of band-crossing points. In particular, the transverse component is given by the Biot-Savart law in a nonlocal way. For a cyclic change of $\vec{Q}$ along a loop C, the linking number between this string and C represents the amount of the pumped charge, which is quantized to be an integer as discussed by Thouless.