Topological Nature of Polarization and Charge Pumping in Ferroelectrics

Dielectric properties of insulators have been one of the most important subjects in condensed matter physics. Recently, the quantum theory of electric polarization in insulators has been developed in terms of the Berry phase, which fully takes account of the covalency. However, the intuitive picture is hidden in sophisticated first-principle band calculations. Here, we provide a clear and comprehensive view on the quantum theory of polarization for two-band models: In the space of external parameters such as displacements of atomic or molecular orbitals, pressure, and electric field, a vector field representing an infinitesimal change of polarization due to that of the parameters is dominantly characterized by a string. This string is a trajectory of the band crossing points and fully describes the singularity of the vector field. The total flux of the string and thus the charge pumped during a cyclic adiabatic change of the external parameters are quantized to an integer multiplied by the electronic charge. Applications of this picture to various organic ferroelectrics and BaTiO$_3$ are discussed.