Topological Nature of Polarization and Charge Pumping in Ferroelectrics

Electric polarization in insulators is represented by the transferred charge through a shift of the Bloch wave functions induced by an adiabatic change of external parameters $\overrightarrow{Q}$. It is found that this covalent/quantum contribution is determined nonlocally by the topological structure in the $\overrightarrow{Q}$ space. The condition for the charge pumping for a cyclic change of $\overrightarrow{Q}$ is also obtained. Applications of this picture to various organic ferroelectrics and ${\mathrm{B}\mathrm{a}\mathrm{T}\mathrm{i}\mathrm{O}}_3$ are discussed.

Figure 1

Change of polarization induced by an adiabatic change of $\overrightarrow{Q}$ is schematically shown in the $\overrightarrow{Q}$ space.

Figure 2

One-dimensional two-band models for molecular orbitals are schematically shown.

Figure 3

Geometrically different string structures of $\overrightarrow{B}$ (red) in the $\overrightarrow{Q}$ space for different classes of models in the text. In (a1), we also show the vector field $\overrightarrow{A}$. The loop $C$ encompasses the flux string. The arrows on the flux strings show the directions of $\overrightarrow{B}$ in the strings. Total flux of each string is unity, except for the strings on the $Q_3$ axis in (b1) and on the $Q_2$ and the $Q_3$ axes in (b2), whose total flux is two.

Figure 4

An ionic or charge-ordering dimer model for TTF-CA or TMTTF.