Anomalous dielectric response of inversion-symmetric systems with the π Zak phase

In the modern theory of polarization, the charge polarization is written in terms of the Zak phase. When the system is invariant under space inversion, the Zak phase is either 0 or π, and when it is π, it indicates existence of half-filled edge states. We expect that such half-filled edge states are sensitive to external electric field, and they exhibit novel electric response. For this purpose, we consider the one-dimensional SSH model, which can have either π or 0 Zak phase by tuning the model parameters. We calculate the electric polarization under the electric field, and we find singular nonlinear response of the electric polarization when the Zak phase is π. We confirm that this anomalous phenomenon is caused by the edge states. We also find this anomalous polarization in higher-dimensional systems with π Zak phase when the dispersion of edge states is negligible.