Polarization and corner charge in Chern insulators

The concept of multipole moments in crystals and their quantization under symmetries has led to the discovery of higher-order topological insulators. These multipole moments are understood from a localized description of electrons in terms of exponentially localized Wannier functions (ELWFs). For instance, the dipole moment can be calculated from the displacement of the ELWF centers from the ionic center; this has allowed for a topological classification of conventional insulators or “atomic limits” under crystalline symmetries. However, such a construction of ELWFs is impossible for Chern insulators (CIs) due to the non-zero Chern number that prevents a smooth gauge choice for the states. This presents a fundamental difficulty in defining multipole moments of the electronic states in CIs.

In this work, we ask: can polarization and corner charge, two concepts that rely on a well-defined dipole moment of electronic states, be meaningful in CIs? We show that these quantities can indeed be meaningful in CIs and are quantized under inversion symmetry. We find that the boundary manifestations of these invariants in CIs are similar to that of atomic limits, i.e. the appearance of fractional charges at the edges and corners of a finite system. However, unlike in atomic limits, these fractional charges only appear between two CIs with the same Chern number but non-zero relative dipole moments. We present our results in a tight-binding setting as well as an experimentally realizable gyro-magnetic photonic crystal.