Corner charge and bulk multipole moment in periodic systems

Recently, various kind of insulators that feature electric charges localized to their hinges and corners have been proposed and studied. In this talk, we discuss a formula for the boundary charges in terms of the bulk quadrupole moment of the insulator. This is an analog of the "modern theory" formula for the surface charge density in terms of the bulk polarization. In two dimensional systems with n-fold rotation symmetry (n = 3, 4, and 6), the quadrupole moment is quantized and is independent of the spread or shape of Wannier orbitals, depending only on the location of Wannier centers of filled bands. In this case, our formula predicts the fractional part of the quadrupole moment purely from the bulk property. The system can contain many-body interactions as long as the ground state is gapped and topologically trivial in the sense it is smoothly connected to a product state limit. In three dimensions, in general, even the fractional part of the corner charge is not fully predictable from the bulk perspective even in the presence of point group symmetry.

Ref: Haruki Watanabe, Seishiro Ono, Phys. Rev. B 102, 165120 (2020)