Corner charge and bulk multipole moment in periodic systems

TL;DR

This paper derives a bulk formula relating the corner charge to the quadrupole moment in 2D periodic systems, showing quantization under certain symmetries and discussing extensions to 3D.

Contribution

It introduces a bulk-based formula for corner charge in 2D systems, linking it to the quadrupole moment and demonstrating quantization under specific rotational symmetries.

Findings

Quadrupole moment is quantized with $n$-fold rotation symmetry.

Corner charge can be predicted from bulk properties in certain symmetric systems.

Extension to 3D systems shows limitations in bulk predictability of corner charge.

Abstract

A formula for the corner charge in terms of the bulk quadrupole moment is derived for two-dimensional periodic systems. This is an analog of the formula for the surface charge density in terms of the bulk polarization. In the presence of an $n$-fold rotation symmetry with $n=3$, $4$, 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. An extension of these results to three-dimensional systems is also discussed. 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.