General corner charge formula in two-dimensional C_n-symmetric higher-order topological insulators

TL;DR

This paper derives a comprehensive formula for quantized fractional corner charges in 2D C_n-symmetric higher-order topological insulators, accounting for various surface conditions and bulk polarization effects.

Contribution

It extends previous corner charge formulas by including general surface conditions and non-zero bulk polarization, broadening the applicability of the theory.

Findings

The corner charge depends on surface termination in certain cases.

The formula applies to systems with reconstructed surfaces and higher Miller index surfaces.

It accounts for non-vanishing bulk polarization effects.

Abstract

In this paper, we derive a general formula for the quantized fractional corner charge in two-dimensional C_n-symmetric higher-order topological insulators. We assume that the electronic states can be described by the Wannier functions and that the edges are charge neutral, but we do not assume vanishing bulk electric polarization. We expand the scope of the corner charge formula obtained in previous works by considering more general surface conditions, such as surfaces with higher Miller index and surfaces with surface reconstruction. Our theory is applicable even when the electronic states are largely modulated near system boundaries. It also applies to insulators with non-vanishing bulk polarization, and we find that in such cases the value of the corner charge depends on the surface termination even for the same bulk crystal with C_3 or C_4 symmetry, via a difference in the Wyckoff position of the center of the C_n-symmetric crystal.