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

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 nonvanishing 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.