LAUSR

In conventional higher-order topological insulators (HOTIs), the emergence of topological states can be explained by using the nonzero bulk polarization index. However, corner states emerge in HOTIs with incomplete boundary unit cells (i.e., boundary defects) even though the bulk polarization is zero, which challenges the conventional understanding of HOTIs. Here, based on a Kekulé-distorted honeycomb lattice with incomplete unit cells, we reveal that incomplete unit cells exhibit fractional charges through the analysis of Wannier centers by developing a compensation method and creating the concept of Wannier center domain (WCD) which is the smallest region that one Wannier center occupies. This method compensates for the missing parts of these boundary incomplete unit cells with additional WCDs to make them complete. The compensated WCDs automatically carry the corresponding charge, and this charge together with that of the incomplete unit cell constitutes the total charge of the complete unit cell after compensation. We conclude that the emergence of corner states is attributed to the filling anomaly, which is a fundamental mechanism. Our results refresh the understanding of HOTIs, especially those with structural discontinuities, and provide a novel design for topological states which have application value in producing optical functional devices.