| Abstract: | We investigate the interplay between local defects and topology in the Haldane model within the framework of the
tenfold classification. The Haldane model realizes a Chern insulating phase characterized by an integer topological
invariant and supports chiral edge states. Introducing vacancies in this model gives rise to localized topological states
at the vacancy sites, characterized by a Z2 invariant: an odd number of vacancies hosts protected zero-energy modes,
while an even number does not. These vacancy-induced states exhibit characteristic dislocations in their wavefunction
profiles, which serve as a probe of defect topology. We contrast this behavior with local adatom defects, which do not
show such dislocations. In addition, we demonstrate the emergence of fractional charge bound to vacancy defects, in
contrast to adatom defects, providing further evidence that vacancy-induced states are topological in origin. Finally,
we analyze current patterns and show that currents circulating around vacancy-induced states flow opposite to those
of edge states, analogous to the behavior of vortices in superconductors. |