Local Defects and the Topology of the Haldane Model

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 ($C=\pm 1$) and supports chiral edge states. Introducing vacancies gives rise to localized states at the defect sites, classified by a $\mathbb{Z}_2$ invariant $\nu = C\cdot m,\mathrm{mod},2$, where $m=N_A-N_B$ is the net sublattice imbalance of the vacancy configuration: an odd imbalance hosts a protected zero-energy mode, whereas an even imbalance does not. We identify three independent experimental signatures that distinguish these topological defect states from trivial (adatom) defects. First, vacancy-induced states exhibit characteristic dislocations in their wavefunction profiles that track the phase winding associated with the defect. Second, a fractional charge of $e/2$ accumulates at vacancy sites, while no such charge appears at adatoms. Third, the probability current circulating around a vacancy-induced state flows in the opposite direction to that of chiral edge states, in direct analogy with the current reversal produced by a vortex in a $p$-wave superconductor. All three signatures are in quantitative agreement with the $\mathbb{Z}_2$ prediction.