Impurity-bound states and Green's function zeros as local signatures of topology

Abstract

We show that the local in-gap Green's function of a band insulator ${\mathbf{G}}_{0}(\ensuremath{\epsilon},{\mathbf{k}}_{\ensuremath{\parallel}},{\mathbf{r}}_{\ensuremath{\perp}}=0)$, with ${\mathbf{r}}_{\ensuremath{\perp}}$ the position perpendicular to a codimension-1 or codimension-2 impurity, reveals the topological nature of the phase. For a topological insulator, the eigenvalues of this Green's function attain zeros in the gap, whereas for a trivial insulator the eigenvalues remain nonzero. This topological classification is related to the existence of in-gap bound states along codimension-1 and codimension-2 impurities. Whereas codimension-1 impurities can be viewed as soft edges, the result for codimension-2 impurities is nontrivial and allows for a direct experimental measurement of the topological nature of two-dimensional insulators.