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

We show that the local in-gap Green's function of a band insulator ${\mathbf{G}}_0(\epsilon ,{\mathbf{k}}_{\parallel },{\mathbf{r}}_{\perp }=0)$, with ${\mathbf{r}}_{\perp }$ 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.

Figure 1

The eigenvalues of the local Green's function ${\mathbf{G}}_0(\epsilon ,k_{\parallel }=0,r_{\perp }=0)$ in the $M/B$ model, relevant for codimension-1 (left) and codimension-2 (right) impurities. In the trivial system (dashed lines, $M/B=-1$) the eigenvalues ${\lambda }_{\pm }$ are nonzero. In the topological system (solid line, $M/B=1$) the eigenvalues are zero for some in-gap energy and hence in-gap bound states always exist.

Figure 2

Typical energy of impurity bound states in a two-dimensional insulator as a function of impurity strength $V$. Here, we discern between the topological regime (solid lines, $M/B=1$) and the trivial regime (dashed line, $M/B=-1$). For strong $V$, the bound state in the trivial phase has disappeared in the conduction band, whereas the bound states in the topological phase remain.