Bound States of Conical Singularities in Graphene-Based Topological Insulators

We investigate the electronic structure induced by wedge disclinations (conical singularities) in a honeycomb lattice model realizing Chern numbers $\gamma =\pm 1$. We establish a correspondence between the bound state of (i) an isolated ${\Phi }_0/2$ flux, (ii) an isolated pentagon ($n=1$) or heptagon ($n=-1$) defect with an external flux of magnitude $n\gamma {\Phi }_0/4$ through the center, and (iii) an isolated square or octagon defect without external flux, where ${\Phi }_0=h/e$ is the flux quantum. Because of the above correspondence, the existence of isolated electronic states bound to disclinations is robust against various perturbations. Hence, measuring these defect states offers an interesting probe of graphene-based topological insulators which is complementary to measurements of the quantized edge currents.

Figure 1

Correspondence between the bound state of (a) and isolated $\pi$ flux in the defect-free case ($n=0$), (b) an isolated pentagon defect ($n=1$) with an external flux $\pi /2$, and (c) an isolated square defect ($n=2$) without external flux. The external flux $\Phi$ was applied through the center marked by a circle. Results are presented for the model Eq. (1) with $t_{2i}/t=0.4$. The local density-of-states (LDOS) were obtained on a site of the central polygon using the Lanczos algorithm.

Figure 2

(a) Continuum version of the cut-and-glue construction with a regularization hole of radius $\rho$ around the origin. $\mathbf{\alpha }(\theta )$ is a closed path around the cone. (b) The boundary conditions for the spinor across the seam have to compensate for the mismatch of the base functions, as indicated for a pair of matching degrees of freedom $A$ and $B^{\prime }$ for the pentagon disclination.

Figure 3

(a) General solution for the bound-state energy from Eq. (11) as function of $\nu$ for different radii of the hole $\rho |m|$. (b) Bound-state energies as function of external flux $\Phi /{\Phi }_0$ for the pentagon defect.

Figure 4

Coupling of chiral edge states across the seam in the cut and glue construction of (a) a $n=1$ and (b) a $n=2$ disclination. (c) Without coupling, the edge states cross at $k_{E}a=\pi$, which is protected by inversion symmetry. Turning on a weak coupling locally opens a gap. The different matching conditions between the edge states on the left- and right-hand side of the defect is described by a $\xi$-dependent mass term $\mu (\xi )$ where $\xi$ is the coordinate along the edge. (d) For $n=2$, $\mu (\xi )$ changes sign, implying a bound state.