Marginality of bulk-edge correspondence for single-valley Hamiltonians

We study the correspondence between the nontrivial topological properties associated with the individual valleys of gapped bilayer graphene (BLG), as a prototypical multivalley system, and the gapless modes at its edges and other interfaces. We find that the exact connection between the valley-specific Hall conductivity and the number of gapless edge modes does not hold in general, but is dependent on the boundary conditions, even in the absence of intervalley coupling. This nonuniversality is attributed to the absence of a well-defined topological invariant within a given valley of BLG; yet, a more general topological invariant may be defined in certain cases, which explains the distinction between the BLG-vacuum and BLG-BLG interfaces.

Figure 1

(Color online) Plot of ${\mathcal{B}}_k=\frac{1}{4\pi }\widehat{\mathbf{g}}\cdot \left[{\partial }_{k_x}\widehat{\mathbf{g}}\times {\partial }_{k_y}\widehat{\mathbf{g}}\right]$ [integrand of Eq. (2)] as a function of momentum $\mathit{k}$ over the entire Brillouin zone, obtained from the tight-binding description of gapped BLG. ${\mathcal{B}}_k$ is nonvanishing only close to the K and ${\text{K}}^{\prime }$ points (valleys), where it is well approximated by the expression for $\widehat{\mathbf{g}}\left(\mathit{k}\right)$ that enters the long-wavelength effective Hamiltonian [see Eq. (6)]. The integral of ${\mathcal{B}}_k$ over the entire Brillouin zone vanishes, owing to the equal and opposite contributions of the two valleys ($=\pm 1$, corresponding to the quantized Hall conductivity of the individual valleys).

Figure 2

(Color online) Illustration of bilayer graphene edges (highlighted by thick lines) terminating all in the zigzag direction but with different sublattices. The number of subgap edge modes is 1 for case (a) and (b), 2 for case (c), and 0 for case (d). A unit cell of the lattice is also shown in (a) in broken lines.

Figure 3

(Color online) Comparison of the dispersion relations for the subgap edge modes obtained by solving the continuum model and exact diagonalization of the tight-binding model, with $\Delta =0.15$. The upper panel case corresponds to the edge shown in Fig. 2a, and contains only one branch of gapless edge modes; the lower panel case corresponds to Fig. 2c and contains two branches of gapless edge modes.

Figure 4

Illustration of the mapping $\widehat{\mathbf{g}}\left(\mathit{k}\right)$ for gapped BLG. Panel (a) shows $\widehat{\mathbf{g}}\left(\mathit{k}\right)$ for a single valley as a vector field on the $R^2$ plane of $\mathit{k}$. Panel (b) shows the solid angle $\Omega$ that $\widehat{\mathbf{g}}$ covers when $\mathit{k}$ runs over the whole plane, the double arrow signifying the double covering of the (upper) hemisphere. Panel (c) shows how two marginal topological mappings can be “glued” to form a well-defined topological mapping, which is the case for a BLG-BLG domain wall with opposite mass signs on the two sides.