Topological proximity effects in a Haldane graphene bilayer system

We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half filling, resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counterpropagating (protected) edge modes with different velocities. In the strong-coupling limit, the energy spectrum shows flat bands. Therefore we build a perturbation theory and address further the bulk-edge correspondence. We also show the occurrence of a topological insulating phase with Chern number one when only the lowest band is filled. We generalize the effect to Haldane bilayer systems with asymmetric Semenoff masses. Moreover, we propose an alternative definition of the topological invariant on the Bloch sphere.

Figure 1

Berry curvature in the Brillouin zone for the Haldane and graphene layers at $r=0$ and small $r$, showing the Berry phase jump effect [38]. Here, $t_1=1$ and $t_2=1/3$. Results in the Haldane layer remain almost unchanged from $r=0$ to $r=0.4$: The pseudospin 1/2 is polarized close to the Dirac point, and the dominant contribution to the Berry curvature occurs close to the highly symmetric $M$ points [38].

Figure 2

Cylinder representations of the Haldane (in gray) and graphene (in green) systems, in the wave-vector space. The green region has 60 unit cells and the gray region has 14 additional unit cells. Band structures for $t_1=1$ and $t_2=1/3$ in the weak- and strong-coupling limits for this cylinder geometry [1]; the lattice spacing is $a=1$. On the left, we zoom on the two low-energy graphene bulk bands. We observe two counterpropagating edge modes with different velocities at zero energy for $r$ of the order of $t_2$. For very strong couplings, on the right, the counterpropagating edge modes at low energy are only linked to the Haldane region, and the central strongly hybridized region becomes identical to the vacuum (or becomes topologically trivial).

Figure 3

Berry curvatures for the two lowest-energy bands at strong coupling $r=7$ ($t_1=1$ and $t_2=1/3$).

Figure 4

Numerical phase diagram for two coupled Haldane models with ${\mathbf{d}}_1={\mathbf{d}}_2={\mathbf{d}}^h$. Evolution of the total Chern number $C$ for the two lowest bands as a function of $r$ and $M_2$ for $M_1=1/\sqrt{3}$ in units of $t_1=1$ and $t_2=1/3$. Illustration of Berry phase “jumps” at the second phase transition.