Localized states, topology and anomalous Hall conductivity on a 30 degrees twisted bilayer honeycomb lattice

We consider 30 • twisted bilayer formed by two copies of Haldane model and explore its evolution with varying interlayer coupling strength. Specifically, we compute the system's energy spectrum, its fractal dimensions, topological entanglement entropy, local Chern markers and anomalous Hall conductivity. We find that at weak interlayer coupling, the system still has a bulk energy gap, topological edge states and retains topological properties of the isolated layers, but at strong interlayer coupling, this energy gap closes. However, at small values of the Haldane mass m, another bulk gap opens. At strong interlayer coupling, the system possesses multiple states localized at various locations of the lattice, including corner states. We emphasize that these corner states do not originate from the topological edge states at the weak coupling, and their location is not necesarily attributed to the bulk gap. We also compute fractal dimensions and establish that the system at large interlayer coupling is multifractal. Finally, we establish that topological entanglement entropy and anomalous Hall conductivity can be used to characterize the system's topological properties in the same way as a local Chern marker. In particular, we establish that anomalous Hall conductivity on a finite system behaves qualitatively in the same way as a local Chern marker. Our results suggest that the bulk gap at the strong interlayer coupling has non-topological origin.