Multifractaility, 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 evolution of its properties 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 remains gapped and retains topological properties of the isolated layers, but at strong interlayer coupling, the system forms a gapless multifractal state, which also hosts modes localized at the corners and at the center of the system. We also establish that anomalous Hall conductivity can be used to characterize the system's topological properties in the same way as a local Chern marker. Both local Chern marker and anomalous Hall conductivity can be used to identify a hidden phase transition invisible from the energy spectrum.