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

We consider $30^{\circ}$ 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. Our results suggest that the bulk gap at the strong interlayer coupling has non-topological origin.

💡 Research Summary

The paper investigates a 30° twisted bilayer honeycomb lattice constructed from two copies of the Haldane model, focusing on how the system evolves as the inter‑layer coupling strength (t_inter) and the Haldane mass (m) are varied. The authors employ exact diagonalization (and Lanczos for larger sizes) to obtain the full energy spectrum, participation‑ratio based fractal dimensions, topological entanglement entropy (TEE), local Chern markers, and the anomalous Hall conductivity computed via the Kubo formula.

In the decoupled limit (t_inter = 0) each layer is a Chern insulator for m < 1 with Chern number ±1, a gapless Dirac point at m = 1, and a trivial insulator for m > 1. When a weak inter‑layer hopping is introduced (t_inter ≲ 1.5 for m = 0) the bulk gap of the combined system remains open, edge states persist, and all three topological diagnostics—local Chern marker, TEE (≈ ln 2), and Hall conductivity (≈ e²/h)—agree, confirming that the bilayer retains the topology of its constituent layers.

Increasing t_inter drives a topological phase transition: the bulk gap closes around a critical t_inter (≈ 1.5 for m = 0) and the edge modes disappear. For small Haldane mass (m ≈ 0) a second bulk gap reopens at larger t_inter (≈ 5–10), but this gap is topologically trivial. In this regime the local Chern marker averages to zero, the TEE vanishes, and the Hall conductivity drops to zero, indicating that the new gap does not stem from a non‑trivial band topology.

Fractal‑dimension analysis reveals that in the weak‑coupling regime the average D₂ ≈ 1 with negligible variance, consistent with fully extended bulk states. In the strong‑coupling regime (t_inter = 10) the average D₂ falls below 1 (≈ 0.6–0.8) and the variance grows, signaling multifractality. The dependence of D_q on q is non‑linear, confirming that eigenstates are neither purely extended nor localized. Lanczos calculations up to linear size L ≈ 800 corroborate these findings, showing that multifractality persists in the thermodynamic limit.

A striking result is the emergence of highly localized states with D₂ ≈ 0. These appear at lattice corners, along edges, at the geometric centre, and at other interior sites. Corner modes are present even when the bulk gap is closed, and their energies shift with t_inter and system size, demonstrating that they are not protected by bulk polarization as in conventional higher‑order topological insulators. Similarly, centre‑localized states arise only in the strong‑coupling regime and are attributed to the lack of translational symmetry in the quasiperiodic structure.

The authors compare three real‑space topological indicators: (i) topological entanglement entropy, (ii) anomalous Hall conductivity, and (iii) the local Chern marker. All three give identical classifications across the phase diagram, establishing that real‑space diagnostics are sufficient to identify topology in non‑periodic systems where momentum‑space invariants are ill‑defined.

Overall, the study maps out a rich phase diagram: (a) a weakly‑coupled topological Chern‑insulator phase, (b) a weakly‑coupled trivial insulating phase, (c) a strongly‑coupled gapless phase, and (d) a strongly‑coupled multifractal phase that can be either gapped (trivial) or gapless. The work highlights that strong inter‑layer hybridization can destroy the bulk gap without erasing the underlying topological character, while simultaneously generating a plethora of localized and multifractal states unique to quasiperiodic bilayers. The findings open avenues for exploring higher‑order topology, disorder‑induced localization, and multifractality in engineered moiré systems and suggest experimental probes such as local spectroscopy and transport measurements to detect the predicted corner and centre modes.