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

Abstract

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.

Figures (22)

• Figure 1: A schematic picture of the $30^{\circ}$ twisted bilayer of two honeycomb lattices. Here we show its top-left corners with $L_{x, y}=20$ (\ref{['Lattice_Lx_20']}), $L_{x, y}=30$ (\ref{['Lattice_Lx_30']}), $L_{x, y}=40$ (\ref{['Lattice_Lx_40']}). The top layer (blue) is formed by sites with coordinates $\vec{r}_{A, B} = \vec{a}_1 n_1 + \vec{a}_2 n_2 + \vec{c}_{A, B}$, where $\vec{a}_1 = (\sqrt{3}, 0)$ , $\vec{a}_2 = (\sqrt{3}/2, 3/2)$, $\vec{c}_{A, B} = (0, \pm 1)$. The bottom layer (red) is obtained by rotating the top layer at 30 degrees. We consider a system with open boundary conditions, whose sites are located inside a square $-L_{x, y} < x,y < L_{x, y}$. The cases (\ref{['Lattice_Lx_20']}, \ref{['Lattice_Lx_30']}) correspond to zigzag-armchair edges, and the case (\ref{['Lattice_Lx_40']}) corresponds to the bearded-armchair edge.
• Figure 2: A schematic picture of couplings in Haldane model in the top (\ref{['Top_layer_couplings']}) and bottom (\ref{['Bot_layer_couplings']}) layers. In each layer, there is a nearest neighbors coupling $t$ between $A$, $B$ sites and a next nearest neighbors coupling $t_2 e^{\pm i \phi}$. The arrows mark the sign of each coupling, namely an arrow pointing from the site $i$ to the site $j$ means that the corresponding term in the Hamiltonian has the form $t_2 e^{+ i \phi} c_i^{\dagger} c_j$. The bottom layer can be obtained by rotating the top layer by $30^{\circ}$ degrees.
• Figure 3: Schematic phase diagram of the $30^{\circ}$ degrees twisted bilayer Haldane model defined by the Eq. \ref{['MainHamiltonian']} obtained by exact diagonalization. At small $t_{inter}$, the model has exactly the same topological/non-topological phases as a single layer Haldane model. At larger $t_{inter}$, the bulk gap closes, and the in-gap edge states disappear. Simultaneously, another bulk gap opens at $m \approx 0$ and $m \gtrsim 1$.
• Figure 4: (\ref{['Loop_over_m']}) Energies of states plotted against $m$ for various values of $t_{inter}$. We choose $L_x = L_y=30$. The color represents their fractal dimensions $D_q$ for $q=2$. At weak interlayer coupling, the spectrum is gapped for $m>1$, whereas for $m <1$ the gap is filled by the edge states, which have smaller fractal dimensions. At strong inerlayer coupling, the edge states disappear, and the spectrum becomes gapless.
• Figure 5: Energies of states plotted against $t_{inter}$ for $m=0$ and $L_{x, y}=20$ (\ref{['Loop_over_t_inter_m_0.0_Lx_20']}), $L_{x, y}=30$ (\ref{['Loop_over_t_inter_m_0.0_Lx_30']}), $L_{x, y}=40$ (\ref{['Loop_over_t_inter_m_0.0_Lx_40']}) as well as $m=0.5$ (\ref{['Loop_over_t_inter_m_0.0_Lx_20']}), $m=1.0$ (\ref{['Loop_over_t_inter_m_0.0_Lx_20']}), $m=1.5$ (\ref{['Loop_over_t_inter_m_0.0_Lx_40']}) at $L_{x, y}=30$. One can see that at small $t_{inter}$, the system retains properties of its counterpart at $t_{inter}=0$, but at large $t_{inter}$, the system either become gapless ($m=0.5, 1$), or a new gap opens ($m=0, 1.5$). The color represents each state's fractal dimension $D_q$ at $q=2$, and we use the same colorbar as on the Fig. \ref{['Loop_over_m_and_t_inter']}. Localized states have the smallest fractal dimensions and thus are shown in blue color, whereas more extended states are shown in green.