Stacking-Induced Large-Chern-Number Quantum Anomalous Hall Phases

TL;DR

This work addresses the challenge of realizing quantum anomalous Hall phases with large Chern numbers in two-dimensional systems. By modeling bilayer hexagonal lattices with both vertical and skew interlayer couplings and introducing Haldane-type complex next-nearest-neighbor hoppings, the authors demonstrate that relative layer sliding shifts Dirac points to generic momenta and, when gapped, yields topological phases with $|C|>2$. Phase diagrams in the interlayer-phase space $(\phi_1,\phi_2)$ reveal regions with $|C|=3$ and $|C|=4$ in several configurations, with skew interlayer hopping playing a crucial stabilizing role. Edge-state calculations in ribbon geometries corroborate bulk results, showing multiple chiral edge modes consistent with the computed Chern numbers, thereby establishing bulk–boundary correspondence and providing a realistic route to engineering high-$|C|$ QAH phases in van der Waals heterostructures. The study highlights how interlayer sliding and symmetry-controlled hybridization can robustly realize non-additive, high-Chern-number QAH states in experimentally accessible platforms, potentially enabling multi-channel, low-dissipation electronic devices.

Abstract

We investigate the interaction between quantum anomalous Hall (QAH) phases hosted by two atomically thin hexagonal lattices and demonstrate the emergence of topological phases with large Chern numbers. Interlayer coupling between two graphene-like lattices produces band crossings, while relative sliding preserves gapless Dirac points located at generic, low-symmetry $\mathbf{k}$ points. The introduction of Haldane-type complex next-nearest-neighbor hoppings gaps these Dirac points, breaks time-reversal symmetry, and generates a sequence of quantum anomalous Hall phases. Depending on the phase angles $φ_1$ and $φ_2$ associated with the two layers, the system exhibits QAH states with Chern numbers $|C|>2$. The nontrivial bulk topology is verified by the presence of the corresponding number of chiral edge modes in ribbon geometries. These high-Chern-number phases originate from the enhanced twisting of the valence-band manifold induced by interlayer stacking.

Figures (6)

• Figure 1: Atomic structures of five representative SBL configurations characterized by typical sliding vectors $\boldsymbol{\tau}$. Configuration C1 corresponds to $\boldsymbol{\tau} = (0,0)$ (AA-stacked), C2 to $\boldsymbol{\tau} = (0.5,0)$, C3 to $\boldsymbol{\tau} = (1,0)$ (AB-stacked), C4 to $\boldsymbol{\tau} = (0,0.5)$, and C5 to $\boldsymbol{\tau} = (0,1)$. The unit cell is indicated by the green rhombus defined by the basis vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. The sliding vector $\boldsymbol{\tau}$ is shown in pink. The corresponding energy surfaces are obtained using Slater-Koster model, which yields $t_2=0.10$ and the following interlayer hopping amplitudes: (C1) $t_u=0.18,\,t_v=t_w=0.08$, (C2) $t_u=t_v=0.14,\,t_w=0.10$, (C3) $t_v=0.18,\,t_u=t_w=0.08$, (C4) $t_u=0.15,\,t_v=t_w=0.12$, and (C5) $t_u=0.10,\,t_v=t_w=0.14$.
• Figure 2: Phase diagrams of five SBL configurations computed using Slater-Koster model. The hopping amplitudes in the top row are identical to those in Fig. \ref{['Fig_1']}. In the bottom row, all interlayer hopping parameters are uniformly scaled by a factor of $2.8$, resulting in the following values: (C1) $t_u=0.50,\,t_v=t_w=0.22$, (C2) $t_u=t_v=0.39,\,t_w=0.28$, (C3) $t_v=0.50,\,t_u=t_w=0.22$, (C4) $t_u=0.42,\,t_v=t_w=0.34$, and (C5) $t_u=0.28,\,t_v=t_w=0.39$.
• Figure 3: Chern number diagrams for four SBL configurations C2, C3, C4, and C5 as functions of the phases $\phi_1$ and $\phi_2$. The C1 (AA-stacked) configuration does not exhibit a phase with $|C|>2$ and is therefore not included. The values of hopping amplitudes are taken to be (C2) $t_2=0.56,\,t_u=0.90,\,t_v=0.50,\,t_w=0.50$, (C3) $t_2=0.10,\,t_u=0.08,\,t_v=0.70,\,t_w=0.60$, (C4) $t_2=0.30,\,t_u=0.90,\,t_v=t_w=0.70$, and (C5) $t_2=0.30,\,t_u=0.50,\,t_v=t_w=0.20$.
• Figure 4: (a) Phase diagrams of two valence manifolds $\mathcal{M}_{V_1}$ and $\mathcal{M}_{V_2}$ corresponding to the two configurations C2 and C5 presented in FIG. \ref{['Fig_3']}. (b) Phase diagrams of the valence manifold $\mathcal{M}_V$ for a representative configuration C3 computed in the two cases: (left panel) all skew hoppings absent with $t_v=0.70,\;t_u=t_w=0$, and (right panel) one skew hopping present with $t_u=0,\;t_v=0.70,\;t_w=0.60$. The color codes here are identical to those in FIG. \ref{['Fig_3']}.
• Figure 5: Energy surfaces and Berry connection fields, $\mathbf{A}(\mathbf{k})$, of valence manifolds $\mathcal{M}_{V_1}$ and $\mathcal{M}_{V_2}$ plotted for a representative configuration C3 in the two cases: (left column) all skew hoppings absent with $t_v=0.70,\;t_u=t_w=0$, and (right column) one skew hopping present with $t_u=0,\;t_v=0.70,\;t_w=0.60$. The Haldane phases in both cases are $(\phi_1,\phi_2)=(-2.60,-0.20)$. The cyan and blue dots indicate the positive and negative signs of Berry curvature, respectively, computed by integrating over vortex regions of $\mathbf{A}(\mathbf{k})$. The black and magenta dashed lines are plotted to denote the Brillouin zone and to align $\mathbf{K},\mathbf{K}^\prime$ valleys, respectively.