我們探討魔角扭轉雙層石墨烯之朗道能階。當我們關掉扭轉雙層石墨烯在AA堆疊區域層與層之間的耦合，系統會呈現額外的手徵對稱性並展現出完美的平帶。朗道能階來自於系統本身被量子化的迴旋軌道對應到的能量，所以朗道能階的範圍通常被侷限在系統還沒有外加磁場的能帶範圍內。然而軌道的磁矩，對應到系統能帶間的耦合，產生額外的迴旋軌道使得異常的朗道能階出現在原本的能帶外。
我們以朗道能階的基底展開連續模型來獲得其朗能階。我們發現滿足手徵對稱性的扭轉雙層石墨烯的能帶間耦合被其手徵對稱性和時間-空間反衍對稱性限制為零。我們提出一量值為純虛數的耦合，發生於上層的A(B)原子和下層的A(B)原子間。其可對應到一具有周期且局域的規範勢存在於扭轉雙層石墨烯的AA堆疊區域。此耦合破壞了時間-空間反衍對稱性，並展現出了異常的朗道能階結構，其結果符合經過考慮軌道磁化的半古典(semi-classical)量子化條件。

We investigate the Landau levels in magic angle twisted bilayer graphene (TBG). In TBG, the system preserves additional chiral symmetry and exhibits an exact flat band structure when interlayer tunneling within the AA stacked region vanishes. Landau levels arise from the energy of quantized cyclotron orbits, which are typically limited by the non-magnetic energy bounds. However, the orbital magnetic moment, which characterizes the inter-band coupling, leads to the formation of additional cyclotron orbits and results in the anomalous spreading of Landau levels outside these non-magnetic energy bounds.
We obtained the Landau levels by expanding the continuum model in the Landau level basis. Anomalous Landau level spreading is prohibited in chiral twisted bilayer graphene (CTBG) since the inter-band coupling is constrained by the chiral and C2T (space-time inversion) symmetry. However, we propose a fully imaginary interlayer tunneling between A(B) atoms in the top layer and A(B) atoms in the bottom layer, which can be induced by a periodic gauge potential localized within AA stacked regions. This interlayer tunneling breaks the C2T symmetry, making the anomalous Landau level spreading visible. Our results are consistent with the semi-classical quantization condition when considering the orbital magnetization.

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