Bistritzer-MacDonald哈密顿量中手征极限之外的精确平带
Exactly flat bands beyond the chiral limit in Bistritzer--MacDonald Hamiltonians
- University of California, Berkeley（加州大学伯克利分校）
- Flatiron Institute（平顿研究所）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究在Bistritzer-MacDonald哈密顿量中构造了非手征隧穿势族，证明其可在魔角外维持精确平带，为开放问题3提供反例，并刻画了标准BM模型无此类平带的特性。
中文摘要 AI 辅助
扭曲双层石墨烯的Bistritzer-MacDonald模型的手征极限在魔角处具有精确平带。我们证明，在通常的对称类中，带有非手征隧穿势的非手征极限之外，精确平带仍可存在。我们构造了一族非手征隧穿势，对于每一个布洛赫动量以及每一个非手征耦合的实数值，该哈密顿量都至少有两个零能态。这为Zworski综述中的开放问题3提供了一族反例。反之，我们证明所有具有该性质的可允许势都属于这一族，从而得到了完整的刻画。相比之下，我们表明标准BM模型（即具有一阶谐波隧穿势的BM哈密顿量）在魔角处不允许精确平带，除非可能在非手征耦合强度的离散集合处，且该集合无有限聚点。为证明这一点，我们引入了一个显式可逆性判据。该判据要求投影扰动在某个布洛赫动量处具有非零标量系数，从而阻碍了一般非手征隧穿的精确平带。局部泰勒展开论证验证了标准BM势满足该判据。然而，对于上述构造的势族，投影扰动在每一个布洛赫动量处都消失，因此可逆性判据失效。
英文摘要
The chiral limit of the Bistritzer--MacDonald model for twisted bilayer graphene has exactly flat bands at magic angles. We show that exact flatness can persist beyond the chiral limit with nonchiral tunnelling potentials in the usual symmetry class. We construct a family of nonchiral tunnelling potentials for which the Hamiltonian has at least two zero-energy states at every Bloch momentum and for every real value of the nonchiral coupling. This gives a family of counterexamples to Open Problem~3 of Zworski's survey \cite{ZworskiSurvey}. Conversely, we prove that every admissible potential with this property belongs to this family, thereby obtaining a complete characterization. By contrast, we show that the standard BM model, namely, BM Hamiltonians with first harmonic tunnelling potentials, does not admit exactly flat bands at the magic angle except possibly at a discrete set of nonchiral coupling strengths, with no finite accumulation point. To prove this, we introduce an explicit invertibility criterion. The criterion requires the projected perturbation to have a nonzero scalar coefficient at some Bloch momentum, obstructing exact flatness for general nonchiral tunnelling. A local Taylor expansion argument verifies the criterion for the standard BM potentials. For the family constructed above, however, the projected perturbation vanishes at every Bloch momentum, so the invertibility criterion fails.
