We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-MacDonald PDE model, which is periodic with respect to the bilayer's moir\'e pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes-Thomas estimates. We then provide extensive numerical computations which clarify the range of validity of the Bistritzer-MacDonald model.

翻译：我们研究了在扭转双层石墨烯中数值计算电子量子动力学的问题。挑战在于，由于不同石墨烯层之间的不可公度性，原子尺度的动力学模型在一般扭转角下呈现非周期性。最近研究表明，Bistritzer-MacDonald偏微分方程模型（该模型以双层石墨烯的莫尔条纹图案为周期）能在特定参数范围内严格描述这些动力学。本文首先证明不可公度扭转双层石墨烯紧束缚模型的动力学可通过有限区域计算近似，该证明的核心要素是基于Combes-Thomas估计建立的传播速度估计。随后我们通过大量数值计算，阐明了Bistritzer-MacDonald模型的适用性范围。