Single-particle continuum models such as the popular Bistritzer-MacDonald model have become powerful tools for predicting electronic phenomena of incommensurate 2D materials and the development of many-body models aimed to model unconventional superconductivity and correlated insulators. In this work, we introduce a procedure to construct continuum models of arbitrary accuracy relative to tight-binding models for moir\'{e} incommensurate bilayers. This is done by recognizing the continuum model as arising from Taylor expansions of a high accuracy momentum space approximation of the tight-binding model. We apply our procedure in full detail to two models of twisted bilayer graphene and demonstrate both admit the Bistritzer-MacDonald model as the leading order continuum model, while higher order expansions reveal qualitative spectral differences.

翻译：诸如广受欢迎的Bistritzer-MacDonald模型等单粒子连续模型，已成为预测非公度二维材料电子现象以及构建旨在模拟非常规超导性和关联绝缘体的多体模型的有力工具。本工作提出了一种构建莫尔非公度双层体系连续模型的流程，该模型相对于紧束缚模型可达到任意精度。这一流程的实现基于以下认识：连续模型产生于对紧束缚模型高精度动量空间近似的泰勒展开。我们将此流程完整应用于两种扭转双层石墨烯模型，并证明二者均以Bistritzer-MacDonald模型作为其主导阶连续模型，而更高阶展开则揭示了二者在能谱上的定性差异。