We investigate the electronic and optical properties of twisted bilayer graphene with arbitrary twist angles $\theta$. Our results are based on a method of evolving in time quantum states in lattice space. We propose an efficient scheme of sampling lattice nodes that helps to reduce significantly computational cost, particularly for tiny twist angles. We demonstrate the continuous variation of the density of states and the optical conductivity with respect to the twist angle. It indicates that the commensurability between the two graphene layers does not play an essential role in governing the electronic and optical properties. We point out that, for the twist angles roughly in the range $0.1^{\circ }<\theta <3^{\circ }$, the density of states in the vicinity of the Fermi energy exhibits the typical W shape with a small peak locating at the Fermi energy. This peak is formed as the merging of two van Hove peaks and reflects the appearance of states strongly localized in the AA-like region of moiré zones. When decreasing the twist angle to zero, the W shape is gradually transformed to the U shape, which is seen as the behavior of the density of states in the limit of $\theta \to 0^{\circ }$.

中文翻译：

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扭曲双层石墨烯的电子结构和光学性质，通过真实空间中状态的时间演化来计算

我们研究了具有任意扭曲角的扭曲双层石墨烯的电子和光学性质 $\theta$。我们的结果基于在晶格空间中演化时间量子态的方法。我们提出了一种采样晶格节点的有效方案，该方案有助于显着降低计算成本，尤其是对于较小的扭曲角度而言。我们证明了状态密度和光导率相对于扭转角的连续变化。这表明两个石墨烯层之间的可比性在控制电子和光学性能方面没有起重要作用。我们指出，对于扭转角，大致在该范围内$0。{\mathrm{1个}}^{\circ }<\theta <3^{\circ }$，费米能量附近的状态密度表现出典型的W形，并且在费米能量处有一个小的峰值。该峰是两个范霍夫峰的合并，形成的峰反映了强烈位于莫尔区AA状区域中的状态的出现。当将扭曲角减小到零时，W形逐渐转变为U形，这被视为在极限范围内状态密度的行为。$\theta \to 0^{\circ }$。