Title: Classification of High-Ordered Topological Nodes towards Moiré Flat Bands in Twisted Bilayers Show Chinese title

Title: 扭曲双层结构中莫尔平带的高阶拓扑节点分类

Abstract: At magic twisted angles, Dirac cones in twisted bilayer graphene (TBG) can evolve into flat bands, serving as a critical playground for the study of strongly correlated physics. When chiral symmetry is introduced, rigorous mathematical proof confirms that the flat bands are locked at zero energy in the entire Moir\'e Brillouin zone (BZ). Yet, TBG is not the sole platform that exhibits this absolute band flatness. Central to this flatness phenomenon are topological nodes and their specific locations in the BZ. In this study, considering twisted bilayer systems that preserve chiral symmetry, we classify various ordered topological nodes in base layers and all possible node locations across different BZs. Specifically, we constrain the node locations to rotational centers, such as {\Gamma} and M points, to ensure the interlayer coupling retains equal strength in all directions. Using this classification as a foundation, we systematically identify the conditions under which Moir\'e flat bands emerge. Additionally, through the extension of holomorphic functions, we provide proof that flat bands are locked at zero energy, shedding light on the origin of the band flatness. Remarkably, beyond Dirac cones, numerous twisted bilayer nodal platforms can host flat bands with a degeneracy number of more than two, such as four-fold, six-fold, and eight-fold. This multiplicity of degeneracy in flat bands might unveil more complex and enriched correlation physics. Show Chinese abstract

Abstract: 在魔角扭曲角度下，扭曲双层石墨烯（TBG）中的狄拉克锥可以演化为平带，这为研究强关联物理提供了一个关键的平台。 当引入手性对称性时，严格的数学证明确认了平带在整个莫尔布里渊区（BZ）内被锁定在零能级。 然而，TBG并不是唯一表现出这种绝对带平度的平台。 这一带平度现象的核心是拓扑节点及其在BZ中的特定位置。 在本研究中，考虑到保持手性对称性的扭曲双层系统，我们对基底层中的各种有序拓扑节点以及不同BZ中的所有可能节点位置进行了分类。 具体而言，我们将节点位置限制在旋转中心，如{\Gamma }和M点，以确保层间耦合在所有方向上保持相同的强度。 利用这一分类作为基础，我们系统地识别了莫尔平带出现的条件。 此外，通过解析函数的扩展，我们提供了平带被锁定在零能级的证明，揭示了带平度的起源。 值得注意的是，除了狄拉克锥之外，许多扭曲双层节点平台可以容纳具有超过二重简并度的平带，例如四重、六重和八重简并。 平带中这种简并度的多样性可能会揭示更复杂和丰富的关联物理。