AI 中文总结
本文提出元胞自然轨道（CNOs）方法，用于解析拓扑能带的相互作用层级，将其应用于魔角双层石墨烯，揭示了不同CNO通道的相互作用特征，为量子材料研究提供了系统框架。
AI 中文摘要
拓扑能带对指数局域且对称的瓦尼尔函数存在阻碍，挑战了用有限范围局域轨道表示投影相互作用的标准范式。为了忠实地捕捉拓扑能带的形状因子和量子几何，我们引入了能带投影密度形状因子的奇异值分解，该方法可实现希尔伯特空间的基于几何的截断方案，揭示了由 underlying 波函数决定的能带投影相互作用的内在层级。这种分解最自然地用元胞自然轨道（Cell Natural Orbitals, CNOs）描述，CNOs是单胞约化单粒子密度矩阵的本征态，其占据数为忠实地表示能带波函数重叠所需的最小轨道复杂度提供了度量。CNO分解系统地确定了重现短程相互作用所需的最小局域轨道数，同时解析了跨CNO通道的相互作用强度层级。将该方法应用于手征极限下的魔角双层石墨烯，我们发现主导CNO以AA位为中心，类似重费米子模型的f费米子；次主导CNO通道携带的相互作用矩阵元逐渐减弱，可在静态平均场水平处理，而主导通道需要动力学自能。该形式主义阐明了单胞内电荷密度的变化如何在CNO包络函数中产生动量依赖性，进而导致单粒子谱函数的色散。更广泛地说，我们的结果确立了CNO作为连接能带拓扑与实空间关联的几何信息桥梁，为分析量子材料中的相互作用和涌现相提供了系统框架。
英文摘要
Topological bands exhibit obstruction to exponentially localized and symmetric Wannier functions, challenging the standard paradigm of representing projected interactions in terms of local orbitals with finite range. To faithfully capture the form factors and quantum geometry of topological bands we introduce a singular-value decomposition of the band-projected density form factors, enabling a geometry-based truncation scheme of the Hilbert space, exposing an intrinsic hierarchy on band-projected interactions that is determined by the underlying wavefunctions. This decomposition is most naturally described in terms of Cell Natural Orbitals (CNOs), as the eigenstates of the unit-cell reduced one-particle density matrix, whose occupation provide a measure of the minimal orbital complexity required to faithfully represent the band wavefunctions overlaps. The CNO decomposition identifies systematically the minimal number of local orbitals needed to reproduce short-ranged interactions while resolving the hierarchy of interaction strengths across CNO channels. Applied to magic-angle twisted bilayer graphene in the chiral limit, we find that the dominant CNO is centered at the AA site, resembling the $f$-fermion of the heavy-fermion model. The subdominant CNO channels carry progressively weaker interaction matrix elements, allowing them to be treated at the static mean-field level, while the dominant channel requires a dynamical self-energy. The formalism illustrates how variations of charge density within the unit cell generate momentum dependence in the CNO envelope function and, consequently, dispersion in the single-particle spectral function. More broadly, our results establish CNOs as a geometry-informed bridge between band topology and real-space correlations, providing a systematic framework for analyzing interactions and emergent phases in quantum materials.
Comments11 pages, 9 figures
