转角双层石墨烯：霍夫施塔特蝴蝶的重复性高阶拓扑

晶体的磁平移对称性在外部磁场存在时表现为能谱的分形形式，类似于重复出现的蝴蝶，被称为霍夫施塔特蝴蝶。通过大规模合成具有宏观尺度原胞的范德华超晶格，能够显著降低产生朗道能级复制体所需的磁场。对于这一关键性发展，实验上已经在石墨烯超晶格、魔角扭转双层石墨烯（TBG）和转角双层石墨烯中成功实现了霍夫施塔特蝴蝶拓扑结构。

Fig. 1 Replica HOTI states in TBG.

最近，已有研究揭示了磁平移对称性与受对称性保护的拓扑物态之间的联系。对于每个原胞中含有多个位点的一般晶格模型，霍夫施塔特能谱随着通量的周期性增加而近似重复出现，从而导致哈密顿量在幺正变换下具有通量平移对称性。值得注意的是，有效的时间反演对称性在半个通量周期下得到恢复，从而允许存在多种受对称性保护的拓扑物态。

Fig. 2 Hofstadter butterflies of the atomistic tight-binding model of TBG.

来自韩国科学技术院物理系的Sun-Woo Kim等，证明了以局部角态为特征的高阶拓扑绝缘体（HOTI）相，作为原始HOTI的复制体而出现，从而实现霍夫施塔特能谱的自相似性。他们证明了在所有相称的角度下，TBG中均存在精确的通量平移对称性。

Fig. 3 Characterization of HOTI states. 

基于这一结果，他们确定了零通量下原始的HOTI相在经过半个通量周期后再次出现，其中有效的二重旋转对称性保持不变。此外，在没有对称性保护的情况下，还发现了许多原始HOTI的复制体。与原始的HOTI一样，HOTI复制体同时具有局域角态和边缘局域的实空间拓扑标记。HOTI复制体起源于TBG中不同的相互作用尺度，即层内耦合和层间耦合。这些结果所揭示的霍夫施塔特蝴蝶拓扑结构突出了量子分形中受对称性保护的拓扑。

Fig. 4 Characterization of replica HOTI states.

该研究为探索具有多重相互作用尺度的一般莫尔多层和莫尔准周期系统在磁场下的重复性拓扑结构铺平了道路。该文近期发布于npj Computational Materials 9: 152 (2023).

Fig. 4 Characterization of replica HOTI states.

Editorial Summary

Twisted bilayer graphene: Replica higher-order topology of Hofstadter butterflies

Magnetic translational symmetry of crystals in the presence of an external magnetic field manifests as a fractal form of the energy spectrum that resembles recurring replicas of butterflies, known as Hofstadter butterflies. The magnetic field required to produce replicas of the Landau levels could be significantly reduced by the large-scale synthesis of a van der Waals superlattice with a macroscopic unit cell. For this crucial development, the Hofstadter butterflies have been experimentally realized in a graphene superlattice, magic-angle twisted bilayer graphene (TBG), and twisted double-bilayer graphene. Notably, the link with the magnetic translational symmetry and symmetry-protected topological phases of matter has been revealed recently. In a general lattice model with multiple sites per unit cell, the Hofstadter energy spectrum becomes approximately replicative under the addition of the flux periodicity, which constitutes the additional flux translational symmetry via the unitary transformation of the Hamiltonian. Remarkably, the effective time-reversal symmetry is restored at a half-flux periodicity, allowing for the existence of diverse topological states of matter protected by symmetries. In this work, Sun-Woo Kim et al. from the Department of Physics, KAIST, Korea,demonstrated that higher-order topological insulator (HOTI) phases, characterized by localized corner states, occur as replicas of the original HOTIs to fulfill the self-similarity of the Hofstadter spectrum. They showed the existence of exact flux translational symmetry in TBG at all commensurate angles. Based on this result, they identified that the original HOTI phase at zero flux is re-entrant at a half-flux periodicity, where the effective twofold rotation is preserved. In addition, numerous replicas of the original HOTIs were found for fluxes without protecting symmetries. Like the original HOTIs, replica HOTIs feature both localized corner states and edge-localized real-space topological markers. The replica HOTIs originate from the different interaction scales, namely, intralayer and interlayer couplings, in TBG. The topological aspect of Hofstadter butterflies revealed in these results highlights symmetry-protected topology in quantum fractals. This result can pave the way for studying replica topology under magnetic field in generic moiré multilayer and moiré quasiperodic systems that host multiple interaction scales. This article was recently published in npj Computational Materials 9: 152 (2023).

原文Abstract及其翻译

Replica higher-order topology of Hofstadter butterflies in twisted bilayer graphene(转角双层石墨烯中霍夫施塔特蝴蝶的重复性高阶拓扑)

Sun-Woo Kim, Sunam Jeon, Moon Jip Park & Youngkuk Kim

Abstract The Hofstadter energy spectrum of twisted bilayer graphene (TBG) is found to have recursive higher-order topological properties. We demonstrate that higher-order topological insulator (HOTI) phases, characterized by localized corner states, occur as replicas of the original HOTIs to fulfill the self-similarity of the Hofstadter spectrum. We show the existence of exact flux translational symmetry in TBG at all commensurate angles. Based on this result, we identify that the original HOTI phase at zero flux is re-entrant at a half-flux periodicity, where the effective twofold rotation is preserved. In addition, numerous replicas of the original HOTIs are found for fluxes without protecting symmetries. Like the original HOTIs, replica HOTIs feature both localized corner states and edge-localized real-space topological markers. The replica HOTIs originate from the different interaction scales, namely, intralayer and interlayer couplings, in TBG. The topological aspect of Hofstadter butterflies revealed in our results highlights symmetry-protected topology in quantum fractals.

摘要转角双层石墨烯（TBG）的霍夫施塔特能谱具有重复性高阶拓扑性质。我们证明了以局部角态为特征的高阶拓扑绝缘体（HOTI）相，作为原始HOTI的复制体而出现，从而实现霍夫施塔特能谱的自相似性。我们证明了在所有相称的角度下，TBG中均存在精确的通量平移对称性。基于这一结果，我们确定了零通量下原始的HOTI相在经过半个通量周期后再次出现，其中有效的二重旋转对称性保持不变。此外，在没有对称性保护的情况下，还发现了许多原始HOTI的复制体。与原始的HOTI一样，HOTI复制体同时具有局域角态和边缘局域的实空间拓扑标记。HOTI复制体起源于TBG中不同的相互作用尺度，即层内耦合和层间耦合。研究结果所揭示的霍夫施塔特蝴蝶拓扑结构突出了量子分形中受对称性保护的拓扑。