Song Zhida, Wang Zhijun, Shi Wujun, Li Gang, Fang Chen, Bernevig B Andrei
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.
Beijing National Research Center for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.
Phys Rev Lett. 2019 Jul 19;123(3):036401. doi: 10.1103/PhysRevLett.123.036401.
We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semimetallic and topological phases. In particular, we are able to prove, using an approximate low-energy particle-hole symmetry, that the gapped set of bands that exist around all magic angles have a nontrivial topology stabilized by a magnetic symmetry, provided band gaps appear at fillings of ±4 electrons per moiré unit cell. The topological index is given as the winding number (a Z number) of the Wilson loop in the moiré Brillouin zone. Furthermore, we also claim that, when the gapped bands are allowed to couple with higher-energy bands, the Z index collapses to a stable Z_{2} index. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: When strongly broken, nontopological phases can appear. Our Letter underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a four-band short-range tight-binding model whose two lower bands have the same topology, symmetry, and flatness as those of the twisted bilayer graphene and which can be used as an effective low-energy model. We then perform large-scale (11000 atoms per unit cell, 40 days per k-point computing time) ab initio calculations of a series of small angles, from 3° to 1°, which show a more complex and somewhat different evolution of the symmetry of the low-energy bands than that of the theoretical moiré model but which confirm the topological nature of the system.
我们表明，小角度扭曲双层石墨烯中低能带的电子结构由一系列半金属相和拓扑相组成。特别是，我们能够利用近似的低能粒子-空穴对称性证明，在所有魔角附近存在的带隙能带集具有由磁对称性稳定的非平凡拓扑结构，前提是在每个莫尔晶胞填充±4个电子时出现带隙。拓扑指数由莫尔布里渊区中威尔逊圈的缠绕数（一个整数）给出。此外，我们还声称，当允许带隙能带与高能带耦合时，Z指数会坍缩为一个稳定的Z₂指数。近似的、涌现的粒子-空穴对称性对于石墨烯的拓扑结构至关重要：当它被强烈破坏时，可能会出现非拓扑相。我们的论文强调了拓扑结构是描述低能石墨烯的关键要素。我们提供了一个四能带短程紧束缚模型，其两个较低能带与扭曲双层石墨烯的能带具有相同的拓扑结构、对称性和平坦度，并可作为一个有效的低能模型。然后，我们对一系列从3°到1°的小角度进行了大规模（每晶胞11000个原子，每个k点计算时间40天）的从头计算，结果表明低能带对称性的演化比理论莫尔模型更复杂且有所不同，但证实了该系统的拓扑性质。
