期刊
PHYSICAL REVIEW LETTERS
卷 123, 期 3, 页码 -出版社
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevLett.123.036401
关键词
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资金
- MOST [2016YFA302400, 2016YFA302600]
- NSFC [11674370, 11421092, 11504117]
- Department of Energy [DE-SC0016239]
- National Science Foundation EAGER Grant [DMR 1643312]
- Simons Investigator Grants [404513]
- ONR [N00014-14-1-0330]
- NSF-MRSEC [DMR-142051]
- Packard Foundation
- Schmidt Fund for Innovative Research
- National Thousand-Young-Talents Program
- CAS Pioneer Hundred Talents Program
- ShanghaiTech University
- Program for Professor of Special Appointment (Shanghai Eastern Scholar)
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 moire unit cell. The topological index is given as the winding number (a Z number) of the Wilson loop in the moire 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 degrees to 1 degrees, which show a more complex and somewhat different evolution of the symmetry of the low-energy bands than that of the theoretical moire model but which confirm the topological nature of the system.
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