TY - JOUR

T1 - All Magic Angles in Twisted Bilayer Graphene are Topological

AU - Song, Zhida

AU - Wang, Zhijun

AU - Shi, Wujun

AU - Li, Gang

AU - Fang, Chen

AU - Bernevig, B. Andrei

N1 - Publisher Copyright:
© 2019 American Physical Society.

PY - 2019/7/16

Y1 - 2019/7/16

N2 - 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 Z2 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.

AB - 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 Z2 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.

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U2 - 10.1103/PhysRevLett.123.036401

DO - 10.1103/PhysRevLett.123.036401

M3 - Article

C2 - 31386469

AN - SCOPUS:85069976698

VL - 123

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 3

M1 - 036401

ER -