paper

All "Magic Angles" Are "Stable" Topological

arXiv:1807.10676 · doi:10.1103/PhysRevLett.123.036401

Abstract

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semi-metallic 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 has what we conjecture to be a stable topological index stabilized by a magnetic symmetry and reflected in the odd winding of the Wilson loop in the Moiré BZ. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: when strongly broken, non-topological phases can appear. Our paper underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a -band short range tight-binding model whose lower bands have the same topology, symmetry, and flatness as those of the twisted graphene, and which can be used as an effective low-energy model. We then perform large-scale ( atoms per unit cell, 40 days per -point computing time) ab-initio calculations of a series of small angles, from to , which show a more complex and somewhat qualitatively different evolution of the symmetry of the low-energy bands than that of the theoretical Moiré model, but which confirms the topological nature of the system. At certain angles, we find no insulating filling in graphene at electrons per Moiré unit cell. The ab-initio evolution of gaps tends to differ from that of the continuum Moiré model.

arXiv comment7+23 pages, 3+12 figures, 2+3 tables; v2: references added, note added

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