All "Magic Angles" Are "Stable" Topological

Understand

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 Moir\'e 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.