Fragile Topological Bands With Stable Index at Half-Filling inTwisted Bilayer Graphene

Using an approximate but crucial particle-hole symmetry, we prove that half-filled graphene has bands that exhibit Wannier (Wilson loop) flow, and hence have to be topological. While this Wannier flow can be interrupted by the addition of certain bands, a topological index remains stable, and characterizes the set of bands. We then show that electron-phonon interaction in this set of bands can cause large superconducting temperatures and predict a certain set of angles where superconductivity should be observed.