Magic Manifold and Stable Symmetry Anomaly in Twisted Bilayer Graphene*

We investigate the single-particle Hamiltonian of Twisted Bilayer Graphene (TBG) model. We provide an analytical perturbative understanding of why the TBG bands are flat over the whole Brillouin zone at the first magic angle, despite it is defined only by vanishing Dirac velocity. We derive a connected "magic manifold": w1=2 \sqrt{1+w0^2} – \sqrt{2+3w0^2}, on which the bands remain extremely flat. We also show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous. The fragile topology of the two flat bands is enhanced to a particle-hole-symmetry-protected stable topology. This stable topology implies 4n+2 Dirac points between the middle two bands. Remarkably, this table topology, as well as the corresponding 4n+2 Dirac points, cannot be realized in lattice models that preserve both C2T and particle-hole symmetries. In other words, the continuous model of TBG is anomalous.