Twisted bilayer graphene. II. Stable symmetry anomaly

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just two active bands) with particle-hole symmetry is anomalous and hence incompatible a lattice model. Previous works, e.g., [Phys. Rev. Lett. 123, 036401], X 9, 021013], B 99, 195455], others [1-4] found flat bands in TBG possess fragile topology protected by $C_{2z}T$ symmetry. 036401] also pointed out an approximate ($\mathcal{P}$) TBG. In this work, we numerically confirm $\mathcal{P}$ indeed good approximation for enhanced to $\mathcal{P}$-protected stable topology. This implies $4l+2$ ($l\in\mathbb{N}$) Dirac points between middle bands. The robust against arbitrary gap closings other further that, remarkably, topology, as well corresponding $4l + 2$ points, cannot be realized models preserve both symmetries. words, on lattices. Two related topics, consequences interacting problem, i.e., choice Chern band basis perfect metal phase so-called second chiral limit, are discussed.