Twisted bilayer graphene. II. Stable symmetry anomaly

Abstract

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works, e.g., [Song et al., Phys. Rev. Lett. 123, 036401 (2019)], [Ahn et al., Phys. Rev. X 9, 021013 (2019)], [Po et al., Phys. Rev. B 99, 195455 (2019)], and others [Kang et al. Phys. Rev. X 8, 031088 (2018), Koshino et al., Phys. Rev. X 8, 031087 (2018), Liu et al., Phys. Rev. B 99, 155415 (2019), Zou et al., Phys. Rev. B 98, 085435 (2018)] found that the two flat bands in TBG possess a fragile topology protected by the C2z T symmetry. [Song et al., Phys. Rev. Lett. 123, 036401 (2019)] also pointed out an approximate particle-hole symmetry (P) in the continuous model of TBG. In this work, we numerically confirm that P is indeed a good approximation for TBG and show that the fragile topology of the two flat bands is enhanced to a P-protected stable topology. This stable topology implies 4l + 2 (l ∈ N) Dirac points between the middle two bands. The P-protected stable topology is robust against arbitrary gap closings between the middle two bands the other bands. We further show that, remarkably, this P-protected stable topology, as well as the corresponding 4l + 2 Dirac points, cannot be realized in lattice models that preserve both C2z T and P symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flat bands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.