Twisted bilayer graphene. II. Stable symmetry anomaly

TL;DR

This paper demonstrates that the continuous model of twisted bilayer graphene with particle-hole symmetry exhibits a stable topology and anomalies that cannot be realized in lattice models, revealing fundamental topological constraints.

Contribution

It shows that the entire continuous model of TBG has a stable topology protected by particle-hole symmetry, which leads to anomalies incompatible with lattice realizations.

Findings

The continuous TBG model has a stable topology with 4l+2 Dirac points.

Particle-hole symmetry is a good approximation in TBG.

The stable topology cannot be realized in lattice models preserving both $C_{2z}T$ and $ ext{P}$.

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 a lattice model. Previous works, e.g., [Phys. Rev. Lett. 123, 036401], [Phys. Rev. X 9, 021013], [Phys. Rev. B 99, 195455], and others [1-4] found that the two flat bands in TBG possess a fragile topology protected by the $C_{2z}T$ symmetry. [Phys. Rev. Lett. 123, 036401] also pointed out an approximate particle-hole symmetry ($\mathcal{P}$) in the continuous model of TBG. In this work, we numerically confirm that $\mathcal{P}$ is indeed a good approximation for TBG and show that the fragile topology of the two flat bands is enhanced to a $\mathcal{P}$-protected stable topology. This stable topology implies $4l+2$ ($l\in\mathbb{N}$) Dirac points between the middle two bands. The $\mathcal{P}$-protected stable topology is robust against arbitrary gap closings between the middle two bands the other bands. We further show that, remarkably, this $\mathcal{P}$-protected stable topology, as well as the corresponding $4l + 2$ Dirac points, cannot be realized in lattice models that preserve both $C_{2z}T$ and $\mathcal{P}$ 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.