Fine structure of flat bands in a chiral model of magic angles

We analyze symmetries of Bloch eigenfunctions at magic angles for the Tarnopolsky--Kruchkov--Vishwanath chiral model of the twisted bilayer graphene (TBG) following the framework introduced by Becker--Embree--Wittsten--Zworski. In particular, we show that vanishing of the first Bloch eigenvalue away from the Dirac points implies its vanishing at all momenta, that is the existence of a flat band. We then study how the multiplicity of the flat band is related to the nodal set of the Bloch eigenfunctions. We also demonstrate that for a generic choice of tunneling potentials, obeying all translational and rotational symmetries, the Hamiltonian only exhibits flat bands of minimal multiplicity. We conclude with two numerical observations about the structure of flat bands.