Emergent D6 symmetry in fully relaxed magic-angle twisted bilayer graphene

We present a tight-binding calculation of a twisted bilayer graphene at magic angle θ∼1.08, allowing for full, in- and out-of-plane, relaxation of the atomic positions. The resulting band structure displays, as usual, four narrow minibands around the neutrality point, well separated from all other bands after the lattice relaxation. A thorough analysis of the miniband Bloch functions reveals an emergent D6 symmetry, despite the lack of any manifest point-group symmetry in the relaxed lattice. The Bloch functions at the Γ point are degenerate in pairs, reflecting the so-called valley degeneracy. Moreover, each of them is invariant under C3z, i.e., transforming like a one-dimensional, in-plane symmetric irreducible representation of an "emergent" D6 group. Out of plane, the lower doublet is even under C2x, while the upper doublet is odd, which implies that at least eight Wannier orbitals, two s-like and two pz-like ones for each of the supercell sublattices AB and BA, are necessary but probably not sufficient to describe the four minibands. This unexpected one-electron complexity is likely to play an important role in the still unexplained metal-insulator-superconductor phenomenology of this system.