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 theta similar to 1.08 degrees, 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 D-6 symmetry, despite the lack of any manifest point-group symmetry in the relaxed lattice. The Bloch functions at the Gamma point are degenerate in pairs, reflecting the so-called valley degeneracy. Moreover, each of them is invariant under C-3(z), i.e., transforming like a one-dimensional, in-plane symmetric irreducible representation of an "emergent" D-6 group. Out of plane, the lower doublet is even under C-2x, while the upper doublet is odd, which implies that at least eight Wannier orbitals, two s-like and two p(z)-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.