Emergent $\text{D}_6$ symmetry in fully-relaxed magic-angle twisted bilayer graphene

TL;DR

This study uses tight-binding calculations with full atomic relaxation to reveal an emergent D6 symmetry in the mini bands of magic-angle twisted bilayer graphene, impacting its electronic properties.

Contribution

It uncovers an emergent D6 symmetry in relaxed twisted bilayer graphene, providing new insights into its band structure and Wannier orbital requirements.

Findings

Four narrow mini bands are well separated after relaxation.

Emergent D6 symmetry is observed despite relaxed lattice.

Eight Wannier orbitals are likely needed to describe the mini bands.

Abstract

We present a tight-binding calculation of a twisted bilayer graphene at magic angle $\theta\sim 1.08^\circ$, allowing for full, in- and out-of-plane, relaxation of the atomic positions. The resulting band structure displays as usual four narrow mini bands around the neutrality point, well separated from all other bands after the lattice relaxation. A thorough analysis of the mini-bands 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$_{3z}$, i.e., transforming like 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 for each of the two supercell sublattices AB and BA are necessary, probably not sufficient, to describe the four mini bands. This unexpected one-electron complexity is likely to play an important role in the still unexplained metal-insulator-superconductor phenomenology of this system.