Emergent $D_6$ symmetry in fully relaxed magic-angle twisted bilayer graphene

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 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 $\mathrm{\Gamma }$ point are degenerate in pairs, reflecting the so-called valley degeneracy. Moreover, each of them is invariant under $C_{3z}$, i.e., transforming like 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.

Figure 1

(a) Two graphene sheets rotated by a small angle (shown here for $\theta \approx 3.{89}^{\circ }$, while remaining calculations will be for $\theta =1.{08}^{\circ }$) with respect to each other. The emerging moiré pattern is highlighted by a gray shaded line, and the predominant characters of the stacking between the two layers, AA and AB (or BA), are indicated by black triangles and circles, respectively. The triangular superlattice vectors ${\mathbf{L}}_1$ and ${\mathbf{L}}_2$ connect different AA zones. (b) Mini Brillouin zone of tBLG. The high-symmetry points $\mathbf{\Gamma },{\mathbf{K}}_1,{\mathbf{K}}_2,\mathbf{M}$ are shown together with the reciprocal lattice vectors ${\mathbf{G}}_1$ and ${\mathbf{G}}_2$.

Figure 2

Pictorial view of the Wannier functions ${\mathrm{\Psi }}^{AB}(\mathbf{r}-{\mathbf{r}}_{AB})$ and ${\mathrm{\Psi }}^{BA}(\mathbf{r}-{\mathbf{r}}_{BA})$ centered at AB and BA sites, respectively. The triangles represent wave function components centered around the AA regions, while the combination of the three triangles defines the Wannier orbital, centered instead around AB (left) or BA (right).

Figure 3

(a) The supercell of a tBLG at $\theta \approx 1.{08}^{\circ }$ used in simulations, obtained upon rotating a bilayer initially in the AA stacking configuration around a vertical C-C bond ($D_3$ structure). Arrows show the primitive lattice vectors, of length $L_M$, of the triangular moiré superstructure. Green, gray, red, and blue circles mark the regions of AA, SP, AB, and BA stacking, respectively. (b) Local structure before and after relaxation around the center of the AA, SP, and AB regions. (c) Displacement field showing the in-plane deformations of the upper layer. The displacement vectors {${\mathbf{u}}_i$} go from the equilibrium position of the carbon atoms in the nonrelaxed configuration to the corresponding position in the fully relaxed structure. Only a few vectors are shown for clarity, magnified by a factor of 10. (d) Colored map showing the local interlayer distance. The colored circles reported in (c) and (d) correspond to the samples in (b).

Figure 4

Band structure at twist angle $1.{08}^{\circ }$ of the relaxed tBLG. (a) A close-up of the band structure showing only the four minibands, where labels indicate their degeneracy at the high-symmetry points. (b) The full band structure. The two circles indicate the $s$ (bottom) and $p_z$ (top) doublets used to construct the Wannier orbitals. (c) Level spectrum and degeneracy at the $\mathbf{\Gamma }$ point. The labels $s$ and $p_z$ refer to the symmetry under $C_{2x}$ (see the text).

Figure 5

Layer (1 and 2) and sublattice (A and B) components of one state within the lowest-energy doublet at $\mathbf{\Gamma }$ in the flat bands. The color of each point indicates its complex phase, while its size is a measure of its square modulus. Each unit cell (black dashed line in the top left panel) has been replicated three times to improve visibility. This eigenstate is invariant under $C_{3z}$, even with respect to $C_{2x}$, and odd under $C_{2z}$.

Figure 6

Layer 1 and sublattice (A and B) components of $s_1\left(\mathbf{r}\right)$ (left) and $s_2\left(\mathbf{r}\right)$ (right). The color of each point indicates its complex phase, while its size is a measure of its square modulus. Each unit cell (black dashed line in the top left panel) has been replicated three times to improve visibility.

Figure 7

Layer 1 and sublattice (A and B) components of $p_1\left(\mathbf{r}\right)$ (left) and $p_2\left(\mathbf{r}\right)$ (right). The color of each point indicates its complex phase, while its size is a measure of its square modulus. Each unit cell (black dashed line in the top left panel) has been replicated three times to improve visibility

Figure 8

Layer and sublattice components in the unit cell of one of the two degenerate Bloch functions at ${\mathbf{K}}_1$ whose Wannier orbitals are centered on AB.