Ferromagnetic Mott state in Twisted Graphene Bilayers at the Magic Angle

We address the effective tight-binding Hamiltonian that describes the insulating Mott state of twisted graphene bilayers at a magic angle. In that configuration, twisted bilayers form a honeycomb superlattice of localized states, characterized by the appearance of flat bands with fourfold degeneracy. After calculating the maximally localized superlattice Wannier wave functions, we derive the effective spin model that describes the Mott state. We suggest that the system is an exotic ferromagnetic Mott insulator, with well-defined experimental signatures.

Figure 1

(a) Moiré pattern of twisted graphene bilayers for a twist angle of $\theta =1.8°$. Each layer has two sublattices, $A$ and $B$. The pattern indicates regions of $AA$, $AB$, and $BA$ alignment. Fourfold degenerate states are observed around the $AA$ stacking regions. (b) Twisted graphene bilayer rotated around $A$ sites ($AA$ region). At those points, the bilayer has $D_3$ symmetry, comprised of a $C_3$ rotation around the $z$ axis and a $C_2^{\prime }$ rotation around the $y$ axis (dashed line) in between two layers. Red and blue dots: top and bottom layer.

Figure 2

(a) Moiré Brillouin zone of the twisted graphene system (blue hexagon), containing the $K$ and $K^{\prime }$ points at the corners. (b) Flat bands in the moiré Brillouin zone for $\theta =1.0845°$, near the magic angle ${\theta }_0\sim 1.1°$. The ${\mathrm{\Gamma }}^{\prime }$ point is at the center of the Moire Brillouin zone. $M^{\prime }$ is the midpoint between $K$ and $K^{\prime }$ points.

Figure 3

Wannier wave function in the moiré superlattice. Amplitude $|{\widehat{W}}_{+}(\mathbf{r}-{\mathbf{R}}_j)|=|{\widehat{W}}_{-}(\mathbf{r}-{\mathbf{R}}_j)|$ of the orbitals centered around (a) $j\in AB$ sites, and (b) $j\in BA$ sites, showing three sharp peaks. The orbitals ${\widehat{W}}_{\alpha }$ have twofold degeneracy per site, and are eigenstates of the $C_3$ rotation operator with eigenvalues $\epsilon$ ($\alpha =+$), and ${\epsilon }^{*}$ ($\alpha =-$). (c) Sketch of the orbitals in the moiré unit cell (green line). Orange: AB centered Wannier orbitals. Blue: $BA$ centered ones. The gray dotted line indicates the honeycomb superlattice formed by the center of the orbitals. Their unusual three peak structure indicates strong overlap between superlattice sites, favoring ferromagnetic ordering at zero temperature.