Model for the metal-insulator transition in graphene superlattices and beyond

We propose a two-orbital Hubbard model on an emergent honeycomb lattice to describe the low-energy physics of twisted bilayer graphene. Our model provides a theoretical basis for studying metal-insulator transition, Landau level degeneracy lifting, and unconventional superconductivity that are recently observed.

Figure 1

Atomic structure and tight-binding model in twisted bilayer graphene. In this figure, rotation angle $\theta =6.{01}^{\circ }$. Blue and red little dots represent the carbon atoms of bottom and top layers, respectively. Green and blue giant dots represent AB and BA spots, which form a honeycomb lattice, and AA spots lie in hexagon centers of the honeycomb lattice. At each giant dot (AB/BA spots), there reside two degenerate orbitals with $p_{x,y}$ symmetries under on-site threefold rotation.

Figure 2

The strategy and organization of this work.

Figure 3

Twisted bilayer graphene with rotation angle $\theta =21.8^{\circ }$. (a) Atomic structure. Blue and green lines represent the lattices of bottom and top layers, respectively. (b) Mini Brillouin zone (MBZ). Blue and green large hexagons correspond to the first Brillouin zone of bottom and top layers, respectively, and thick small-hexagon to the MBZ. In MBZ, open and filled circles are two inequivalent $K$ points, red dots are equivalent points of $\mathrm{\Gamma }$ point, and blue dots are equivalent points of $M$ point.

Figure 4

Real space probability distributions $|{\mathrm{\Psi }}_e^{\mathrm{\Gamma }}{\left(\mathit{x}\right)|}^2={\left|{\mathrm{\Psi }}_h^{\mathrm{\Gamma }}\left(\mathit{x}\right)\right|}^2$ of lowest eigenstates at $\mathrm{\Gamma }$ point with angular momentum $L_z=+1$. Eigenenergies are shown in (a) in unit of $2v_F/\left(\sqrt{3}a\right)$, where the lowest energies are in red. $|{\mathrm{\Psi }}_{e,h}^{\mathrm{\Gamma }}{\left(\mathit{x}\right)|}^2$ in layers 1 and 2 are shown in (b) and (c), respectively. The hexagon is the unit cell of graphene superlattice and $O$ is origin. Details can be found in the Supplemental Material [24].

Figure 5

(a) Hoppings between the nearest neighbors $t_1,t_1^{\prime }$ and fifth-nearest neighbors $t_2,t_2^{\prime }$, as shown in Eqs. (17, 18, 19). (b) Real space representation of Eq. (18) in the green sublattice. Along arrow directions the hopping terms are $\mp i{t}_2^{\prime }$ for orbital $c_{\pm }$. Here we consider hopping terms associated with the central site as an example, and hopping terms associated with other sites can be obtained by lattice translation. The blue sublattice has the same hopping pattern. (c) Band structure of the tight-binding model $H_{tb}$ with $t_1=1,t_1^{\prime }=0,t_2=0.1,t_2^{\prime }=0.4,\mu =0$. (d) Band structure with the same parameters in (c) except $t_1^{\prime }=0.1$.