Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase

We propose that the electronic structure of twisted bilayer graphene (TBG) can be understood as Dirac fermions coupled with opposite pseudomagnetic fields generated by the moiré pattern. The two low-energy flat bands from each monolayer valley originate from the two zeroth pseudo Landau levels of Dirac fermions under such opposite effective magnetic fields, which have opposite sublattice polarizations and carry opposite Chern numbers $\pm 1$, giving rise to helical edge states in the gaps below and above the low-energy bulk bands near the first magic angle. We argue that small Coulomb interactions would split the eightfold degeneracy (including valley and physical spin) of these zeroth pseudo Landau levels, and may lead to insulating phases with nonvanishing Chern numbers at integer fillings. Besides, we show that all the high-energy bands below or above the flat bands are also topologically nontrivial in the sense that for each valley the sum of their Berry phases is quantized as $\pm \pi$. Such quantized Berry phases give rise to nearly flat edge states, which are dependent on truncations on the moiré length scale. Our paper provides a complete and clear picture for the electronic structure and topological properties of TBG, and has significant implications on the nature of the correlated insulating phase observed in experiments.

Figure 1

(a) Left: A top view of the moiré pattern of twisted bilayer graphene for $m=5$ ($\theta \approx 6.{01}^{\circ }$). The solid and dashed lines represent lattice truncations through the $AA$ regions and the $AB/BA$ regions, respectively. The two arrows denote the lattice vectors. Right: Illustration of atomic corrugations. (b) The Brillouin zones of the top monolayer, bottom monolayer, and the moireé supercell are plotted in red, blue, and black lines, respectively.

Figure 2

The band structure of twisted bilayer graphene at $\theta \approx 1.{08}^{\circ }$ in a ribbon geometry with the open boundary condition. The red and black lines represent the edge states from the two edges of the ribbon and blue lines represent the bulk states.

Figure 3

(a) The Wilson loops of the four low-energy bands of twisted bilayer graphene at $m=15$ ($\theta \approx 2.{13}^{\circ }$). The blue circles, squares, diamonds, and plus signs represent the microscopic configurations with $D_6$ symmetry, $D_3$ symmetry, $D_3$ symmetry with staggered sublattice potential $V_s=0.04\mathrm{eV}$, and $D_3$ symmetry with vertical electric field $V_e=0.04\mathrm{eV}$, respectively. (b) A schematic illustration of the different microscopic configurations.

Figure 4

(a) Bulk band structure of TBG at $(m,r)=(30,1)$ calculated from the microscopic tight-binding model (blue) and the continuum model (red), including effects of atomic corrugations. (b) The total Berry phases of all the bands below (above) the four flat bands at $(m,r)=(30,1)$ are denoted by ${\beta }_o\left(k_2\right) \left[{\beta }_u\left(k_2\right)\right]$, and the Wilson loops of the four flat bands at $(m,r)=(30,1)$ denoted by $w\left(k_2\right)$.

Figure 5

Edge states in twist bilayer graphene at $m=15$: (a), (b) the edge states below the four low-energy bands and (c), (d) above the four low-energy bands. (a), (c) The system truncated through the $\mathit{AA}$ region; (b), (d), the system truncated through the $\mathit{BA}$ and $\mathit{AB}$ regions. The inset shows the bulk band structure for $(m,r)=(15,1)$, where energy windows for the edge states are marked in light blue shadow.

Figure 6

(a) Color map of the indirect gap above the four flat bands in TBG, in units of eV. The horizontal axis is the corrugation strength parameterized by $d_1$ [see Eq. (1)], and the vertical axis is the integer $m$ characterizing the rotation angle. (b) Color map of the direct gap above the four flat bands at ${\mathrm{\Gamma }}_s$. The dashed black lines in (a) and (b) mark the actual atomic corrugation strength.