Recent experiments have observed strongly correlated physics in twisted bilayer graphene (TBG) at very small angles, along with nearly flat electron bands at certain fillings. A good starting point in understanding the physics is a continuum model (CM) proposed by Lopes dos Santos et al. [Phys. Rev. Lett. 99, 256802 (2007)] and Bistritzer et al. [Proc. Natl. Acad. Sci. USA 108, 12233 (2011)] for TBG at small twist angles, which successfully predicts the bandwidth reduction of the middle two bands of TBG near the first magic angle ${\theta }_0=1.{05}^{\circ }$. In this paper, we analyze the symmetries of the CM and investigate the low energy flat band structure in the entire moiré Brillouin zone near ${\theta }_0$. Instead of observing flat bands at only one “magic” angle, we notice that the bands remain almost flat within a small range around ${\theta }_0$, where multiple topological transitions occur. The topological transitions are caused by the creation and annihilation of Dirac points at either $\text{K},{\text{K}}^{\prime }$, or $\mathrm{\Gamma }$ points or along the high symmetry lines in the moiré Brillouin zone. We trace the evolution of the Dirac points, which are very sensitive to the twist angle, and find that there are several processes transporting Dirac points from $\mathrm{\Gamma }$ to K and ${\text{K}}^{\prime }$. At the $\mathrm{\Gamma }$ point, the lowest energy levels of the CM are doubly degenerate for some range of twisting angle around ${\theta }_0$, suggesting that the physics is not described by any two-band model. Based on this observation, we propose an effective six-band model (up to second order in quasimomentum) near the $\mathrm{\Gamma }$ point with the full symmetries of the CM, which we argue is the minimal model that explains the motion of the Dirac points around $\mathrm{\Gamma }$ as the twist angle is varied. By fitting the coefficients from the numerical results, we show that this six-band model captures the important physics over a wide range of angles near the first “magic” angle.

• Received 8 August 2018

DOI:https://doi.org/10.1103/PhysRevB.99.035111