Multiple topological transitions in twisted bilayer graphene near the first magic angle

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. [PNAS 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 moir\'e 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 nodes at either $\text{K}$, $\text{K}^\prime$, or $\Gamma$ points, or along the high symmetry lines in the moir\'e Brillouin zone. We trace the evolution of the nodes and find that there are several processes transporting them from $\Gamma$ to $\text{K}$ and $\text{K}^\prime$. At the $\Gamma$ 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 quasi-momentum) near the $\Gamma$ point with the full symmetries of the CM, which we argue is the minimal model that explains the motion of the Dirac nodes around $\Gamma$ 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.