Multiple topological transitions in twisted bilayer graphene near the first magic angle (1808.01568v2)

Abstract: 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.