Origin of Magic Angles in Twisted Bilayer Graphene

Twisted Bilayer graphene (TBG) is known to feature isolated and relatively flat bands near charge neutrality, when tuned to special magic angles. However, different criteria for the magic angle such as the vanishing of Dirac speed, minimal bandwidth or maximal band gap to higher bands typically give different results. Here we study a modified continuum model for TBG which has an infinite sequence of magic angles $\theta$ at which, we simultaneously find that (i) the Dirac speed vanishes (ii) absolutely flat bands appear at neutrality and (iii) bandgaps to the excited bands are maximized. When parameterized in terms of $\alpha\sim 1/\theta, $ they recur with the simple periodicity of $\Delta \alpha \simeq 3/2$, which, beyond the first magic angle, differs from earlier calculations. Further, in this model we prove that the vanishing of the Dirac velocity ensures the exact flatness of the band and show that the flat band wave functions are related to doubly-periodic functions composed of ratios of theta functions. Also, using perturbation theory up to $\alpha^8$ we capture important features of the first magic angle $\theta\approx1.09^{\circ}$ ($\alpha \approx 0.586$), which precisely explains the numerical results. Finally, based on our model we discuss the prospects for observing the second magic angle in TBG.