Twisted bilayers of graphene have shown strikingly rich phenomenology while also posing significant theoretical challenges. As a consequence, many numerically discovered phenomena have defied analytical understanding to date. A famous such example is the set of so-called magic twist angles, where the velocity of charge carriers vanishes. They have been observed numerically in a nonperturbative regime which has not been accessed analytically. Here we identify an analytically solvable, nonperturbative limit of twisted graphene bilayers in the presence of a perpendicular electric field. We find phenomena very similar to the ones previously studied numerically, but our theory now provides analytical insights and it affords exact predictions. For example, we demonstrate the emergence of myriad Lifshitz transitions--abrupt discontinuities in physical observables--and find a regime in which a universal set of these transitions appear. This reveals a deep connection between seemingly disparate systems--resonant graphene bilayers with vastly different twist angles. Most prominently, we demonstrate that graphene bilayers in our resonant limit also have "magic twist angles" where charge carriers have zero velocity and infinite mass at zero Dirac wavevector. We prove that there are infinitely many such angles and we provide exact, analytical expressions for them.
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MATIN Development Team