A Spectral Gap for Complex Magic Angles

Twisting two layers of graphene by certain "magic" angles produces flat electronic bands linked to effects like superconductivity. In the standard idealized model, these angles correspond to a set of mostly complex-valued parameters. We prove that the first real magic parameter, which gives the most important magic angle, is cleanly separated in size from all the others. The proof uses the model's rotational symmetry and computer-assisted checks done in exact arithmetic. As a result, we resolve an open problem posed by Zworski and obtain a rigorous method for computing the first magic angle to any desired accuracy.