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We consider a lattice model of twisted bilayer graphene (TBG) for incommensurate twist angles, focusing on the role of large-momentum-transfer umklapp terms. These terms, which nearly connect the Fermi points of different layers, are typically neglected in effective continuum descriptions but could, in principle, destroy the Dirac cones; they are indeed closely analogous to those appearing in fermions within quasiperiodic potentials, where they play a crucial role. We prove that, for small but finite interlayer coupling, the semimetallic phase is stable provided the angles belong to a fractal set of large measure (which decreases with the hopping strength) characterized by a number-theoretic Diophantine condition. In particular, this set excludes the (zero measure) commensurate angles. Our method combines a renormalization group (RG) analysis of the imaginary-time, zero-temperature Green's functions, with number theoretic properties, and it is similar to the technique used in the Lindstedt series approach to Kolmogorov-Arnold-Moser (KAM) theory. The convergence of the resulting series allows us to rule out nonperturbative effects. The result provides a partial justification of the effective continuum description of TBG in which such large-momentum interlayer hopping processes are neglected.
Jauslin et al. (Wed,) studied this question.