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arXiv · 2510.12918

Incommensurate Twisted Bilayer Graphene: emerging quasi-periodicity and stability

Ian Jauslin·Vieri Mastropietro
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Abstract

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 quasi-periodic 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 non-perturbative effects. The result provides a partial justification of the effective continuum description of TBG in which such large-momentum interlayer hopping processes are neglected.

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Ian Jauslin, Vieri Mastropietro. 2025-10-14. Incommensurate Twisted Bilayer Graphene: emerging quasi-periodicity and stability. https://doi.org/10.1103/t4j9-3c26

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