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Incommensurate twisted bilayer graphene: Emerging quasiperiodicity and stability

Ian Jauslin*

Vieri Mastropietro†

  • *Contact author: ian.jauslin@rutgers.edu
  • †Contact author: vieri.mastropietro@uniroma1.it
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Phys. Rev. B 113, 115142 – Published 19 March, 2026

DOI: https://doi.org/10.1103/t4j9-3c26

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 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.

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Phys. Rev. B 113, 115142– Published 19 March, 2026
  • Received 23 October 2025
  • Revised 19 February 2026
  • Accepted 25 February 2026
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