  Allotropes of carbon, including one-dimensional carbon nanotubes and
two-dimensional graphene sheets, continue to draw attention as promising
platforms for probing the physics of electrons in lower dimensions. Recent
research has shown that the electronic properties of graphene multilayers are
exquisitely sensitive to the relative orientation between sheets, and in the
bilayer case exhibit strong electronic correlations when close to a magic twist
angle. Here, we investigate the electronic properties of a carbon nanotube
deposited on a graphene sheet by deriving a low-energy theory that accounts
both for rotations and rigid displacements of the nanotube with respect to the
underlying graphene layer. We show that this heterostructure is described by a
translationally invariant, a periodic or a quasi-periodic Hamiltonian,
depending on the orientation and the chirality of the nanotube. Furthermore, we
find that, even for a vanishing twist angle, rigid displacements of a nanotube
with respect to a graphene substrate can alter its electronic structure
qualitatively. Our results identify a promising new direction for strong
correlation physics in low dimensions.
