  The screening of Coulomb interaction controls many-body physics in carbon
nanotubes, as it tunes the range and strength of the force that acts on charge
carriers and binds electron-hole pairs into excitons. In doped tubes, the
effective Coulomb interaction drives the competition between Luttinger liquid
and Wigner crystal, whereas in undoped narrow-gap tubes it dictates the Mott or
excitonic nature of the correlated insulator observed at low temperature. Here,
by computing the dielectric function of selected narrow- and zero-gap tubes
from first principles, we show that the standard effective-mass model of
screening systematically underestimates the interaction strength at long
wavelength, hence missing the binding of low-energy excitons. The reason is
that the model critically lacks the full three-dimensional topology of the
tube, being adapted from graphene theory. As ab inito calculations are limited
to small tubes, we develop a two-band model dielectric function based on the
plane-wave expansion of Bloch states and the exact truncated Coulomb cutoff
technique. We demonstrate that our -- computationally cheap -- approach
provides the correct screening for narrow-gap tubes of any size and chirality.
A striking result is that the screened interaction remains long-ranged even in
gapless tubes, as an effect of the microscopic local fields generated by the
electrons moving on the curved tube surface. As an application, we show that
the effective electron-electron force that is felt at distances relevant to
quantum transport experiments is super Coulombic.
