Ultraclean, undoped carbon nanotubes are observed to be always insulating, even when the gap predicted by band theory is zero: the residual band gap is then thought to have a many-body origin. Here we theoretically show that the correlated insulator is excitonic in all stable narrow-gap tubes irrespective of their size, thus extending our previous claim, limited to gapless (armchair) tubes [D. Varsano, et al., Nat. Commun. 8, 1461 (2017)]. We derive the scaling law of the exciton binding energy with the tube radius and chirality, and compute self-consistently the fundamental transport gap of the excitonic insulator, by enhancing the two-band model with an accurate treatment of screening validated from first principles. Our findings point to the broader connection between the exciton length scale, dictated by structure, and the stability of the excitonic phase.

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• Excitons
• First-principles calculations
• Nanotubes
• GW method
• k dot p method