  We show theoretically that an undoped carbon nanotube might be an excitonic
insulator---the long-sought phase of matter proposed by Keldysh, Kohn and
others fifty years ago. We predict that the condensation of triplet excitons,
driven by intervalley exchange interaction, spontaneously occurs at equilibrium
if the tube radius is sufficiently small. The signatures of exciton
condensation are its sizeable contributions to both the energy gap and the
magnetic moment per electron. The increase of the gap might have already been
measured, albeit with a different explanation [Deshpande et al., Science 323,
106 (2009)]. The enhancement of the quasiparticle magnetic moment is a
pair-breaking effect that counteracts the weak paramagnetism of the
ground-state condensate of excitons. This property could rationalize the
anomalous magnitude of magnetic moments recently observed in different devices
close to charge neutrality.
