Fermi level quantum numbers and secondary gap of conducting carbon nanotubes

For the single-wall carbon nanotubes conducting in the simplest tight binding model, the complete set of line group symmetry based quantum numbers for the bands crossing at Fermi level are given. Besides linear (k), helical (k̃ and angular momenta, emerging from roto-translational symmetries, the parities of U axis and (in the zig-zag and armchair cases only) mirror planes appear in the assignation. The helical and angular momentum quantum numbers of the crossing bands never vanishes, what supports proposed chirality of currents. Except for the armchair tubes, the crossing bands have the same quantum numbers and, according to the noncrossing rule, a secondary gap arises, as it is shown by the accurate tight-binding calculation. In the armchair case the different vertical mirror parity of the crossing bands provides substantial conductivity, though kF is slightly decreased.