Asymmetric velocities of Dirac particles and Vernier spectrum in metallic single-wall carbon nanotubes

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Left- and right-going Dirac particles near the $K$ (or ${K}^{\ensuremath{'}}$) point in the two-dimensional Brillouin zone have different velocities in metallic single-wall carbon nanotubes. The metallic linear energy band is tilted by a curvature effect of the nanotube. The difference of the velocities is calculated by a numerical calculation and by an effective mass theory. Tilting as well as warping of the Dirac cones explains the diameter and chirality dependences of the velocities quantitatively. The asymmetric velocities give a ``vernier'' spectrum of an electron in a nanotube quantum dot with a sharp confinement potential containing two different sequences of levels, which is relevant to the lifting of the fourfold degeneracy of the energy levels.