JETP Letters: issues online

When electron states in carbon nanotubes are characterized by two-dimensional wave vectors with the components K₁ and K₂ along the nanotube circumference and cylindrical axis, respectively, then two such vectors symmetric about a M-point in the reciprocal space of graphene are shown to be related by the time-reversal operation. To each carbon nanotube there correspond five relevant M-points with the following coordinates: $K_1^{(1)}={\cal N}/2R$ , K₂⁽¹⁾=0; $K_1^{(2)}={\cal M}/2R$ , K₂⁽²⁾=-π/T; $K_1^{(3)}=(2 {\cal N} -{\cal M})/2R$ , K₂⁽³⁾=π/T; $K_1^{(4)}=({\cal M} +{\cal N})/2R$ , K₂⁽⁴⁾=-π/T, and $K_1^{(5)}=({\cal N} -{\cal M})/2R$ , K₂⁽⁵⁾=π/T, where ${\cal N}$ and ${\cal M}$ are the integers relating the chiral, C_h, symmetry, R, and translational, T, vectors of the nanotube by ${\cal N} {\bf R}={\bf C}_h + {\cal M} {\bf T}$ , T=| T|, and R is the nanotube radius. We show that the states at the edges of the one-dimensional Brillouin zone which are symmetric about the M-points with $K_2=\pm \pi/T$ are degenerate due to the time-reversal symmetry.