Implications of time-reversal symmetry for band structure of single-wall carbon nanotubes

When electron states in carbon nanotubes are characterized by two-dimensional wave vectors with the components $K_1$ and $K_2$ along the nanotube circumference and cylindrical axis, respectively, then two such vectors symmetric about a ${\bf M}$-point in the reciprocal space of graphene are shown to be related by the time-reversal operation. To each nanotube there correspond five relevant ${\bf M}$-points with the following co\"ordinates: $K_1^{(1)}={\cal N}/2R$, $K_2^{(1)}=0$; $K_1^{(2)}={\cal M}/2R$, $K_2^{(2)}=-\pi/T$; $K_1^{(3)}=(2 \, {\cal N} -{\cal M})/2R$, $K_2^{(3)}=\pi/T$; $K_1^{(4)}=({\cal M} +{\cal N})/2R$, $K_2^{(4)}=-\pi/T$, and $K_1^{(5)}=({\cal N} -{\cal M})/2R$, $K_2^{(5)}=\pi/T$, where ${\cal N}$ and ${\cal M}$ are the integers relating the chiral, ${\bf C}_h$, symmetry, ${\bf R}$, and translational, ${\bf T}$, vectors of the nanotube by ${\cal N} \, {\bf R}={\bf C}_h + {\cal M} \, {\bf T}$, $T=|{\bf T}|$, 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 ${\bf M}$-points with $K_2=\pm \pi/T$ are degenerate due to the time-reversal symmetry. Explicit expressions are obtained for the co\"ordinates of the ${\bf K}$-points in the reciprocal space of graphene relevant to a given nanotube.