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 ${\mathbf M}$-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 ${\mathbf M}$-points with the following coordinates: $K_1^{(1)}=\mathcal N/2R$, $K_2^{(1)}=0$; $K_1^{(2)}=\mathcal M/2R$, $K_2^{(2)}=-\pi/T$; $K_1^{(3)}=(2\mathcal N-\mathcal M)/2R$, $K_2^{(3)}=\pi/T$; $K_1^{(4)}=(\mathcal M+\mathcal N)/2R$, $K_2^{(4)}=-\pi/T$, and $K_1^{(5)}=(\mathcal N-\mathcal M)/2R$, $K_2^{(5)}=\pi/T$, where $\mathcal N$ and $\mathcal M$ are the integers relating the chiral, $\mathbf C_h$, symmetry, $\mathbf R$, and translational, $\mathbf T$, vectors of the nanotube by $\mathcal N\, \mathbf R=\mathbf C_h+\mathcal M\,\mathbf T$, $T=|\mathbf T|$, 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 $\mathbf M$-points with $K_2=\pm \pi/T$ are degenerate due to the time-reversal symmetry.