Topological classification of the single-wall carbon nanotube

The single-wall carbon nanotube (SWNT) can be a one-dimensional topological insulator, which is characterized by a $\mathbb{Z}$-topological invariant, winding number. Using the analytical expression for the winding number, we classify the topology for all possible chiralities of SWNTs in the absence and presence of a magnetic field, which belongs to the topological categories of BDI and AIII, respectively. We find that the majority of SWNTs are nontrivial topological insulators in the absence of a magnetic field. In addition, the topological phase transition takes place when the band gap is closed by applying a magnetic field along the tube axis, in all the SWNTs except armchair nanotubes. The winding number determines the number of edge states localized at the tube ends by the bulk-edge correspondence, the proof of which is given for SWNTs in general. This enables the identification of the topology in experiments.

Figure 1

(a) The mapping of the $(n,m)$-SWNT to a graphene sheet. The chiral vector, ${\mathit{C}}_{\mathrm{h}}=n{\mathit{a}}_1+m{\mathit{a}}_2$, indicates the circumference of the tube with ${\mathit{a}}_1$ and ${\mathit{a}}_2$ being the primitive lattice vectors of graphene. The three vectors ${\mathbf{\Delta }}_j(j=1,2,3)$ connect the nearest-neighbor atoms. The $d$-fold symmetry around the tube axis (helical symmetry) corresponds to the translational symmetry of ${\mathit{C}}_{\text{h}}/d(\mathit{R}=p{\mathit{a}}_1+q{\mathit{a}}_2)$, where $d=gcd(n,m)$ and integers $p$ and $q$ are given by Eq. (3). This figure shows the case of $(n,m)=(6,3)$ with $d=3,p=1$, and $q=0$. (b) A 1D lattice model in which $A$ and $B$ lattice sites are aligned in the axial direction.

Figure 2

$f_{\mu }\left(k\right)$ on the complex plane for the (6, 4)-SWNT with (a) $\mu =0$ and (b) $\mu =1$. $d=2$ and we choose $p=q=-1$. It draws a flower-shaped trajectory centered at $z=1$, with petal $j=1$ to 5. The distance from $z=1$ takes its maximum, 2, when the argument measured from $z=1$ is ${\mathrm{\Phi }}_j$ in Eq. (17), whereas it is 1 for ${\mathrm{\Phi }}_j\pm \mathrm{\Delta }/2$.

Figure 3

(a) Number of edge states in the absence of a magnetic field $\left(B=0\right)$ and (b) critical magnetic field $B^{*}$ of topological phase transition at which the number of edge states changes discontinuously. A hexagon from the leftmost one indicates the chiral vector ${\mathit{C}}_{\mathrm{h}}=n{\mathit{a}}_1+m{\mathit{a}}_2$, where ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are the primitive lattice vectors of graphene, shown in Fig. 1.

Figure 4

(a) An extension of Fig. 1 to include the hopping to the second-nearest-neighbor atoms. The three vectors ${\mathbf{\Delta }}_j(j=1,2,3)$ connect the nearest-neighbor atoms, whereas the six vectors ${\mathbf{\Delta }}_j^{\left(2\right)}(j=1,2,...,6)$ connect the second-nearest-neighbor ones. (b) An extended 1D lattice model to describe the metallic SWNTs.

Figure 5

(a) $f_{\mu ,s}\left(k\right)$ on the complex plane for $\mu =0$ in a metallic SWNT with chirality $(7,1)$. $d=1,p=1$, and $q=0$. Petals $j=3$ and 5 pass the origin in the absence of curvature effects, whereas petal $j=4$ rounds the origin. (b) $f_{\mu ,s}\left(k\right)$ around the origin on the complex plane for $\mu ={\mu }_{+}$ which passes the $K$ point, in metallic SWNTs with chiral angle $\theta$. $r=(\mathrm{\Delta }{k}_c+s\mathrm{\Delta }{k}_{\mathrm{SO}}+\mathrm{\Delta }{k}_{\varphi })(\sqrt{3}a/2)>0$. The dotted line intersects the real axis orthogonally at $r$. The trajectory for a zigzag SWNT $\left(\theta =0\right)$ is obtained by rotating it by $-2\pi /3$ around the origin (blue line), whereas that for an armchair SWNT $\left(\theta =\pi /6\right)$ is obtained by rotating it by $-\pi /2$ (orange line). Therefore the intercept of the real axis is always negative (pink segment) for SWNTs of $0\le \theta <\pi /6$.

Figure 6

Zero-energy modes at $B$ sublattice in the $(6,4)$-SWNT on the complex plane of ${\lambda }^{\prime }$ in Eq. (B5). Circles and squares correspond to modes of $\mu =0$ and 1, respectively. The solid line shows Eq. (B6), whereas the broken line shows a unit circle centered at the origin. They are crossed at ${\lambda }_{\pm }^{\prime }=e^{i2\pi /3}$.