Topological Phase Transition in Metallic Single-Wall Carbon Nanotube

The topological phase transition is theoretically studied in a metallic single-wall carbon nanotube (SWNT) by applying a magnetic field $B$ parallel to the tube. The $\mathbb{Z}$ topological invariant, winding number, is changed discontinuously when a small band gap is closed at a critical value of $B$, which can be observed as a change in the number of edge states owing to the bulk-edge correspondence. This is confirmed by numerical calculations for finite SWNTs of $\sim$ 1 $\mu$m length, using a one-dimensional lattice model to effectively describe the mixing between $\sigma$ and $\pi$ orbitals and spin-orbit interaction, which are relevant to the formation of the band gap in metallic SWNTs.

ation of topology [12,13]. Here, we examine the winding number and edge states in metallic SWNTs of general types. For this purpose, we make a 1D lattice model to effectively describe the σ-π mixing and SO interaction, which are relevant to the formation of a small band gap, as an extension of the effective model proposed in Ref. [9]. Our model is applicable to the metallic SWNTs of length L NT = a few µm, which enables the calculation of edge states with a decay length of 10 2 − 10 3 nm. This is advantageous compared with a tight-binding model with all σ and π orbitals at each carbon atom [14], which can be used for L NT 10 2 nm by moderate computers.
1D lattice model-We construct an effective 1D lattice model for electrons around the Fermi level E F in a metal-arXiv:1610.05034v2 [cond-mat.mes-hall] 5 Dec 2016 lic SWNT, starting from the Hamiltonian of k · p theory in the continuum limit [15]. For (n, m)-SWNT, the diameter is given by d t = |C h |/π = a √ n 2 + nm + m 2 /π with the lattice constant a = 0.246 nm in graphene. The chiral angle θ is defined as the angle between C h and a 1 . For 0 ≤ m ≤ n, 0 ≤ θ ≤ π/6 with θ = 0 and π/6 for types of zigzag and armchair, respectively. In a magnetic field B in the axial direction, the Hamiltonian in the vicinity of K and K points reads whereσ x ,σ y , and1 are the Pauli matrices and identity operator, respectively, in the sublattice space of σ = A or B. s = ±1 is the spin in the axial direction, whereas τ = ±1 is the pseudo-spin to represent the K or K valley. v F = 8.32 × 10 5 m/s is the Fermi velocity and g s 2 is the spin g-factor. k c and k z are the circumference and axial components of wavenumber measured from K or K points, respectively: k c is discretized in units of 2π/|C h | while k z is continuous.
In H s,τ (k), the hybridization between π and σ orbitals results in the shift of Dirac points from K or K points, with β = 0.0436 nm and ζ = −0.185 nm. ∆k c opens a small gap E g between the conduction and valence bands around K and K points, except θ = π/6 (armchair). The curvature-enhanced SO interaction yields with α 1 = 8.8×10 −5 meV −1 and α 2 = −0.045 nm. V so = 6 meV is the SO interaction for 2p orbitals in carbon atom. ∆k so gives a correction to E g . The Aharonov-Bohm (AB) phase by magnetic field B appears as The band gap is closed at B * when τ ∆k c +s∆k so +∆k φ = 0. The last term in H s,τ (k) yields the energy shift from E F = 0, which is assumed to be small compared with the band gap except in the vicinity of B = B * . We construct a 2D lattice model to reproduce H s,τ (k) around the Dirac points [16]. The model involves the hoppings not only to the first nearest-neighbor atoms but also to the second ones. The former connects A and B atoms, which are depicted by three vectors ∆ (1) j (j = 1, 2, 3) in Fig. 1(a), whereas the latter connects the same species indicated by six vectors ∆ (2) j (j = 1, 2, · · · , 6).
Finally, the effective 1D lattice model is derived from the 2D lattice model, along the lines of Ref. [9], to utilize the helical-angular construction [17]. (n, m)-SWNT has the d-fold symmetry around the tube axis, where d = gcd(n, m) is the greatest common divisor of n and m. It also has the helical symmetry with translation a z = √ 3da 2 /(2πd t ) along the tube axis with a rotation around it [see Fig. 1(a)]. Thanks to these symmetries, the Hamiltonian is block-diagonalized into the subspace of orbital angular momentum µ = 0, 1, 2, . . . , d − 1 and spin s = ±1, as H = c µ,s σ, This is a 1D lattice model in which A and B atoms are aligned in the axial direction with the lattice constant a z . c µ,s σ, is the field operator of an electron with angular momentum µ and spin s at atom σ of site index . The hopping to the first [second] nearest-neighbor atoms in Fig. 1(a) gives rise to the hopping to the sites separated by ∆ (2) j ], as illustrated in Fig. 1(b). The hopping integral γ (1) s,j is given by where φ s,j stems from the SO interaction as For the bulk states, the Fourier transformation of H µ,s yields a subband labeled by µ and s in the first Brillouin zone, −π/a z ≤ k < π/a z . It is expressed as The system is an insulator when | (0) µ,s (k)| < |f µ,s (k)| in the whole Brillouin zone. Then positive and negative µ,s (k) form the conduction and valence bands, respectively. The edge states are obtained by the diagonalization of H µ,s for a finite system of N sites Our effective 1D lattice model is justified as follows. First, the bulk states in Eq. (8) coincide with those calculated from the Hamiltonian in Eq. (1) around K and K points. Second, we compare the bulk and edge states obtained by our model and those by the tight-binding model with all σ and π orbitals [14] for a system of 50 nm [16]. They are in good agreement.
Winding number-The phase of f µ,s (k) in Eq. (10) determines the winding number, for subband (µ, s) [6][7][8][9]. This is meaningful for an insulator only [20]. The winding number is related to the number of edge states, N edge , by the bulk-edge correspondence, when the tube is cut by a broken line in Fig. 1(a) [9]. The case of the other boundaries is discussed later. Although the energy levels of edge states are deviated from E F = 0, they are within the band gap as long as the winding number is well-defined.
Numerical results-Now we calculate the winding number and edge states in metallic SWNTs, using the effective 1D lattice model for finite systems of N sites. The metallic SWNTs with mod (2n + m, 3) = 0 are categorized to metal-1 or metal-2 according to the angular momenta of the Dirac points when the tube curvature and SO interaction are disregarded. In general, the subband of Metal-1 corresponds to the latter while metal-2 corresponds to the former. The wavenumber at the Dirac point is given by k K/K = ±(2π/3a z ) mod(2p + q, 3), where p and q are integers satisfying mp − nq = d. [9].
As an example of metal-1, Fig. 2 presents the calculated results for (15, 12)-SWNT. For d = gcd(15, 12) = 3, we have six subbands with µ = 0, 1, 2 and s = ±1. µ K = 2 and µ K = 1 while k K = k K = 0. Figure 2 σ = A, B), for the upper state of (e) µ K and (f) µ K with s = +1. The solid and dotted lines indicate the amplitudes at A and B atoms, respectively. |ψσ(z)| 2 is normalized by its maximum value at a sharp peak around an edge (not seen in this length scale). B = 0 (they seem spin-degenerate in this energy scale). A small band gap around k K = k K = 0 is given by E g = 2.71 ± s 0.32 meV for µ K /µ K and spin s = ±1. The band gap of subband (µ K , s = ±1) is closed at B = B * = 3.55 + s 0.41 T and reopened at B > B * while that of (µ K , s = ±1) is always finite, as shown in panel (b). Figure 2(c) depicts arg f µ,s as a function of wavenumber k for µ K and µ K , at B = 0 and 7.10 T. At B = 0, the behavior of arg f µ,s is almost the same for µ K and µ K . They rapidly but continuously change around k = 0 and increase by 2π as k runs through the whole Brillouin zone. Hence Eq. (11) yields w µ,s = 1. At 7.10 T, on the other hand, arg f µ,s behaves differently for µ K and µ K : w µ,s = 0 for µ K and 1 for µ K . This clearly indicates a topological phase transition at B = B * for subbands (µ K , s = ±1). For subbands (µ = 0, s = ±1), we always find w µ,s = 0 (not shown).
Accompanied by this phase transition, the number of edge states changes from 8 to 4 at B = B * according to Eq. (12). Four states of µ K and s = ±1 are changed from edge to bulk states. Figure 2(e) depicts the probability amplitude of the upper state of µ K and s = 1 along the tube axis. It is localized at the edges (large amplitude at A sites around an edge and at B sites around the other edge) at B = 0, whereas it is delocalized through the tube at B = 7.10 T. The states of µ K remain localized at the edges, as shown in Fig. 2(f).
For metal-2, numerical results for (13, 10)-SWNT are given in Fig. 3. For d = gcd(13, 10) = 1, we have two subbands of µ = 0 and s = ±1. µ K = µ K = 0 while k K = −k K = −2π/3. The band gap is given by E g = 4.36±s 0.37 meV around k K and k K at B = 0. The band gap around k K is closed at B = B * = 6.71 + s 0.57 T while that around k K is always finite [panel (b)].
The winding number changes from two to one at B = B * , as seen in the behavior of arg f µ,s in panel (c). This topological phase transition changes the number of edge states from 8 to 4: Four of eight states within the band gap at B = 0 go out of the band gap around B = B * and the rest remain within the band gap, as shown in panel (d). The former change from edge states to bulk states around B = B * , whereas the latter are always edge states [panels (e) and (f)]. Summary and comments-We have studied the topological phase transition in metallic SWNTs by magnetic field B parallel to the tubes, using an effective 1D lattice model to describe the mixing between σ and π orbitals and spin-orbit interaction. We have demonstrated that a change in winding number is accompanied by a change in the number of edge states in (15, 12)-and (13, 10)-SWNTs, as examples of metal-1 and metal-2, respectively.
Besides these SWNTs, we have examined several metallic SWNTs of other chiralities. The topological phase transition takes place in all the SWNTs of metal-1 that we have studied. The phase transition is also commonly seen in the case of metal-2 except for SWNTs of armchair type. In armchair SWNTs, the band gap is opened by the SO interaction only since ∆k c = 0 in Eq.
(2). The closing of the band gap is not accompanied by the change in winding number in such SWNTs, which will be explained elsewhere.
A comment should be made on the boundary condition, which is important for the edge states in 1D topological insulators. Our numerical calculations have been performed for finite systems in which a SWNT is cut by a broken line in Fig. 1(a) (angular momentum µ is a good quantum number in this case). This is a "minimal boundary edge," where every atom at the ends has just one dangling bond [21]. The relation in Eq. (12) holds only for such edges. Some other boundary conditions result in different numbers of edge states, as discussed in Ref. [9]. Even in these cases, we find the change in the number of edge states at the topological phase transition. We speculate that this statement is true in general with any boundaries and also in the presence of impurities inside the bulk. However, this problem would require further studies. Note that our 1D lattice model is useful for numerical studies on this problem.
[20] Mathematically, the topological argument is valid only when the sublattice symmetry holds, that is, we obtain H s,τ AB . We then have A 1 = 2/3. In a similar manner, we obtain A 2 = 1/ 3 √ 3 . ξ ∼ 0.4 a approximately reproduces A 1 and A 2 simultaneously. Next, we consider the magnetic field B. The spin-Zeeman term, s,σ,rσ 1 2 g s µ B Bs c s † rσ c s rσ , is added. For the orbital magnetism, we take into account the Aharonov-Bohm (AB) effect only, neglecting a small deformation of atomic orbitals. When an electron circulates around the tube axis, it acquires the AB phase 2πBS(−e)/h = −πeBd 2 t /(4 ). The hopping from a B atom to a nearest A atom corresponds to the rotation of −∆ (1) j ·C h /|C h | 2 = −a CC cos φ (1) j / πd t around the axis. Therefore, we substitute γ → γ exp −i∆k φ a CC cos φ (1) j in Eq. (17) with ∆k φ = −eBd t /(4 ). Then, Finally, we obtain the Hamiltonian, Note that for B = 0, SWNTs have the C 2 symmetry. In addition, zigzag and armchair SWNTs have two mirror planes which include the tube axis and are perpendicular to it.
where Q 1 = −qb 1 + pb 2 and Q 2 = (m/d)b 1 + (n/d)b 2 are the reciprocal lattice vectors conjugate to C h /d and R, respectively. µ and k are the orbital angular momentum and wavenumber in the axial direction. By performing the partial Fourier transformation along Q 1 direction to a linear combination of atomic orbitals, we obtain eigenstates of angular momentum, Substituting Eq. (25) into Eq. (23), we obtain the 1D lattice model in Eq. (5) in the main material.
Comparison between the effective one-dimensional model and extended tight-binding model We compare the calculated results for a metallic SWNT by the effective 1D lattice model and extended tight-binding model (ETB) with all σ and π orbitals at each carbon atom. 3) In ETB, hopping and overlap integrals are taken into account between atoms within the distance of 10 a B , where a B is the Bohr radius. Their values are evaluated by the ab initio calculations. 4) The optimization of atomic positions is also performed. Each dangling bond at the ends is terminated by a hydrogen atom. Here, we consider (7, 4)-SWNT, which is a metal-2 nanotube with diameter d t = 0.76 nm. Figure 4(a) shows the energy eigenvalues, i , obtained by the both methods for a tube with length L NT = 50 nm and minimal boundary edges at both ends. The magnetic field is B = 10 T. Here, i is an index of energy eigenstates in ascending order, with i = 0 and 1 being the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), respectively. The origin of energy is chosen so as to ( 0 + 1 )/2 = 0. Triangles and inverted triangles indicate spin-up (s = +1) and down (s = −1) in the axial direction, respectively. Precisely speaking, an energy eigenstate is not an eigenstate of electron spin in ETB. However, the probability amplitude of s = +1 (−1) is more than 99.9999 % in spin-up (-down) states shown in Fig. 4(a). The results by the two methods are in agreement semiquantitatively. Especially, eight edge states in the energy gap are observed by the both methods. The effective 1D model overestimates the energy for the valence band electrons. This is because the overlap integrals are not taking account, which enhances the width of valence band. The spin configuration in the edge states are partially inconsistent, however these states are almost degenerate. Figures 4(b) and (c) show the probability amplitude for i = 0 and 5 states, respectively. The shapes of edge state (i = 0) in panel (b) and traveling mode (i = 5) in panel (c) show good agreement between these calculations, respectively.