Topologically Protected Correlated End Spin Formation in Carbon Nanotubes

For most chiralities, semiconducting nanotubes display topologically protected end states of multiple degeneracies. We demonstrate using density matrix renormalization group based quantum chemistry tools that the presence of Coulomb interactions induces the formation of robust end spins. These are the close analogs of ferromagnetic edge states emerging in graphene nanoribbons. The interaction between the two ends is sensitive to the length of the nanotube, its dielectric constant, and the size of the end spins: for $S=1/2$ end spins, their interaction is antiferromagnetic, while for $S>1/2$, it changes from antiferromagnetic to ferromagnetic as the nanotube length increases. The interaction between end spins can be controlled by changing the dielectric constant of the environment, thereby providing a possible platform for two-spin quantum manipulations.

Figure 1

(a) Topologically protected spins are formed at both edges of most semiconducting nanotubes. (b) Band structure of a semiconducting nanotube in the absence of interactions. Topological end states (red lines) appear in the gap. (c) Many-body spectrum at finite interaction. For ferromagnetic end spin coupling, the ground state has a total spin $S_T$ equal to the number of edge states $N_{\mathrm{edge}}$. Spin excitations appear at low energies due to coupling between end spins.

Figure 2

(a) Mapping of an infinite carbon nanotube to an effective 1D ladder-like lattice model with $d$ decoupled chains for a chirality $\chi =(6,2)$ and $d=2$. Arrows indicate hoppings between carbon atoms. (b) The number of edge states, $N_{\mathrm{edge}}$, as a function of the chirality $\chi =(n,m)$. (c) Band structure and the corresponding winding numbers for a (6,2) nanotube.

Figure 3

Effective exchange interaction $J_{\mathrm{eff}}$ between the localized spins at the two ends of the nanotube as function of its length. When $N_{\mathrm{edge}}=1,J_{\mathrm{eff}}$ is always positive, indicating an antiferromagnetic exchange, while for $N_{\mathrm{edge}}\ge 2$ an antiferromagnetic to ferromagnetic transition occurs. As the inset shows, for appropriate nanotube length, the sign of the interaction can be changed by changing the dielectric constant of the environment.

Figure 4

Extension of the wave function of the additional spectrum as a function of $\epsilon$ in a (7, 5) nanotube of length $L=30\mathrm{nm}$ with $N_{\mathrm{edge}}=1$. For $\epsilon <{\epsilon }_C\approx 2.5$, the added charge delocalizes along the nanotube, while for $\epsilon >{\epsilon }_C$ the charge is added to the topological quantum dots (it is delocalized between them in the ground state). As the coloring indicates, the two end states live on different sublattices. The localization length of the added electron diverges as $\epsilon$ approaches ${\epsilon }_C$.