Antiferromagnetic nuclear spin helix and topological superconductivity in ${}^{13}\text{C}$ nanotubes

We investigate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting ${}^{13}\text{C}$ carbon nanotubes. Using the Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in noninteracting carbon nanotubes, and it leads to an antiferromagnetic nuclear spin helix in finite-size systems. The transition temperature reaches up to tens of mK, due to a strong boost by a positive feedback through the Overhauser field from ordered nuclear spins. Similar to GaAs nanowires, the formation of the helical nuclear spin order gaps out half of the conduction electrons, and is therefore observable as a reduction of conductance by a factor of 2 in a transport experiment. The nuclear spin helix leads to a density wave combining spin and charge degrees of freedom in the electron subsystem, resulting in synthetic spin-orbit interaction, which induces nontrivial topological phases. As a result, topological superconductivity with Majorana fermion bound states can be realized in the system in the presence of proximity-induced superconductivity without the need of fine tuning the chemical potential. We present the phase diagram as a function of system parameters, including the pairing gaps, the gap due to the nuclear spin helix, and the Zeeman field perpendicular to the helical plane.

Figure 1

Backscattering processes on the Fermi surface. $K_{\gamma }$ indicates the two Dirac points with the valley index $\gamma =\pm$. The red (green) arrows correspond to the intravalley backscattering ${\psi }_{\ell ,\gamma ,\sigma }^{†}{\psi }_{\overline{\ell },\gamma ,{\sigma }^{\prime }}$ (intervalley backscattering ${\psi }_{\ell ,\gamma ,\sigma }^{†}{\psi }_{\overline{\ell },\overline{\gamma },{\sigma }^{\prime }})$ processes. The arrows are mutually independent and the spins are not shown.

Figure 2

RKKY interaction from Eq. (37) in momentum space. The interaction has peaks at $q=\pm 2k_F$ with the width $\sim \frac{2\pi }{{\lambda }_T{k}_F}$. The parameters used here are $K_{cS}=0.2,K_{cA}=K_{sS}=K_{sA}=1$ [75, 76, 77, 78, 79, 80], $I=\frac{1}{2},A_0=6.0\mu \text{eV}$ [24, 25, 26, 51, 52, 68, 69], $v_F=8.0\times {10}^5$ m/s, $k_F=4.0\times {10}^8$ ${\text{m}}^{-1},a=2.46\text{Å}$ [38], $L=1.0\mu \text{m}$, and $N_{\perp }=12$ [23]. For the purpose of illustration, we choose an unrealistically short thermal length, corresponding to an unrealistically high temperature $T=100$ K, to demonstrate the RKKY peaks. For realistic temperatures, the peaks will be much sharper.

Figure 3

A sketch of the antiferromagnetic nuclear spin helix in the (a) original $(\widehat{x},\widehat{y},\widehat{z})$ and (b) rotated $({\widehat{e}}_j^1,{\widehat{e}}_j^2,{\widehat{e}}_j^3)$ coordinates. The black and red arrows indicate the spins on the sublattice sites A and B, respectively. For simplicity, we do not plot the actual honeycomb lattice here.

Figure 4

Magnon spectrum of the antiferromagnetic helix. The parameters used here are the same as in Fig. 2. The blue solid and red dashed lines correspond to the ${\omega }_q^{\left(1\right)}$ and ${\omega }_q^{\left(2\right)}$ energy bands, respectively. As in Fig. 2, here we use an unrealistically high temperature to illustrate the dips in the spectrum. The region of the linear dispersion is given by the width of the RKKY peaks $\sim \frac{2\pi }{{\lambda }_T{k}_F}$, which is much narrower for realistic temperatures.

Figure 5

The antiferromagnetic helix corresponds to the intravalley backscattering processes (a), gapping out half of conduction electrons (b). The up and down spins are marked in black and red colors, respectively. The chemical potential is plotted with the blue dashed lines. The green arrows describe the scattering processes. The dispersions for up and down spins are slightly shifted for clarity.

Figure 6

Exponents $g_{\mu }$ and ${\tilde{g}}_{\mu }$ as a function of the Luttinger-liquid parameter, $K_{cS}$. The blue solid and red dotted lines give the modified exponents ${\tilde{g}}_x={\tilde{g}}_y$ and ${\tilde{g}}_z$ in the presence of the feedback, respectively. The green dashed curve describes the exponent $g_{\mu }$ without the feedback. The vertical black dashed line marks the value we use to evaluate, $K_{cS}=0.2$. The other parameters used here are the same as in Fig. 2.

Figure 7

Ratio of the RKKY peak values, $R_p\equiv |{\tilde{J}}_{\alpha \beta }^x(q=2k_F)/J_{\alpha \beta }^x(q=2k_F)|$ at $T=50$ mK as a function of the Luttinger-liquid parameter, $K_{cS}$. The other parameters are the same as in Fig. 2. The black dashed line marks the value $K_{cS}=0.2$.

Figure 8

Temperature dependence of the order parameter. Top: the order parameter in the presence of the feedback, ${\tilde{m}}_{2k_F}\left(T\right)$, from Eq. (90). Bottom: the order parameter without (red dashed) and with (blue solid) the feedback from Eqs. (56) and (90), respectively, in the logarithmic scale. The parameters used here are the same as in Fig. 2.

Figure 9

RKKY peak value at zero temperature as a function of the pairing gap in the presence of the feedback. The parameters used here are the same as in Fig. 2.

Figure 10

Phase diagram on the ${\mathrm{\Delta }}_m\text{−}{\mathrm{\Delta }}_Z$ plane. The black solid curves are marked as $C_{+}^{\left(1\right)},C_{-}^{\left(1\right)},C_e^{\left(2\right)}$, and $C_i^{\left(2\right)}$, whereas the black dashed curve is marked as $C^{\left(3\right)}$. The intercepts of these curves on the axes are also labeled. The yellow shaded region corresponds to the MF regime. In the blue region the bulk spectrum is gapless.