Nuclear Magnetism and Electronic Order in ${}^{13}\mathrm{C}$ Nanotubes

Single wall carbon nanotubes grown entirely from ${}^{13}\mathrm{C}$ form an ideal system to study the effect of electron interaction on nuclear magnetism in one dimension. If the electrons are in the metallic, Luttinger liquid regime, we show that even a very weak hyperfine coupling to the ${}^{13}\mathrm{C}$ nuclear spins has a striking effect: The system is driven into an ordered phase, which combines electron and nuclear degrees of freedom, and which persists up into the millikelvin range. In this phase the conductance is reduced by a universal factor of 2, allowing for detection by standard transport experiments.

Figure 1

Illustration of the helical nuclear magnetism (indicated by the blue ribbon) of the single wall ${}^{13}\mathrm{C}$ nanotube (SWNT), which emerges below a critical temperature through strong renormalization of the hyperfine coupling by electron correlations. The nuclear spins (red arrows) order ferromagnetically on a cross section of the SWNT and rotate along the SWNT axis with a period $\pi /k_F$ in the spin $xy$ plane (chosen here arbitrarily orthogonal to the SWNT axis).

Figure 2

Sketch of the RKKY interaction $J_q$ given by Eq. (3).

Figure 3

(a) Magnetization $m_{2k_F}(T)$ [Eq. (5)]. Dashed line: without feedback. Solid line: with feedback. Parameters for the curves are [14, 15, 23, 24] $E_F=0.1\mathrm{eV}$, $A_0={10}^{-7}\mathrm{eV}$, $v_F=8\times {10}^5\mathrm{m}/\mathrm{s}$, $a=2.46\AA$, $K_s=1$, $K_c=0.2$ (leading to $g=0.6$, $g^{\prime }=0.33$), and $L=2\mu \mathrm{m}$. The vertical lines mark the temperatures written next to them. (b) Characteristic temperature $T^{*}$ [Eq. (9)] as a function of the hyperfine constant $A_0$. The curve follows a power law $T^{*}\propto A_0^{2/(3-2g^{\prime })}=A_0^{0.86}$, and is plotted up to the self-consistency limit $T^{*}\approx v_F/L{k}_B=3\mathrm{K}$.