Enhanced NMR relaxation of Tomonaga-Luttinger liquids and the magnitude of the carbon hyperfine coupling in single-wall carbon nanotubes

Recent transport measurements [Churchill $\mathit{\text{et}}\mathit{\text{al.}}$ Nat. Phys. $\mathbf{\text{5}}$, 321 (2009)] found a surprisingly large, 2-3 orders of magnitude larger than usual ${}^{13}$C hyperfine coupling (HFC) in ${}^{13}$C enriched single-wall carbon nanotubes (SWCNTs). We formulate the theory of the nuclear relaxation time in the framework of the Tomonaga-Luttinger liquid theory to enable the determination of the HFC from recent data by Ihara $\mathit{\text{et}}\mathit{\text{al.}}$ [Ihara $\mathit{\text{et}}\mathit{\text{al.}}$ EPL $\mathbf{\text{90}}$, 17004 (2010)]. Though we find that $1\mathrm{/}{T}_1$ is orders of magnitude enhanced with respect to a Fermi-liquid behavior, the HFC has its usual, small value. Then, we reexamine the theoretical description used to extract the HFC from transport experiments and show that similar features could be obtained with HFC-independent system parameters.