Electron Spin Resonance as a route to Spin-Gap detection in Carbon Nanotubes

The recent observation of a charge-neutral excitation gap in ultraclean carbon nanotubes\footnote[1]{V. V. Deshpande {\it{et al.}}, Science {\bf 323}, 106 (2009)} raises the intriguing possibility of a phase with gapless charge spectrum and gapped spin spectrum: the Luther-Emery liquid. We note that ESR would be an ideal probe to directly test whether the observed gap is a spin-gap, as it probes the non-local correlations of conduction electron spins. We focus on the Luther-Emery point ($K_s=1/2$, also known as free fermion point) where an explicit calculation of relevant spin-spin correlation function is possible, to calculate the ESR signal in a Luther-Emery liquid. At high frequencies of $\omega>2 \Delta_s$ where $\Delta_s$ is the spin-gap, the ESR signal of the Luther-Emery liquid will exhibits a second peak at magnetic fields away from the resonance condition of $B=\omega/\mu_B g K_s$. We discuss how to measure the spin-gap from the location of this additional peak as a function of applied field strength.