Acoustic phonon exchange, attractive interactions, and the Wentzel-Bardeen singularity in single-wall nanotubes

We derive the effective low-energy theory for interacting electrons in metallic single-wall carbon nanotubes, taking into account phonon exchange due to twisting, stretching, and breathing modes within a continuum elastic description. In many cases, the nanotube can be described as a standard Luttinger liquid with possibly attractive interactions. We predict surprisingly strong attractive interactions for thin nanotubes. Once the tube radius reaches a critical value $R_0\approx 3.6\pm 1.4\mathrm{\AA },$ the Wentzel-Bardeen singularity is approached, accompanied by strong superconducting fluctuations. The surprisingly large $R_0$ indicates that this singularity could be reached experimentally. We also discuss the conditions for a Peierls transition due to acoustic phonons.