Theory of superconducting fluctuations in the thinest carbon nanotubes

The low-energy electronic Hamiltonian for the thinest zigzag carbon nanotube, embedded into a dielectric host, is derived and its phase diagram is discussed. The specific multi-band structure and the microscopic form of the electron-electron interaction in this systems is considered. The interband repulsive interaction, which is almost unscreened, leads to a retarded intraband attraction between the electrons and can stabilize the Cooper pairs. For a dielectric constant of the host $\epsilon_d\sim 2-4$, the theory predicts that the superconducting fluctuations should develop, which is in agreement with the experiment. For $\epsilon_d \gtrsim 8$, the density wave fluctuations should be amplified. Between the two phases, there is a metallic state where all two-particle fluctuations are suppressed.