  We investigate a metallic zigzag carbon nanotube by means of a Hubbard model
which includes both on-site and nearest neighbour interactions. Assuming weak
interactions, a renormalization group analysis of the equivalent two-leg ladder
followed by bosonization and refermionization results in a Gross-Neveu model
with an enlarged symmetry relative to the original Hamiltonian. For the undoped
case the symmetry of the Gross-Neveu model is SO(8), but for the doped case the
particle-hole symmetry is broken and the symmetry reduces to SO(6). Four ground
state phases are found in the undoped carbon nanotube with repulsive
interactions, a d-wave Mott insulator, an s-wave Mott insulator, a p-density
wave and a charge density wave. The doped case has two ground state phases, a
d-wave superconductor and a phase where a p-density wave and a charge density
wave co-exist. We also explore the global phase diagram with a general
interaction profile and find several additional states, including a chiral
current phase where current flows around the nanotube along the zigzag bonds.

Strongly Correlated Electrons