In this paper we report the applicability of the density matrix renormalization group (DMRG) approach to the cylindrical single wall carbon nanotube (SWCN) for the purpose of its correlation effect. By applying the DMRG approach to the
t+U+V model, with
t and
V being the hopping and Coulomb energies between the nearest neighboring sites, respectively, and
U the on-site Coulomb energy, we calculate the phase diagram for the SWCN with chiral numbers (
n1=3,
n2=2), which reflects the competition between the correlation energy
U and
V. Within reasonable parameter ranges, we investigate possible correlated ground states, the lowest excitations, and the corresponding correlation functions in which the connection with the excitonic insulator is particularly addressed.
We derive a low-energy effective model of metallic zigzag carbon nanotubes at half filling. We show that there are three important features characterizing the low-energy properties of these systems: the long-range Coulomb interaction, umklapp scattering, and an explicit dimerization generated by interactions. The ratio of the dimerization induced gap and the Mott gap induced by the umklapp interactions is dependent on the radius of the nanotube and can drive the system upon increasing dimerization strength from a Haldane spin-liquid phase through a quantum phase transition with
SU(2)1 quantum symmetry to a dimerized phase. We consider the physical properties of the phases on either side of this transition, which should be relevant for realistic nanotubes.