Correlation effects of single-wall carbon nanotubes in weak coupling

Single-wall carbon nanotubes with on-site interaction $U\ll t$ are studied by a controlled renormalization group method. When formulated as a system of $N_f$ flavors of interacting Dirac fermions, the effective model at generic filling resembles a set of quasi-1D Hubbard chains. For the undoped ${(N}_y{,N}_y)$ armchair and the ${(N}_x,-N_x)$ zigzag with $N_x=3n,$ specific mappings to the two-chain Hubbard model at half filling with effective interaction $u_{\mathrm{eff}}{=U/N}_y$ and ${2U/N}_x$ are found from this more general equivalence. The phase diagrams of the (3,3) armchair and the $(6,-6)$ zigzag at generic fillings are studied and compared as an example.