According to band theory, an ideal undoped (n,n) carbon nanotube is metallic. We show that the electron-electron interaction causes it to become Mott insulating with a spin gap. More interestingly, upon doping it develops superconducting fluctuations. 71.10Hf, 71.10.Pm, 78.66.Tr Carbon nano-structures such as C 60 [1] and nanotubes [2] have attracted considerable interest recently. The latter is a graphite sheet wrapped into a cylinder form. A a pair of integers (n, m) specifies the wrapping. Starting from a graphite sheet with the primitive lattice vectors a, b making an angle of 60 degrees, the (n, m) tube is a cylinder with axis running perpendicular to n a + m b, so that atoms separated by n a + m b are wrapped onto each other.Considerable efforts have gone into studying the bandstructure of carbon nanotubes [3]. The purpose of this work is to address the effects of electron-electron interaction on the low energy properties of them. In one-electron tight-binding description where one retains a single π orbital per atom and keeps only the nearest neighbor hopping, all n = m tubes are band-insulators with gaps generally scaling inversely with the radius [3]. For these tubes a sufficiently weak electron-electron interaction is not expected to change the low energy properties qualitatively. The same can not be said of the (n,n) tubes, which are band metals. To be precise, for the latter two out of 4n bands intersect the Fermi level to form two Dirac points (Fig.1). Due to the low dimensionality and the presence of gapless excitations, the effects of electronelectron interaction must be examined more carefully.The low-energy band structures of (n,n) tubes. "L" and "R" labels the Dirac points, "+", "-" labels the right and left movers. The dashed line denotes the Fermi level in the doped case.In this paper we perform perturbative renormalization group (RG) calculations to analyze the asymptotic low energy behavior of the (n,n) tubes [4]. The microscopic Hamiltonian is consisted of a nearest-neighbor tight-binding model for the π orbital on each carbon atom (the bonding topology is illustrated in Fig. 2) and a Hubbard U plus a nearest neighbor V for electron correlations. To build up an effective model for low temperature, we first discard all bands that do not intersect the Fermi level. Second, we regard the two remaining bands as linearly dispersing, i.e., we ignore the energy dependence of the band velocity. Due to these approximations, a upper energy cutoff E c has to be imposed on our subsequent discussions [5]. In the Hilbert space of the two bands, the original interactions U and V give rise to twelve independent scattering amplitudes g abcd ijkl .2n FIG. 2. The bonding structure of (n,n) tubes.

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We study a simple model for the metallic stripes found in La1.6−xN d0.4SrxCuO4: two chain Hubbard ladder embedded in a static antiferromagnetic environments. We consider two cases: a "topological stripe", for which the phase of the Neel order parameter shifts by π across the ladder, and a "non-topological stripe", for which there is no phase shift across the ladder. We perform one-loop renormalization group calculations to determine the low energy properties. We compare the results with those of the isolated ladder and show that for small doping superconductivity is enhanced in the topological stripe, and suppressed in the non-topological one. In the topological stripe, the superconducting order parameter is a mixture of a spin singlet component with zero momentum and a spin triplet component with momentum π. We argue that this mixture is generic, and is due to the presence of a new term in the quantum Ginzburg-Landau action. Some consequences of this mixing are discussed.

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