Figure 1

(a) The linearized single electron-band structure near Fermi level. A and A′ bands are degenerate, with opposite Fermi velocity to that in the B band. (b) The Brillouin zone. The horizontal axis is $k$ along the tube axis, while the vertical axis is angular momentum around the tube circumference. The black dots denoted by A±, B± are the Fermi points in the 1D band. (c)–(h) show the schematic diagrams for the six intratube channels. (i)–(k) illustrate $g_{1\parallel }$, $g_{1\perp }$, and $g_2$ in each channel.Reuse & Permissions

Figure 2

The logarithmically divergent diagrams, (a), (b), (c), and (d), in the perturbative expansion of the interaction vertex associated with the second-order RG. Nondivergent diagrams and higher-order diagrams are not shown. Here a solid line denotes the right-moving electron, while a dashed line denotes the left-moving electron.Reuse & Permissions

Figure 3

The diagrams that give rise to ln(ω/$E_0$) for ${\Gamma }_1^2$. The first four diagrams belong to the first-order RG diagrams, while others belong to the second-order RG diagrams. The down-triangle diagrams are similar to the up-triangle ones and hence not listed.Reuse & Permissions

Figure 4

The diagrams for the second-order self-energy correction to the Green function. The first five diagrams (top row) are related to the A/A′ band, while the last five diagrams (bottom row) are related to the B band.Reuse & Permissions

Figure 5

The crossover of the CDW and SS orders. Only the two most divergent response functions of CDW and SS are shown.Reuse & Permissions

Figure 6

The phase diagram of the most dominant response function of the (5,0) electronic system plotted as a function of the dielectric constant $\kappa$ of the embedding matrix, as analyzed by the second-order renormalization group calculations. The case of a single (5,0) CNT is shown in (a), and in (b) the case of a thin array of (5,0) CNTs, comprising one central nanotube surrounded by six others with a separation of 1 nm, is shown. Here the $Y$-axis denotes the scaling factor −ln(ω/$E_0$). Increasing $Y$ implies decreasing energy/temperature. Different phases are delineated by different colors/shades of gray, with the number following the abbreviation for each phase denoting the divergent channel. For example, “CDW4” in the central part of Fig. 6(a) means that in this region, the largest response function is the CDW order in the fourth channel, etc. The vertical lines at κ $=$ 1.11 and κ $=$ 1.38 mark the division between different fixed points. Within the array configuration, the SS order is favored over a large range of dielectric constant κ. In particular, for the AFI zeolite with κ $=$ 6, superconducting ground state is favored.Reuse & Permissions

Figure 7

(a) and (b) are the calculated differential resistance versus the energy-scaling variable in channels 1 and 4. Both are characterized by a peak in differential resistance at the Fermi level.Reuse & Permissions

Figure 8

Differential resistance plotted as a function of the bias current. At temperatures above 2.5 K, a zero-bias resistance peak is clearly seen. At low temperatures, a dip in resistance appears that indicates the existence of supercurrent. Inset is the corresponding R-T curve under 0 and 4 T magnetic field. No difference is seen. The temperature dependence of the resistance and its magnetic field independence clearly demonstrate that the overall behavior is that of 1D superconductivity.Reuse & Permissions