Influence of electron-phonon coupling on the low-temperature phases of metallic single-wall carbon nanotubes

We investigate the effect of electron-phonon coupling on low-temperature phases in metallic single-wall carbon nanotubes. We obtain low-temperature phase diagrams of armchair and zigzag type nanotubes with screened interactions with a weak-coupling renormalization group approach. In the absence of electron-phonon coupling, two types of nanotubes have similar phase diagrams. A $D$-Mott phase or $d$-wave superconductivity appears when the on-site interaction is dominant, while a charge-density wave or an excitonic insulator phase emerges when the nearest neighbor interaction becomes comparable to the on-site interaction. The electron-phonon coupling, treated by a two-cutoff scaling scheme, leads to different behavior in two types of nanotubes. For strong electron-phonon interactions, phonon softening is induced and a Peierls insulator phase appears in armchair nanotubes. We find that this softening of phonons may occur for any intraband scattering phonon mode. On the other hand, the effect of electron-phonon coupling is negligible for zigzag nanotubes. The distinct behavior of armchair and zigzag nanotubes against lattice distortion is explained by analysis of the renormalization group equations.

Figure 1

(a) Graphene lattice. Filled (empty) circles represent $A\left(B\right)$ sites. (b) The upper band of a graphene tight-binding dispersion in units of $J_0$ in Eq. (3). The dashed horizontal line corresponds to the gapless mode in armchair nanotubes, and the dotted vertical lines to the ones in zigzag nanotubes.

Figure 2

Effective two-leg ladder models obtained by considering only the gapless modes of the tight-binding Hamiltonian for armchair and zigzag nanotubes.

Figure 3

Weak-coupling phase diagrams of armchair nanotubes without e-ph interactions for undoped (left) and doped (right) systems.

Figure 4

Weak-coupling phase diagrams of zigzag nanotubes without e-ph interactions for undoped (left) and doped (right) systems.

Figure 5

Schematic presentation of phases appearing in armchair and zigzag nanotubes at half-filling. Blue circles represent the $s$-wave pairing, orange ellipses the $d$-wave pairing, red dots charge accumulations, and arrows currents (we follow the notation of Ref. [24]).

Figure 6

Phase diagrams for armchair nanotubes with e-ph coupling for undoped (left) and doped (right) cases. We take ${\kappa }_1={\kappa }_2=\kappa$ in Eq. (28).

Figure 7

Parameter dependence of the asymptotic solutions in Eq. (35). Among the three possible behaviors, only region I gives the phonon-driven distinct phases.