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98 CHAPTER 4. HONEYCOMB LATTICE
possibility of hole doping of the tubes by the gold metallic contacts [119].
Very interestingly, the nanotubes allow the investigation of superconductivity
in the 1D limit which was never explored before. We propose in this section to
study the t − J model on wrapped honeycomb sheet in order to investigate how
our variational wavefunction behaves in this limit. The variational wavefunction
supports, as in the previous section, both antiferromagnetism and superconductivity
order parameters.
Nevertheless, in 1D systems it is known that no continuous symmetry can be
broken and therefore we do not expect to stabilize real long-range order in the true
quantum ground state. The continuous symmetry breaking is indeed necessary
for superconductivity (broken U(1) symmetry) and also for antiferromagnetism
(broken SU(2) symmetry). Although these phases cannot form the true groundstate
of the t − J model, the fluctuations around them can still be important for
determining its physics. For example, one could imagine that such fluctuations
stabilize a superconducting state in ropes of nanotubes. In view of this argument,
we believe that it is pertinent to study the tendency toward superconductivity in
nanotubes with our wavefunctions.
We therefore use the variational approximation for different chiralities T 1 =
(l 1 ,l 2 ) (the notation was introduced in section 4.3). The vector T 1 fixes the
chirality of the nanotube, and T 2 is chosen to be orthogonal to T 1 . We have
considered one armchair nanotube of chirality (2, 2), zig-zag nanotubes (3, 0) and
(4, 0), and the other tubes (2, 1), (3, 1) and (3, 2). We used periodic boundary
along T 1 , such that the tube is wrapped. The geometry of a carbon nanotube
imposes that the tube is open along T 2 . However, to minimize the finite size
effect, we have also used periodic boundary conditions in this direction. The
tube was studied in the limit when ‖T 2 ‖≫‖T 1 ‖. Let us emphasize that in
experiments the nanotubes have bigger diameter than the one considered here:
the size in experiments would rather correspond to (10, 10) wrapping for the
armchair nanotube. Besides, the size of the nanotubes in experiments is not the
same for all of the tubes, and a distribution of sizes is expected.
Eventually, the t−J model describes the physics of a doped Mott insulator.
When we consider this model for nanotubes, the question whether the undoped
compounds are still close to the Mott-insulating phase is relevant. However,
the spin-gap obtained by QMC calculation for the different nanotubes is finite,
although its value decreases when the diameter of the tube is larger. Therefore,
it is expected that close to half-filling the ground state of the nanotubes is not a
Fermi liquid. Superconductivity, if present in nanotube, might therefore be driven
by strong correlation in these compounds, as opposed to a BCS phonon-mediated
mechanism.