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102 CHAPTER 4. HONEYCOMB LATTICE
0.4
0.35
2D
CNT (3,0)
CNT (2,2)
CNT (2,1)
CNT (4,0)
CNT (3,1)
CNT (3,2)
M AF
0.3
0.25
0.2
0 0.01 0.02
1/N
Figure 4.11: Scaling of the staggered magnetization per site in the Heisenberg
model (t − J model at half-filling) for our best variational wavefunction.
The antiferromagnetic order depends also on the wrapping of the tube. The
amplitude of the spin-spin correlations versus the diameter of the tube is shown
in Fig.4.12. We find that it decreases very fast when the diameter reaches the
diameter of the 2-leg ladder case. In this limit, the variational magnetism is
totally suppressed.
4.5.2 Doping Carbon nanotubes
Our variational wavefunction has the advantage to allow hole doping. Therefore,
since it reproduces well the Quantum Monte-Carlo results at half-filling, we have
studied the effect of hole doping.
We observe that not only the amplitude of superconductivity, but the phase
of the pairing on each nearest neighbors link depends on the wrapping of the
tube (see Fig.4.13). We have measured the phase after projection of the BCS
pairing in the different tubes (see also Table 4.3). We observe that the phases of
the pairing observable is moving from the d x 2 −y 2 + id xy symmetry in the case of
the 2 dimensional lattice towards intermediate value and converge to the d-wave
symmetry in the case of the 2-leg ladder, which is also the smallest nanotube
that can be wrapped with a 2-site unit-cell. In conclusion, we find both the
suppression of the magnetism when the diameter is small (see Fig. 4.12) and
reaches the limit of the 2-leg ladder, and we find an enhancement of the pairing