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4.5. VARIATIONAL MONTE-CARLO APPLIED TO THE CARBON NANOTUBES99
4.5.1 Néel-like fluctuations in carbon nanotubes at halffilling
When minimizing the energy of our wavefunction on different nanotubes, we find
that antiferromagnetism is still stabilized. The stabilization of finite long-range
AF order in our quasi-1D system is obviously an artefact of our calculations,
since the Mermin-Wagner theorem [120] implies that no continuous symmetry
of the hamiltonian can be broken at T =0K in a 1D system. Nevertheless,
we expect that our results give a qualitative answer on the strength of the Néel
correlations. For instance, we expect that when the variational magnetic order is
strong, then the true ground will certainly have strong Néel like fluctuations as
well. Furthermore, our variational results might hold for experiments done on a
finite set of neighboring nanotubes, where the inter-tube coupling might restore
the possibility for true long-range magnetic order.
We show a benchmark the variational calculations at half-filling in Table 4.2.
We have first carried on calculations at half-filling, to be able to compare with
the exact results obtained with the Quantum Monte-Carlo method.
Because the carbon nanotubes break the 120 ◦ symmetry which was present in
the 2D honeycomb lattice, we expect that the exchange energy is also anisotropic.
In Table 4.2 we depict the nearest-neighbor exchange energy in the three different
directions of the lattice a i .
Interestingly, the variational approximation reproduces remarkably well the
Quantum Monte-Carlo results. The comparison of the different wavefunction
shows, on one hand, that the Gutzwiller wavefunction catches already approximatively
the good symmetry in some of the nanotubes, on another hand, in addition
the introduction of the AF variational parameter allows to correct slightly
the exchange energy so that we get similar results to the QMC. The anisotropy
of the exchange energy which is already present in the Gutzwiller wavefunction
sheds light on the fact that the projection of the t−J model plays a subtle role,
that allows to already catch some of the features of the true ground state.