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4.6. CONCLUSION 107
order parameter in the same limit (see Fig.4.13). This can be interpreted as the
signature that quantum fluctuations become much larger when the tube reaches
the one dimensional limit, and our variational (but non-physical) magnetic parameter
is no longer stabilized when these fluctuations become too strong. At
the same time, it is interesting to note that the pairing survives very well in the
1D limit, though we do not expect any real pairing in a quasi-1D model. Indeed,
in this limit the ground state will be a Luttinger liquid. The fact that the pairing
is large in our variational approach might be a signature that the true Luttinger
liquid ground state is not a Fermi liquid and does not have magnetic order.
4.6 Conclusion
We have determined the ground state phase diagram of the t−J model (with a
generic value of J =0.4) on the 2D honeycomb lattice using VMC calculations.
Our results are summarized in Fig. 4.15. At half-filling, we have found antiferromagnetism
and superconducting variational parameters that coexist in the
variational wavefunction. The alternating magnetization is 66% of the classical
value, which is slightly higher than the 50% obtained with exact quantum Monte
Carlo. However, the energy obtained is very close to the exact value: we find
an Heisenberg energy per site with VMC of 0.5430(1) which is only 0.3% higher
than the QMC value. A phase of coexistence of the two order parameters is found
in the range δ =[0, 0.07], and superconductivity is suppressed at the van Hove
singularity. Therefore the range of superconducting order is δ =]0, 1 [. The amplitude
and range of existence of the superconducting parameter are four times
8
smaller than in the square lattice. We found good agreement between the VMC
calculations and an the RVB MF theory in the superconducting phase, namely
the same d x 2 −y 2 + id xy symmetry and a similar amplitude of the pairing order
parameter. Moreover the spinon excitation spectrum is gapped for 0