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4.4. RESULTS AND DISCUSSION 93
lattice FS best VMC QMC
triangular −0.3547(2) −0.5330(5) [112] −0.5450(1) [82]
square −0.4270(2) −0.6640(1) [32] −0.6692(1) [113]
honeycomb −0.5275(2) −0.5430(2) −0.5440(10) [114]
Table 4.1: We compare the Heisenberg energy per site in the thermodynamic
limit (in units of J) from VMC (Gutzwiller projected FS, the best available trial
state) and from exact quantum Monte Carlo simulations.
towards antiferromagnetism: QMC calculations give a lower value for the magnetic
order (53% of the classical value [106]). The magnetic order is slightly more
renormalized in the honeycomb lattice than in the square lattice, where QMC
gives a magnetic order of 60% of the classical value [111]). This is expected as
the fluctuations should be larger in the honeycomb lattice due to a lower coordinance
number. The over estimation of magnetism at half-filling seems to be a
general feature of this type of wavefunction, since the same discrepancy occurs
also for the triangular and square lattice [32, 112].
As can be seen from table 4.1, the Gutzwiller projected Fermi sea state gives
a value surprisingly close to the exact value in the honeycomb lattice. This is
not the case for the square and the triangular lattices. However from figure
4.5 it becomes clear that there is no fundamental difference in the projected FS
state on different lattices. The spin-spin correlations decrease with distance very
rapidly to zero for all lattices with the only difference that the nearest neighbor
correlations are substantially larger on the honeycomb lattice.
4.4.3 Magnetism and Superconductivity
We find that superconductivity is observed in the small range of doping δ =]0, 1 8 [.
The BCS pairing is suppressed at the doping which corresponds to the van Hove
singularity in the free electron density of states. We note also that a coexistence of
aNéel phase and superconductivity is present in the range [0, 0.07] (see figure 4.6).
The VMC simulations and the self-consistent MF calculations predict d x 2 −y 2+id xy
symmetry for the superconducting order parameter. Also the amplitude of the
mean-field pairing order parameter is in good agreement with the variational
calculations in the relevant range of doping ]0, 1/8[. There is a strong reduction
of the order parameter close to 1/8 doping, and the MF solutions show a long
tail falling down at δ =0.4. The long MF tail for dopings larger than 1/8 has no
relevance since we have shown in our VMC calculations that superconductivity
is completely suppressed by SDW and ferromagnetic instabilities in this region.
Interestingly, the excitations of the quasi-particles in the MF scheme are gapless
at half-filling, but the excitation gap rises up to a maximum value of approx-