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4.4. RESULTS AND DISCUSSION 91
where
ξ(k) =t ( e i(α 2−α 1 ) + e iα 2
+1 ) ,
B(k) =∆/2 ( e −iφ 3
cos(α 2 − α 1 )+e −iφ 2
cos α 2 + e ) −iφ 1
.
α 1 and α 2 are the projections on the reciprocal vectors: k = α 1 /2π b 1 +α 2 /2π b 2 .
We emphasize that ∆ is not the superconducting 〈 〉 〈 order parameter 〉 〈 which 〉 is
actually given in our approximation by c † i↑ c† j↓
= b i f † i↑ b jf † j↓
≃〈b i b j 〉 f † i↑ f † j↓
.
〈 〉
At zero temperature we have c † i↑ c† j↓
≃ δ∆.
4.4 Results and discussion
4.4.1 VMC approach
The VMC approach allows the exact evaluation (up to statistical error bars) of
the energy expectation value of the Gutzwiller projected trial functions defined
previously. The goal is now to find the lowest energy state at each doping level.
To compare the energy expectation values of the different states we define the
condensation energy e c
e c = e var − e FS , (4.13)
which is the energy gain per site with respect to the FS state, the Gutzwiller
projected Fermi sea. The results for the 72 and the 144 sites cluster are shown
in figures 4.3 and 4.4. We have checked that our variational subspace is large
enough by minimizing the trial function (4.1) on the 72 sites cluster with respect
to all parameters of the Hamiltonian (1.23) simultaneously, not allowing however
the breaking of translational symmetry. By comparing the results for the two
clusters we see that our results are rebust with respect to size effects.
For all filling factors we have found that the isotropic hopping term t ij ≡ 1
and θ ij ≡ 0 always give the minimal energy. Moreover for the symmetry of the
RVB state we found d x 2 −y 2+id xy symmetry in the minimal energy configuration at
all dopings, i.e. the phases in equation (4.2) are given by φ 1 =2π/3, φ 2 =4π/3,
and φ 3 = 0. Of course one can interchange these phases and also add or subtract
2π to each of them and still the same energy.
In the reminder of this section we will describe in detail the different phases
appearing in the honeycomb lattice from our VMC simulations.
4.4.2 Half-filling
At half-filling we found that the optimal energy wavefunction is the RVB/AF
mixed state. This state has a considerable fraction (66%) of the classical Néel
magnetism. However, our variational ansatz seems to overestimate the tendency