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162 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
the simulations with periodic boundary conditions. We emphasize that no condition
is present at the variational level that would ensure the current conservations
in the variational wavefunction which is proposed as Ansatz for the three-band
Hubbard model.
Finally, we conclude that the JA/FLUX wavefunction is stabilized also in the
case when it cannot have a non-physical flux through the boundary. Nevertheless,
the open boundary conditions are also expected to introduce severe finite-size
effects, that could also stabilize non-physical phases.
6.10.3 Magnetism and superconductivity
According to previous VMC evaluations for the U d = ∞ three-band Hubbard
model [140], the antiferromagnetic region extends up to 50% hole doping and
the d-wave superconducting phase exists only in the infinitesimally small region
near the boundary of the antiferromagnetic phase. Thus, previous VMC results
concluded that the chance for d-wave superconductivity is small in the threeband
Hubbard model. It was concluded that the parameters of the Hubbard
Hamiltonian should be tuned such that the anfiferromagnetic phase shrinks to a
smaller range of doping. We propose to study in this section the stabilization of
the RVB and magnetic phase with our Jastrow wavefunction, which is expected
to treat correctly the correlations.
Therefore, besides the orbital current instabilities, we considered also the possibility
for Néel magnetic long-range order and superconductivity. We considered
as a first approximation only the Q =(π, π) pitch vector for the spin density
wave. We would although expect that the pitch vector is doping dependent.
This issue was addressed for the three-band Hubbard model in Ref. [59]. The
possibility for stripes was also considered by variational calculations [139]. In
our work, we expect that the long-range correlations contained in the Jastrow
factor will allow a correct treatment of the spin √ correlations. We find indeed that
the magnetic order parameter M = lim r→∞ 〈S
z
i Si+ z 〉) is for our best variational
wavefunction 66% of the classical value (see Table 6.2). Using this wavefunction
as a guiding function for the fixed node calculations, we find a slightly higher
magnetic order with 69%. This value can be compared with the 60% obtained
by quantum Monte Carlo in the one-band Heisenberg model. However, in the
three-band Hubbard model, the magnetic instability is strongly dependent on the
oxygen-oxygen hopping integral t pp . Since this hopping frustrates the geometry,
we find that the magnetic order is destroyed when t pp ≈ 2eV .
Nevertheless, the magnetic instability is overestimated when compared to the
cuprates phase diagram where it vanishes for a small hole doping of approximately
x = 2%. The spin density wave is however very likely to be stabilized in
variational calculations, since the alternating magnetization allow to avoid double
occupancy in the uncorrelated part of the wavefunction. The presence of
magnetic order obviously costs kinetic energy, but it it does a better job than a