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174 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
metry is θ 2 like, and when the hopping signs lead to 4 triangles with positive
hopping transfer, the orbital current symmetry is θ 1 like. Current-current correlations
measured in the corresponding free Fermi sea wavefunction show that
short-range correlations with the respective symmetry are present with corresponding
θ 1,2 for these two particular choices of the hopping signs. Nevertheless,
these choices of the transfer integrals are arbitrary and do no correspond to the
real physical model, which is expected to describe the cuprates. Applying the
Variational Monte Carlo method, we found in our best variational Ansatz the
presence of small currents for doping close to x =0.08. The symmetry of the
currents measured in the projected wavefunction consists of lines of currents,
as in the θ 2 phase, but with reversed diagonal currents. We get a net macroscopic
current running through the periodic boundary of the lattice, which is
clearly forbidden in the thermodynamic limit and suggests that the presence of
currents at the variational level might be an artefact. On the other hand, the
variational parameters of the variational function are exactly those of the θ 2
phase. The other competing instabilities are the spin-density wave and the resonating
valence bond wavefunctions. Antiferromagnetism is found to be stable
for x =[0., 0.11], and the Néel magnetism is reduced at half-filling down to 68%
of the classical value by the quantum fluctuations. When the doping is close
to 0.25%, we find superconductivity with a d-wave like symmetry, although the
energy gain of the superconducting wavefunction is small, and it is difficult to
get a definitive conclusion on the exact nature of the RVB phase. We proposed
additionally two other models that might shed light on the presence of orbital
currents: (1) We extended first the three-band Hubbard model by considering
the additional out-of-plane apical oxygens and the copper d 3z2−r2 orbitals, (2)
We considered an additional correlated hopping term in the three-band Hubbard
model. In the former model we found interestingly that when the transfer integral
connected to the apical oxygen are slightly enhanced, then strong orbital currents
start to develop with an in-plane magnetic moment, in agreements with neutron
experimental results. In the latter model, we found that the orbital currents
are strongly stabilized when the correlated hopping V 2 ≈ 0.4. The symmetry of
the current pattern in this case is θ 2 -like. As an outlook, we would suggest to
study further the orbital current present in the three-band Hubbard model when
open boundaries are considered. In this particular case, the flux running through
the boundary would at least be removed, and the stability of the charge current
phase could be better studied. Finally, it has also been argued that holes doped
in the antiferromagnetically correlated spin systems induce incommensurate spin
correlations in the ground state for the one- and three-band Hubbard models
within mean-field approximations. Therefore, it would be interesting to compare
the energies of these phases with our Jastrow projected wavefunction, and see if
these latter phases are still present.