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146 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
and ∆ v par is the renormalized energy difference between the d and p atomic levels
in the variational wavefunction. The parameters χ ij are independent within one
copper unit-cell (this represents 8 complex parameters) with both an amplitude
and a phase. When χ ij is complex, the order parameter is associated with an
external flux which leads to the circulation of the holes. The operator c † iσ creates
a hole in the orbitals d x2−y2 , p x and p y .
Moreover, we consider pairing parameters between nearest oxygen neighbors
but also a pairing between oxygen sites with |i − j| < 3 (this gives 106 complex
parameters). This allows to have a first approximation of the pairing between
Zhang-Rice singlets, which is expected to lead to the d-wave superconducting
instability in the t−J model [12].
The chemical potential in the mean-field Hamiltonian is fixed such that the
non-projected wavefunction has a mean-number of holes that is consistent with
the hole doping which is considered. To simplify the calculation, we will consider
independently each of the instabilities, and denote by FLUX/SDW/RVB the part
of the wavefunction that was considered.
Furthermore, we introduce a correlated part with a spin and charge Jastrow
factor:
( ) ( )
∑
∑
J =exp vij c n in j exp
(6.21)
i,j=1,N
i,j=1,N
v S ij Sz i Sz j
where all v c ij and v S ij are considered as free variational parameters. We impose
however the symmetry of the lattice T×P,whereP is the point-group symmetry
of the lattice, and T are the translations which are consistent which the unitcell
of the wavefunction (we assume in our case a 2-copper unit-cell to allow
Néel magnetism). In what follows, we will denote by Ja/{Flux/RVB/SDW} a
wavefunction that contains the Jastrow.
6.6.1 Jastrow factor obtained after optimization
Interestingly enough, we find after calculations that the charge Jastrow factor
is slowly decreasing with the distance (see Fig. 6.9), and the nearest-neighbor
charge repulsion is not negligible. We find this result even when the true nearest
neighbor repulsion is zero : V dp = 0. This certainly means that the on-site
repulsion U d generates via second-order process a first neighbor repulsion that
is captured by the Jastrow factor. This is in agreement with the fact that our
results are generally only weakly dependent on V dp .
6.7 Minimization of the Energy
The minimization of the variational parameters for a three-band model is not a
simple task, and a difficult problem to overcome is the fact that the minimization