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6.6. VARIATIONAL WAVEFUNCTION 145
Once the self-consistency is achieved, we measure the order parameter 〈c † kσ c lσ〉 =
|〈c † kσ c lσ〉|e iθ kl
in the mean-field wavefunction. The current operator is then defined
by:
〈jkl
MF 〉 = ∑ σ
it MF
kl 〈c † kσ c lσ〉 + c.c. (6.15)
When the self-consistency condition is satisfied, t MF
kl
is given by:
t MF
kl = t kl − V dp 〈c † lσ c k〉 (6.16)
In terms of the phase θ kl , this finally gives:
〈jkl
MF 〉 = ∑ σ
i
(
)
t kl |〈c † kσ c lσ〉|e iθ kl
− V dp |〈c † kσ c lσ〉||〈c † lσ c kσ〉|
(6.17)
and we get:
〈jkl
MF 〉 =2 ∑ σ
t kl |〈c † kσ c lσ〉| sin iθ kl (6.18)
Eventually we find that the expectation value of the mean-field current operator
jkl
MF , when measured in the mean-field ground-state, is also equal to the
expectation value of the true current operator defined for the Hubbard model :
j kl = ∑ σ
it kl 〈c † kσ c lσ〉 + c.c. (6.19)
We emphasize that the relation 〈jkl
MF 〉 = 〈j kl 〉 breaks down when the calculations
are not self-consistent.
We find that for large nearest-neighbors Coulomb repulsion V dp ≈ 2(seeFig.
6.7) the orbital currents start to develop for the hole doping part of the phase
diagram. The orbital current phase is however stable for unrealistic large hole
doping range. A more sophisticated treatment of the Coulomb repulsion terms
is certainly called for to avoid the artefact of the mean-field calculations.
6.6 Variational wavefunction
The wavefunction that we consider throughout this chapter is defined by the
usual BCS like mean-field hamiltonian (1.23):
H MF = ∑ 〈i,j〉
χ ij c † iσ c j,σ +∆ var
p
∑
ˆn pσ + µ ∑ ˆn i
p,σ
i
+ ∑ )
(∆ i,j c † i,↑ c† j,↓ + c.c. + ∑
〈i,j〉
i
h i .S i (6.20)
Where χ ij ,∆ var
p and ∆ ij are complex variational parameters. The order parameter
h i allow to consider Néel magnetism. Finally, µ is the chemical potential