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6.5. MEAN-FIELD CALCULATIONS 143
Assuming that the fluctuations about the mean values are small we can write the
self-consistent mean-field equations :
( )
Hij ∆
H ij =
ij
(6.9)
∆ ∗ ij
−H ∗ ij
In the fully self-consistent Bogoliubov De Gennes equation the normal state
Hamiltonian H ij is given by :
And :
H ij =(t ij + 1 2 U jiχ ji )+(ɛ i − µ) δ ij (6.10)
∑
j
H ij
( u
n
j
υ n j
)
= E n
( u
n
i
υ n i
)
(6.11)
We then perform the Bogoliubov canononical transformation enabling us to obtain
u ni and v ni , the particle and hole amplitudes at site i, associated with an
eigen-energy E n and where ∆ ij is the possibly non-local pairing potential or
gap function. This allows us to find the self-consistent equations to be satisfied
[35, 36, 37, 38, 39, 40]:
∆ ij = −U ij F ij (6.12)
with :
F ij = 〈c iσ c j−σ 〉 = ∑ n
(
u
n
i
(
υ
n
j
) ∗
(1 − f (E n )) − (υ n i )∗ ( u n j
)
f (En ) (6.13)
and:
χ ij = ∑ σ
〈 〉
c † iσ c jσ =2 ∑ n
(
(u
n
i ) ∗ u n j f (E n)+υ n i
(
υ
n
j
) ∗
(1 − f (E n ) ) (6.14)
f(E n ) is the usual Fermi-Dirac distribution. A solution to the above system of
equations will be fully self-consistent provided that both the χ ij and ∆ ij potentials
are determined consistently. We turn now to the results for the three-band
Hubbard model. We have carried out mean-field calculations by solving the selfconsistent
Bogololiubov equations. It was argued that this mean-field consistent
frame [142] insures current conservation. We assume a 2-copper unit-cell (6 sites)
and solve the equations on a 12 × 12 copper lattice (496 sites). We have iterated
the equations until the observables were converged up to 10 −4 , which is basically
achieved in a few hundred of steps. We checked that minimizing all the parameters
on smaller lattices was leading to the same result. We did not consider
a further spin decoupling that would lead to antiferromagnetism, since we are
mostly interested in time-reversal symmetry breaking.