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1.4. MICROSCOPIC MODELS FOR THE CUPRATES 27
Figure 1.3: The projected BCS wave-function is a superposition of valence-bond
states (resonating valence-bond state). A valence-bond state consist of a paving
of the lattice with short-range dimers.
BCS wave-function is expected to be a good starting point to describe a doped
Mott insulator. However, at half-filling the t−J model reduces to the Heisenberg
model, and it was pointed out that the magnetic order is not destroyed by the
quantum fluctuations, though it is renormalized down to 60% of the classical
value on the square lattice [16]. Therefore, we have to take into account an additional
mean-field decoupling that allows to put back the long-range magnetic
correlation in the wave-function. This can be done by considering additionally a
spin decoupling of the exchange term in the t−J model:
H t−J ≈−t ∑
〈i,j〉σ
( )
c † iσ c jσ + c.c. + J 2
⎛
⎝ ∑ 〈i,j〉
⎞
〈S j 〉S i + S j 〈S i 〉 ⎠
⎛
⎞
− J 2
⎝ ∑ 〈i,j〉
〈b † i,j 〉b i,j + b † i,j 〈b i,j〉 ⎠ (1.22)
As a matter of fact, the more general quadratic mean-field hamiltonian can be
obtained after a mean-field decoupling of the t−J model, where the decoupled
exchange energy leads to the χ ij ,∆ i,j and h i order parameters:
H MF = −t ∑
〈i,j〉σ
(
)
χ i,j c † iσ c jσ + h.c. +
∑ (
〈i,j〉σ i σ j
∆ σ iσ j
i,j
)
c + iσ i
c † jσ j
+ h.c. + ∑ i
h i S i − µ ∑ iσ
n iσ (1.23)
The first term in Hamiltonian (1.23) is a renormalized hopping term, that is allowed
to take a complex phase. The phases are associated with a vector potential