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1.4. MICROSCOPIC MODELS FOR THE CUPRATES 25
To illustrate our future strategy, let us recall in more details the simple basis
of the RVB theory. Anderson propose to write the exchange part of the t−J
model in terms of singlet creation operators:
b + i,j |0〉 = c+ i↓ c+ j↑ − c+ i↑ c+ j↓
√
2
|0〉 = 1 √
2
(|↓ i ↑ j 〉−|↑ i ↓ j 〉) (1.9)
This leads to the following equivalent Hamiltonian :
H t−J = −t ∑
b † i,j b i,j (1.10)
〈i,j〉σ
c † iσ c jσ − J ∑ 〈i,j〉
In this language, the exchange term of the t−J model can be seen as a term
that condenses singlets. A simple mean-field decoupling of the singlet creation
operator was proposed by P.W. Anderson:
∆ i,j = 〈b i,j 〉 0
(1.11)
The mean-field hamiltonian is:
⎛
⎞
H MF = −t ∑
′ c † iσ c jσ − J ⎝ ∑ 〈b † i,j 〉b i,j + b † i,j 〈b i,j〉 + 〈b † i,j 〉〈b i,j〉 ⎠ (1.12)
〈i,j〉σ
〈i,j〉
And if the quantum fluctuations around the mean-field value are small, the final
mean-field Hamiltonian reads:
H MF = ∑ )
)
ε k
(c † kσ c kσ + c.c. +
(∆ k c † k↑ c† −k↓ + c.c. (1.13)
k
Therefore the BCS mean-field theory is recovered as a mean-field theory of the
t−J model. The pairing order parameter assumed above is defined in real
space. For example, when ∆ is isotropic the Fourier transform of the pairing
leads to the so called s-wave symmetry, and for the square lattice we get
∆ k =∆(cos(k x )+cos(k y )). In the case when ∆ ij has alternating sign on the
square lattice (+1 on the horizontal bonds and −1 on the vertical bonds) the
Fourier transform gives the d-wave pairing symmetry ∆ k =∆(cos(k x ) − cos(k y )).
The BCS wave-function has the peculiar property that it is a superposition of
different numbers of particle states. Even more relevant, the full projection that
forbids doubly occupied sites in the t−J model is not taken into account by the
simple mean-field decoupling. Therefore, we have to remove the doubly occupied
site of the wave-function. As a first step, the wave-function is written in real
space :
|ψ〉 = ∏ ( ∏
(
uk + v k c + k↑ −k↓) c+ |0〉 ∝ 1+ v )
k
c + k↑
u c+ −k↓
|0〉 (1.14)
k
k∈BZ
k∈BZ