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26 CHAPTER 1. INTRODUCTION
This can be expanded in power of c + k↑ c+ −k↓ ,andweget:
|ψ〉 = ∏ (
1+ v ( ) ( )
2
k
c † k↑
u c† −k↓ + vk
∑
c † v k
k↑
k u c† −k↓
+ ..)
=exp c † k↑
k u c† −k↓
k
k∈BZ
k
(1.15)
In this representation, we can project the wave-function on a state with fixed
number of particles. The variance of the number of particles in the BCS wavefunction
behaves like 1/ √ N, and the projection of the wave-function on a state
with a fixed number of particle is not inducing drastic changes when the system
is large. Hence, the wave-function that we consider is:
|ψ ′ 〉 = P N exp
{ ∑
i,j
( ∑
k
v k
u k
e ik(R i↑−R j↓ )
)
c † i↑ c† j↓
And u k and v k are respectively the particle and the hole densities :
}
|0〉 (1.16)
u k = √ 1
√ √√√ γ k
1 −
(1.17)
2
√γ k + |∆ k | 2
v k = 1 √
2
√ √√√
1+
γ k
√γ k + |∆ k | 2 (1.18)
υ k /u k =∆ k / ( )
ξ k + ξk 2 +∆ 2 1/2
)
k
(1.19)
Very interestingly, the obtained projected BCS wavefunction can be written as a
superposition of dimer paving of the lattice:
|ψ ′ 〉 = P N |ψ BCS 〉 =
( ∑
i,j
f(i, j)c + i↑ c+ j↓) N/2
|0〉 (1.20)
where f ij depends on the choice of the order parameters. In the case of BCS
projected wave-function, we have:
f BCS (i, j) = ∑ k
v k
u k
e ik(R i↑−R j↓ )
(1.21)
The f ij plays the role of a pairing amplitude between a pair of electrons. The
wave-function (1.20) is actually a superposition of many valence-bond configurations
(see Fig.1.3), and was named resonating valence-bond state (RVB). To
summarize, at zero temperature, the RVB theory can be formulated in terms of a
variational wavefunction obtained by applying the so called Gutzwiller projector,
that removes the doubly occupied site of the BCS wavefunction. The projected