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Appendix B
A Simple variational
wavefunction for a pair of
particles
We consider in this Appendix a simple variational wavefunction for the threeband
Hubbard model on a simple CuO 4 cluster. We study the more general
variational wavefunction for a S z = 0 pair of holes, that is :
( ( ) )
∑
|ψ〉 = αd † ↑ d† ↓ + β j d † ↑ p† j↓ − γ jd † ↓ p† j↑
+ λ ij p † i↑ p† j↓
|0〉 (B.1)
j
where d stands for the d x2−y2 orbital and p for the p σ orbitals. The current
operator measured in this wavefunction is given for a d − p link by:
(
〈j d−px 〉 = it d−px α † (γ x + β x )+ ∑ (
β † j λ xj + γ † j jx) )
λ + c.c. (B.2)
j
and the current operator measured along a p x − p y link by:
(
〈j px−py 〉 = it px−py β xβ † y + γ xγ † y + ∑ (
λ † xj λ yj + λ † jx jy) )
λ + c.c. (B.3)
j
Similar equations hold for bonds obtained by 90 ◦ rotations. The current will
therefore be finite when α, β, γ and λ ij are complex numbers: α = R a e ia , β x,y =
Re ibx,y , γ x,y = Re icx,y and λ ij = L ij e il ij
.
〈j d−px 〉 =2t d−px
(R a R sin(c x − a)+R a R sin(b x − a)
+ ∑ )
(RL xj sin(l xj − b j )+RL jx sin(l jx − c j ))
j
(B.4)
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