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36 CHAPTER 2. NUMERICAL METHODS
pairing also when tight-binding wavefunctions are considered. Therefore, for open
shells, we take f i,j with a small pairing in order to split the bare degeneracy, the
resulting wavefunction being still a singlet. Finally, to avoid spurious and peculiar
effects in the calculations, we impose that the non-projected wavefunction has
the same number of particle than the projected wavefunction.
2.1.2 Pfaffian variational Monte Carlo
In this section we address the issue of collinear magnetism or triplet pairing in
the mean-field Hamiltonian (1.23). In both cases, the exponent in (2.8) becomes
spin dependant:
⎛
|ψ MF 〉 =exp⎝ 1 2
∑
f σ i,σ j
i,j
i,j,σ i, σ j
c † i,σ i
c † j,σ j
⎞
⎠ |0〉 (2.22)
We emphasize that |ψ MF 〉 has neither a fixed number of particles due to the
presence of the pairing, nor a fixed total S z due to the non-collinear magnetic
order. Thus in order to use it for the VMC study we apply to it the following
projectors: P N which projects the wavefunction on a state with fixed number
of electrons, and P S z, which projects the wavefunction on the sector with total
S z =0.
Expanding (2.22) we get:
|ψ〉 = P N |ψ MF 〉 (2.23)
=
{
∑
i,j
λ(i, j)c + i↓ c+ j↓ + ω(i, j)c+ i↑ c+ j↑ + θ(i, j)c+ i↓ c+ j↑ + χ(i, j)c+ i↑ c+ j↓} N/2
(2.24)
=
{
∑
i,j,σ i ,σ j
D (i,j,σi ,σ j )c + iσ i
c + jσ j
} N/2
(2.25)
|ψ〉 =
∑
(R 1 ..,R N/2) ( )
R ′ 1 ..R′ N/2
}
{D (R1 ,R ′ 1 ) ..D (RN/2 ,R ′ N/2 ) c † R 1
c † R
..c † ′ R 1 N/2
c † R ′ N/2
(2.26)
where we used the notations R i =(x i ,σ i ). Then the projection on the real basis
state 〈α| = 〈0| c R1 ..c RN is given by :
〈α|ψ〉 = ∑ {
}
D (PR1 ,P R2 )...D (PRN−1 ,P RN ) (−1) Signature(P) (2.27)
P
We introduced the notation P(R i )=R k ,k = P(i).
∑ {
}
D (PR1 ,P R2 )...D (PRN−1 ,P RN ) (−1) Signature(P) = P f (A) (2.28)
P