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