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3.3. VARIATIONAL MONTE CARLO 57
sites by applying the complete Gutzwiller projector P G . The wavefunction we
use as an Ansatz for our variational study is thus:
⎧
⎫N/2
⎨ ∑
⎬
|ψ var 〉 = P G P S zP N |ψ MF 〉 = P G P S z a
⎩ (i,j,σi ,σ j )c † iσ i
c † jσ j
|0〉 (3.3)
⎭
i,j,σ i ,σ j
Although the wavefunction (5.8) looks formidable, it can be reduced to a form
suitable for VMC calculations. Using
〈α| = 〈0| c k1 ,σ 1
...c kN ,σ N
, (3.4)
we find that
〈α | ψ var 〉 = P f (Q)
Q i,j = a (ki ,k j ,σ i ,σ j ) − a (kj ,k i ,σ j ,σ i )
(3.5)
where P f (Q) denotes the Pfaffian of the matrix Q. Using this last relation, the
function (5.8) can now be evaluated numerically using a Monte Carlo procedure
with Pfaffian updates, as introduced in Ref. [47]. In the particular case where
a k,l,↑,↑ = a k,l,↓,↓ =0andatS z = 0 (this happens if the BCS pairing is of singlet
type and the magnetic order is collinear), the Pfaffian reduces to a simple determinant,
and the method becomes equivalent to the standard Variational Monte
Carlo [42] technique.
The above mean field Hamiltonian and wavefunction contain the main physical
ingredients and broken symmetries we want to implement in the wavefunction.
In order to further improve the energy and allow for out of plane fluctuations of
the magnetic order we also add a nearest-neighbor spin-dependent Jastrow [85]
term to the wavefunction:
⎛ ⎞
P J =exp⎝α ∑ Si z Sz j
⎠, (3.6)
〈i,j〉
where α is an additional variational parameter. Our final wavefunction is thus:
|ψ var 〉 = P J P S zP N P G |ψ MF 〉 (3.7)
When α