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38 CHAPTER 2. NUMERICAL METHODS
Where the matrix elements of B are the f ij matrix of equation (2.8), and we get
back the usual determinantal variational Monte Carlo calculations. We emphasize
that the matrix that we need to update in the Pfaffian Monte Carlo simulations
has linear sizes twice larger than in the calculations with determinants. In conclusion,
the Pfaffian Monte Carlo procedure is nothing else but an extension of the
usual variational wavefunction method. This procedure allows to treat generally
every order parameter contained in the mean-field Hamiltonian (1.23). However,
we considered up to now only the uncorrelated part of the wavefunction. It will
likely contain many doubly occupied sites that will cost a lot of energy within
the Hubbard model. To treat correctly the correlations, we need to introduce an
additional projection that takes care of the doubly occupied site contained in the
wavefunction.
2.2 Jastrow factors, Gutzwiller projection
In the simplest approximation of the ground-state of the Hubbard model, a simple
Fermi sea can be considered, and a simple so-called Gutzwiller projection can be
used to treat the on-site repulsion of the Hubbard model. This gives one of the
simplest possible variational Ansatz :
|ψ〉 = P G |ψ〉 (2.34)
where ψ is the one-body wavefunction and the Gutzwiller projection P G is defined
as [49]:
P G = Π
i
(1 − (1 − g)ˆn i↑ˆn i↓ ) (2.35)
The variational parameter g is running from 0 to unity and i labels the sites of the
lattice in real space. The Gutzwiller projection is very well suited by the variational
Ansatz described in the last section, since the numerical variational Monte
Carlo samples the wavefunction in the real-space fermionic configurations, and
therefore the Gutzwiller projector is a diagonal operator in this representation.
The Gutzwiller projection can be extended by the incorporation of a Jastrow
factor [50] which provides an additional powerful way to tune more precisely the
correlations of the wavefunction. Some of the possible choices of the Jastrow
term are the long-range charge Jastrow :
(
)
1 ∑
J d =exp u ijˆn iˆn j (2.36)
2
i,j
Another long-range spin Jastrow can also be considered :
(
)
1 ∑
J s =exp v ij Ŝi z 2
Ŝz j
i,j
(2.37)