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2.3. STOCHASTIC MINIMIZATION 39
The full variational wavefunction is, when all these projections are considered:
|ψ VMC 〉 = J s J d P N P S z =0|ψ MF 〉 (2.38)
Where P N projects the wavefunction on a state with fixed number of particles
N, and P S z =0 projects the wavefunction on the Stot z = 0 sector. We expect
that the projections allow to improve significantly the wavefunction. Let us
note that in the t−J model the situation is somehow simplified, since the full
projection contained in the model forbids any doubly occupied site. Therefore,
the local Gutzwiller projection is not needed. Nevertheless, the Jastrow term
between different site of the lattice might still play an important role. Finally,
we emphasize that the Jastrow parameters change the wavefunction’s energy on
much larger scales than the variations of the parameters in the one-body part.
2.3 Stochastic minimization
Quantum Monte Carlo methods are some of the most accurate and efficient methods
for treating many-body systems. The success of these methods is in large
part due to the flexibility in the form of the trial wavefunctions, discussed in the
previous section, that results from doing integrals by Monte Carlo calculations.
Since the capability to efficiently optimize the parameters in trial wavefunctions
is crucial to the success of the variational Monte Carlo (VMC), a lot of
effort has been put into inventing better optimization methods. Two recent
works [51, 52] have put the limitations in the minimization procedure to even
further grounds.
In our work, we are mainly interested in the energy minimization, instead of
the possible variance minimization. Both minimization are not identical. The
variance of the wavefunction measures basically its distance to a true eigenstate
of the Hamiltonian, that might be an excited state and not the ground-state.
Therefore, one typically seeks the lowest energy by minimizing either the onebody
uncorrelated part of the variational wavefunction, that consist mainly of a
determinant or a Pfaffian of a matrix, or by minimizing the Jastrow variational
parameter, that consists of the strongly correlated part of the wavefunction.
To carry out the minimization, we start therefore by expanding in quadratic
order the wavefunction in a neighborhood of the parameters.
[
|ψ α+γ 〉≃ 1+ ∑ ]
γ k (O k −〈O k 〉)+ β ∑
γ k γ k ′ (O k −〈O k ′〉) |ψ α 〉 (2.39)
2
k
k,k ′
where ψ α (x) is the variational wavefunction, with set of variational parameters
α =(α 1 ...α m ), projected on the real space fermionic configuration x. The O k
operators are the logarithmic derivative of the wavefunction with respect to the