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148 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
process falls most of the time in local minima. For example, using the simple
Fermi sea projected with a local Gutzwiller factor, we find mainly two local
minima by minimizing the variational parameters : (i) a minimum where the
gutzwiller projection is very strong, which will lead to a very bad kinetic energy,
and once the kinetic energy is very poor, the system will finally minimize the
charge transfer energy by using a larger ∆ var
p , (ii) a local minimum where the
wavefunction optimizes its kinetic energy by using a small ∆ var
p , and a weak
Gutzwiller projection. These local minima are likely to be found if the longrange
Jastrow factor is not used.
Moreover, we use in this chapter a stochastic minimization procedure [52,51]
to minimize both the parameters of the uncorrelated part of the wavefunction
and the Jastrow parameters at the same time. This method allows one to deal
with a large number of parameters, since the gradients are calculated all at the
same time during a simulation. The new parameters are then calculated using
the obtained gradients, and the procedure is iterated until the parameters are
converged.
Once the final wavefunction is optimized, we can apply finally one further
Lanczos step on the wavefunction. If the energy changes qualitatively, this means
that the parameters are either not converged, or more generally that the wavefunction
is not good enough to catch the low energy physics of the ground state.
Moreover, the wavefunction can be used as a guiding function for the Green
function Monte Carlo (GFMC) procedure which allow to correct the correlations
of the observable, or it can also be used as an input for further Auxiliary-Field
Quantum Monte Carlo (AFQMC) calculations. Starting the simulation by assuming
random variational parameters, we find after usually a hundred iterations
a convergence of both the variational energy and the variance. However, since the
calculation is variational, we cannot rule out the possibility that this minimum
is only local. A typical Variance/Energy profile is shown in Fig. 6.10. Once our
wavefunction is minimized, we measure every observable, and we normalize every
quantity by the number of copper atoms in the lattice.
6.8 Current-current correlations in a small cluster
: Lanczos
Before doing the variational calculation, we have first considered the currentcurrent
correlations in small 8 copper lattice (24 sites) with respectively 9 and
10 holes (the corresponding doping are x =0.125% and x =0.25%). We considered
periodic boundary conditions. Such small clusters can be studied by exact
diagonalization (Lanczos). We found that the ground state is in the sector of the
Hilbert space S z = 0. By considering rotational (3 rotations), translational (7
translations) and the mirror symmetries, we can reduce the Hilbert space of 10