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6.9. BENCHMARK OF VMC 153
6.9 Comparison VMC/AFQMC/Lanczos
Exact-diagonalization calculations are interesting for comparison with non-exact
techniques, like variational Monte Carlo, though they are restricted to very small
clusters and can only give limited information on the long-range properties. Consequently,
we propose in this section a benchmark of the quality of the wavefunction
(6.20). We first compare energies for a small 8 copper cluster (24 sites)
with 10 holes (see Table 6.1). Interestingly, we find that our variational wavefunction
is very close in energy to the true ground state when the full Jastrow
and the RVB parameters are considered (wavefunction RVB/JA) . The energy of
the best variational wavefunction can be systematically improved by minimizing
the new wavefunction |ψ ′ 〉 =
(
1+αĤ
)
|ψ〉, whereα is a variational parameter
(this procedure is called Lanczos Step). We obtain an improved energy (wavefunction
1LS/RVB/JA, see Table 6.1), which has an energy similar to what can
be obtained with the Green function Monte Carlo method (GFMC). The latter
method suffers from the minus sign problem, and we have to use the fixed
node (FN) approximation to overcome this problem, using the long-range Jastrow
wavefunction as a guiding function. This latter procedure gives improved
variational energies. The GFMC, though it improves drastically the energy, can
only be used in our implementation for real wavefunction 3 , and therefore is not
suited to improve the circulating current wavefunction. Another problem of the
GFMC is that the calculations of non-diagonal observables needs a very large
amount of computer time. The auxiliary-field quantum Monte Carlo (AFQMC)
technique allows also to improve the wavefunction. AFQMC considers the improved
wavefunction |ψ ′ 〉 = e λ 1 ˆK e µ 1Û × ... × e λm ˆK e µmÛ |ψ〉, whereλ 1 ...λ m and
µ 1 ...µ m are variational parameters, m =1, .., 5 is the number of iterations. In
AFQMC |ψ〉 is allowed to be complex, and ˆK and Û are respectively the operators
of the kinetic and on-site repulsion parts of the Hubbard Hamiltonian (we
apply the AFQMC method on the orbital current instability). The limitation of
AFQMC is that it cannot deal with the long-range Jastrow factor that optimizes
our variational wavefunction. The reason for this is related to the well known
sign problem which occurs in Quantum Monte Carlo simulations, though it is
related in AFQMC to the choice of the Hubbard-Stratanovitch transformation
that maps the correlated problem to uncorrelated fermions coupled to an external
fluctuating field. Hence, in the framework of AFQMC calculations, we drop out
the full Jastrow factor and keep only the local Gutzwiller projection. The energy
for different iterations of the AFQMC method are shown in Fig. 6.13 and Table
6.10.1, and convergence is apparently obtained for m =5.
In conclusion, each quantum method has some advantages and some restrictions,
but our best simple variational wavefunction leads to energies that are
3 The so-called fixed phase approximation should be used to use the GFMC with complex
wavefunctions.