pdf, 9 MiB - Infoscience - EPFL
pdf, 9 MiB - Infoscience - EPFL
pdf, 9 MiB - Infoscience - EPFL
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44 CHAPTER 2. NUMERICAL METHODS<br />
The variational energy is given by the lowest eigenvalue of the eigen-system:<br />
∑<br />
H ij c j = E ∑ S ij c j (2.63)<br />
j<br />
j<br />
The expectation value of other observables is given by:<br />
∑<br />
c † i c jO ij<br />
〈Ô〉 =<br />
ij<br />
∑<br />
c † i c (2.64)<br />
jS ij<br />
ij<br />
where O ij = 〈φ i |Ô|φ j〉. Moreover, we can follow similar steps as in section 2.3.3<br />
to calculate the overlap matrix S ij and the matrix element O ij :<br />
〈ψ 2 |Ô|ψ 1〉 = ∑ ( )<br />
∑<br />
|〈ψ 1 |x〉| 2 〈x|O|y〉 〈y|ψ 1〉 〈x|ψ 2 〉<br />
(2.65)<br />
〈x|ψ<br />
x<br />
y<br />
1 〉 〈x|ψ 1 〉<br />
By running a simulation for ψ 1 , we can generate the configurations {x} obtained<br />
by a Metropolis algorithm with the probability p = min(1, |〈x|ψ 1 〉| 2 ), and the<br />
matrix element 〈ψ 2 |Ô|ψ 1〉 are then easily obtained through the above equation.<br />
2.4 Optimization of the wavefunction : Lanczos<br />
Step<br />
Once the energy of the wavefunction is satisfactorily optimized, we can systematically<br />
[57] improve the quality of the ( energy)<br />
and of the variance by considering<br />
the extended wavefunction |ψ ′ 〉 = 1+λĤ |ψ〉. This procedure is similar to<br />
what is done in a Lanczos calculation, though we apply here only one step of<br />
the full Lanczos calculation. The wavefunction ψ ′ can be sampled by using the<br />
following relations:<br />
ψ α ′ (x′ )<br />
ψ α ′ (x) = 〈x′ |(1 + αH)| ψ〉<br />
〈x |(1 + αH)| ψ〉 = ψ ( )<br />
(x′ ) 1+αEx ′<br />
(2.66)<br />
ψ (x) 1+αE x<br />
ψ α (x ′ )/ψ α (x) is the only necessary quantity that we need to perform the variational<br />
calculations. These calculations are however relatively heavy, and the cpu<br />
time for further Lanczos steps increases very fast, such that a two-Lanczos step<br />
calculation can only be done on small ≈ 20 site clusters. The parameter α is<br />
an additional variational parameter, and a very powerful method was proposed<br />
in Ref. [58] to find the optimal α for one Lanczos step calculations. The one<br />
lanczos step applied on the optimized variational function is a further test for<br />
the reliability of the method. Indeed, the variational results will certainly not<br />
be well converged if the energy changes significantly after applying one Lanczos<br />
iteration onto the optimized wavefunction.