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