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42 CHAPTER 2. NUMERICAL METHODS
The following change in the eigenvalues are obtained with first order perturbation
theory:
λ i = λ 0i + x T 0i ([δK]) x 0i, (2.53)
and the eigenvectors are given by:
x i = x 0i + ∑ j≠i
x T 0i ([δK]) x 0j
λ 0i − λ 0j
x 0j (2.54)
The derivatives are easily obtained from these formulae. In the case of degenerate
eigenvectors, the above formulae are no longer valid, and a more general
degenerate perturbation theory should be considered.
Research on the derivative of degenerate eigenvectors has been an area of significant
interest in recent years. Engineer have focused on the study of the so
called sensitivities, that are simply the derivatives of eigenvalues and eigenvectors
with respect to design parameters that make the matrix evolve. Degenerate eigenvalues
are unavoidable in practice, and, for instance, appear in the calculation of
the derivatives of the eigenvectors of the BCS Hamiltonian.
We sketch only here the main results obtained for the derivate of a degenerate
eigenvector (for more details the reader is referred to Ref. [53,54,55,56]). Let us
assume without loss of generality that the first eigenvalue λ 1 is degenerate with
multiplicity r. We define Φ as the matrix containing all the eigenvectors, Φ 1 as
the matrix of the degenerate eigenvectors, and Φ 2 as the matrix containing the
other vectors. With this definitions we have the trivial identity:
[ ]
Φ T λ1 1
K 0 (p)Φ=Λ= r 0
, (2.55)
0 Λ 2
It can be proved that the derivative of the degenerate eigenvectors φ 1 is :
Φ ′ 1 =Φ 2 (λ 1 I − Λ 2 ) −1 Φ T 2 D jΦ 1 (2.56)
Knowing the derivative of the U and V matrices allows to measure the O k observables
of equation (2.40) and the stochastic minimization procedure is now
well defined.
2.3.3 Correlated measurement minimization
One limitation of the stochastic minimization method, in our present implementation,
is that the formulae obtained for the derivation of a determinant are not
valid for the more general Pfaffian wavefunction, since there is no simple formula
for the derivative of a Pfaffian (see equation (2.48)). Therefore, in the case
of a Pfaffian Monte Carlo simulations, the gradients of the energy with respect
to the variational parameters have to computed numerically. This task is very
heavy and limits the minimization to a few parameters (about 10 parameters at