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34 CHAPTER 2. NUMERICAL METHODS
Since P (α) is a positive function, the sum can be sampled by Monte Carlo calculations
and no spurious sign problem occurs. The probability of a transition
from state X toanewstateX ′ is given by:
(
)
P (X → X ′ )=min 1, |det(A(X′ ))| 2
|det(A(X))| 2 (2.20)
Therefore, during a simulation the main task consists in calculating ratios of
determinants. Although the brute force calculation of a determinant scales like
N 3 numerical operations, we can reduce it to N operations [42] by considering
optimized formulae that allow to compute directly ratios of determinants of two
matrices A ′ /A, where the two matrices A and A ′ differ only the column j:
w j = det(A′ )
det(A) = ∑ k
A −1
j,k v k (2.21)
where v is the column j of the new matrix A ′ . Similar formula can be obtained
when several lines and columns are updated at the same time in the new matrix
A ′ (for further details see ref. [45]). However, the formula (2.21) involves the
inverse of the matrix A. The matrix A −1 can be updated when a set of rows
and/or columns are changed in the matrix A 1 each time that a Monte Carlo
move is accepted, and the update can be done in N 2 numerical operation [42].
In the simple implementation of the Metropolis algorithm, the moves are often
rejected and therefore the formula 2.21 is the bottleneck of the simulation. A
even much improved algorithm, that updates directly the weights w j and does
not compute the inverse of the matrix A, is currently used by Sandro Sorella and
collaborators. This latter algorithm computes the weight w j in a single operation.
However, the implementation of such an algorithm is much more involved and is
beyond the scope of this dissertation.
2.1.1 Degenerate open shell
As discussed in the previous section, the variational wavefunction is built by
piling the quasi-particle states up to the Fermi energy, in the case of a tightbinding
wavefunction, or up to the chemical potential µ when the BCS pairing
is considered. When the energy of the single-particle state is non-degenerate
the resulting wavefunction is well defined. However, the wavefunction becomes
ambiguous when the energies are degenerate. This happens in finite size clusters,
due to the symmetry of the lattice: The k points are lying on shells of same energy.
This gives artificial degeneracies related to the different filling possibilities.
1 The hopping of one particle involves a change of row or a column in A, andtheswapof
an up and down particles involves the change of both a row and a column in A.