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48 CHAPTER 2. NUMERICAL METHODS
with φ † m = ∑ j
c † jm and Φ′ = e U Φ, where U is a square matrix whose elements
are given by U ij and B = e U is therefore an N × N square matrix as well. In
conclusion, the operation of ˆB on |φ〉 simply involves multiplying an N × N
matrix by an N × M matrix. This allows to apply the exponential of the kinetic
term e λK on the Salter determinant |ψ MF 〉. Furthermore, for two states |L〉 and
|R〉 represented by Slater determinants L and R respectively, the single-particle
right and left Green functions are written as :
〈
〉
∣
L ∣c †
G L ij ≡ i c j∣ R [
= R ( L T R ) ]
−1
L
T
(2.85)
〈L |R〉
ij
〈
〉
∣
L ∣c i c † G R ij ≡ j∣ R [
= δ ij − R ( L T R ) ]
−1
L
T
(2.86)
〈L |R〉
ij
In order to evaluate all the expectation values of the observables, we generate a
Monte Carlo sample by using the importance sampling with the following weight
function :
ω = det ( )
ψMF T eV (u1) e λ1K e V (u2) e λ2K ...e λ2K e V (u2m−1) e λ1K e V (u2m) ψ MF (2.87)
where {u i } are the Ising spins of the different species i =1, .., 2m lying on the
lattice. The Monte Carlo algorithm consists to update the Ising variables, from
the old s i to the new one s ′ i , the and the ratio r = ω′ /ω is calculated to determine
whether to accept or reject the new configuration. In the process updating the
spin, the Green function can be used to calculate very fast the ratio of the the
weights r:
r = det (1 + G L ∆) (2.88)
where ∆ = ( e V ′ −V − 1 ) and the two states L and R (in the notations of equation
(2.85)) that define G L are :
L = ψMF T eV (u1) e λ1K e V (u2) e λ2K ...e V (u i)
(2.89)
R = e λiK ...e V (u2m−1) e λ1K e V (u2m) ψ MF (2.90)
u i is the spin species that is changed during the move. When a change of the
spin of the corresponding species is accepted, the Green function is updated like
:
G ′ L = G L − G L∆(1 − G L )
(2.91)
1+(1 − G L )∆
Moreover, Steve White and collaborators have proposed a very efficient way to
carry on the simulations [62] by proposing moves successively for each of the
species of spin, that allows for a fast sweep over all the site of the lattice and over
all the species of spin. Finally, since the AFQMC is a non-interacting theory,