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2.1. VARIATIONAL MONTE CARLO 33
and additionally:
φ ij = − ∑ λ
(
U
−1 ) iλ V λj = − ( U −1 V ) ij
(2.13)
The Monte Carlo algorithm by Ceperley [42] is applicable to wavefunction of
the form (2.7). These functions were extensively studied by variational Monte
Carlo [43, 44, 18, 19, 21, 32].
To handle these functions, we have first to write them in terms of the real
space fermionic configurations. A real space configuration of a fix number of
fermions is defined as :
and we project the wavefunction in this basis:
|α〉 = c + R 1 ↑ ...c+ R Ne/2 ↑ c+ R ′ 1 ↓.....c+ R ′ Ne/2 ↓ (2.14)
|ψ〉 = ∑ α
〈α|ψ〉|α〉 (2.15)
The projection of the wavefunction on a configuration state |α〉 is a sum of all
the permutations of the fermions in this configuration:
( ) Ne/2
∑
|ψ〉 = a(i, j)c + i↑ c+ j↓
|0〉 =
i,j
∑ {
a(i1 ,j 1 )...a(i } Ne/2,j Ne/2) c + i 1 ↑ c+ j 1 ↓ ......c+ i Ne ↑ c+ j Ne ↓
|0〉 (2.16)
i 1 ...i Ne/2 j 1 ...j Ne/2
It is straightforward to prove that the coefficient is indeed a determinant (see
Appendix A):
〈α|ψ〉 =det{Q = {a i,j }} (2.17)
Now that the wavefunction is written in the real-space representation, the observables
can be easily computed using Monte Carlo calculations. For instance,
the variational energy is calculated as follows:
〈ψ|H|ψ〉
〈ψ|ψ〉
= ∑ α,β
〈ψ|α〉
〈α|H|β〉〈β|ψ〉 (2.18)
〈ψ|ψ〉
We can separate the sum in terms of a positive probability P (α) and of the local
energy e(α):
⎛
⎞
〈ψ|H|ψ〉
〈ψ|ψ〉
= ∑ α
|〈α|ψ〉| 2
∑
〈α|H|β〉 〈β|ψ〉
〈ψ|ψ〉 ⎜
〈α|ψ〉 ⎟
} {{ } ⎝ β
} {{ }
⎠
P (α)
e(α)
(2.19)