pdf, 9 MiB - Infoscience - EPFL
pdf, 9 MiB - Infoscience - EPFL
pdf, 9 MiB - Infoscience - EPFL
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2.1. VARIATIONAL MONTE CARLO 33<br />
and additionally:<br />
φ ij = − ∑ λ<br />
(<br />
U<br />
−1 ) iλ V λj = − ( U −1 V ) ij<br />
(2.13)<br />
The Monte Carlo algorithm by Ceperley [42] is applicable to wavefunction of<br />
the form (2.7). These functions were extensively studied by variational Monte<br />
Carlo [43, 44, 18, 19, 21, 32].<br />
To handle these functions, we have first to write them in terms of the real<br />
space fermionic configurations. A real space configuration of a fix number of<br />
fermions is defined as :<br />
and we project the wavefunction in this basis:<br />
|α〉 = c + R 1 ↑ ...c+ R Ne/2 ↑ c+ R ′ 1 ↓.....c+ R ′ Ne/2 ↓ (2.14)<br />
|ψ〉 = ∑ α<br />
〈α|ψ〉|α〉 (2.15)<br />
The projection of the wavefunction on a configuration state |α〉 is a sum of all<br />
the permutations of the fermions in this configuration:<br />
( ) Ne/2<br />
∑<br />
|ψ〉 = a(i, j)c + i↑ c+ j↓<br />
|0〉 =<br />
i,j<br />
∑ {<br />
a(i1 ,j 1 )...a(i } Ne/2,j Ne/2) c + i 1 ↑ c+ j 1 ↓ ......c+ i Ne ↑ c+ j Ne ↓<br />
|0〉 (2.16)<br />
i 1 ...i Ne/2 j 1 ...j Ne/2<br />
It is straightforward to prove that the coefficient is indeed a determinant (see<br />
Appendix A):<br />
〈α|ψ〉 =det{Q = {a i,j }} (2.17)<br />
Now that the wavefunction is written in the real-space representation, the observables<br />
can be easily computed using Monte Carlo calculations. For instance,<br />
the variational energy is calculated as follows:<br />
〈ψ|H|ψ〉<br />
〈ψ|ψ〉<br />
= ∑ α,β<br />
〈ψ|α〉<br />
〈α|H|β〉〈β|ψ〉 (2.18)<br />
〈ψ|ψ〉<br />
We can separate the sum in terms of a positive probability P (α) and of the local<br />
energy e(α):<br />
⎛<br />
⎞<br />
〈ψ|H|ψ〉<br />
〈ψ|ψ〉<br />
= ∑ α<br />
|〈α|ψ〉| 2<br />
∑<br />
〈α|H|β〉 〈β|ψ〉<br />
〈ψ|ψ〉 ⎜<br />
〈α|ψ〉 ⎟<br />
} {{ } ⎝ β<br />
} {{ }<br />
⎠<br />
P (α)<br />
e(α)<br />
(2.19)