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2.6. AUXILIARY-FIELD QUANTUM MONTE CARLO 49
Wick’s theorem holds for one spin configuration, and the observables can be
extracted
∑
from the Green function. Any expectation value of an observable Ô =
O ijkl c † i c† j c kc l taken in a spin state s =(s 1 , ..., s 2m )isgivenby:
ijkl
〈〈Ô〉〉 s = ∑ ijkl
O ijkl (G ′ jk G′ il − G′ ik G′ jl ) (2.92)
The average is then obtained by sampling the spin configurations :
∑
〈〈Ô〉〉 sω(s)
〈Ô〉 = s
∑
(2.93)
ω s
The sign problem occurs when ω s is not positive, or even worse when ω s is complex.
However, the phase of ω s can be put in the observable:
∑ (
)
〈〈Ô〉〉 ssign(ω s ) |ω(s)|
〈Ô〉 = s
∑
(2.94)
sign(ω s )|ω s |
s
The sign problem is nevertheless not entirely solved, since when the lattice becomes
large the denominator is strongly fluctuating and ∑ sign(ω s ) ≈ 0. In this
s
case the error bars become very large, and the simulation cannot lead to any
converged observables.
One further outcome of AFQMC is that the Ising spin configuration generated
during one simulation can also play the role of a good variational basis [72].
Therefore, for a given set of Ising configuration S = {σ i }, and once the corresponding
Hamiltonian matrix elements H λ,σ are known, we can find the variational
energy by solving the eigenvalue problem ∑ H στ c τ = E ∑ R στ c τ ,where
τ∈S
τ∈S
c τ are the eigenvectors, E is the variational energy, and R στ is the overlap matrix
of the states τ and σ.
2.6.1 Particle-hole transformation
To allow the possibility of a BCS slater determinant for ψ MF , it is more convenient
within the AFQMC frame to do a particle-hole transformation of the down spin
species:
d i =
d † i =
s
c † −i↓
c −i↓
After the transformation, the number of particles is written:
N e = N + ∑ 〈 〉
c † i c i − d † i d i
i
(2.95)