CASINO manual - Theory of Condensed Matter

where {R i } i=M
i=1 is a set of configurations distributed according to |Ψ(R)| 2 .
The estimate of the value of the rotational average of a function f(r) in the bth bin Ω b , is
∫ rb
[∫
r
f b =
b−1
f(r ′ )δ(|r ′ | − r)dr ′] ∫
dr
Ω
∫ rb
= b
f(r ′ )dr ′
∫ = 1 ∑m b
f(r ′
r b−1
S(r)dr
Ω b
dr ′ m
i) , (399)
b
where {r ′ i } is a set of points uniformly distributed in the simulation cell, and m b is the number of
such points falling into the bth bin.
Hence the estimate of the translational-rotational average of the OBDM is
and for the TBDM we have
ρ (1)T R
α (r b ) ≈ N α
ΩMm b
ρ (2)T R
αβ
(r b ) ≈ N α(N β − δ αβ )
Ω 2 Mm b
∑
∑
M m b
i=1 j=1
∑
∑
M m b
i=1 j=1
i=1
Ψ(r i 1 + r ′ j )
Ψ(r i 1 ) , (400)
Ψ(r i 1 + r ′ j , ri 2 + r ′ j )
Ψ(r i 1 , ri 2 ) . (401)
In order to improve the statistics, one can take advantage of the antisymmetry of the wave function.
33.5.3 Condensate fraction
A well-known limit of the fermionic TBDM for α ≠ β and N α = N β is [86, 87]
lim
|r|→∞ ρ(2) αβ (r 1, r 2 ; r 1 + r, r 2 + r) = cN α |ϕ(|r 2 − r 1 |)| 2 , (402)
where c is the condensate fraction and ϕ(r) is a function such that ∫ |ϕ(r)| 2 dr = 1/Ω. The OBDM is
zero in this limit.
Applying the limit to Eq. (397) and substituting, we can estimate c as
c = Ω2
lim
N α r→∞ ρ(2)T R
αβ
(r) . (403)
33.5.4 Improved estimators
Consider a system of two distinguishable particles with wave function Ψ(r 1 , r 2 ) = φ(r 1 )φ(r 2 ). The
two particles are independent from one another. Equations (390) and 391 can combined, giving
ρ (2)
αβ (r 1, r 2 ; r 1 + r ′ , r 2 + r ′ ) = ρ (1)
α (r 1 ; r 1 + r ′ )ρ (1)
β (r 2, r 2 + r ′ ) , (404)
from which we can see that in the independent-particle case the TBDM does nothing but reflect the
one-body properties of the system.
An opposite example is a system of two completely paired particles with wave function Ψ(r 1 , r 2 ) =
δ ɛ (r 1 − r 2 ), where δ ɛ (r) is a localized function of range ɛ that tends to the Dirac delta as ɛ → 0. In
this case, the OBDM is zero for all |r ′ | > ɛ, whereas the TBDM is
ρ (2)
αβ (r 1, r 2 ; r 1 + r ′ , r 2 + r ′ |Ψ(r 1 , r 2 )| 2
) = ∫
|Ψ(r1 , r 2 )| 2 . (405)
dr 1 dr 2
In this case the value of the TBDM is completely due to two-body effects.
To improve the estimation of the condensate fraction c it is necessary to eliminate the one-body
effects of the form of Eq. (404), while keeping the pure two-body effects of Eq. (405) intact. One
such approach [88] is to subtract ρ (1)
α (r 1 ; r 1 + r ′ )ρ (1)
β (r 2, r 2 + r ′ ) from ρ (2)
αβ (r 1, r 2 ; r 1 + r ′ , r 2 + r ′ ). It is
clear from Eq. (404) that this removes one-body effects, and does not bias the value of c because the
OBDM is zero when Eq. (402) holds. The condensate fraction could then be estimated as
c = Ω2
N α
{
lim
r→∞
ρ (2)T R
αβ
[
(r) −
194
ρ (1)T R
α
]}
(r)ρ (1)T R
β
(r) , (406)