CASINO manual - Theory of Condensed Matter

[
where the approximation
ρ (1)T
α
(r)ρ (1)T
β
(r)
] R
≈ ρ
(1)T R
Another method, proposed here, is to define a modified TBDM,
˜ρ (2)
αβ (r 1, r 2 ; r ′ 1, r ′ 2) = N α (N β − δ αβ )
and compute the condensate fraction as
α
∫
|Ψ(r1 , r 2 )| 2 [ Ψ(r ′ 1 ,r′ 2 )
(r)ρ (1)T R
β
(r) has been used.
Ψ(r − Ψ(r′ 1 ,r2) Ψ(r 1,r ′ 2 )
1,r 2) Ψ(r 1,r 2) Ψ(r 1,r 2)
∫
|Ψ(R)| 2 dR
]
dr 3 . . . dr N
, (407)
c = Ω2
lim
N α r→∞ ˜ρ(2)T R
αβ
(r) , (408)
which also achieves the same purposes, but benefits from correlated sampling and is somewhat cheaper
to evaluate, as the wave function updates required for the OBDM can be re-used in the evaluation of
the TBDM.
The three condensate fraction estimators have been computed for a two-dimensional electron–hole
bilayer (r s = 5, d = 1, N e = N h = 58), and are represented in the figure below. From top to bottom,
the TBDM estimator [Eq. (403)], the TBDM-OBDM estimator [Eq. (406)], and the modified-TBDM
estimator [Eq. (408)]. The advantages of Eq. (408) are evident in the short-range region, while the
long-range region seems to display a slightly noisier behaviour than the other two.
0.5
(Ω 2 /N α
) ρ αβ
(2)TR
(r)
Condensate fraction estimators
0
0.5
0
0.5
(Ω 2 /N α
) [ρ αβ
(2)TR
(r)-ρα
(1)TR
(r)ρβ
(1)TR
(r)]
(Ω 2 /N α
) ρ’ αβ
(2)TR
(r)
0
0 1 2 3 4 5 6 7
r/r s
33.6 Momentum density
K eyword: mom den
The momentum density is the Fourier transform of the one-body density matrix, and is explicitly
calculated as such. The k-vectors of the transformation are the reciprocal-lattice vectors of the simulation
cell. Notice that for a homogeneous system these vectors are affected by keyword k offset,
hence running different accumulations with different k offset values allows evaluating the momentum
density on an arbitrarily fine grid.
33.7 Localization tensor
K eyword: loc tensor
An early success of quantum mechanics was to explain the distinction between metal and non-metal
using band theory. The system is metallic if the conduction and valence band overlap and more than
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