2.6M - 1. Institut für Theoretische Physik - Universität Stuttgart

Summary
The acceptance probability is given by equation (2.55). Due to the presence of an
external magnetic field, the ground state wavefunction will be complex-valued, just as the
local energy. To overcome the fixed-phase approximation [1] and also to ensure that the
simulation takes place in real space, the phase factor can be considered as a further weight
of the branching term (see equation (2.68)). The phase factor ϕ( � R ′ , ∆τ) = e i∆τ Im EL(�R ′ )
is a complex weight factor Υ (see equation (2.69)) and is multiplied by the local energy
to calculate the statistical average. This technique is called released-phase DQMC [10].
The ground state energy is given by:
E0 =
b
1 �max
bmax − bs + 1
b=bs
1
s max
s� max
s=1
� jmax
j
j =1 EL (� R ′ ) · Υj (s)
�jmax j =1 Υj (s)
. (9)
The realisation of the computer algorithm is displayed in a Nassi-Shneiderman diagram
in figure 1.
Guiding Wavefunction: The Hamiltonian of a many-electron atom in an external magnetic
field reads:
ˆH =
N�
i=1
�
− 1
� 2 ∂
2 ∂x 2 +
i
∂2
∂y 2 i
+ ∂2
∂z 2
� �
− iβ
i
with � B = B�ez, β = B/B0 and B0 = 2α2 m 2 e c 2
xi
∂
∂yi
− y ∂
�
∂xi
+ β2 (x 2
i + y 2 i )
+ βˆσzi −
2
Z
|�ri|
�
+ 1
2
N�
i,j =1
j �=i
e� ∼ = 4.7 · 10 5 T, α = e2
4πɛ0�c
1
|�ri − �rj |
(10)
. It is employed
an ansatz for the guiding wavefunction ΨG as a product of a guiding wavefuntion Ψ ad
determined by adiabatic approximation and the Jastrow factor Ψ JF . The adiabatic
guiding wavefunction Ψad itself is based on an approach with a Slater determinant
Ψ = 1
�
�
� ψ1(�r1)
�
� ψ2(�r1)
√ �
N ! �
�
.
�ψN
(�r1)
ψ1(�r2)
ψ2(�r2)
.
ψN (�r2)
· · ·
· · ·
. ..
· · ·
�
ψ1(�rN ) �
�
ψ2(�rN ) �
�
� ,
. �
�
ψN (�rN ) �
(11)
composed of single-particle wavefunctions ψi. Those single-particle wavefunctions are
assumed as products of a z-dependent, not yet known wavefunction Pνm(z) of the longitudinal
Coulomb excitation along the z axis (ν stands for the number of nodes), a ρ
and ϕ dependent Landau state Φnm(ρ, ϕ) with the energy levels En = �ωc(n + 1),
with
2
n = 0, 1, 2, . . . and projection of the angular momentum m (m = 0, −1, −2, . . . ) on the
z axis, and a spinor state χ(¯s):
128
ψi(�r, ¯s) = ψνnm(z, ρ, ϕ, ¯s) = Pνm(z) · Φnm(ρ, ϕ) · χ(¯s) . (12)