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Dressed Multi-Particle Electron Wave Functions

​
show
ψ
2
sign(ψ)
Fermi wave vector
k
F
1
backflow parameters:
a
1
r
0
1
The detailed structure of many-particle wave functions of electrons (such as in any conductor, semiconductor, or superconductor) is a fascinating current physics problem that still contains many unexplored aspects. For noninteracting fermions, the stationary wave function can be represented as a Slater determinant
Ψ(
r
1
,
r
2
,…,
r
N
)∼det
expi
k
i
r
j
i=1,…,N
j=1,…,N
where
N
denotes the number of electrons and the
k
i
are the wave vectors from within the (discretized) Fermi surface/sphere. Many-particle effects (due to the electron–electron Coulomb interaction) can be taken into account semi-phenomenologically through backflow effects, realized through a change to collective variables
r
i
⟶
r
i
, where
r
i
=
r
i
+
N
∑
j=1
j≠i
η
r
i
-
r
j
(
r
i
-
r
j
)
.
Here, the simple parametrization
η(r)=
3
a
3
r
+
3
r
0
has desirable asymptotic properties.
If in the resulting function
Ψ
(
r
1
,
r
2
,…,
r
N
)
the positions
r
1
,
r
2
,…,
r
N-1
of the first
N-1
particles are fixed, a real-valued reduced wave function of
r
N
results that can be visualized in the 2D case as a contour plot.
The wave vectors within the Fermi sphere are discretized through Born–von Karman boundary conditions applied to the wave function. As a result, there is a one-to-one correspondence between the number of electrons and the Fermi wave vector and at certain
k
F
the wave functions will change their appearance due to the addition of new electrons.
This Demonstration visualizes the square of the absolute value of the reduced wave function as a contour plot on the regions where the reduced wave function is positive or negative. The nodal lines of the resulting one-particle function are shown as blue curves and the movable white points indicate the fixed electrons at position
r
1
,
r
2
,…,
r
N-1
. Depending on the backflow parameters, the resulting wave function can develop regions with intricate fractal structure.
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