Chemistry Reference
In-Depth Information
overlap charge density between any two single particle states corresponding
to different valleys are identically zero.
Using Laughlin's plasma analogy, the expression
[
82
]
for energy per
electron is found to be
e
2
0
r
62
e
2
0
B
g
↑
ττ
(
U
FM
≈ π
(
r
)
−
1)
rdr ≈−
0
.
(26)
We will argue that we can get a lower energy in our composite fermi sea.
The wave function for the composite fermi sea we are proposing is given by
P
LLL
e
−
i<jστ σ
τ
f
(
|
r
iστ
−
r
jσ
τ
|
)
]=
ˆ
Ψ
F
G
[
z
i
k
·
r
στ
i<j
e
z
iστ
− z
jστ
)
2
e
−
i
|z
iστ
|
2
4
(
(27)
στ
There are 4 composite fermi sea of electrons, corresponding to two spin and
two valley indices. Each composite fermi sea has a filling fraction
=
2
ν
containing
N
φ
2
electrons. There is a common fermi momentum for each
fermi sea given by the expression:
k
F
≡
√
4
is electron density.
We have also introduced a two particle short range Jastrow factor
through the function
πρ
.Here
ρ
) between any two electrons. This introduces cor-
relation between two electrons having different valley and spin quantum
numbers. Since we have only a half filled Landau level valley exchange
and spin exchange scattering within n = 0 levels are not completely Pauli
blocked and they build short distance holes in the respective g(r), reduce
the coulomb repulsion energy.
Our short range Jastrow correlation for electrons with different quan-
tum numbers is unusual in quantum Hall situation. Long ranged Laughlin-
Jastrow correlation are normally used, as they also take care of antisym-
metry automatically. To keep a gapless fermi sea and at the same time
maintain half filling in each Landau sub band, short range Jastrow func-
tion seems essential in the present case. When we attempt to use Laughlin-
Jastrow factor, it either changes the mean density (expands the Laughlin
drop significantly) or we seem to get incompressible states. Since one im-
plements lowest Landau level projection at the end, in principle our short
range Jastrow factor is allowed. We have no proof that our short range f(r)
maintains the mean density per Landau level to be half.
In the above variational wave function, the projection to n = 0 Landau
level is done by
ˆ
f
(
r
P
LLL
.
This projector effectively replaces ¯
z → ∂
z
.For
≡
k
2
i
(
kz
+
kz
)
2
i
(
kz
+
k∂
z
)
and
˜
1
1
e
i
k
·
r
=
e
i
k
·
r
ij
→
example,
e
→ e
f
(
r
ij
)
f
(
k
)
k
2
i
(
kz
ij
+
k∂
z
ij
)
.Here
˜
1
f
(
k
)
e
k ≡ k
x
+
ik
y
.
