Chemistry Reference
In-Depth Information
×
valley). That is, a plane wave state is either filled with four electrons with
four possible spin and valley quantum numbers or is completely empty. In
view of this it satisfies Fock's cyclic boundary condition. The above SU(4)
singlet state is consistent with a two body interaction which is very nearly
SU(4) symmetric for graphene in the n = 0 Landau level [72].
As before, interaction energy of an electron with quantum numbers
Our composite fermi sea wave function is a global SU(4) singlet (spin
στ
with rest of the electrons, in the composite fermi state is given by
e
2
0
r
ττ
σσ
(
U
CF
=
π
(
g
r
)
−
1)
rdr
(28)
σ
τ
We will estimate the energy of this state in two approximations made
|
2
. As the first approximation we put f(r) = 0. We are left
with products of four independent composite fermi sea, each with a filling
ν
Ψ
F
G
on
|
[
z
]
. The energy of this state has been calculated in the literature
[
60;
62
]
for spin polarized
=
2
=
2
ν
composite fermi state as
46
e
2
0
B
U
CF
≈−
.
0
(29)
This energy is about 25 percent higher than the ferromagnetic integer Hall
state (Eq. (3)). We will show approximately that inclusion of a short range
Jastrow correlation can give more than 25 percent energy gain and make
our composite fermi sea stable. Jastrow factor, by construction is capable
of generating hole in the radial distribution function
g
σσ
). Physically
this possibility arises from possibility of virtual interband scattering be-
tween electrons having different
ττ
(
r
quantum numbers within the n = 0
Landau levels. Fortunately these scattering process are not completely
Pauli blocked, because each Landau sub band is only half filled.
Without making a detailed calculation, we can estimate that a 25 per-
cent improvement of energy is possible, by the following rough argument.
From two particle scattering point of view, we have both direct and ex-
change scattering among two electrons with parallel spins. The correspond-
ing matrix elements are comparable, and decay in a Gaussian fashion for
localized gaussian orbitals separated beyond magnetic length
στ
B
.Forparti-
cles with different quantum numbers the exchange term is absent. That is,
for our interband scattering, exchange scattering, half of the total processes
are absent. Further, the interband scattering can take place to empty single
particle states which are only half of the total number of states available
(the rest are Pauli blocked). In view of this we approximate the amount
