What about the spin degree of
freedom?
Other explanations for the fractional
quantized Hall effect
Fractions other than 1/m or their particlehole
conjugates
Unitary transformation
(Generalized gauge transformation)
Ψ
tr
{
r
j
}
=
Ψ
elec
{
r
j
}
×
Chern Simons magnetic field:
b
i
(
r
) =
∇
×
a
i
(
r
) = 2
π
ρ
i
(
r
)
MeanField Approximation
(Hartree Approximation)
Replace true ChernSimons magnetic field
b
by its
average value
<b> = 4
π
n
e
.
Replace
Coulomb interaction
by the
average
electrostatic
potential
, which is just a
constant
for a homogeneous
system. (We can choose constant = 0).
Get free fermions in an effective magnetic field
Δ
B = B  4
π
n
e
.
Define effective filling
p
≡
2
π
n
e
/
Δ
B
.
Compare to true
filling factor
f
= 2
π
n
e
/ B
.
Find
p
1
=f
1
2 ,
or:
f = p / (2p+1).
Examples
What about the spin degree of
freedom?
If the electron filling fraction is f=1/2
Meanfield ground state = filled Fermi Sea
k
F
= (4
π
n
e
)
1/2
If this is correct, then there is
no energy gap
,
no QHE
.
Should be able to calculate all properties of
ν
=1/2 state using
perturbation theory
, starting from the mean field state.
Perturbations include effects of v(
r
i

r
j
) and
fluctuations in the
Chern Simons field
Δ
b
i
≡
b
i
 < b >
.
Fluctuations are crucial in calculating transport and dynamic
properties, as well as for understanding the energy scale for
excitations
2 flux quanta of actual magnetic field per electron.
Effective magnetic field = B  <b> = 0.
Why should perturbation theory
work?
For the
quantized Hall states
, where the mean field theory predicts an
energy gap
between the ground state and excited states, it is
reasonable that perturbation theory should converge:
if perturbation is
not too strong, will not destroy the energy gap or change the character
of the ground state.
We find that fluctuations reduce the energy gap
significantly, but do not generally drive it to zero.
But what about
ν
=1/2, where there is no energy gap?
Landau
’
s theory of Fermi liquids
ZhETF
30
, 1058 (
1956
);
32
, 59 (
1957
);
35
, 97 (
1958
).
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 Fall '13
 BertrandI.Halperin
 Physics, Electron, Quantum Field Theory, Fundamental physics concepts, Condensed matter physics, Fermi