Download Lecture 6: The Fractional Quantum Hall Effect Fractional quantum

Lecture 6: The Fractional Quantum Hall Effect
Website with slides and handouts:
http://home.uchicago.edu/ matthewroberts/compton/index.html
Fractional quantum Hall plateaus
Experiments have demonstrated that there are quantum Hall plateaus that can not be described
by the integer quantum Hall problem described last week. It must be the case that interactions
split individual Landau levels into separated bands such that we can find insulating intermediate
fillings.
402
H. L. Stormer
I Two-dimensional
electron correlation in high magnetic fields
2
/
--
,/
_---
-
/
n
l/3
I
MAGNETIC
Fig. 1. Low-temperature
diagonal
dimensional
electron system [2].
resistivity,
R, and
Hall
FIELD (Tesla)
resistivity,
R,, of the
high-mobility
(IL - 1.3 x 10” cm’/Vs)
two-
Figure 1:
The Hall
primitive plateaus
states in the lowest
Landau level
Left: Fractional quantum
appear
at low temperature and high field. Center: A
l/q and their electron/hole
symmetric states
1
1
as
well
as
equivalent
states
i
*
1
single Landau level with no interactions. Right: inInteractions split a Landau level into many
higher Landau levels have found a beautiful
interpretation
in terms up
of a novel,
separated bands.
Adding
the many-particle
number of all electrons would give ν = 1.
and many other fractions such and p/9, p/11,
p/l3
and even p/15 are emerging in p,, . These
finer structures are expected to develop into
truly quantized states in yet higher quality 2D
systems.
Experiments of the past years have elucidated
several properties of these quantum states:
1. They occur at fractional filling v - p/q of
Landau levels, as deduced from their position
on the B axis (v is the Landau level filling
factor).
2. They are associated with a quantum number
f = v = p/q, as determined from the concomitant quantization of the Hall resistance.
3. They are sensitive to disorder. Low-mobility
samples do not show a FQHE.
4. The FQHE has a characteristic energy scale
of only a few degrees kelvin.
5. There is a tendency for quantum states with
higher denominators to show weaker transport features.
6. Higher magnetic fields promote the observation of the FQHE.
/q
/q
ground state that can be expressed in an extraordinarily concise wave function first proposed
by Laughlin [4-61. The wave function turns out
to be exact for short range interactions and still
an excellent approximation
for the case of
Coulombic interaction. This is corroborated by
many sophisticated numerical few-particle calculations. This approach has been very successful
in explaining the most distinct implications of the
FQHE: the existence of energy gaps. The manyelectron state has the following properties:
1. It is a stable state at primitive filling factors
v=llm.
2. A case can also be made for v = i +- 1 im,
i = 1,2,3, . . . .
3. Its pair-correlation function indicates that it is
a quantum fluid.
4. Its elementary excitations are separated from
If we fill up one of these sub-bands we will find a fractional plateau. Exactly solving the problem
of interacting electrons is extremely difficult.
The lowest Landau level and holomorphy
Single-particle states in the lowest Landau level take a special form: they are holomorphic, that is
(up to a fixed prefactor) they are forced to be a function of only z = x + iy and not z̄ = x − iy. This
x2 +y 2
greatly constraints the structure of the wave-functions. Explicitly, ψ(x, y) = f (x + iy)e− 2 . This
can be generalized to multi-particle wave-functions in the lowest Landau level, which then must be
2
of the form f (z1 , z2 , . . .)e−|zi | /4 .
Laughlin’s Wave-function
Laughlin’s brilliant insight was that he could right down a very good guess:
Y
2
ψ=
(zi − zj )3 e−|zi | /4 .
i<j
Because the wave-function vanishes more quickly as electrons approach each other, the electrons
Figure 2:
The Laughlin space has electrons on average further spread out due to repulsive interactions.
are on average further apart: 1/3 as dense. In particular, the number of electrons we find is
one third of the regular filled Landau level, and so we must describe a ν = 1/3 state. Direct
h
calculations demonstrate that this guess correctly reproduces RH = (1/3)e
2 , RXX = 0, agreeing
with experiment.
Quasi-holes
These special states have very strange excitations:
topological
defects which have fractional charge
Q
Q
2
+e/3. Adding a defect gives the state ψ = (zi − ζ) i<j (zi − zj )3 e−|zi | /4 . If we remove a single
electron from the system the “hole” of missing charge will fractionalize into three defects of charge
+e/3 each. They also have fractional statistics, unlike bosons or fermions. Braiding two defects
around each uther takes the state
ψ → (−1)2/3 ψ,
and therefore we need to braid three times to return to our original configuration.
Figure 3:
Left: A single electron hole fractionalizes into three defects with fractional charge each. Right:
We have to exchange defects three times to return to our original configuration.