1. QHE discovered by Von Klitzing in 1980

2. Simplified explanation of IQHE
Why 2D?
Only the electrons at the edge of 2D electron gas are conducting in strong magnetic field and it’s ballistic transport at low temperature ~1K
The theory of Landau levels in magnetic field require z-direction confinement to achieve highly separated eigenstates
Plateau of quantum Hall conductance from Anderson localization caused by impurities
The upper line should have reverse orbital direction.
PhysRevB

3. Role of flux quantum in QHE
The degeneracy of a Landau level
Filling factor
If (an integer),
i Landau levels are fully occupied
Each electron is related with ONE flux quantum in IQHE

4. Lower Temperature / Higher Magnetic Field / Less Impurities
Discovery of FQHE
Lower Temperature / Higher Magnetic Field / Less Impurities

5. Difficulty in explanation FQHE under the framework of electron Landau levels in IQHE
In IQHE theory the filling factor has to be an integer, because the Fermi level must be at the localized states between Landau levels

6. Laughlin’s Approach by creating new quasi-particles
He use quasi-electrons and quasi-holes with fractional charge to explain the fractional plateaus of the experiment. (for v= 1/3)
Genius instinct: ‘report … wave functions that I feel are consistent with all the experimental facts’ in his PRL paper.
These quasi-particles were not observed until the 80s since they require strong interaction between electrons (Low K, pure 2DEG)

7. More theories based on Laughlin’s framework
Hierachy states by Haldane
Laughlin’s approach only explain states like 1/3, 1/5. Others like 2/5 need a new theory.
( e.g. m=3, p=1 …… )
Simply speaking, just like Laughlin’s quasi-electron generates from strongly correlated electrons, a hierarchy daughter state generates from its previous hierarchy state.
(Like iterations in programming)

8. More theories based on Laughlin’s framework
Composite fermions by Jain
Each electron is paired with several vortices of flux quantum (also a quasi-particle) to form a composite quasi-particle.
e.g. For 1/3, a electron is paired with 3 vortices
The increase of magnetic field can be described by creation of a flux quantum vortex. Such an unpaired vortex moving in the uncorrelated composite particles behaves as the IQHE of flux quantum, but the equivalent charge is q/3 in such a case.

9. Even denominators in FQHE
Discovered in 1987 by D. C. Tsui et al.
No Hall resistance plateau observed.
Explained by Pfaffian wave functions