In 1982, Tsui etc. discovered fractional quantum Hall effect(FQHE).The FQHE occurs when the Landau level filling factor
where p and m are integers. m can be either odd or even numbers. Figure 8 is a typical result.
Fractionally quantized Hall resistance is not possible for non-interaction electrons. By introducing an interaction potential , Eq. (13) becomes
By restricting our discussion to the extreme quantum limit in which the Landau level degeneracy is large enough that all electrons can be accommodated within the lowest Landau level, the single-partilce wavefunctions in the lowest Landau level can be written as
where z=x+iy. Note that these wavefunctions describe electrons located within one magnetic length of a circle centered on the origin and enclosing an area
Any many-electron wave function formed entirely within the lowest Landau level must be a products of one-electron orbitals for each coordinate which are of the form given by Eq. (25). According to Laughlin, the wave function can be written as
This wave function corresponds to a bound liquid droplet. For a large but finite
number of electrons, N, the maximum power to which 
(or any other coordinate) in 
is
and hence the area occupied by the wave function, according to Eq. (26), is
It follows from Eq. (29) that in the thermodynamic limit A/2
. Thus as the electron density is increased at constant magnetic field so that the filling factor crosses 
we go from a regime where it is possible to form states with 
to a regime where 
vanishes only as 
. This qualitative change in the ability of electrons to avoid each other causes a jump in the chemical potential when the filling factro crosses 1/m and, invoking Eq. (20), also causes the Hall conductance to be quantized at 
at filling factor 1/m.
Detailed discussions on FQHE and fractional charge quasiparticles can be found on the listed references.