Fractional charge in the fractional quantum hall system

The fractional quantum Hall effect (FQHE) has uncovered a number of unique and unexpected quantum behaviors of two-dimensional electrons. For the integer quantum Hall effect (IQFE), the conductance of the bulk is quantized as σxy = n(e2/h) for integer n. This behaviors can be explained as the complete filling of a Landau level in the 2D free electron system with an external magnetic field. Since there is no energy gap when the system is in a half-filled Landau level, but a gap between 2 different Landau levels, it is easy to understand the integer quantization of the hall conductance in the quantum picture. On the other hand, the FQHE is comes from the strongly interacting elections such that the conductance of the bulk is quantized as σ = ν(e2/h) for a fractional number ν (e.g. ν=1/3, 1/5)). The fractional number of ν has a direct consequence that the system is stable when the Landau level is partially-filled (i.e. only ν of a level is filled). And this effect can not be explained in the weakly interacting electron picture and thus a new picture of strongly correlated electrons is required. Similar to the BCS theory in superconductivity, a variational functional called Laughlin state is used as an functional state of the simpel FQHE (simple means ν = 1/n for odd integer n>1, for example ν=1/3). The laughlin wavefunction ψ0 is: