# Lauglin model

The system consist of 2 dimensional electron gas inserted into very strong uniform magnetic field at the temperature near to 0K. In sucha a system the fractional quantum hall effect (FQHE) is observed. The ground state proposed by Lauglin:$\Psi_N^{\nu}(z_1,..,z_N)=\Pi_{i<j}(z_i-z_j)^m e^{\frac{-\sum_{i}|z_i|^2}{4}},$

where*z* = *x* + i*y*, is an icompressive liquid and its elementary collective excitations: Π*i* = 1*N*(*z**i* − *z*0)Ψ*N**ν*(*z*1, .., *z**N*), behave like anyons placed in *z*0with the fractional phase: $\theta = \frac {2\pi}{\nu}$and they have fractional charge $\frac e {\nu}$. In order to see it we remove one electron and the wave function will be:$\Psi_N^{\nu}(z_2,..,z_N)=\Pi_{i=1}z_i\Pi_{i<j}(z_i-z_j)^\nu e^{\frac{-\sum_{i}|z_i|^2}{4}}$.This is an elementary excitation. Removing three electrons would bring us back to the ground state (with one of the electrons in the origin). So one quasi particle which in this case is in fact quasi-hole has an electric charge of $-\frac e \nu$.