DescriptionThis dissertation studied the homogeneous electron systems in different regimes.In the low density regime, it was found that Wigner crystal phase doesn’t exist in the thermodynamic limit. A new method based on renormalization group equation was proposed to approach the low density limit. In the large N limit, the Wigner crystal phase doesn’t exist either. We also argued the existence of colloidal phase in the intermediate density regime, using several different methods. Finally, we put the system on a lattice and applied magnetic field. We defined a lattice version of BF theory and used that to derive Jain’s fractions.