Fractional quantum Hall effect and Chern-Simons gauge theories

We present a theory of the fractional quantum Hall effect (FQHE) based on a second-quantized fermion path-integral approach. We show that the problem of interacting electrons moving on a plane in the presence of an external magnetic field is equivalent to a family of systems of fermions bound to an even number of fluxes described by a Chern-Simons gauge field. The semiclassical approximation of this system has solutions that describe incompressible-liquid states, Wigner crystals, and solitonlike defects. The liquid states belong to the Laughlin sequence and to the first level of the hierarchy. We give a brief description of the FQHE for bosons and anyons in this picture. The semiclassical spectrum of collective modes of the FQHE states has a gap to all excitations. We derive an effective action for the Gaussian fluctuations and study the hydrodynamic regime. The dispersion curve for the magnetoplasmon is calculated in the low-momentum limit. We find a nonzero gap at ${\mathrm{\omega }}_{\mathit{c}}$. The fractionally quantized Hall conductance is calculated and argued to be exact in this approximation. We also give an explicit derivation of the polarization tensor in the integer Hall regime and show that it is transverse.