Precise coincidence of effective potentials in the integral and fractional quantum Hall effects

Abstract

The exact n-body distribution functions are calculated for a two-dimensional, noninteracting quantum electron gas in an external magnetic field for any temperature and density. At low temperature and filled lowest-Landau-level, these functions are identical to the exact distribution functions obtained by Jancovici [Phys. Rev. Lett. 46 (1981) 386] for the classical two-dimensional one-component plasma at the special plasma parameter Γ=2, establishing that the quantum state with filling factor ν=1, associated with the integral quantum Hall effect, is precisely described by an effective classical potential φ( r)=−2ln r. Further, this Boltzmann factor exactly matches that constructed by Laughlin [Phys. Rev. Lett. 50 (1983) 1395] to account for the fractional quantum Hall effect.