Effective potentials in the integral quantum Hall effect

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, 386 (1981)] for the classical two-dimensional one-component plasma (2DOCP) at the special plasma parameter $\ensuremath{\Gamma}=2.$ This establishes that the quantum state with filling factor $\ensuremath{\nu}=1,$ associated with the integral quantum Hall effect, is precisely described by an effective classical potential $\ensuremath{\varphi}(r)=\ensuremath{-}2\mathrm{ln}r,$ so that a classical Boltzmann factor of 2DOCP form can replace the quantum Slater sum. Further, this Boltzmann factor exactly matches that constructed by Laughlin [Phys. Rev. Lett. 50, 1395 (1983)] to account for the fractional quantum Hall effect. Additional effective potentials for higher filling factors $\ensuremath{\nu}=2,3,\dots{}$ are obtained semianalytically from the exact Yvon-Born-Green integral equation and numerically from the approximate hypernetted-chain integral equation. They have the asymptotic form $\ensuremath{\varphi}(r)\ensuremath{\sim}\ensuremath{-}(2/\ensuremath{\nu})\mathrm{ln}r.$