Composite Fermion hierarchy of quantum-Hall states

Abstract

Abstract In Jain's composite Fermion (CF) approach to the fractional quantum-Hall effect, flux tubes carrying an even number, 2 p 0 , of flux quanta are attached to each electron. In the mean-field approximation (MFA) the CFs have an effective filling factor ν * 0 related to the electron filling factor ν 0 by ( ν * 0 ) −1 =( ν 0 ) −1 −2 p 0 . If ν * 0 is equal to an integer n 1 , then ν 0 = n 1 (1+2 p 0 n 1 ) −1 describes an incompressible state belonging to the principal Jain sequence. If ν * 0 is not an integer, one can write ν 0 *= n 1 + ν 1 where ν 1 describes the quasiparticle occupancy of the partially filled shell. Ignoring all but this partially filled shell, the CF transformation can be applied to these quasiparticles to obtain ( ν * 1 ) −1 =( ν 1 ) −1 −2 p 1 . If ν * 1 is not an integer, we can repeat the process to obtain ν −1 l =2 p l +( ν l +1 + n l +1 ) −1 . If all n l are equal to unity, this relation generates Haldane's continued fraction. The CF hierarchy picture not only predicts the values of the degeneracy of the lowest Landau level at which L =0 incompressible fluid ground states occur, it predicts the low energy states of the spectrum for any degeneracy. The picture rests on the assumption that the MFA gives an adequate description of the CFs at the hierarchy level being considered. This assumption can be tested by comparing its predictions with numerical calculations for finite-size systems, and it shows that the MFA fails for most fractions outside the Jain sequence.