On the absence of higher generations of incompressible daughter states of composite Fermion quasiparticles

Jain [1] introduced a simple mean-field (MF) composite Fermion (CF) picture by attaching to each electron in a quantum Hall system a flux tube producing a Chern- Simons magnetic field b(r) = 2pϕ₀ Σ_iδ (r - r_i)z-^ . Here ϕ₀ = hc/e is the quantum of flux, and the sum is over all electron coordinates r _i. He then averaged the total flux and the total charge (electronic plus positive background) over the entire sample. This MF picture gave a system of noninteracting CFs in an effective magnetic field B ^* ₀ = νB ₀. It predicted incompressible quantum liquid (IQL) states at filling factors ν = n (1 + 2 pn )^-1 for integral values of n. Chen and Quinn [2] demonstrated that Jain's MF CF picture predicted the total angular momentum values of the lowest energy band of states for any value of the applied magnetic field B₀. Justification of when the MFCF picture was valid was given by Wojs and Quinn [3], who extended the CF hierarchy scheme of Sitko et al. [4]. The CF hierarchy gave the Jain states for integrally filled CF Landau levels (CF LLs), and the Haldane hierarchy of all odd denominator fractions when quasiparticles in the highest (partially filled) CF angular momentum shell had interactions sufficiently similar to the Coulomb interactions of electrons in the lowest Landau level. Sitko et al. showed that the predictions of the CF hierarchy scheme were not always correct. By using a simple pair angular momentum identity and the concept of fractional grandparentage, Wojs and Quinn showed that higher generations of CFs could result from the interactions of the original CF quasiparticles only if their interaction energy V _QP( L ₂) as a function of their pair angular momentum L ₂ increased with increasing L ₂ faster than L₂(L₂ + 1). For Laughlin quasielectrons of the ν = 1/3 IQL state this condition was not satisfied. Therefore, no second generation of CFs could occur. The observed IQL at electron filling factor ν = 4 / 11 can not be attributed to a daughter IQL state at ν _QE = 1/3 in a totally spin polarized system. Rather, it is an IQL state of electron pairs with pair angular momentum ι _p = 2 ι – 1. A description of the IQL state of electron pairs and its low energy excitations will be presented, and the possibility of a spin flip quantum phase transition as a function of well width and applied magnetic field will be discussed. (paper)