1. If we completely fill the lowest Landau level with spin up electrons and with spin down electrons (imagine ν = 2 and zero Zeeman energy), then spin up and spin down electrons are uncorrelated, g⇓⇑(r) = 1. It is not a claim of composite fermion theories that the same is true if we do the same with CF Landau levels. The attachment of flux quanta introduces correlations between the originally uncorrelated (n = 0, ⇑) and ( n = 0, ⇓) levels: spin up CFs do not feel the spin down CFs (owing to LL mixing neglect) but they do feel fluxes attached to the spin down CFs.
2. In fact, there are some analytical results. Very appealing schemes how to evaluate energy and correlation functions were suggested by Girvin [24] Takano and Isihara [70]. Interesting extension of the former work was presented by Görbig (Sect. 1.2.2. in [27]). All these schemes however present closed formulae neither for energy nor for correlation functions.
3. Consider the action of S– (the lowering operator for the z-component of spin) on the ν = 2 (or νCF = 2) ground state |Ψ, Sz = 0〉 at zero Zeeman energy (0 ⇑and 0 ⇓ LLs are filled). On one hand, the state S–|Ψ, Sz = 0〉 may not contain any particles in higher LLs (up to Zeeman energy, it should have the same energy as |Ψ, Sz = 0). On the other hand, there is no room for an extra spin down in the lowest LL which is completely filled and therefore flipping a spin ⇑⇒⇓ (as contained in S–) must annihilate the state. Finally, S– |Ψ, Sz = 0 = 0 implies that |Ψ, Sz = 0 is a S2 = 0 state.
4. Going once around an s-fold vortex gives phase 2πs. Exchange of two particles corresponds to one half of such a loop (for ψ(r1, r2) ⇒ψ (r2, r1) corresponds toψrel(ϱ) ⇒ ψrel(–ϱ) with ϱ = r1–r2 in the relative part of the WF; ϱ ⇒ –ϱ is half the way of going around zero). Thus exchanging two particles with s attached vortices, the wavefunction acquires phaseπs. For two fermions with s attached vortices, it is π(s + 1). Thus the wavefunction changes sign at exchange of two particles when s is even and does not change the sign when s is odd.
5. The magnetic field described by the vector potential in (16) is proportional to electron density, Ψ†(r1)Ψ(r1). In other words: the magnetic field felt by an electron at r is only non-zero if r= r1, or, an electron at r sees magnetic field consisting of delta–functions located at positions of other electrons. However, these points in space are inaccessible to the electron by virtue of the Pauli principle.