What's the difference of flux tube and vortex in FQHE (especially in <PERSON> wavefuntion)
In the book Composite Fermion by <PERSON>, he mentioned the motivation of <PERSON> wavefunction: attach flux tube of 2p flux quantum to fermions to form composite fermions. Naively, this is done by gauge transformation \begin{align} \Psi^{MF}=\Phi\prod_{i<j}\left(\frac{z_i-z_j}{|z_i-z_j|}\right)^{2p} \end{align} $MF$ means mean field. $\Psi^{MF}$ are wave functions under magnetic field $B$ and $\Phi$ is wave function of composite fermions under magnetic field $B^=B-2p\rho\phi_0$, thus should be n filled <PERSON> wavefunction, denoted as $\Phi_n(B^)$. so \begin{align} \Psi^{MF}=\Phi_n(B^)\prod_{i<j}\left(\frac{z_i-z_j}{|z_i-z_j|}\right)^{2p} \end{align} However, this wave function suffers several problems like that it does not gives Laughling state at $n=1$, does not gives correlation from Coulomb interaction sinces its wave functions absolute value is every where same like LL wavefunctions and has higher landau component. The last problem is resolved by projection to lowest Landau level(which seems wierd to me too). But what's much more strange to me is the first step: we drop the denominator $|z_i-z_j|$ and change the $\Phi_n(B^)$ to $\Phi_n(B)$.In the following section(5.8.2), <PERSON> said that
We earlier defined composite fermions in two somewhat different ways: as bound states of electrons and flux quanta, and as bound states of electrons and quantized vortices. In going from Eq. (5.30) to Eq. (5.31), we have gone from the former to the latter. 346
The factor \begin{align} \prod_{i<j}\left(\frac{z_i-z_j}{|z_i-z_j|}\right)^{2p} \end{align} binds, by construction, a point flux of strength $2p\phi_0$ to each electron.On the other hand, the Jastrow factor \begin{align} \prod_{j<k}(z_j − z_k )^{2p} \end{align} binds 2p vortices to each electron. (More precisely, each electron sees 2p vortices onevery other electron.) Throwing away the denominator converts the flux tubes into vortices.14 In other words, we are postulating that flux quanta turn into vortices during the adiabatic process in going from Fig. 5.5(b) to 5.5(c), and electron–flux composites evolve into electron–vortex composites.
where Eq. (5.30) to Eq. (5.31) is the gauge transformation I defined above
Here are my questions: 1. How to understand the difference between flux tube and vortices? and how this evolution reflected in the two expression? 2. (less important) How to write $\Phi_n(B)$? Since we know that $\Phi_n(B^)$ are n-filled Landau level wave functions at magnetic field $B^$, when we change $B^*$ to $B$ but keep all Landau level fully filled, the degeneracy of Landau level is not changed, so the total electron number will change do, which is not possible. Here I suggest that this is because we multiple them by $\prod_{j<k}(z_j − z_k )^{2p}$ so the allow filled Landau level maximal angular momentum changes. But I hope someone could tell me if this is right and a detailed analysis. 649