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In the book Composite Fermion by Jainendra K.Jain, he mentioned the motivation of Jain 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 Landau 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), Jain 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. 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. enter image description here

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.
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Answer to question 1: The idea behind composite fermions (CFs) when describing the FQHE is that a system of electrons exhibiting the FQHE can be modelled as a system of CFs exhibiting the integer QHE. This is useful because we know how to find wave functions for the low-lying IQHE states (filled Landau levels).

So, how do we go from one to the other? This is what is depicted in Figure 5.5.

  1. Start with electrons exhibiting the IQHE at $B^*$. The wave function is $\Phi(B^*)$.
  2. Attach a (purely theoretical, of course) flux tube to each electron. You now have CFs at $B^*$, or equivalently, electrons at $B$. However, the field is per now very inhomogeneous. The wave function is your initial $\Psi^{MF}$.
  3. Smear the magnetic field lines out so that $B$ becomes uniform (as appropriate for a system of electrons at external field $B$). The wave function thus obtained still needs vortices, since Pauli+Coulomb needs electrons to avoid eachother, and that is taken care of by the Jastrow factor. But since the (theoretical) flux tubes are now gone, the magnetic field attributed to the bare Jastrow factor is now different. To compensate for this, the $\Phi$ factor of the wave function must also be taken at a different field.
  4. Finally, an electronic FQHE state must reside in the lowest Landau level (LLL) but the construction above does not guarantee this. Jain argues that "most of" the wave function lives in the LLL before projection, but this point has been contested by e.g. Steve Simon (don't have access to the reference atm). So, LLL projection is done to get a "working" wave function.

Answer to question 2: You have the gist of the approach correct. For me it has always been easier to visualize a quantum dot here, where there is some harmonic potential confining the electrons. This extra potential changes the eigenvalues of the single particle states but not the single-particle wave functions. Then, we know that e.g. the ground state must have zero angular momentum $L=0$. Thus, contruct a/the Slater determinant that accomplishes this when multiplied by the Jastrow factor and then projected to the LLL. For the main Jain sequenc(es), this Slater determinant is unique.

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  • $\begingroup$ Thanks! So when we change from $\Phi_n(B^*)$ to $\Phi_n(B)$, the polynomial part do not change but only let $l_{B}$ goes to $l_{B^*}$, am I right? $\endgroup$ Apr 6, 2022 at 15:55
  • $\begingroup$ Besides, I think I have a hand-waving picture but I still hard to take the result for granted. The third point of you is just like what Jain said in his book. What's the relation between the spread of magnetic field and throwing out the $|z_i-z_j|$ in denominators? $\endgroup$ Apr 6, 2022 at 16:00
  • $\begingroup$ To your first point: yes, I agree. For the second one I believe I have a text that goes into more detail one the smearing aspect. I'll look for it. However: the result (the LLL wave function) is a product of a phenomenological statement anyway that (as Jain often points out) can only be verified or falsified by experiment (computer or lab). It is in particular not a true eigenstate of the Coulomb Hamiltonian so the picture cannot be 100% correct. $\endgroup$ Apr 6, 2022 at 20:35

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