# Composite fermions: Flux attachment non-relativistic fermion v.s. emergent Dirac fermion

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The question concerns the composite fermion as electrons in strong magnetic fields in quantum Hall states; in particular, I am comparing two scenarios:

non-relativistic fermion with flux attachment v.s. Dirac fermion as the vortex dual.

Recently Son proposed an interesting composite fermion picture as Dirac particle (vortex) distinct from the usual flux attachment procedure: http://journals.aps.org/prx/pdf/10.1103/PhysRevX.5.031027.

Son tackles the un-easiness and uncomfort of using flux attachment, by giving an example, the Jain sequence quantum Hall states which are particle-hole conjugate pairs with the filling fractions of original electron $\nu=\frac{n}{2n+1}$ and $\nu=\frac{n+1}{2n+1}$. However, using the flux attachment picture, one attach 2 flux quanta to the original electron to form the composite fermion. In this case, we send:

$$\nu=\frac{n+1}{2n+1} \to \nu_{CF}=n$$

$$\nu=\frac{n}{2n+1} \to \nu_{CF}=-(n+1)$$

Namely, the filling fractions of original electron $\nu$ as fractional quantum Hall state is sent to the filling fractions of composite fermion $\nu_{CF}$ with integer filling fractions as integer quantum Hall state.

In the new picture Son proposed,

$$\nu=\frac{n+1}{2n+1} \to \nu_{CF,Dirac}=n+\frac{1}{2}$$

$$\nu=\frac{n}{2n+1} \to \nu_{CF,Dirac}=-(n+\frac{1}{2})$$

Namely, the filling fractions of original electron $\nu$ as fractional quantum Hall state is sent to the filling fractions of composite fermion $\nu_{CF,Dirac}$ with integer+1/2 filling fractions as quantum Hall state of Dirac particle.

Questions: Since the original Son's proposal is attacking the 1/2-filling Landau level, the compressible un-quantized gapless metallic state, the $n\to \infty$ is taken. And this is not a gapped quantum Hall state. However, from the examples motivated his thinking, the gapped fractional quantum Hall Jain sequence, he seemed not to be satisfactory on the flux attachment on the gapped quantum Hall state as well. Thus, I ask how strong is that Son's attack on flux attachment non-relativistic fermion is incorrect or improper, for just (a) 1/2-filled Landau level gapless metallic state, or (b) also for all the gapped Jain's sequence quantum Hall, or (c) even more general for any composite fermion, as comparing to the new scenario of the Dirac fermion he newly proposed? Namely, does Son mean that the flux attachment on non-relativistic fermion is fundamentally problematic?

However, if Son's attack is a stronger claim, I suppose that there are certain interesting physics (such as Moore-Read Pfaffian state) can still be explained by the flux attachment. I suppose that Son's theory can recover every correct physics of the $\nu=1/2$-filled Landau level state. But does Son's Dirac particle story also work for generic quantum Hall state as well?

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