Recently it has been realized that there is a duality (see, e.g. http://arxiv.org/abs/1605.03582) between a free Dirac fermion in an external EM field (which I call Theory A) and a Dirac fermion coupled to a gauge field (which I call Theory B, also commonly called QED$_3$). Below I list the actions of the two theories (reproduced from the aforementioned paper)

Theory A: $\mathcal{S}_A=i\bar\Psi\gamma^\mu(\partial_\mu-iA_\mu)\Psi$, and

Theory B: $\mathcal{S}_B=i\bar{\tilde{\Psi}}\gamma^\mu(\partial_\mu-ia_\mu)\tilde\Psi+\frac{\epsilon^{\mu\nu\rho}}{4\pi}A_\mu\partial_\nu a_\rho+\mathcal{S}_{a,{\rm Maxwell}}+...$

This duality was proposed by Son (http://arxiv.org/abs/1608.05111 for a review) in an attempt to resolve the issue of particle-hole symmetry in the half-filled Landau level problem.

One of the goals (albeit not the main one) of Son's is to reproduce the famous Jain's sequence $$\nu=n/(2n+1)$$ for fractional quantum Hall effect in theory A by the integer quantum hall effect in theory B. After identifying corresponding quantities, the relation between filling factors in the two theories is not difficult to obtain.

However, with theory A being a free theory, how can there be any fractional quantum hall effect, to begin with? Without interaction, I think at any fractional filling the system is gapless (compressible).


First of all, obtaining Jain's sequence is not his objective in his new proposal. Second, the duality you mentioned was proposed for describing the low energy physics of the state at half filling fraction, at which a single layer QHE system has found to be gapless in most of experiments. If the filling fraction is not half, then in theory A there is a nonzero magnetic field. The system is NOT gapless.

  • $\begingroup$ First, while Jain's sequence was first obtained from the simple flux attachment picture, Son's Dirac composite fermion also reproduces Jain's sequence (arxiv.org/abs/1608.05111). Second, I don't agree that the duality only exists when electrons half-fill the LL. Actually the physical electron density in the dual theory is given by $\rho=b/4\pi$, where $b$ is the magnetic field of the statistical gauge field. To describe physical electrons away from half filling, one just need, in the dual theory, have a nonzero $b$ field. $\endgroup$ Aug 24 '16 at 18:27
  • $\begingroup$ One can propose that in theory B there is an integer quantum hall states of field $\vec b$. This, after simple algebra of duality relation (see Son's paper), corresponds to the filling factors in Jain's sequence in theory A. Here comes the puzzle -- theory A is free, gapless, and hence cannot have Jain's sequence, while you see at corresponding filling theory B is gapped by integer quantum hall effect! $\endgroup$ Aug 24 '16 at 18:34
  • $\begingroup$ Maybe I did not state it clearly. I agree that Jain's sequence can be reproduced in this new particle hole symmetric Dirac CF picture(otherwise Son's proposal can not have any merits). But this does not need to be his objective/intention of writing that paper. The major issue he wants to resolve is an emergent so called particle hole symmetry in this problem, which exists only precisely at half filling. That is why his attention is focused on the gapless non interacting theory. $\endgroup$ Aug 25 '16 at 17:59
  • $\begingroup$ I have made some wrong statement in my original answer. Now it has been corrected. Away from the half filling point, the system described by theory A is not gapless. $\endgroup$ Aug 25 '16 at 18:15
  • $\begingroup$ I agree that Son's main goal was to restore PH symmetry. While I was aware of that, this question is concerned with Jain's sequence in his Dirac CF theory. $\endgroup$ Aug 25 '16 at 18:42

Let me give a tentative answer to my own question. If the duality is exact, then the only explanation is that theory B, a strongly interacting theory, does not have IQH in terms of the $b$ field either. Then it is consistent with theory A does not have FQH because it is a free theory.

Son's paper (http://arxiv.org/abs/1502.03446) seems to agree with this. In that paper he has a Maxwell term $F\wedge F/e^2$ in both theories (let me call them theories A' and B'). Now theory A' is interacting and can have FQH.

In page 7 there is a remark that the Maxwell term $F\wedge F/e^2$ in theory B' renormalizes the fermi velocity $v_F$ of the Dirac composite fermion from the bare value of 1 to $e^2$. The consequence of this is that the gap of the IQH of theory B' is $\omega_c\sim v_F \sqrt{b}\sim e^2 \sqrt{\rho_e}\sim e^2/r_e$, where $r_e$ is the typical separation of physical electrons. This agrees with the typical energy scale in the FQH state.

Now if one formally takes $e\to 0$ limit, theory A' becomes free (the case this question is concerned with, theory A). We see that in theory B'$\to$ B, according to Son's claim, $v_F\sim e^2 \to 0$, i.e., should be some sort of non-Fermi liquid. Then for the energy gap in theory B we have $\omega_c\sim v_F \sqrt{b} \to 0$, which means no IQH in theory B!

However, I do not know how $v_F\sim e^2$ in theory B', as claimed by Son, is obtained.


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