Spin of fractional quantum Hall neutral modes and "missing states" on a sphere

This is a succinct, clear, and beautiful paper, providing a new and very interesting argument for the missing states at some angular momenta in neutral excitation modes of fractional quantum Hall (FQH) systems. While the "missing states" has been known before, this paper interprets it from a novel aspect based on Berry phase. I like the Berry-phase based analogy between a neutral quasiparticle with intrinsic angular momentum on a sphere and a charged particle on a sphere with a fictitious magnetic monopole at the center. I would recommend the publication in SciPost Physics, if my following comments can be properly addressed. The other two referees also raised important questions.

(*) The dispersion of the neutral excitation is assumed to be quadratic in Eq. (2.1). To what extent does the argument in Sec. 3 depend on this assumption? Is it possible that in the long wavelength limit some FQH neutral excitation modes do not obey the quadratic dispersion with k=0 as a maximum/minimum? I suggest the authors to discuss this issue. By the way, the effective mass in Eq. (2.1) can be negative, right? At least the magnetoroton mode of the \nu=1/3 Laughlin state in the lowest Landau level goes up when k->0, suggesting a negative mass.

(*) In the last paragraph of Sec. 2, the authors mentioned the difference between the low-energy excitations on the torus and on the sphere. This reminds me one thing which I didn’t understand well. For the \nu=1/3 Laughlin state, the magnetoroton mode on the sphere starts from angular momentum L=2 -- no states at L=1 and L=0 in this mode. So the long wavelength limit of this mode, the so called spin-2 graviton, should be carried by the L=2 state for finite systems. As the “momentum” on the sphere is often defined as L/R, where R is the sphere’s radius, this graviton actually has a finite momentum for finite systems on the sphere. As the Laughlin ground state has L=0, the graviton on the sphere has different L and (for finite systems) different “momentum” from the ground state. By contrast, spin-2 spectral function calculations on the torus, like those done in Refs. [5] and [6], found evidence of the spin-2 graviton exactly in the momentum k=0 sector even for finite systems. As the ground state is also in the k=0 sector on the torus, this means the graviton and the Laughlin ground state already have the same momentum for finite systems on the torus. This is quite different from the situation on the sphere. Why does not the graviton appear at a finite momentum (nonzero nx or ny) on the torus for finite systems? Can the authors comment on this discrepancy between different geometries?

Ask for minor revision

This is a terrific paper. It is short, simple, and beautiful. The result is clear and clearly presented. It should certainly be in SciPost.

I have a few minor comments. I found some typos. And I have a few suggestions.

Let me start with the minor things:

(1) Suggestion. Just after eq. 2.1, it is worth saying a few more words about why only neutral particles have a dispersion. I had to think for a while to understand what you meant and why this is true.

(2) Change of language. I think about 7 lines up from the bottom of page 2, the word "should" would be better as "could", since in the very next sentence you say this is not always the case.

(3) Missing word. 6 lines below the start of the conclusion the word "is" is missing. (but that state IS annihilated)

Now a larger suggestion for discussion of some additional phyiscs
which I think would add to the paper nicely:

I think it is worth comparing your result, at least for the case of
the jain series, to the spectra expected from composite fermion
excitons. Maybe this should be done in an appendix, maybe in the
discussion section, or some combination. For nu=1/3, just by
constructing all possible excitons --- a single particle in an excited
CFLL and a single hole in the valence CF LL --- one obtains 1 state at L=1, 2 states at L=2, 3 states at L=3 and so forth in the thermodynamic limit. (This is trivial to show just by counting since the valence level has some angular momentum L0 and the first empty level has angular momentum L0+1 then next has L0+2 etc, and you want to add the hole in L0 to the electron in L0+M to generate some general L.) In the Jain picture, the L=1 exciton state vanishes under projection but it is unclear exactly why. In the HLR picture it is more clear (using some mass renormalized version of HLR --- see the reference from the other referee, thank you for the citation!). Here the Kohn mode is part of the spectrum, so it grabs one state at each L. So the low energy part of the spectrum then gets 0 states at L=1, 1 state at L=2, 2 states at L=3 and so forth. In your language would you say that we are seeing one spin-2 branch, one spin-3 branch and so forth? At nu=2/5, just by counting excitons again, you get 1 state at L=1, 3 states at L=2, 5 states at L=3 and so forth. Again one mode of each L is pushed up to the Kohn mode leaving 0 states at L=1, 2 states at L=2, 4 states at L=3 and so forth. So you would say that we have 2 modes of spin 2, 2 more modes of spin 3 and so forth? Is this correct? If so, it then suggests that one should go looking for these in numerics. However, one thing that seems special about the magnetoroton is that, at least under some circumstances, it doesn't get buried in the continuum. Is there any hope that these higher spin modes could also be out of the continuum and could be observed?

Because I was a bit slow in writing this report (just a few days!), I did have the benefit of reading one of the reports from another referee. That referee says that this work is "thin on results." I would actually disagree. While I would not object to extending this work or addressing some of the above issues (which overlaps with the issues raised by the other referee), I think the paper really does stand on its own. The paper makes a clear point and is an important point to make. It reminds me of the good old days of PRL (1950s) when a "letter" was often one page or less --- and these papers are very memorable. Now, much to our detriment, people feel obliged to extend such papers to include more, and often end up diluting the main message.

Publish (surpasses expectations and criteria for this Journal; among top 10%)

In this brief note, the authors provide an a posteriori explanation for the angular momentum structure of collective modes in fractional quantum Hall states, namely the fact that the dispersions of these modes often start from a well-defined, non-zero value of angular momentum. The argument is based on the Berry phase contribution of a fictitious magnetic monopole that is due to the intrinsic spin of the excitation. Overall, the argument looks plausible and it seems to account well for a few well-known cases in the literature.

However, the manuscript feels a bit "thin" on the content and I am not sure if it tells the whole story. For example, the role of LLL projection is not very transparent. The absence of L=1 state is not limited to the Laughlin or composite fermion states -- this state vanishes simply due to the fact that the ground state is a singlet (L=0) and the LLL-projected density operator essentially acts as a raising operator (see https://journals.aps.org/prb/pdf/10.1103/PhysRevB.50.1823). Based on this argument, I would expect that if we relax the condition of LLL projection, we can have situations where L=1 excitation becomes possible. For example, what would happen if one took nu=2/5 Jain state in the lowest two Landau levels? With the condition that the two LLs are degenerate in energy, this state has the exact parent Hamiltonian (V1 pseudopotential) and, presumably, a well-defined collective mode exists. What do the authors predict for such a case, would the collective mode start from L=1 or not? Similarly, one could ask what they predict for higher order Read-Rezayi states?

I am also not convinced by the claim that higher spin excitations in nu=n/(2n+/-1) Jain states should start from increasingly higher values of L. Such modes have been studied in CF theory framework, e.g., in https://www.nature.com/articles/nphys1275 but it does not seem that any states are missing (apart from L=1). How does this numerical observation fit in with the authors' arguments?

Another general question is what happens if one considers a different type of geometry, such as a (finite) cylinder or disk. One should be able to directly map states between these geometries and the sphere (e.g., stereographically), but it is less clear to me if the authors' argument would work in these cases.

The paper makes an interesting observation, but I think it needs to be further strengthened by addressing the questions raised above before it can be reconsidered for publication.

Ask for major revision