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Spectrum Degeneracies from Quantum Evanescence

by Weiguang Cao, Xiaochuan Lu, Tom Melia

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Authors (as registered SciPost users): Weiguang Cao · Xiaochuan Lu · Tom Melia
Submission information
Preprint Link:  (pdf)
Date submitted: 2024-02-23 10:20
Submitted by: Melia, Tom
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


We identify degeneracies in the energy spectra of quantum theories that occur at integer values of the continuous parameter $N$ of an analytically continued global $O(N)$ symmetry. The degeneracies are enforced by the requirement that evanescent states fall out of the spectrum. They occur between different irreducible representations of the analytic continuation of $O(N)$ and hold non-perturbatively. We give examples in the spectra of the critical $O(N)$ model.

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Reports on this Submission

Anonymous Report 3 on 2024-3-27 (Invited Report)


pretty clear


I doubt there is much original here


There is little actual physics in this paper, it is mostly about representations of SU(N) and SO(N). As $N$ is decreased to low
values various representations disappear or in the SO(N) case become
identical. Any formula for anomalous dimensions needs to respect this and the authors are able to show this in a few cases in their Tables III and IV. I do not think that this can be described as degeneracy (as this is usually understood) but this is a necessary consistency condition which should provide a useful check on formulae for general N. In this respect the results might be useful but this is not the emphasis of the paper.

The authors might like to look at the Racah Speiser algorithm which explains how representation cancel. (See J. Fuchs and C Schweigert, Symmetries, Lie Algebras and Representations, Cam bridge University Press, Cambridge, 1997.)

As things stand I do not there is any real justification for publication.

Requested changes

I do not think there are any specific changes that would change my recommendation.

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Author:  Tom Melia  on 2024-04-16  [id 4425]

(in reply to Report 3 on 2024-03-27)

We thank the referee for spending time reading our paper and providing feedback. However, we strongly disagree with the referee's comments.

(1) The referee claims that our analytic continuation analysis on the representation theory of SU(N) and O(N) groups "is usually understood" and there is not much original in our paper. The referee further suggests that the Racah Speiser algorithm explains how representations cancel and hence covers our results on the O(N) continuation. Regarding this claim, we would like to highlight an important nontrivial feature of the O(N) continuation compared to the SU(N) case. As N is decreased to low values, O(N) representations do not just disappear or become identical - they could become "negative" representations sometimes; see e.g. the result in our Eq.(B14).

To the best of our knowledge, our clipping prescription detailed in Appendix B.2 is a completely novel algorithm that can efficiently yield these rules. We would like to think that this point alone might even justify publication. If the referee disagrees with us, we kindly ask for a reference, or an explicit demo, on how the Racah Speiser algorithm would yield the result shown in our Eq.(B14) , with comparable efficiency, if at all. (We do not see any result similar to our Eq.(B14) in e.g. J. Fuchs and C Schweigert, Symmetries, Lie Algebras and Representations, Cambridge University Press, Cambridge, 1997.)

(2) We do agree with the referee that this requirement can be viewed as a consistency check on formulae for general N. But we emphasise that when the continued O(N) symmetry is an actual symmetry of a physical theory (as opposed to just an auxiliary continuation for purposes of formulae handling), our results go beyond a consistency check: they provide non-perturbative information about how the spectrum at non-integer N is behaving as it approaches an integer N. 

That is, we think that the characterization of the latter scenario as a consistency check is at odds with how we usually think about the physical consequences of symmetry. The situation at hand is analogous to an analysis of the Hydrogen atom in an external magnetic field. The external field splits the degeneracy at a fixed principle quantum number; we would not characterize the fact that the energies approach each other as the magnetic field is tuned back to zero as a consistency check on our calculation of energy levels. Rather we would interpret this behaviour as a physical consequence of the SO(4) symmetry when B=0.

Anonymous Report 2 on 2024-3-22 (Invited Report)


The subject of this paper is quantum theories with a global $O(N)$ symmetry. The paper seeks to argue that performing an analytic continuation in $N$, it is possible to deduce the presence of degeneracies at special integer values of $N$, and that the method produces non-perturbative results.

In general, non-perturbative statements about quantum theories are either highly non-trivial or elementary. I am inclined to think the statements of the present paper falls into the latter category.

Let me attempt to reconstruct the reasoning of the paper as I understand it. At large (but still finite) values of $N$, the Weyl characters of $O(N)$ assume a fixed form as functions of $N$, while at low values of $N$ there are degeneracies, and some characters instead reduce to zero or to ($\pm$) other characters. The authors write down a type of partition function in (15) using the large $N$ values for all characters. They then perform an analytic continuation of this large $N$ partition function to finite $N$. It is then implicitly stated (that's what I gather from them dropping the overline on $Z$ in going from (15) to (17)), whether as a conjecture or rigorous result I'm not sure, that this continued answer equals the actual finite $N$ partition function, which does not contain negative terms, so that the negative terms from the continued large $N$ partition function must cancel.

The paper does not contain any new calculations, but to illustrate the method, the authors point out operators for which the previously computed scaling dimensions in the $\epsilon$ expansion can be checked to become identical for $N=1$ and $N=2$.

Allow me to remark that the analytically continued path integral the authors write down also requires the energy levels to be analytically continued. The authors only discuss how to continue the Weyl characters, they do not discuss how to continue the energies, but they do reference the paper by Binder and Rychkov that discusses how to make sense of non-integer $N$ using Deligne categories.

My above outline may be misguided, in truth I don't understand the role played by analytic continuation. It does not seem to me that the ability to smoothly vary $N$ or to assign it non-integer value is ever needed (unless it somehow forms part of the authors' reasoning for equating $Z$ and $\overline{Z}$).

In fact, I think the examples given in the paper can be accounted for much more simply, without invoking Deligne categories, analytic continuation, the partition function, or even Weyl characters. It seems to me simply to be an instance of operators that are distinct at large $N$ but become linearly related at small $N$. If two operators which are distinct for $N>1$, say, become identical at $N=1$, then naturally their scaling dimensions too should agree at $N=1$.

To me, it would be more natural to think of the specializations of the Weyl characters and the matchings of scaling dimensions at low values of $N$, as both being consequences of linear relations among irreps, rather than the matchings being something that can surprisingly be deduced from character tables.

I don't agree with the usage of the word ``degeneracy" in this paper, and I find the analogy with the hydrogen atom deceiving. True degeneracy is a physically observable phenomenon where different states have the same energy. But now look at the first half of Table III, say. Here two states with four scalar fields and two derivatives are said to be degenerate. But when $N=1$, there is only a single such Lorentz scalar. There is only one actual state, and so there is no degeneracy.

The fact that operators that are distinct at large $N$ become linearly related at small $N$ does, it is true, provide constraints on functions like $\Delta_{\lambda,i}(N)$. But whether these constraincts can be usefully leveraged to compute $N$-dependent CFT data remains to be seen.

I hope the authors will forgive me if I have misconstrued their paper entirely. I would be pleased if my objections were refuted and new non-perturbative results were indeed established.

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Author:  Tom Melia  on 2024-04-16  [id 4424]

(in reply to Report 2 on 2024-03-22)

We thank the referee for the detailed comments. We find that we actually agree with many of the referee's arguments at the level of technical detail. However, there seem to be some key misunderstandings that have led to our disagreement on the overall evaluation/interpretation/attribution of the results and claims in our paper. In the below, we try to summarize the main objections from the referee, and make clarifications accordingly.

(1) The referee says that the role played by analytic continuation is not clear, and the ability to smoothly vary N or to assign it non-integer value is never needed (unless it somehow forms part of the authors' reasoning for equating Z and \bar{Z}). The referee also calls into question the level of rigour in our statement that the continued partition function \bar{Z} in Eq.(15) equals the actual finite N partition function Z in Eq.(17).

We believe this is the key miscommunication between us and the referee. For the O(N) group, we know that there is an algebra at each integer N (encoded in our Eq.(2)), but they are different at different values of integer N. From our point of view, finding an analytic continuation of the O(N) representation theory means that we need to find an algebra that holds also for non-integer values of N, such as N=3.99 or N=4.01. Any integer N version of our Eq.(2) would not satisfy this requirement, but one can take its large integer N limit to get our Eq.(4), which is N independent and therefore also holds for N=3.99 or N=4.01. In summary, the ability to smoothly vary N or to assign it non-integer value forces us to consider a unified algebra, which holds at all real values of N in the same way. So it is indeed our reasoning for equating Z and \bar{Z}.

(2) The referee claims that our results can be accounted for much more simply, as consequences of linear relations among irreps --- an instance of operators that are distinct at large N become linearly related at small N. One can view our results this way. Say, at N=4, some operators are linearly related. However, the point is that as soon as we are at N=4.001, these operators are distinct operators. Again, it is the ability to smoothly vary N that forces us to view these linearly related operators as intrinsically distinct operators. From this point of view, they just "happen to be" linearly related at N=4.

(3) The referee disagrees with the use of the word "degeneracy", because it is supposed to describe relations between different physical states, while in our cases, there are no different states at integer N, due to the linear relations mentioned in (2) above. Similar to the clarification above in (2), states that are linearly related at an integer N need to be viewed as intrinsically distinct states, if one requires the ability to smoothly vary N. Therefore, we believe it is a legitimate use of "degeneracy" in the usual sense.

Anonymous Report 1 on 2024-3-21 (Invited Report)


1. An interesting approach to the question of how analytically continued O(N) theories behave in the N->positive integer limit.
2. Some checks of this in 4-$\epsilon$ dimension


1. The presentation is a little confusing at times
2. A bit more could be said about connections with CFTs


The paper offers a new approach in identifying which operators should be degenerate with each other when we take a CFT with O(N) symmetry for real N, and send N to a positive integer. The idea relies on folding some O(N) Young Tableaux and character properties of the $O(N)$ irreps

Requested changes

1. First page, third paragraph: I do not agree with the statement that N is supposed to be considered as a coupling constant, because if that was the case then it would run under RG flow. This is not the case, see for example [1]
2. First page, second paragraph of right column: the theory becomes unitary only in a specific sense. For example, it's known that the $N \to N_0$ limit is larger than the $N=N_0$ theory, with $N_0$ some positive integer, see e.g. [2]. The $N->N_0$ limit is not unitary, but rather logarithmic. The idea of having a "unitary subsector" was discussed in [3]
3. I find Section 2 being called "analytic continuation" a bit confusing, given that N is integer and there is no analytic continuation to real N. I would also reiterate that all equations here, e.g. (6), refer to integer $N$.
4. Just above eq (9). I think the vector for $O(N\to 1)$ becomes the fundamental, not the trivial irrep.
5. It's claimed that the representation (1,1) is not good for N=2,3. I thought (and I could be wrong) that if we continue an operator transforming in the (1,1) for generic N down to N=3 we get a pseudovector, so it looks like a `good' representation to me and I don't understand why we make the identification of it with the vector. Why is considering \bar{\chi} necessary for this case?
6. Similarly, on page 4, it's mentioned that via the cross product $(1,1) \to (1)$ for $N=3$. But given that we have O(N) and not SO(N) the epsilon tensor is not an invariant tensor, so I don't think we are allowed to do this.
7. The authors mention the possibility of a larger symmetry group than just O(N). Let me point out that in two dimensions there are indeed many degeneracies at \textit{generic} N between scaling dimensions of different irreps of O(N). It was proposed that this is because of a non-invertible symmetry in [3], and a microscopic proposal for this was then put forward in [4]
8. As a note, I think the authors could also make some checks in two dimensions, where the partition function is known nonperturbatively [5]. At least the N->1,2 limit could be checked here. Also the connection with the requiring of finiteness of CFTs as explained in [2] could be mentioned.

[1] 1911.07895
[2] 1302.4279
[3] 2005.07708
[4] 2305.05746
[5] P. di Francesco, H. Saleur & J. B. Zuber, Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models

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  • formatting: perfect
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Author:  Tom Melia  on 2024-04-16  [id 4423]

(in reply to Report 1 on 2024-03-21)

We thank the referee for the positive review of the paper.

Clarifications regarding the overview part of the report:

(1) Our discussions apply to, but are not limited to CFTs.

(2) Our key idea of claiming degeneracies is that for the continued O(N) group, some irreps will specialize as negative ordinary irreps of an O(N) group at integer N. Such irreps must be degenerate with other states (at the corresponding integer N), so that the partition function is free of negativity upon the specialization. The referee is correct that to figure out when this negative specialization occurs, our technology relies on using our clipping rules (or folding rules) and studying the character properties.

Responses to the requested changes:
(1) We agree with the referee, and thank the referee for pointing this out.

(2) We thank the referee for clarifying this point.

(3) We believe the referee has missed an important point in our Section 2. In particular, we disagree with the referee's comment that there is no analytic continuation to real N and that all equations refer to integer N. In our point of view, the meaning of continuing the O(N) group to non-integer N is to find an algebra that holds for both integer and non-integer N. In section 2, we show that one way to do so is to start with the integer-N algebra in our Eq.(2), and then consider the large integer-N limit of it. This limit leads to the N-independent algebra in our Eq.(4), which holds for all real N, not just integer N, and therefore serves as a natural analytic continuation. To reiterate, our Eqs.(4) and (5) hold for all real N, not just integer N. On the other hand, the referee is correct that our Eq.(6) refer only to integer N, because it is talking about "specialization", which by definition, happens only at integer N, as we did state already in the paragraph above Eq.(6).

(4) We are not sure what the referee means here. In Eq.(9), we are showing the character of the continued vector (or fundamental) irrep of O(N). Our comment above Eq.(9) is to explain that if one take N=1 strictly, then the group specializes as O(1)=Z2, and the irrep in Eq. (9) specializes as the trivial irrep of Z2. Is the referee claiming that the irrep should specialize as the "fundamental irrep of Z2"?

(5) The group O(2) or O(3) has rank 1, and therefore all their "good" irreps are labeled by partitions of length not exceeding 1. The partition (1,1) has length 2, and is not on the list. It is an irrep for the continued group, i.e. O(N) at non-integer N, so we need to consider \bar\chi. Our Eqs.(12) and (13) shows how \bar\chi specializes as characters of `good' irreps at N=3 and N=2.

(6) In fact, we are only looking at the character of the O_+ branch of the O(N) groups. Therefore, the character function for real vs pseudo scalars, vectors, tensors, etc. are the same.

(7) We thank the referee for this informative comment.

(8) We thank the referee for these constructive suggestions.

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