SciPost Submission Page

Weak integrability breaking and level spacing distribution

by D. Szász-Schagrin, B. Pozsgay, G. Takács

Submission summary

As Contributors: Gabor Takacs
Arxiv Link: (pdf)
Date submitted: 2021-03-23 08:50
Submitted by: Takacs, Gabor
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


Recently it was suggested that certain perturbations of integrable spin chains lead to a weak breaking of integrability in the sense that integrability is preserved at the first order of the coupling. Here we examine this claim using level spacing distribution. We find that the volume dependent cross-over between integrable and chaotic statistics is markedly different between weak and strong breaking of integrability, supporting the claim that perturbations by generalised currents only break integrability at higher order. In addition, for the massless case we find that the critical coupling as a function of the volume $L$ scales with a $1/L^{2}$ law for weak breaking as opposed to the previously found $1/L^{3}$ law for the strong case.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2103.06308v2 on 23 March 2021

Reports on this Submission

Report 2 by Aaron Friedman on 2021-4-23 Invited Report


1. The authors clearly state the problem to be investigated in their work: namely they conjecture that perturbing an integrable spin chain by one of the model's conserved currents breaks the underlying integrability in a weaker sense than generic perturbations.

2. The authors' theoretical approach is sound: level statistics for integrable and chaotic systems have been studied at length and are a reliable diagnostic of integrability versus chaos.

3. The authors' numerical evidence is both thorough and convincing.

4. Integrability breaking is highly topical in the quantum dynamics and statistical mechanics communities.

5. The authors' abstract provides an exceptionally clear summary of the results.

6. The authors provide plenty of details to explain their methods and allow others to reproduce their results with ease. However, the numerical evidence is quite convincing and the analysis thereof thorough.

7. Despite the intimidating number of figures and their imposing size, they are quite easy to understand and support the authors' claims nicely. This paper can basically be understood from the abstract, figures, and a few equations, which I appreciate.

8. The paper's scientific merit is clear.


1. The paper would benefit from revision throughout to make some of the language more clear. Many of the longer sentences are confusing, with ambiguous pronouns and dangling modifiers. Shortening sentences and clearly indicating "former" vs. "latter" or using punctuation to indicate clearly which statements go together (and general clarification) would be highly beneficial. This mostly applies to the introduction and conclusion.

2. The discussion of level statistics is quite pedagogical, e.g., while the introduction and background on integrability and integrability breaking assume a higher level of knowledge on the part of the reader than is reasonable. Even as someone with knowledge of integrability, I found that background on the models to which these results apply was lacking. The beginning of Sec. 3 is fine as written; my point here is that it would be nice if the introduction were written at a similar level.

3. The paper could more clearly contextualize the results in the context of the field at large. Who should care about this work and why?

As currently written, the paper provides a clear answer to a highly particular question; it is not clear that this will be of direct interest to more than a handful of people (namely, the authors Refs. 8-11). This can be remedied by providing more context for the results: Do the results have relevance to experiments? Do these results have implications for generalized hydrodynamics and integrability breaking? Is there any implication for chaos or other topics in quantum dynamics?

In the introduction, for example, the authors point out that using the GGE and GHD formalisms requires knowledge of the expectation values of infinitely many currents, but this is not possible in generic models. However, in the models with weakly broken integrability, this remains possible. Highlighting this claim and drawing a connection to ongoing work in closely related fields (e.g. quantum thermalization), could improve the paper's overall quality and appeal.

4. I would suggest the paper cite several recent works on integrability breaking, e.g. by R. Vasseur's group. However, as a coauthor of one of these works, I can't ethically insist on this point. However, the paper does benefit from the topical nature of integrability breaking and it would be reasonable to cite these works. Ideally, some discussion of how these results fit in with recent work on integrability breaking would improve the paper's quality and overall appeal.

5. I cannot say confidently that this paper meets one of the four expected Acceptance Criteria. However, with a bit of clarification and greater discussion of the results in the context of integrability more generally, I believe conditions 1 and 2 would be met. I think if the authors can make some of the proposed changes, this paper will be fine for SciPost.

6. It isn't clear if there is a "fit parameter" for the Poisson distribution. Naïvely, I would think e^{-s} has no fit parameter, but the use of the word "fitted" in the caption of one of Fig 2.

7. It might be helpful to state more clearly what is meant by weakly broken integrability. My takeaway was that weak breaking corresponds to a truncated version of the long range deformations that preserve integrability, where the truncation destroys the integrability, but only in a manner that preserves properties of the underlying integrable model.

8. Rather than saying that these perturbations "only break integrability at higher order", I would perhaps state that the "integrability is perturbatively stable to terms proportional to generalized currents". This is not a huge distinction, but seems like a safer claim since the paper only identifies the crossover strength, rather than a particular order (e.g. g^2) at which integrability breaks.

9. I found the discussion of the level crossing analytic confusing and had to read it several times. It would help to identify all the couplings (e.g. a1(L) ) that appear in the equations.

11. In the first paragraph of 3.4.1, a radius "r" is mentioned...what value is used?


This is a solid paper that convincingly addresses a reasonable question. I think the paper could benefit from some clarification of the wording and a few minor points listed above.

Overall, the research presented seems thorough; apart from some unclear sentences, all of the General Acceptance criteria are met.

What is slightly less clear to me is the extent to which the Expected Acceptance Criteria are met. I think by providing more context for their work as it relates to the field of integrability and quantum dynamics more broadly would remedy my concern.

If another referee does not share this concern, then I think it would be reasonable to accept the paper as is. Otherwise, I think with slight revision this paper ought to be accepted.

Requested changes

1. Please clarify the wording throughout. There are too many examples to list explicitly. In general, the longer sentences are difficult to understand or unclear.

2. Please clarify whether a fit parameter is used for the Poisson line in Fig 2, and the value of the radius r.

3. It would be helpful to comment on whether or not the same contrast between the J3 and NNNI perturbations is as stark if we use the naïve coupling, g, rather than g_{eff}.

4. Clarification of the variables in the equations in Sec. 3.3 would be useful. This discussion can be cleaned up somewhat as well.

5. Providing broader context for these results in the introduction and conclusion would greatly improve the paper in my opinion.

  • validity: high
  • significance: good
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: good

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