SciPost Submission Page
Geometric expansion of fluctuations and average shadows
by Clément Berthiere, Benoit Estienne, Jean-Marie Stéphan, William Witczak-Krempa
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Clément Berthière · Jean-Marie Stéphan |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2408.08364v2 (pdf) |
| Date submitted: | May 30, 2025, 7:07 p.m. |
| Submitted by: | Clément Berthière |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this letter, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction $1/2$, by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-8-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2408.08364v2, delivered 2025-08-28, doi: 10.21468/SciPost.Report.11819
Strengths
Weaknesses
The structure could be improved slightly: - the convention to compute volumes below Eq. (29) needs to be given explicitly earlier in the paper - in section III maybe the flow would be more natural by first deriving the formulae and then giving and analysing the formula including the special cases (there is indeed already a section for the main results)
Report
They derive expressions for the cumulants that clearly separate the geometric contribution (which are described in details) to the physical contributions.
The paper derives new and interesting results that will benefit the community working on these topics. I therefore recommend the paper for publication in Scipost Physics after some minor changes.
Requested changes
See suggestions in "weaknesses"
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 1) on 2025-8-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2408.08364v2, delivered 2025-08-15, doi: 10.21468/SciPost.Report.11706
Strengths
- Solid analytical results.
- Coherent links to other quantities and fields.
- The individual results are clearly stated.
Weaknesses
- The global structure of the paper is a very meandering and would be helped by a minor refactoring.
- By structure, it focuses on gapped phases (or at least on terms which are not affected by the gaplessness of the Hamiltonian). Anomalous contributions to correlators in gapless systems have seen a lot of interest.
Report
The derivation is well-explained, and universal, thanks to a clear factorization between physical observable and geometrical factors. Crucially, it only relies on global rotation and translation invariance, and therefore also applies to strongly interacting systems.
The results are solid and provide interesting directions to study a wide variety of models thanks to the explicit formula derived in this paper. As such I would recommend publication in Scipost Physics after minor revisions.
Indeed, I did find the paper sometimes hard to follow in its logic. While each point is in itself quite understandable, the flow keeps going back and forth. I would strongly recommend a refactoring of the paper.
As an example: Eq. 9 and Eq. 11 appear to differ by a factor of 2. It is only below Eq. 29 that the counting convention is made explicit.
Additionally, there is not really any discussions of the logarithmic corrections that can arise in gapless models, despite the significant interest they raise (as the authors very well know). While I understand the generalization to this type of models might be out of scope for this paper, at least a discussion of what and where the approach fails, as well as some general connecting comments would be appreciated.
Requested changes
1) Refactoring the structure of the paper would be a plus, to streamline the presentation and avoid some repetitions (typically results for variance and skewness are mentioned several times throughout the paper).
2) There is no connection to the anomalous logarithmic corrections, which I would appreciate.
3) While the geometrical contributions are extremely well described, there are no discussions of the physical meaning of the observable part.
Minor change: - "letter" should be changed in the introduction
Recommendation
Ask for minor revision