SciPost Submission Page
Emergent Area Operators in the Boundary
by Ronak M Soni
Submission summary
| Authors (as registered SciPost users): | Ronak M. Soni |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2511.01382v2 (pdf) |
| Date submitted: | Nov. 9, 2025, 9:44 a.m. |
| Submitted by: | Ronak M. Soni |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
- Diagram generation (tikz formatting).
- Literature searches.
Main ideas and all the writing done by humans.
Abstract
In some cases in two and three bulk dimensions without bulk local degrees of freedom, I look for area operators in a fixed boundary theory. In each case, I define an exact quantum error-correcting code (QECC) and show that it admits a central decomposition. However, the area operator that arises from this central decomposition vanishes. A non-zero area operator, however, emerges after coarse-graining. The expectation value of this operator approximates the actual entanglement entropy for a class of states that do not form a linear subspace. These non-linear constraints can be interpreted as semiclassicality conditions. The coarse-grained area operator is ambiguous, and this ambiguity can be matched with that in defining fixed-area states.
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 #3 by Anonymous (Referee 3) on 2026-1-19 (Contributed Report)
The referee discloses that the following generative AI tools have been used in the preparation of this report:
Used to refine grammar of report.
Weaknesses
Report
Due to the above-mentioned reasons, I recommend that the author publish this manuscript in SciPost Physics Lecture Notes.
Recommendation
Accept in alternative Journal (see Report)
Strengths
Weaknesses
Report
Requested changes
I have various suggestions for the author:
1) A very closely related idea has been discussed in arXiv:2509.21438. It would be appropriate to cite this work and explain the connection. 2) There is a typo in Eq 2.2, the relation between c and b should involve a square to be consistent with the later estimate at small b. 3) Eq 3.11 discusses that the code subspace involves all states of the form $|E\rangle|E\rangle$, it might be good to move that fact forward to Section 3.1 since it is a simple description of the code subspace. 4) Below Eq 4.24, we have $E_\alpha- E(\beta)=O(\sqrt{c})$, which is valid if we only consider the peak of the spectrum. This is sufficient to explain the entanglement entropy, but would be insufficient for the Renyi entropies. It would be good to explicitly state that, since for example arXiv:2509.21438 discusses a coarse-graining where the Renyi FLM formula is also approximately true after coarse-graining. 5) Below Eq 4.25, there is a discussion of the exact code not having a factorisation problem, but I didn't understand this since this code doesn't have states of the form $|E\rangle|E'\rangle$. 6) It would also be appropriate to cite arXiv:1910.06329 when crediting previous papers for fixed-area states since this paper clarified it in great detail. 7) The single interval discussion seems very related to the previous discussion if one applies the Casini-Huerta-Myers map. This may be useful to clarify. 8) I have a more general comment which is that the discussion in this paper is more on the boundary and looks very naturally related to fixed-energy states as opposed to fixed-area states. Another related comment is that there can be excitations in the bulk in this code subspace as long as they are on both sides, unlike how it seems to be stated in the discussion. The distinction between fixed-area and fixed-energy was raised in arXiv:2010.12592 and clarified in arXiv:2203.04973. It would be good to clarify this in the current draft.
Recommendation
Accept in alternative Journal (see Report)
Strengths
Weaknesses
Results and arguments are known in the literature
Report
However, the central result-- the statistical description of the coarse-graining-- in the paper has been understood in the community. Though it might not be written down in a general form in existing literature due to its transparency and simplicity, writing a small piece of the known result down is not enough to qualify as a scientific paper. This draft is better suited for pedagogical journals like SciPost lecture notes. Thus I suggest the author to consider this type of journals instead of journals reporting novel scientific results.
Recommendation
Accept in alternative Journal (see Report)
Author: Ronak Soni on 2026-01-21 [id 6256]
(in reply to Report 2 on 2026-01-16)I thank the referee for the detailed report.
I thank the referee for pointing me to arXiv:2509.21438. I had not looked at it and I will update the manuscript to include a comparison.
I will fix this typo.
In my mind, the main motivation for this code space is that it contains reasonable path integral states, and one of the points I wished to make was that such a code space naturally contains fully factorised states. Flipping the definition would not be natural, in my mind.
If we allow energy fluctuations to scale with $\sqrt{c} \sim 1/\sqrt{G_N}$, that captures all but exponential tails of the wavefunction. The 'tensor network' approximation that gives a flat entanglement spectrum is when the fluctuations scale as $c^0$.
This is a good point, and I will update the manuscript to refine this statement.
I will add the citation.
I am not sure what the connection to CHM is. They taught us how to calculate the entropy in the vacuum, but didn't identify an area operator or a coarse-graining. The regulator used is also different, I believe.
In fact, the coarse-grained states are fixed-area and not fixed-energy. The easiest way to see this is that the states have $\mathcal{O} (c)$ entropy. In this simple code, this coarse-graining is the only thing needed to go from fixed-energy to fixed-area. In fact, if one looks at the inner product, it more closely matches the fixed-S states of arXiv:1809.08647 than the fixed-E ones. In upcoming work, we will show more concrete relations between these sorts of coarse-grained states and fixed-area states. I will update the manuscript to make these points more explicit.
Anonymous on 2026-01-21 [id 6261]
(in reply to Ronak Soni on 2026-01-21 [id 6256])4) Yes, but the point I was trying to make was that such an approximation would get the Renyi entropies of the semiclassical state you are interested in wrong, since it misses the exponential tails. A fixed-area/tensor network state would indeed have $O(c^0)$ fluctuations and have a flat Renyi spectrum. In this case, there are no exponential tails, whereas there are in your case of interest.
7) I wasn't claiming CHM did what you did. I just meant to say using the CHM map relates your single interval analysis to your thermofield double analysis, unifying your paper.
8) When I say fixed-energy states, I mean microcanonical states that include a window of energy say $O(c^0)$ and so are also coarse-grained. They also do have $O(c)$ entropy. However, there is a difference in the bulk, since one of them imposes a condition at the asymptotic boundary whereas the other imposes a deep in the bulk condition. The example you give is a case where there is no matter and there is thus, no difference between fixing the area and fixing the energy. In a general holographic CFT, there will be matter fields in the bulk and there will be a distinction between these two. My claim was what you are doing is fixed-energy and not fixed-area.
Author: Ronak Soni on 2026-01-21 [id 6262]
(in reply to Anonymous Comment on 2026-01-21 [id 6261])Thanks for the further comments.
4) Look at eqn 2.23 of 1812.01171 for example. They show that $O(1/\sqrt{G_N})$ fluctuations are sufficient to reproduce the Renyi entropies.
7) Thanks, this is true and I apologise for missing the point earlier.
8) Notice that we are fixing only on the primary weights, not the energy. There are two ways to explain the significance of this difference.
a. The primary weight is in the centre of the algebra whereas the energy is not. Fixing an operator in the centre of the code algebra must have an effect spacelike to the EFT excitations, which in this case are the boundary gravitons. So it cannot be the ADM energy that is being fixed.
b. The Verlinde loop in the CFT is the same as a particular Wilson loop in the bulk. In the boundary this loop has eigenvalues $\cosh 2\pi b P$, with $P$ the momentum of the primary, and in the bulk the Wilson loop has eigenvalues $\cosh l$, where $l$ is the length of the extremal surface in the given homotopy class. See e.g. sec 4.3 of 1412.5205.
I will update the manuscript to explain this more refined argument. I thank the referee for the useful questions.