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Fractal decompositions and tensor network representations of Bethe wavefunctions
by Subhayan Sahu, Guifre Vidal
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Subhayan Sahu |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2412.00923v2 (pdf) |
| Date submitted: | July 14, 2025, 1:36 p.m. |
| Submitted by: | Subhayan Sahu |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate the entanglement structure of a generic $M$-particle Bethe wavefunction (not necessarily an eigenstate of an integrable model) on a 1d lattice by dividing the lattice into L parts and decomposing the wavefunction into a sum of products of $L$ local wavefunctions. We show that a Bethe wavefunction accepts a fractal multipartite decomposition: it can always be written as a linear combination of $L^M$ products of $L$ local wavefunctions, where each local wavefunction is in turn also a Bethe wavefunction. Building upon this result, we then build exact, analytical tensor network representations with finite bond dimension $\chi=2^M$, for a generic planar tree tensor network (TTN), which includes a matrix product states (MPS) and a regular binary TTN as prominent particular cases. For a regular binary tree, the network has depth $\log_{2}(N/M)$ and can be transformed into an adaptive quantum circuit of the same depth, composed of unitary gates acting on $2^M$-dimensional qudits and mid-circuit measurements, that deterministically prepares the Bethe wavefunction. Finally, we put forward a much larger class of generalized Bethe wavefunctions, for which the above decompositions, tensor network and quantum circuit representations are also possible.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-7-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2412.00923v2, delivered 2025-07-22, doi: 10.21468/SciPost.Report.11620
Strengths
1- The paper has a generalization of Bethe states, which is very useful for theoretical studies. I think that the generalization in Section VII did not appear previously in any work, definitely not in this form. This is a valuable addition to the literature.
Weaknesses
1- The authors are not familiar with many earlier results.
Report
The later parts are fine. I think the TTN part is completely new. And while the authors do not solve the state preparation with polynomially complex unitary circuits (which might or might not be possible), they give useful new results.
Requested changes
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Refs [7-9] are not really reviews about Algebraic Bethe Ansatz. The authors state that "for recent reviews" we should consult these papers. But [7] is about the notion of integrability (very general!), [8] is about quench action (no nested Bethe Ansatz), and [9] is a good introductory material, but again, it does not really review new results about Algebraic Bethe Ansatz. Actually I don't know a good "review", but for example https://arxiv.org/abs/1804.07350 does talk about Algebraic Bethe Ansatz in depth, and in fact it explains some old results that are relevant. For nested Bethe Ansatz I would suggest perhaps https://arxiv.org/abs/1911.12811 The authors can choose to include these references or similar ones, and/or just replace the wording "for recent reviews" because those are not recent reviews about Algebrai Bethe Ansatz.
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The authors write in the Introduction: "In spite of the solvable character of integrable models, the exact com- putation of their long-range and high order correlation functions and of dynamical properties remains a chal- lenging task." There should be some references. I understand that the authors are not experts in this area, nevertheless they should make an effort. If I had to collect just one document, then I would cite the habilitation thesis of Karol Kozlowski: https://arxiv.org/abs/1508.06085 This was 10 years ago, and it includes lots of exact results about long range correlation functions in the ground state. Maybe there are more recent reviews too. And if I had the chance to include one more document, I would add https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.257201 This has lots of information also about generic higher point correlation functions, including finite temperature situations. But this group of authors has many many more recent works, unfortunately no reviews, as far as I know. And for excited state correlation functions see https://arxiv.org/abs/1605.09347 So actually a lot is known, but these computations can indeed be challenging. Something should be added, it is up to the authors whether they add these references or other ones. But the statement without references is definitely not good.
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In the intro the authors write: "we uncover a surprising property of Bethe wavefunctions" This result that the authors have is actually well known. What they do later, namely the tree tensor network, is their own. But the fact that the Bethe wave function can be decomposed in this particular way, is very well known since a long time. It is a new result for the generalization that they have, but for previously studied Bethe wave functions this is not new. The authors acknowledge this is a "Note added". However, this is not sufficient for publication in a journal. I understand that perhaps the authors wanted to push the paper to the arxiv, for a variety of reasons. But now when they have the time to prepare the submission, they should make an effort, and add comments wherever appropriate. The paper should not appear this way, that there is just a single Note at the end, explaining that so many results were known since the 80's.
Recommendation
Ask for minor revision
Report
Recommendation
Publish (meets expectations and criteria for this Journal)