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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/2412.00923v3 (pdf) |
| Date submitted: | Aug. 27, 2025, 2:26 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. Using the fact that a Bethe wavefunction accepts a \textit{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 -- we then build \textit{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 \textit{generalized} Bethe wavefunctions, for which the above decompositions, tensor network and quantum circuit representations are also possible.
Author comments upon resubmission
List of changes
- Modified the abstract and the inroduction to highlight the fact that the first part of the paper reviews known facts about decomposition of Bethe wavefunctions, and contextualize our contribution for the tensor network constructions in that regard.
- Added more references for Algebraic Bethe ansatz.
- Corrected a few typos.
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Requested changes
The authors really made an effort to make the paper readable, and to refer to previous literature. However, one issue remains.
In writing down Theorem 3.1, there is no reference to previous work. This might be a matter of style, but in my point of view, when the authors claim that this is their main new result, and if in some special cases it appeared in the literature before, then the authors should give a quick citation also at that point.
The precise statement is that Theorem 3.1 is NEW, I agree with the authors. It is new, because it concerns the most general Bethe Ansatz wave function. However, specifically for Bethe states of the XXZ model, given by ABA, this theorem was known.
Therefore, I would like to ask the authors, to just give a quick footnote or something, that in the special case of XXZ spin chain, computed via Algebraic Bethe Ansatz, this theorem was known. Reference could be the QISM book [6], or some concrete paper. (any equivalent acknowledgement of the known result is also good)
Recommendation
Ask for minor revision