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Exact Solution for Two $δ$-Interacting Bosons on a Ring in the Presence of a $δ$-Barrier: Asymmetric Bethe Ansatz for Spatially Odd States
by Maxim Olshanii, Mathias Albert, Gianni Aupetit-Diallo, Patrizia Vignolo, Steven G. Jackson
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Maxim Olshanii |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2508.17371v1 (pdf) |
| Date submitted: | Aug. 26, 2025, 4:28 a.m. |
| Submitted by: | Maxim Olshanii |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this article, we apply the recently proposed Asymmetric Bethe Ansatz method to the problem of two one-dimensional, short-range-interacting bosons on a ring in the presence of a $\delta$-function barrier. Only half of the Hilbert space--namely, the two-body states that are odd under point inversion about the position of the barrier--is accessible to this method. The other half is presumably non-integrable. We consider benchmarking the recently proposed $1/g$ expansion about the hard-core boson point [A. G. Volosniev, D. V. Fedorov, A. S. Jensen, M. Valiente, N. T. Zinner, Nature Communications 5, 5300 (2014)] as one application of our results. Additionally, we find that when the $\delta$-barrier is converted to a $\delta$-well with strength equal to that of the particle-particle interaction, the system exhibits the spectrum of its non-interacting counterpart while its eigenstates display features of a strongly interacting system. We discuss this phenomenon in the "Summary and Future Research" section of our paper.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2025-10-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2508.17371v1, delivered 2025-10-05, doi: 10.21468/SciPost.Report.12069
Strengths
- construction of exact eigenstates of the Hamiltonian of an interesting few-body problem: two repulsively interacting bosons on a ring scattering on a static delta barrier potential
Weaknesses
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the model and the method is the same as in Refs. [12,13] (although, these references were for attractive bosons), so it is not entirely clear where is the innovation in this new work
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not all the eigenstates are obtained, only the ones corresponding to an integrable subsector. Not much is said about the eigenstates that are not of this form
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the writing could be improved: some notations are not clear, some pages of the manuscript look merely like working notes and should still be polished
Report
The paper is interesting and will be useful for experts working on this type of problems. I believe that, after appropriate revision, it can be suitable for Scipost Physics Core. However the present version suffers from several issues that should be adressed first, see list below.
Requested changes
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As the model and method have been introduced in previous works, e.g. Refs. [12,13], the distinction between what was already known and what is new in this work should be made more explicit.
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I think the eigenstates of the model that are not of the asymmetric Bethe Ansatz form should at least be discussed. Can anything be said about those? Do we have any idea about how many such eigenstates there are, and where they would be in the spectrum? Perhaps the spectrum could be shown, highlighting the eigenstates that are of the form studied in this paper, and the ones that are not? Or is this the purpose of the strangely positioned figure 2, which appears after the bibliography, and is barely referred to in the text?
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The notations should be clarified. Formulas from page 4 to page 7 are difficult to read. (For instance: How is $\psi^{0<x_1<x_2}$ defined the first time it appears at the top of page 5?)
More minor points:
(i) I suggest to use conventions where $\hbar = m=1$. Also, please avoid introducing unnecessary notations such as '$\mu = m/2$' (no need for a new notation just for a factor $1/2$). By the way, the second formula (11) is probably wrong and should be (in the right units) '$a_B = -1/g_B$', not '$-1/g$'.
(ii) The use of references is a bit sloppy. The authors refer to 'the book [6]', but then it turns ref. [6] is not a book. Same for ref. [15]
Recommendation
Ask for minor revision