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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
Submission summary
| Authors (as registered SciPost users): | Maxim Olshanii |
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| Preprint Link: | https://arxiv.org/abs/2508.17371v3 (pdf) |
| Date accepted: | Nov. 3, 2025 |
| Date submitted: | Oct. 8, 2025, 6:46 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.
Author comments upon resubmission
List of changes
>> 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.
The second to last paragraph of Sec. 6 addresses this explains why the periodic boundary conditions and the unbounded spectrum are so important. In short, our manuscript is the stage one of a plan to address appearance of quantum chaos in systems that a classically regular.
>> I think the eigenstates of the model that are not of the asymmetric Bethe Ansatz form should at least be discussed.
We address them in the same paragraph (second to last paragraph of Sec. 6).
>> 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?
The even sector requires hard numerics. We don't need it for the odd states because of the integrability. Incidentally, such a numerical calculation is the very subject of our current research, almost ready for submission. In the new version, we cite this text under Ref. 20.
>> The notations should be clarified. Formulas from page 4 to page 7 are difficult to read. (For instance: How is ψ0<x1<x2 defined? The first time it appears >> at the top of page 5?)
We changed the notations completely.
>> More minor points:
>> (i) I suggest to use conventions where ℏ=m=1. >> Also, please avoid introducing unnecessary notations such as 'μ=m/2' (no need for a new notation just for a factor 1/2).
==
While the scattering length 'a' governs the relative motion of two particles, the scattering length 'a_B' is responsible for the cartesian motion of a single particle. Hence, the reduced mass μ in the former case and the true mass m in the latter case. In both cases, the corresponding scattering length is the position of the first node of the even scattering wave in the limit of zero energy. For that reason, it is important to keep 'hbar' and 'm'.
>> By the way, the second formula (11) is probably wrong and should be (in the right units) 'aB=−1/gB', not '−1/g
Corrected.
>> (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]
Corrected
Published as SciPost Phys. Core 8, 083 (2025)
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