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Composite-boson formalism applied to strongly bound fermion pairs in a one-dimensional trap
by Martín D. Jiménez, Eloisa Cuestas, Ana P. Majtey and Cecilia Cormick
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Submission summary
| Authors (as registered SciPost users): | Cecilia Cormick · Ana P. Majtey |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202207_00047v1 (pdf) |
| Date submitted: | 2022-07-29 14:18 |
| Submitted by: | Cormick, Cecilia |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We analyze a system of fermions in a one-dimensional harmonic trap with attractive delta-interactions between different fermion species, as an approximate description of experiments involving atomic dimers. We solve the problem of two fermion pairs numerically using the so-called “coboson formalism” as an alternative to techniques which are based on the single-particle basis. This allows us to explore the strongly bound regime, approaching the limit of infinite attraction in which the composite particles behave as hard-core bosons. Our procedure is computationally inexpensive and illustrates how the coboson toolbox is useful for ultracold atom systems even in absence of condensation.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2022-10-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202207_00047v1, delivered 2022-10-10, doi: 10.21468/SciPost.Report.5861
Strengths
- interesting (semi-) analytical ansatz
- clearly written
Weaknesses
- very limited accessible system size
- larger systems numerically easily accessible using other established methods
- significance and impact of results remain unclear
Report
The work addresses a problem of four particles in one dimension, using a newly proposed semi-analytical ansatz wavefunction. The paper is clearly written, and the presented science is sound and well explained.
However, the main problem I see is that the type of problem addressed here, even with significantly larger particle numbers, can be addressed using established numerical approaches, such as for example diffusion Monte Carlo methods or the multi configuration self-consistent field method. Hence I find it difficult to understand how the present work significantly advances the current knowledge of the field, and which important questions the paper addresses (being restricted to just four particles - can/did the authors go beyond N=4?). Before the authors convince me of their motivation and the significance of their results, I cannot conclude that the manuscript meets the required acceptance criteria.
Report #1 by Anonymous (Referee 4) on 2022-9-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202207_00047v1, delivered 2022-09-14, doi: 10.21468/SciPost.Report.5696
Strengths
Science preformed and presented in the paper is clear, accurate and sound.
Weaknesses
Motivation is lacking, and discussion/connection to the broader community is limited.
Report
The authors find the ground state of 4 interacting fermions in 1D. Two fermions are of type "a" and the other two are of type "b", and fermions of different species are strongly attractive and form a tightly bound state. Such a tightly bound state allows for analytic approximations, and an hilbert space truncation that increases numerical efficiency. The results presented are sound and representative of good science, and the presentation is accurate and clear. I therefore recommend publication after a the minor changes are made as described below.
In addition, the authors can improve the quality of the paper by addressing the the following concerns:
1) The paper appears to fit in a broader context then discussed in the introduction. Specifically, they are studying the ground state of 4 fermions, and the methods they use should be similar to related calculations in quantum chemistry.
2) It seems like there should be integrable models of the form similar to the ones studied by the authors such as those discussed in Guan et. al. Rev. Mod. Phys. 85, 1633 (2013). A discussion on this would be helpful to put the paper in a broader context.
3) What do the authors think for N>2 states? Will similar physics hold in a many body limit?
4) Adding the coboson ansatz line to fig 5a could be helpful even though it is trivial.
Requested changes
1) The authors refer to a true ground state |GS>. I think this is the ground state computed using the basis in equation 10, but it is not clear from the text. The authors should clarify this.
2) The extent to which |GS> is the "true" ground state depends on the validity of approximation in eq 4. This should be clarified, and the conditions should be specified for when this approximation is valid.
3) Specify N=2 for the coboson ansatz when discussing equation 12
4) The authors should explain why the infinite interaction strength approximation seems to work so well in fig 2, but less so in fig 1.
Also a few issues on language:
1) This is a missing period between eq 11 and 12
2) In eq 14, the sum over j is confusing because j is already a fixed index.
3) There is a "this" that refers to equation 9, but I think the authors intended it to refer to something else.
4)"quarctic" above equation 15
5) The phrase "one-pair eigenstate" at the bottom of page 2 was not clear on first reading. It is also inconsistent with "single-pair" basis which is also used.
6)The sentence above equation 8 is confusing. On first reading it seemed to suggest the index n was indexing the excited internal states.
7) Above equation 15, there is a cryptic sentence about the one body term not playing a role in the energy. This is not true, and the statement in the "( ...)" is not clear enough to explain.
8) Add more context to equation 18. As is, it was presented out of no where, was it proved in reference 7?
9) The authors should explain why this wave function has interaction energy \hbar \omega, or give more justification then is present.
10) In the last paragraph on page 7, the authors refer to "the coefficients". Which coefficients are they referring to?
11) The second paragraph of page 9 refers to N=2 without context as to what N is.