SciPost Submission Page
Completeness of the Bethe states for the rational, spin-1/2 Richardson--Gaudin system
by Jon Links
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Jon Links |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1603.03542v4 (pdf) |
| Date submitted: | May 26, 2017, 2 a.m. |
| Submitted by: | Jon Links |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson--Gaudin system it will be argued that, for generic values of the system's coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2017-6-20 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1603.03542v4, delivered 2017-06-20, doi: 10.21468/SciPost.Report.169
Strengths
1- Very well written and easy to follow 2- Original and simple approach to the completeness problem for a Bethe Ansatz solvable model 3- Offers great potential for generalisation to other Richardson-Gaudin systems.
Weaknesses
1 - The claim that "general position" is a generic setting appears to be the weakest part of the work, at least from a formal point of view.
They do prove it in limit of large | \alpha |, they recognise that there are instances for which it is not the case: certain specific coupling values (with matching singularities) and the generic limit \alpha=0. They appeal to numerical studies to claim that those matching singularities points are indeed non-generic. In this particular work, this appeal to numerics seems completely sufficient to validate this point, but this could become problematic for generalisations of this work, which might then require case by case numerics.
2- It focuses on a single model while possible generalisations are mentioned throughout the paper: section 2.1 mentions a generalised problem on which no further claims seem to be made.
Report
The submission deals with the lack of spurious solutions, the capacity to regularise potential "null" eignevectors, and uses the fact that the problem has a simple spectrum (one-dimensional eigenspaces) in relation with the operator identities found in proposition 1, which links together the conserved charges of this integrable model. Therefore, it present a complete proof of the completness of the BEthe Ansatz for generic parameters of the system
The results are presented very clearly and in a way which makes this work accessible to both the mathematics and physics communities.
Provided the requested minor changes are addressed, this work deserves publication in scipost.org.
Requested changes
1- The author insists on the technique not relying on string hypothesis or solution counting and claims that not much has been published in the literature concerning this problem. There is at least one reference (not mentioned by the author):
E. Mukhin, V. Tarasov and A. Varchenko, Glasgow Math. J. 51A 137 (2009) doi:10.1017/S0017089508004850 or https://arxiv.org/abs/0712.0981
which also adresses the same problem and reaches the same conclusions, it would be crucial that the author makes explicit reference to this work and adresses clearly how his work adds to and differs from theirs.
2- Following proposition 1, the author should make reference to the similar identity found in [30] (as was the case in the earlier versions of this submission (v1; v2)). Such a fact is now only acknowledged in the conclusion of this submission.
3- There is a typo in the sentence ending the proof paragraph of page 11: "The can be no spurious solutions" should read " There can be ... "
Report #1 by Anonymous (Referee 1) on 2017-6-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1603.03542v4, delivered 2017-06-14, doi: 10.21468/SciPost.Report.165
Strengths
1.The article be well written, easy to read. 2. The method used is new and functional. 3. the model considered is simple. 4. The results are of interest to both physicists and mathematicians.
Weaknesses
Report
Requested changes
non Changes