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Ground-state energy of a Richardson-Gaudin integrable BCS model
by Yibing Shen, Phillip S. Isaac, Jon Links
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Submission summary
| Authors (as registered SciPost users): | Phillip Isaac · Jon Links |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1912.05692v1 (pdf) |
| Date submitted: | 2019-12-13 01:00 |
| Submitted by: | Links, Jon |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate the ground-state energy of a Richardson-Gaudin integrable BCS model, generalizing the closed and open p+ip models. The Hamiltonian supports a family of mutually commuting conserved operators satisfying quadratic relations. From the eigenvalues of the conserved operators we derive, in the continuum limit, an integral equation for which a solution corresponding to the ground state is established. The energy expression from this solution agrees with the BCS mean-field result.
Current status:
Reports on this Submission
Anonymous Report 3 on 2020-1-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1912.05692v1, delivered 2020-01-10, doi: 10.21468/SciPost.Report.1440
Strengths
1. Exact result for ground-state energy of an integrable system.
2. Detailed proof of result contained.
Weaknesses
1. No discussion of the physical interpretation of the model.
2. Limited scope of result.
3. Few results of link to previously studied systems.
4. The authors show that the obtained result is identical to the ground-state result in the mean-field treatment, but they should add an argument why this also implies that they have obtained the true ground-state energy.
Report
The authors derive the ground-state energy of a generalised BCS model and show that it agrees with the finding of a mean-field analysis. Overall the results seem sound and suitable for publication in SciPost Core once the requested changes have been addressed.
Requested changes
1. The model (1) should be discussed in terms of its physical interpretation. Also it should be clarified under which conditions the model simplifies to the previously studied ones (eg, p+ip model).
2. After (2) it is required that \beta>0, but after (5) the authors set \beta=0.
3. It is unclear what the term "open" refers to since I cannot identify anything like a heat or particle bath in the Hamiltonian (1).
4. The authors should extend the discussion of how the x_i introduced in Sec. 4 enter, eg, in (7). I am confused since I cannot see any notion of a length in (1). so I do not understand how the density is introduced.
5. Similarly in spirit, the authors should extend the discussion of how the q_i of (7) become the functions Q(\epsilon) in (13).
6. Clarify the conclusion "this solution corresponds to the ground state of the model" in Proposition 1 and its relation to the mean-field analysis.
Anonymous Report 2 on 2019-12-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1912.05692v1, delivered 2019-12-17, doi: 10.21468/SciPost.Report.1396
Strengths
1- Shows that the ground state's energy of a generalisation of the p+ip BCS pairing model can, in the continuum limit, be found exactly without specifying the Bethe equations and/or the Bethe roots, by making use of the quadratic equations linking the eigenvalues.
2- It proves the validity of the mean-field result in the description of the model's GS energy.
3- The submission is remarkably clear.
Weaknesses
1 - The paper is very close in spirit to previous work by the authors and, as such, does not necessarily introduce new techniques but simply a new application of a previously established approach.
2- The physical relevance of this particular generalisation of the BCS model is not discussed which might limit the appeal of the result to the "integrable-models community".
Report
The work presented in this submission is mathematically correct and presents a new result concerning the exact solution of a generalised p+ip BCS model.
The authors find the explicit ground state solution, in the continuum limit, of the quadratic equations characteristic of spin-1/2 Richardson-Gaudin models for the "non-skew symmetric"/"XYZ with field" model.
This demonstration provides an interesting new result further extending the usability of the method developed in their previous work (doi:10.1016/j.nuclphysb.2018.08.015).
This further proof of usability of a promising method should, in my opinion, warrant publication. I feel that the Expectations of Scipost physics (groundbreaking, etc) are not met by the submission and therefore recommend publication in SciPost physics Core.
Requested changes
No necessary changes. Could be published as is in Scipost Physics Core.