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Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb-Liniger and Yang-Gaudin models
by Guillaume Lang
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Submission summary
| Authors (as registered SciPost users): | Guillaume Lang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_201908_00007v1 (pdf) |
| Date submitted: | Aug. 9, 2019, 2 a.m. |
| Submitted by: | Lang, Guillaume |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb-Liniger Bose gas and the Yang-Gaudin Fermi gas with contact interactions. We identify patterns in weak- and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2019-9-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_201908_00007v1, delivered 2019-09-11, doi: 10.21468/SciPost.Report.1164
Strengths
1) Provides an interesting set of conjectures on the weak and strong coupling expansions of two important integrable systems in many-body theory.
2) These conjectures might stimulate further research and improve the analytic study of the model.
Weaknesses
Report
Requested changes
In his reply to another referee report, the author says that "The radius of convergence of the weak-coupling expansion is readily understood from the poles of the partial resummations, as pointed out by the referee". This seems to imply that the radius of convergence is finite. However, this contradicts one of the main findings of refs. [50,51], which have shown numerically that the weak-coupling expansion diverges factorially (as in many other quantum-mechanical models) and therefore has zero radius of convergence. In view of this, the author should address this issue more carefully.
Report #2 by Anonymous (Referee 2) on 2019-8-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_201908_00007v1, delivered 2019-08-26, doi: 10.21468/SciPost.Report.1132
Strengths
Weaknesses
Report
Although systematic algorithms now exist in some cases for both strong and weak coupling expansions, there is no closed formula for either expansion up to arbitrary order. The author was able to formulate plausible conjectures about the structure of these expansions, and even perform a complete resummation of some portions of the expressions for the groundstate energy, paving the way to a possible combinatorial solution for the full expansions.
Requested changes
1) It would be useful to add a discussion of the radius of convergence of the strong and weak coupling expansions. Can they be explained by the location of poles in your various resummations ? This would give a clearer motivation for the paper.
2) At several places, e.g. in conjecture V when comparing to [66], it is not clear up to which order the conjectures have been checked. This must be said more explicitly.
3) Maybe you could comment on the advantages of the algorithm from [40] for the strong coupling expansion, if any, compared with the more standard inversion of the integral operator as explained in e.g. [46].
4) Equation (19): it should be $O(\gamma^{(n+1)/2})$ and not $O(\gamma^{(k+1)/2})$.
5) Below (12), the standard notation for the floor function is $\lfloor ... \rfloor$.
Report #1 by Anonymous (Referee 1) on 2019-8-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_201908_00007v1, delivered 2019-08-13, doi: 10.21468/SciPost.Report.1113
Strengths
Weaknesses
and time dependence.
The author do not know the literature on the subject.
Report
1) The relation of integrability and number theory was explained in the paper Quantum Correlations and Number Theory by H. E. Boos, V. E. Korepin, Y. Nishiyama, M. Shiroishi, Journal of Physics A Math. and General, vol 35, pages 4443-4452, 2002
2) Correlation functions were calculated in Lieb-Liniger model in the tex book Quantum Inverse Scattering Method and Correlation Functions by V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Cambridge University Press , 1993.
Requested changes
The author has to compare his results to the literature [see report] and resubmit the paper.
Author: Guillaume Lang on 2019-08-19 [id 580]
(in reply to Report 1 on 2019-08-13)
We thank the referee for pointing out these two relevant references, and apologize for omitting them. They have been added to the bibliography in next version of the article.
We are fully aware that 'the most important in correlation functions is space
and time dependence', and have already published several papers on this topic. However, they are beyond the scope of this very article, and we do not wish to address them here. A few comments on their importance have been added, though, and they are now mentioned in outlook.
The conjecture in reference 1) suggested by the reviewer is indeed very similar to ours, though both the model and observable differ. The appearance of ln(2) in one case and not the other is an important fact, that we had not envisaged at first. Comments have been added to the conjecture in accordance.
The results in reference 2) are exact and very important for mathematical physics, but not explicit beyond first order. It is thus more relevant to compare ours to the most accurate expansions available to date: the exact result for g1, order 20 in 1/gamma for g2 and g3 and first non-zero order for gM.
Resubmission will follow soon.
Author: Guillaume Lang on 2019-08-28 [id 586]
(in reply to Report 2 on 2019-08-26)We answer the referee's queries point by point:
1) It would be useful to add a discussion of the radius of convergence of the strong and weak coupling expansions. Can they be explained by the location of poles in your various resummations ? This would give a clearer motivation for the paper. -> The strong-coupling expansion in its standard form is valid for \alpha>2, i.e. for \gamma>4.53. Once partially resummed, the radius of convergence becomes infinite. -> The radius of convergence of the weak-coupling expansion is readily understood from the poles of the partial resummations, as pointed out by the referee. -> Comments on these points have been added.
2) At several places, e.g. in conjecture V when comparing to [66], it is not clear up to which order the conjectures have been checked. This must be said more explicitly. -> Details have been provided more explicitly everywhere in the new version.
3) Maybe you could comment on the advantages of the algorithm from [40] for the strong coupling expansion, if any, compared with the more standard inversion of the integral operator as explained in e.g. [46]. -> Already at third order, the terms of the Neumann series are too complicated, thus they have never been computed analytically. The method of ref.[40] yields exact coefficients at very high orders in comparison.
4) Equation (19): it should be O(γ(n+1)/2) and not O(γ(k+1)/2). -> Modified as requested.
5) Below (12), the standard notation for the floor function is ⌊...⌋ -> Modified as requested.