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Conjectures about the structure of strong and weakcoupling expansions of a few groundstate observables in the LiebLiniger and YangGaudin 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:  20190809 02:00 
Submitted by:  Lang, Guillaume 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the LiebLiniger Bose gas and the YangGaudin Fermi gas with contact interactions. We identify patterns in weak and strongcoupling series expansions of the groundstate 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
Anonymous Report 3 on 2019911 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_201908_00007v1, delivered 20190911, 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 manybody theory.
2) These conjectures might stimulate further research and improve the analytic study of the model.
Weaknesses
The results of this paper do not seem to have a strong bearing on the physics of these models.
Report
In this paper, a series of conjectures are made on the structure of the weak and strong coupling expansions of the ground state energy in the LiebLiniger and the GaudinYang models, as well as on other observables, based on the most up to date analytic results for these expansions. The author had already made an interesting contribution to the subject in his Ph.D. by identifying some numerical results in terms of analytic expressions featuring zeta functions, and now, based on these recent results, he presents some guesses about partial structures in these expansions which have been validated to very high order. Although these conjectures are somewhat "numerological", and they do not lead to strong intuitions on the underlying physics, they might be useful in future studies of these models.
Requested changes
In his reply to another referee report, the author says that "The radius of convergence of the weakcoupling 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 weakcoupling expansion diverges factorially (as in many other quantummechanical models) and therefore has zero radius of convergence. In view of this, the author should address this issue more carefully.
Anonymous Report 2 on 2019826 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_201908_00007v1, delivered 20190826, doi: 10.21468/SciPost.Report.1132
Strengths
Timely paper supplementing nicely very recent results on the subject.
Weaknesses
The conjectures given in the paper only address very specific contributions to the groundstate energy without any particular physical significance.
Report
The author studies the groundstate energy for the celebrated LiebLiniger and YangGaudin models, which are integrable systems of Bosons and Fermions in one dimension with $\delta$ interaction of strength $\gamma$. Known exact results involving integral equations have been derived more than 50 years ago. While the strong coupling expansion $\gamma\to\infty$ can be easily extracted in a systematic way from the integral equations, the weak coupling expansions $\gamma\to0$ are significantly harder, with only the very first few terms known before a systematic approach was devised [50,51] in 2019.
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 $\lfloor ... \rfloor$.
Anonymous Report 1 on 2019813 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_201908_00007v1, delivered 20190813, doi: 10.21468/SciPost.Report.1113
Strengths
Important models. Attempt to relate integrability to number theory.
Weaknesses
The author do not know that the most important in correlation functions is space
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 44434452, 2002
2) Correlation functions were calculated in LiebLiniger 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 20190819 [id 580]
(in reply to Report 1 on 20190813)
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 nonzero order for gM.
Resubmission will follow soon.
Author: Guillaume Lang on 20190828 [id 586]
(in reply to Report 2 on 20190826)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 strongcoupling 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 weakcoupling 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.