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Resurgence and renormalons in the onedimensional Hubbard model
by Marcos Marino, Tomas Reis
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Submission summary
Authors (as registered SciPost users):  Tomas Reis 
Submission information  

Preprint Link:  https://arxiv.org/abs/2006.05131v2 (pdf) 
Date submitted:  20210902 09:35 
Submitted by:  Reis, Tomas 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We use resurgent analysis to study nonperturbative aspects of the onedimensional, multicomponent Hubbard model with an attractive interaction and arbitrary filling. In the twocomponent case, we show that the leading Borel singularity of the perturbative series for the groundstate energy is determined by the energy gap, as expected for superconducting systems. This singularity turns out to be of the renormalon type, and we identify a class of diagrams leading to the correct factorial growth. As a consequence of our analysis, we propose an explicit expression for the energy gap at weak coupling in the multicomponent Hubbard model, at nexttoleading order in the coupling constant. In the twocomponent, halffilled case, we use the Bethe ansatz solution to determine the full transseries for the ground state energy, and the exact form of its Stokes discontinuity.
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Reports on this Submission
Anonymous Report 1 on 2021105 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2006.05131v2, delivered 20211005, doi: 10.21468/SciPost.Report.3621
Strengths
This paper studies the onedimensional Hubbard model and uses different approaches to study the perturbative and nonperturbative contributions to its ground state energy. The main strengths of the paper can be summarised as:
1. The relation between the nonperturbative contributions and the energy gap due to renormalons: in the twocomponent case the use of integrability and resurgence/transseries techniques allowed the prediction of the nonperturbative contribution to the ground state energy, associated to its divergent formal perturbative expansion. This nonperturbative exponential and more its prefactor were then obtained by studying the zero and the halffilled limiting cases. The authors then show that these results are consistent with the existence of certain renormalon diagrams.
2. A conjecture for the full ground state energy in the multicomponent case for arbitrary filling fractions, by using all the results from different limiting cases.
3. Using tools of resurgence and transseries to determine renormalon ring diagrams contributing nonperturbatively: this analysis is particularly interesting and elucidating at the halffilling limit as many results can be obtained exactly.
Weaknesses
The paper is generally well written and well presented, well referenced and detailed, and without any major weaknesses. The only issue I would say this paper currently has is:
1. In section 2, the authors discuss several different limits/cases of the multicomponent Hubbard model, but they are not clearly separated. It would definitely help the reader if the authors distinguished them more clearly, or summarized them in a paragraph at the beginning.
Report
This paper presents novel theoretical results and conjectures as well as their physical interpretation for the one dimensional Hubbard model. It also shows the strengths of different methods in addressing this problem and how to use them effectively. It is my opinion that this paper meets the expectations and all the acceptance criteria, and it should be published in SciPost after some minor revision.
Requested changes
There are a few minor changes that I would suggest the authors should take into consideration:
1. as mentioned above, in section 2, the authors discuss several different limits/cases of the multicomponent Hubbard model, but they are not clearly separated. It would definitely help the reader if the authors distinguished them more clearly, or summarized them in a paragraph at the beginning.
2. In section 2.1 a general reference to the Hubbard model and its main features is missing
3. Equation (2.6) is a different representation of (2.4), so it would be useful to add H_I= also in the latter.
4. Just after equation (2.20) the authors state that the Fermi momentum is also kept fixed. How is this achieved given the definition (2.8)?
5. Can the authors provide a reference for equation (2.23)?
6. Just after figure 2, when the coefficient E_2(n,kappa) is mentioned, it would be useful to recall where it was defined (equation (2.10) or (2.17)).
7. In section 2.3 the parameter B first appearing in equation (2.44) was never defined.
8. In section 3.2 it is mentioned that other models have been studied using Volin's method. The authors should add references to some of these papers.
9. equation (4.5) looks like a single parameter transseries, although the authors have added multiple parameters. Usually one would have c^ell instead of c_ell for this shape of a transseries, why is this different?
10. Just after equation (4.13), what is RPA?
11. equation (4.66) is somewhat misleading. The median summation has this form because it fixes the transseries parameters such as the lateral summations of the full transseries match. The authors should add a note to this effect. In the current case these lateral summations are very simple and this cancellation must happen automatically, is that the case?