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Resurgence and renormalons in the one-dimensional Hubbard model

by Marcos Marino, Tomas Reis

This is not the current version.

Submission summary

As Contributors: Tomas Reis
Arxiv Link: (pdf)
Date submitted: 2021-09-02 09:35
Submitted by: Reis, Tomas
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We use resurgent analysis to study non-perturbative aspects of the one-dimensional, multicomponent Hubbard model with an attractive interaction and arbitrary filling. In the two-component case, we show that the leading Borel singularity of the perturbative series for the ground-state 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 multi-component Hubbard model, at next-to-leading order in the coupling constant. In the two-component, half-filled case, we use the Bethe ansatz solution to determine the full trans-series for the ground state energy, and the exact form of its Stokes discontinuity.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 2006.05131v3 on 18 November 2021

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Submission 2006.05131v2 on 2 September 2021

Reports on this Submission

Anonymous Report 1 on 2021-10-5 (Invited Report)


This paper studies the one-dimensional Hubbard model and uses different approaches to study the perturbative and non-perturbative contributions to its ground state energy. The main strengths of the paper can be summarised as:

1. The relation between the non-perturbative contributions and the energy gap due to renormalons: in the two-component case the use of integrability and resurgence/trans-series techniques allowed the prediction of the non-perturbative contribution to the ground state energy, associated to its divergent formal perturbative expansion. This non-perturbative exponential and more its prefactor were then obtained by studying the zero and the half-filled 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 multi-component case for arbitrary filling fractions, by using all the results from different limiting cases.

3. Using tools of resurgence and trans-series to determine renormalon ring diagrams contributing non-perturbatively: this analysis is particularly interesting and elucidating at the half-filling limit as many results can be obtained exactly.


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 multi-component 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.


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 multi-component 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 trans-series, although the authors have added multiple parameters. Usually one would have c^ell instead of c_ell for this shape of a trans-series, 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 trans-series parameters such as the lateral summations of the full trans-series 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?

  • validity: high
  • significance: high
  • originality: top
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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