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Resurgence and renormalons in the one-dimensional Hubbard model

by Marcos Marino, Tomas Reis

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Authors (as registered SciPost users): Tomas Reis
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-11-18 17:40
Submitted by: Reis, Tomas
Submitted to: SciPost Physics
Ontological classification
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

Reports on this Submission

Anonymous Report 1 on 2022-9-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2006.05131v3, delivered 2022-09-05, doi: 10.21468/SciPost.Report.5644


This paper studies the 1d multicomponent Hubbard model.
A recent conjecture proposed by the same authors for Fermi systems with attractive interactions relates the gap in the spectrum, which is non-perturbative in the coupling constant, to the large order behaviour of the perturbative expansion.

For the two component Hubbard model the authors use integrability to obtain the perturbative expansion of the ground state energy, and verify their conjecture connecting the large order behaviour of this expansion to the energy gap.

Very interestingly the authors are able to predict the asymptotic gap for the non-integrable multicomponent case using resurgence analysis.


There are no serious weaknesses. The paper is sometimes slightly difficult to read, in particular in the first sections where it refers to numerous models and results obtained in the literature.


The results and the new computational strategies developed in this paper are very interesting and are very likely to lead to many other important results.

This paper meets all the criteria to be published in SciPost after some very minor revisions.

Requested changes

I have only two minor suggestions for the authors.

1) Hartree Fock and Ring diagrams are mentioned in section 2.1 and sketched in Figure1. Perhaps the authors can elaborate a bit more on this, for example adding to the caption an explanation of the notation for dashed vs continuous lines in the diagrams.

2)The authors should clarify the relation between E^ring_l(n,k) in eq. 2.23 and E_l(n,k) in 2.14. What else can contribute to E_l(n,k)?

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

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Tomas Reis  on 2022-01-07  [id 2076]

We would like to thank the referee for their attentive reading of our manuscript. We have incorporated all the suggestions in this revision of the paper, and included the requested clarifications.

Perhaps we can comment more particularly on point (9) raised by the referee. In trans-series arising from ordinary differential equations, the trans-series parameters are all related, and for equations of first order, the coefficient of the \ell-th “instanton" correction is indeed of the form C^\ell. However, in trans-series appearing in QFT, we do not have any reason to believe that this will be also the case (or at least we are not aware of any result going into that direction). That’s why we have decided to write a more generic trans-series form.

The point (11) has been also addressed. It turns out that all the series appearing in the trans-series (except the perturbative one) are Borel summable along the positive real axis, so the median resummation is much simplified.