Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models

  • Marcos MariñoEmail author
  • Tomás Reis


We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb–Liniger and Gaudin–Yang models, starting from the Bethe ansatz solution. This makes it possible to calculate explicitly the coefficients of these series and to study their large order behavior. We find that both series diverge factorially and are not Borel summable. In the case of the Gaudin–Yang model, the first Borel singularity is determined by the non-perturbative energy gap. This provides a new perspective on the Cooper instability.


Quantum gases Bethe ansatz Nonperturbative effects 



We would like to thank Sylvain Prolhac, Wilhelm Zwerger and specially Thierry Giamarchi and Félix Werner for useful discussions and comments on the manuscript. This work is supported in part by the Fonds National Suisse, subsidies 200021-156995 and 200020-141329, and by the NCCR 51NF40-141869 “The Mathematics of Physics” (SwissMAP).


  1. 1.
    Lieb, E.H., Liniger, W.: Exact analysis of an interacting Bose gas. 1. The general solution and the ground state. Phys. Rev. 130, 1605 (1963)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Gaudin, M.: Un systeme à une dimension de fermions en interaction. Phys. Lett. A 24, 55 (1967)ADSCrossRefGoogle Scholar
  3. 3.
    Yang, C.-N.: Some exact results for the many body problems in one dimension with repulsive delta function interaction. Phys. Rev. Lett. 19, 1312 (1967)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Jiang, Y.-Z., Chen, Y.-Y., Guan, X.-W.: Understanding many-body physics in one dimension from the Lieb-Liniger model. Chin. Phys. B 24, 050311 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    Guan, X.-W., Batchelor, M.T., Lee, C.: Fermi gases in one dimension: from Bethe ansatz to experiments. Rev. Mod. Phys. 85, 1633 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Cazalilla, M.A., Citro, R., Giamarchi, T., Orignac, E., Rigol, M.: One dimensional bosons: from condensed matter systems to ultracold gases. Rev. Mod. Phys. 83, 1405 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Popov, V.N.: Theory of one-dimensional Bose gas with point interaction. Theor. Math. Phys. 30, 222 (1977)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Iida, T., Wadati, M.: Exact analysis of \(\delta \)-function attractive fermions and repulsive Bosons in one-dimension. J. Phys. Soc. Jpn. 74, 1724 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    Tracy, C.A., Widom, H.: On the ground state energy of the\(\updelta \)-function Bose gas. J. Phys. A 49, 294001 (2016a)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Tracy, C.A., Widom, H.: On the ground state energy of the delta-function Fermi gas. J. Math. Phys. 57, 103301 (2016b)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Tracy, C.A., Widom, H.: In: Kielanowski, P., Odzijewicz, A., Previato, E. (eds.) Geometric Methods in Physics, pp. 201–212. Springer, New York (2018)Google Scholar
  13. 13.
    Prolhac, S.: Ground state energy of the \(delta\)-Bose and Fermi gas at weak coupling from double extrapolation. J. Phys. A 50, 144001 (2017)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Lang, G.: Correlations in Low-Dimensional Quantum Gases. Springer, New York (2018)CrossRefGoogle Scholar
  15. 15.
    Ristivojevic, Z.: Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion. Phys. Rev. B 100, 081110 (2019)ADSCrossRefGoogle Scholar
  16. 16.
    Volin, D.: From the mass gap in O(N) to the non-Borel-summability in O(3) and O(4) sigma-models. Phys. Rev. D 81, 105008 (2010). arXiv:0904.2744 [hep-th]ADSCrossRefGoogle Scholar
  17. 17.
    Volin, D.: Quantum integrability and functional equations: applications to the spectral problem of AdS/CFT and two-dimensional sigma models. J. Phys. A 44, 124003 (2011). arXiv:1003.4725 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Samaj, L., Bajnok, Z.: Introduction to the Statistical Physics of Integrable Many-Body Systems. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  19. 19.
    Ristivojevic, Z.: Excitation spectrum of the Lieb-Liniger model. Phys. Rev. Lett. 113, 015301 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Hutson, V.: The circular plate condenser at small separations. Math. Proc. Camb. Philos. Soc. 59, 211 (1963)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kostov, I., Serban, D., Volin, D.: Functional BES equation. JHEP 08, 101 (2008). arXiv:0801.2542 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Bender, C.M., Wu, T.T.: Anharmonic oscillator. 2: a study of perturbation theory in large order. Phys. Rev. D 7, 1620 (1973)ADSCrossRefGoogle Scholar
  23. 23.
    Mariño, M.: Lectures on non-perturbative effects in large \(N\) gauge theories, matrix models and strings. Fortschr. Phys. 62, 455 (2014). arXiv:1206.6272 [hep-th]MathSciNetCrossRefGoogle Scholar
  24. 24.
    Aniceto, I., Basar, G., Schiappa, R.: arXiv:1802.10441 [hep-th]
  25. 25.
    Baker, G.A.: Singularity structure of the perturbation series for the ground-state energy of a many-fermion system. Rev. Mod. Phys. 43, 479 (1971)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Rossi, R., Ohgoe, T., Van Houcke, K., Werner, F.: Resummation of diagrammatic series with zero convergence radius for strongly correlated fermions. Phys. Rev. Lett. 121, 130405 (2018)ADSCrossRefGoogle Scholar
  27. 27.
    Mariño, M.: Instantons and Large \(N\). An Introduction to Non-perturbative Methods in Quantum Field theory. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  28. 28.
    Mariño, M., Reis, T.: arXiv:1905.09569 [hep-th]
  29. 29.
    Parisi, G.: Asymptotic estimates in perturbation theory with fermions. Phys. Lett. 66B, 382 (1977)ADSCrossRefGoogle Scholar
  30. 30.
    Baker Jr., G.A., Pirner, H.J.: Asymptotic estimate of large orders in perturbation theory for the many-fermion ground state energy. Ann. Phys. 148, 168 (1983)ADSCrossRefGoogle Scholar
  31. 31.
    Krivnov, V.Y., Ovchinnikov, A.: One-dimensional Fermi gas with attraction between the electrons. J. Exp. Theor. Phys. 40, 781 (1975)ADSGoogle Scholar
  32. 32.
    Fuchs, J.N., Recati, A., Zwerger, W.: Exactly solvable model of the BCS-BEC crossover. Phys. Rev. Lett. 93, 090408 (2004)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de Physique Théorique et Section de MathématiquesUniversité de GenèveGenevaSwitzerland

Personalised recommendations