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

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Abstract

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.

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References

  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)

    Article  ADS  MathSciNet  Google Scholar 

  2. Gaudin, M.: Un systeme à une dimension de fermions en interaction. Phys. Lett. A 24, 55 (1967)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  7. Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008)

    Article  ADS  Google Scholar 

  8. Popov, V.N.: Theory of one-dimensional Bose gas with point interaction. Theor. Math. Phys. 30, 222 (1977)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  10. Tracy, C.A., Widom, H.: On the ground state energy of the\(\updelta \)-function Bose gas. J. Phys. A 49, 294001 (2016a)

    Article  MathSciNet  Google Scholar 

  11. Tracy, C.A., Widom, H.: On the ground state energy of the delta-function Fermi gas. J. Math. Phys. 57, 103301 (2016b)

    Article  ADS  MathSciNet  Google Scholar 

  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. Prolhac, S.: Ground state energy of the \(delta\)-Bose and Fermi gas at weak coupling from double extrapolation. J. Phys. A 50, 144001 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  14. Lang, G.: Correlations in Low-Dimensional Quantum Gases. Springer, New York (2018)

    Book  Google Scholar 

  15. Ristivojevic, Z.: Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion. Phys. Rev. B 100, 081110 (2019)

    Article  ADS  Google Scholar 

  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]

    Article  ADS  Google Scholar 

  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]

    Article  ADS  MathSciNet  Google Scholar 

  18. Samaj, L., Bajnok, Z.: Introduction to the Statistical Physics of Integrable Many-Body Systems. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  19. Ristivojevic, Z.: Excitation spectrum of the Lieb-Liniger model. Phys. Rev. Lett. 113, 015301 (2014)

    Article  ADS  Google Scholar 

  20. Hutson, V.: The circular plate condenser at small separations. Math. Proc. Camb. Philos. Soc. 59, 211 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  21. Kostov, I., Serban, D., Volin, D.: Functional BES equation. JHEP 08, 101 (2008). arXiv:0801.2542 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  22. Bender, C.M., Wu, T.T.: Anharmonic oscillator. 2: a study of perturbation theory in large order. Phys. Rev. D 7, 1620 (1973)

    Article  ADS  Google Scholar 

  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]

    Article  MathSciNet  Google Scholar 

  24. Aniceto, I., Basar, G., Schiappa, R.: arXiv:1802.10441 [hep-th]

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  27. Mariño, M.: Instantons and Large \(N\). An Introduction to Non-perturbative Methods in Quantum Field theory. Cambridge University Press, Cambridge (2015)

    Book  Google Scholar 

  28. Mariño, M., Reis, T.: arXiv:1905.09569 [hep-th]

  29. Parisi, G.: Asymptotic estimates in perturbation theory with fermions. Phys. Lett. 66B, 382 (1977)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  31. Krivnov, V.Y., Ovchinnikov, A.: One-dimensional Fermi gas with attraction between the electrons. J. Exp. Theor. Phys. 40, 781 (1975)

    ADS  Google Scholar 

  32. Fuchs, J.N., Recati, A., Zwerger, W.: Exactly solvable model of the BCS-BEC crossover. Phys. Rev. Lett. 93, 090408 (2004)

    Article  ADS  Google Scholar 

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Acknowledgements

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).

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  1. Département de Physique Théorique et Section de Mathématiques, Université de Genève, 1211, Geneva, Switzerland

    Marcos Mariño & Tomás Reis

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  1. Marcos Mariño
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  2. Tomás Reis
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Correspondence to Marcos Mariño.

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Communicated by Hal Tasaki.

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Mariño, M., Reis, T. Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models. J Stat Phys 177, 1148–1156 (2019). https://doi.org/10.1007/s10955-019-02413-1

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