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Bose–Einstein Condensation in the Luttinger–Sy Model with Contact Interaction

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Abstract

We study bosons on the real line in a Poisson random potential (Luttinger–Sy model) with contact interaction in the thermodynamic limit at absolute zero temperature. We prove that generalized Bose–Einstein condensation (BEC) occurs almost surely if the intensity \(\nu _N\) of the Poisson potential satisfies \([\ln (N)]^4/N^{1 - 2\eta } \ll \nu _N\lesssim 1\) for arbitrary \(0 < \eta \le 1/3\). We also show that the contact interaction alters the type of condensation, going from a type-I BEC to a type-III BEC as the strength of this interaction is increased. Furthermore, for sufficiently strong contact interactions and \(0< \eta < 1/6\), we prove that the mean particle density in the largest interval is almost surely bounded asymptotically by \(\nu _NN^{3/5+\delta }\) for \(\delta > 0\).

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References

  1. Bolte, J., Kerner, J.: Many-particle quantum graphs and Bose–Einstein condensation. J. Math. Phys. 55(6), 061901 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bolte, J., Kerner, J.: Instability of Bose–Einstein condensation into the one-particle ground state on quantum graphs under repulsive perturbations. J. Math. Phys. 57(4), 043301 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Casimir, H.: On Bose–Einstein condensation. Fundam. Probl. Stat. Mech. 3, 188–196 (1968)

    MATH  Google Scholar 

  4. Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27, 642–669 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Smedt, P.: The effect of repulsive interactions on Bose–Einstein condensation. J. Stat. Phys. 45, 201–213 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  6. Einstein, A.: Quantentheorie des einatomigen idealen Gases. Sitzber. Kgl. Preuss. Akad. Wiss. 261–267 (1924)

  7. Einstein, A.: Quantentheorie des einatomigen idealen Gases, II. Abhandlung, Sitzber. Kgl. Preuss. Akad. Wiss. pp. 3–14 (1925)

  8. Girardeau, M.: Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1(6), 516–523 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967)

    Article  ADS  Google Scholar 

  10. Jaeck, T., Pulé, J.V., Zagrebnov, V.A.: On the nature of Bose–Einstein condensation enhanced by localization. J. Math. Phys. 51(10), 103302 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kingman, J.: Poisson Processes. Clarendon Press, New York (1993)

    MATH  Google Scholar 

  12. Kennedy, T., Lieb, E.H., Shastry, B.S.: The XY model has long-range order for all spins and all dimensions greater than one. Phys. Rev. Lett. 61, 2582–2584 (1988)

    Article  ADS  Google Scholar 

  13. Kirsch, W., Simon, B.: Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators. Commun. Math. Phys. 97(3), 453–460 (1985)

    Article  ADS  MATH  Google Scholar 

  14. Lieb, E.H., Liniger, W.: Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Lenoble, O., Pastur, L.A., Zagrebnov, V.A.: Bose–Einstein condensation in random potentials. C. R. Phys. 5(1), 129–142 (2004)

    Article  ADS  Google Scholar 

  16. Luttinger, J.M., Sy, H.K.: Bose–Einstein condensation in a one-dimensional model with random impurities. Phys. Rev. A 7(2), 712 (1973)

    Article  ADS  Google Scholar 

  17. Luttinger, J.M., Sy, H.K.: Low-lying energy spectrum of a one-dimensional disordered system. Phys. Rev. A 7, 701–712 (1973)

    Article  ADS  Google Scholar 

  18. Lieb, E.H., Seiringer, R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002)

    Article  ADS  Google Scholar 

  19. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and Its Condensation, Oberwolfach Seminars, vol. 34. Birkhäuser Verlag, Basel (2005)

  20. Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61(4), 043602 (2000)

    Article  ADS  Google Scholar 

  21. Lieb, E.H., Seiringer, R., Yngvason, J.: One-dimensional behavior of dilute, trapped Bose gases. Commun. Math. Phys. 244(2), 347–393 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Lauwers, J., Verbeure, A., Zagrebnov, V.A.: Proof of Bose–Einstein condensation for interacting gases with a one-particle gap. J. Phys. A 36, 169–174 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Landau, L.J., Wilde, I.F.: On the Bose–Einstein condensation of an ideal gas. Commun. Math. Phys. 70(1), 43–51 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  24. Massart, P.: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18(3), 1269–1283 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Michelangeli, A.: Reduced density matrices and Bose–Einstein condensation. SISSA 39 (2007)

  26. Pastur, A.L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  27. Penrose, O., Onsager, L.: Bose–Einstein condensation and liquid helium. Phys. Rev. 104, 576–584 (1956)

    Article  ADS  MATH  Google Scholar 

  28. Seiringer, R., Yngvason, J., Zagrebnov, V.A.: Disordered Bose–Einstein condensates with interaction in one dimension. J. Stat. Mech. Theory Exp. 2012(11), P11007 (2012)

    Article  Google Scholar 

  29. Seiringer, R., Yngvason, J., Zagrebnov, V.A.: Disordered Bose–Einstein condensates with interaction. In: Proceedings of XVIIth International Congress on Mathematical Physics. World Scientific, pp. 610–619 (2013)

  30. van den Berg, M.: On condensation in the free-boson gas and the spectrum of the Laplacian. J. Stat. Phys. 31, 623–637 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  31. van den Berg, M., Lewis, J.T.: On generalized condensation in the free boson gas. Phys. A Stat. Mech. Appl. 110(3), 550–564 (1982)

    Article  MathSciNet  Google Scholar 

  32. van den Berg, M., Lewis, J.T., Lunn, M.: On the general theory of Bose–Einstein condensation and the state of the free boson gas. Helv. Phys. Acta 59(8), 1289–1310 (1986)

    MathSciNet  Google Scholar 

  33. van den Berg, M., Lewis, J.T., Pulé, J.V.: A general theory of Bose–Einstein condensation. Helv. Phys. Acta 59(8), 1271–1288 (1986)

    MathSciNet  Google Scholar 

  34. Zagrebnov, V.A.: Bose–Einstein condensation in a random media. J. Phys. Stud. 11(1), 108–121 (2007)

    Google Scholar 

  35. Zagrebnov, V.A., Bru, J.B.: The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350(5–6), 291–434 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, FernUniversität in Hagen, 58084, Hagen, Germany

    Joachim Kerner, Maximilian Pechmann & Wolfgang Spitzer

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  1. Joachim Kerner
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  2. Maximilian Pechmann
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  3. Wolfgang Spitzer
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Correspondence to Joachim Kerner.

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Communicated by Vieri Mastropietro.

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Kerner, J., Pechmann, M. & Spitzer, W. Bose–Einstein Condensation in the Luttinger–Sy Model with Contact Interaction. Ann. Henri Poincaré 20, 2101–2134 (2019). https://doi.org/10.1007/s00023-019-00771-w

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