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Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1037–1090 | Cite as

The Bogoliubov Free Energy Functional I: Existence of Minimizers and Phase Diagram

  • Marcin Napiórkowski
  • Robin Reuvers
  • Jan Philip Solovej
Article

Abstract

The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove the existence of a phase transition in this model and provide its phase diagram.

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References

  1. 1.
    Angelescu N., Verbeure A.: Variational solution of a superfluidity model. Physica A 216, 386–396 (1995)ADSCrossRefGoogle Scholar
  2. 2.
    Angelescu N., Verbeure A., Zagrebnov V.: Superfluidity III. J. Phys. A: Math. Gen. 30, 4895–4913 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold P., Moore G.: BEC transition temperature of a dilute homogeneous imperfect Bose gas. Phys. Rev. Lett. 87, 120401 (2001)ADSCrossRefGoogle Scholar
  4. 4.
    Bach V., Breteaux S., Knörr H.K., Menge E.: Generalization of Lieb’s variational principle to Bogoliubov–Hartree–Fock theory. J. Math. Phys. 55, 012101 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bach V., Lieb E.H., Solovej J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balibar, S.: Looking back at superfluid helium. Proceedings of the Conference “Bose–Einstein Condensation”, (Eds. J. Dalibard, B. Duplantier, and V. Rivasseau) Birkäuser, 2004Google Scholar
  7. 7.
    Bogoliubov N.N.: On the theory of superfluidity. J. Phys. (USSR) 11, 23 (1947)MathSciNetGoogle Scholar
  8. 8.
    Bräunlich G., Hainzl C., Seiringer R.: Translation-invariant quasi-free states for fermionic systems and the BCS approximation. Rev. Math. Phys. 26, 1450012 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Critchley R.H., Solomon A.: A variational approach to superfluidity. J. Stat. Phys. 14, 381–393 (1976)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Davis K.B., Mewes M.O., Andrews M.R., van Druten N.J., Durfee D.S., Kurn D.M., Ketterle W.: Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995)ADSCrossRefGoogle Scholar
  11. 11.
    Dereziński, J., Napiórkowski, M.: Excitation spectrum of interacting bosons in the mean-field infinite-volume limit. Annales Henri Poincaré 15, 2409–2439 (2014); Erratum: Annales Henri Poincaré 16, 1709–1711 (2015)Google Scholar
  12. 12.
    Erdos L., Schlein B., Yau H.-T.: Ground-state energy of a low-density Bose gas: a second-order upper bound. Phys. Rev. A 78, 053627 (2008)ADSCrossRefGoogle Scholar
  13. 13.
    Frank R.L., Hainzl C., Naboko S., Seiringer R.: The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17, 559–567 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Giuliani A., Seiringer R.: The ground state energy of the weakly interacting Bose gas at high density. J. Stat. Phys. 135, 915–934 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grech P., Seiringer R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 322, 559–591 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hainzl C., Hamza E., Seiringer R., Solovej J.P.: The BCS functional for general pair interactions. Commun. Math. Phys. 281, 349–367 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hainzl C., Seiringer R.: The BCS critical temperature for potentials with negative scattering length. Lett. Math. Phys. 84, 99–107 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kashurnikov V.A., Prokof’ev N.V., Svistunov B.V.: Critical temperature shift in weakly interacting Bose gas. Phys. Rev. Lett. 87, 120402 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Landau L.: Theory of the superfluidity of helium II. Phys. Rev. 60, 356–358 (1941)ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Lenzmann E., Lewin M.: Minimizers for the Hartree–Fock–Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152, 257–315 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lewin M., Nam P.T., Serfaty S., Solovej J.P.: Bogoliubov spectrum of interacting Bose gases. Commun. Pure Appl. Math. 68(3), 413–471 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose as and Its Condensation. Oberwolfach Seminars, Birkhäuser, 2005Google Scholar
  23. 23.
    Lieb E.H., Seiringer R., Yngvason J.: Justification of c-number substitutions in bosonic hamiltonians. Phys. Rev. Lett. 94, 080401 (2005)ADSCrossRefGoogle Scholar
  24. 24.
    Nam P.T., Seiringer R.: Collective excitations of bose gases in the mean-field regime. Arch. Ration. Mech. Anal. 215, 381–417 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Napiórkowski M., Nam P.T., Solovej J.P.: Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations. J. Funct. Anal. 270, 4340–4368 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Napiórkowski, M., Reuvers, R., Solovej, J.P.: Bogoliubov free energy functional II. The dilute limit. Commun. Math. Phys. (2017). https://doi.org/10.1007/s00220-017-3064-x
  27. 27.
    Napiórkowski, M., Reuvers, R., Solovej, J.P.: Calculation of the Critical Temperature of a Dilute Bose Gas in the Bogoliubov Approximation (2017). arXiv:1706.01822
  28. 28.
    Nho K., Landau D.P.: Bose–Einstein condensation temperature of a homogeneous weakly interacting Bose gas: PIMC study. Phys. Rev. A 70, 053614 (2004)ADSCrossRefGoogle Scholar
  29. 29.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional analysis. Academic Press, 1972Google Scholar
  30. 30.
    Seiringer R.: Free energy of a dilute Bose gas: lower bound. Commun. Math. Phys. 279, 595–636 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Seiringer R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Seiringer R.: Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation. J. Math. Phys., 55, 075209 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Solovej J.P.: Upper bounds to the ground state energies of the one- and two-component charged Bose gases. Commun. Math. Phys. 266, 797–818 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Solovej, J.P.: Many-Body Quantum Mechanics. ESI Vienna, Lecture Notes, 2014Google Scholar
  35. 35.
    Yin J.: Free energies of a dilute Bose gases: upper bound. J. Stat. Phys. 141, 683–726 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zagrebnov V.A., Bru J.-B.: The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350, 291–434 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Science and Technology AustriaKlosterneuburgAustria
  2. 2.Department of Mathematical Methods in Physics, Faculty of PhysicsUniversity of WarsawWarsawPoland
  3. 3.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  4. 4.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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