Skip to main content
SpringerLink
Log in

Nonperturbative Renormalization Group and Bose-Einstein Condensation

  • Chapter
  • First Online:
Renormalization Group and Effective Field Theory Approaches to Many-Body Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 852))

Abstract

These lectures are centered around a specific problem, the effect of weak repulsive interactions on the transition temperature \(T_c\) of a Bose gas. This problem provides indeed a beautiful illustration of many of the techniques which have been discussed at this school on effective theories and renormalization group. Effective theories are used first in order to obtain a simple hamiltonian describing the atomic interactions: because the typical atomic interaction potentials are short range, and the systems that we consider are dilute, these potentials can be replaced by a contact interaction whose strength is determined by the \(s\)-wave scattering length. Effective theories are used next in order to obtain a simple formula for the shift in \(T_c\): this comes from the fact that near \(T_c\) the physics is dominated by low momentum modes whose dynamics is most economically described in terms of classical fields. The ingredients needed to calculate the shift of \(T_c\) can be obtained from this classical field theory. Finally the renormalization group is used both to obtain a qualitative understanding, and also as a non perturbative tool to evaluate quantitatively the shift in \(T_c\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that depending on the choice of the regulator, not all fluctuations may be suppressed when \(\kappa =\Lambda \). However, for renormalisable theories, and if \(\Lambda \) is large enough, the effects of these remnant fluctuations can be absorbed into a redefinition of the parameters of the classical action.

References

  1. Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)

    Google Scholar 

  2. Pethick, C.J., Smith, H.: Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  3. Dalfovo, F., Giorgini, S., Stringari, S., Pitaevskii, L.: Rev. Mod. Phys. 71, 463 (1999)

    Article  ADS  Google Scholar 

  4. Legget, A. J.: Bose-Einstein condensation in the alkali gases: some fundamental concepts. Rev. Mod. Phys. 73, 307 (2001)

    Google Scholar 

  5. Andersen, J.O.: Rev. Mod. Phys. 76, 599 (2004)

    Article  ADS  MATH  Google Scholar 

  6. Baym, G., Blaizot, J.-P., Holzmann, M., Laloë, F., Vautherin, D.: Phys. Rev. Lett. 83, 1703 (1999)

    Article  ADS  Google Scholar 

  7. Baym, G., Blaizot, J.-P., Zinn-Justin, J.: Europhys. Lett. 49, 150 (2000)

    Article  ADS  Google Scholar 

  8. Baym, G., Blaizot, J.-P., Holzmann, M., Laloë, F., Vautherin, D.: Eur. Phys. J. B 24, 107 (2001)

    Article  ADS  Google Scholar 

  9. Holzmann, M., Baym, G., Blaizot, J.-P., Laloë, F.: Phys. Rev. Lett. 87, 120403 (2001)

    Article  ADS  Google Scholar 

  10. Holzmann, M., Fuchs, J.N., Baym, G., Blaizot, J.-P., Laloë, F.: Comptes Rend. Phys. 5, 21 (2004)

    Google Scholar 

  11. Blaizot, J.P., Mendez Galain, R., Wschebor, N.: Europhys. Lett. 72(5), 705–711 (2005)

    Google Scholar 

  12. Blaizot, J.P., Mendez-Galain, R., Wschebor, N.: Phys. Rev. E 74, 051116 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  13. Blaizot, J.P., Mendez-Galain, R., Wschebor, N.: Phys. Rev. E 74, 051117 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  14. Blaizot, J.P., Mendez Galain, R., Wschebor, N.: Phys. Lett. B 632, 571 (2006)

    Google Scholar 

  15. Blaizot, J.P., Mendez-Galain, R., Wschebor, N.: Eur. Phys. J. B 58, 297–309 (2007). doi:10.1140/epjb/e2007-00223-3

  16. Ziff, R.M., Uhlenbeck, G.E., Kac, M.: Phys. Rep. 32, 169 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  17. Holzmann, M., Baym, G., Blaizot, J.-P., Laloë, F.: Proc. Natl. Acad. Sci. 104, 1476–1481 (2007)

    Google Scholar 

  18. Giorgini, S., Pitaevskii, L., Stringari, S.: Phys. Rev. A 54, 4633 (1996)

    Article  ADS  Google Scholar 

  19. Schafer, T.: arXiv:nucl-th/0609075

    Google Scholar 

  20. Kaplan, D.B.: Lectures Delivered at the 17th National Nuclear Physics Summer School 2005, Berkeley, CA, 6–17 June 2005

    Google Scholar 

  21. Braaten, E., Hammer, H.W., Hermans, S.: Phys. Rev. A 63, 063609 (2001)

    Article  ADS  Google Scholar 

  22. Braaten, E., Nieto, A.: Phys. Rev. B 55, 8090–8093 (1997). doi:10.1103/PhysRevB.55.8090

    Google Scholar 

  23. Braaten, E., Nieto, A.: Phys. Rev. B 56, 14745–14765 (1997). doi:10.1103/PhysRevB.56.14745

    Google Scholar 

  24. Toyoda, T.: Ann. Phys. N.Y. 141, 154 (1982)

    Article  ADS  Google Scholar 

  25. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems, vol. 28. McGraw-Hill, New York (1971)

    Google Scholar 

  26. Baym, G., Grinstein, G.: Phys. Rev. D 15, 2897 (1977)

    Article  ADS  Google Scholar 

  27. Kadanoff, L.P., Baym, G.: Quantum Statistical Mechanics. Benjamin, New York (1962)

    Google Scholar 

  28. Abrikosov, A.A., Gorkov, L.P., Dzyaloshinski, I.E.: Methods of Quantum Field Theory in Statistical Physics. Prentice Hall, Englewood Cliffs (1963)

    Google Scholar 

  29. Blaizot, J.P., Ripka, G.: Quantum Theory of Finite Systems. MIT Press, Cambridge (1986)

    Google Scholar 

  30. Patashinskii, A.Z., Pokrovskii, V.L.: Fluctuation Theory of Phase Transitions. Pergamon Press, Oxford (1979)

    Google Scholar 

  31. Ginsparg, P.: Nucl. Phys. B 170, 388 (1980)

    Google Scholar 

  32. Appelquist, T., Pisarski, R.D.: Phys. Rev. D 23, 2305 (1981)

    Google Scholar 

  33. Nadkarni, S.: Phys. Rev. D 27, 917 (1983); Phys. Rev. D 38, 3287 (1988)

    Google Scholar 

  34. Arnold, P., Tomasik, B.: Phys. Rev. A 64, 053609 (2001)

    Article  ADS  Google Scholar 

  35. Moshe, M., Zinn-Justin, J.: Phys. Rept. 385, 69 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Zinn-Justin, J.: Quantum field theory and critical phenomena. Int. Ser. Monogr. Phys. 113, 1 (2002)

    Google Scholar 

  37. Arnold, P., Moore, G.: Phys. Rev. Lett. 87, 120401 (2001); Phys. Rev. E 64, 066113 (2001)

    Google Scholar 

  38. Kashurnikov, V.A., Prokof’ev, N.V., Svistunov, B.V.: Phys. Rev. Lett. 87, 160601 (2001); Phys. Rev. Lett. 87, 120402 (2001)

    Google Scholar 

  39. Arnold, P., Tomasik, B.: Phys. Rev. A 62, 063604 (2000)

    Article  ADS  Google Scholar 

  40. Kneur, J.-L., Neveu, A., Pinto, M.B.: Phys. Rev. A 69, 053624 (2004)

    Article  ADS  Google Scholar 

  41. de Souza Cruz, F., Pinto, M.B., Ramos, R.O.: Phys. Rev. B 64, 014515 (2001); Phys. Rev. A 65, 053613 (2002)

    Google Scholar 

  42. Kastening, B.: Phys. Rev. A 68, 061601 (R) (2003); Phys. Rev. A 69, 043613 (2004)

    Google Scholar 

  43. Arnold, P., Moore, G.D., Tomasik, B.: Phys. Rev. A 65, 013606 (2002)

    Article  ADS  Google Scholar 

  44. Wetterich, C.: Phys. Lett. B 301, 90 (1993)

    Google Scholar 

  45. Ellwanger, U.: Z. Phys. C 58, 619 (1993)

    Google Scholar 

  46. Tetradis, N., Wetterich, C.: Nucl. Phys. B 422, 541 (1994)

    Article  ADS  Google Scholar 

  47. Morris, T.R.: Int. J. Mod. Phys. A 9, 2411 (1994)

    Google Scholar 

  48. Morris, T.R.: Phys. Lett. B 329, 241 (1994)

    Google Scholar 

  49. Bagnuls, C., Bervillier, C.: Phys. Rept. 348, 91 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Berges, J., Tetradis, N., Wetterich, C.: Phys. Rept. 363, 223–386 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Delamotte, B., Mouhanna, D., Tissier, M.: Phys. Rev. B 69, 134413 (2004); Canet, L., Delamotte, B.: Condensed Matter Phys. 8, 163–179 (2005)

    Google Scholar 

  52. Gies, H.: arXiv:hep-ph/0611146

    Google Scholar 

  53. Bijlsma, M., Stoof, H.T.C.: Phys. Rev. A 54, 5085 (1996)

    Article  ADS  Google Scholar 

  54. Andersen, J.O., Strickland, M.: Phys. Rev. A 60, 1442 (1999)

    Article  ADS  Google Scholar 

  55. Ball, R.D., Haagensen, P.E., Latorre, J.I., Moreno, E.: Phys. Lett. B 347, 80 (1995)

    Google Scholar 

  56. Comellas, J.: Nucl. Phys. B 509, 662 (1998)

    Google Scholar 

  57. Litim, D.: Phys. Lett. B 486, 92 (2000); Phys. Rev. D 64, 105007 (2001); Nucl. Phys. B 631, 128 (2002); Int. J. Mod. Phys. A 16, 2081 (2001)

    Google Scholar 

  58. Canet, L., Delamotte, B., Mouhanna, D., Vidal, J.: Phys. Rev. D 67, 065004 (2003)

    Google Scholar 

  59. Ellwanger, U., Wetterich, C.: Nucl. Phys. B 423, 137 (1994)

    Article  ADS  Google Scholar 

  60. Tetradis, N., Litim, D.F.: Nucl. Phys. B 464, 492–511 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. D’Attanasio, M., Morris, T.R.: Phys. Lett. B 409, 363–370 (1997)

    Article  ADS  Google Scholar 

  62. Sun, X.: Phys. Rev. E 67, 066702 (2003)

    Article  ADS  Google Scholar 

  63. Ledowski, S., Hasselmann, N., Kopietz, P.: Phys. Rev. A 69, 061601(R) (2004); Phys. Rev. A 70, 063621 (2004)

    Google Scholar 

Download references

Acknowledgements

Most of the material of these lectures is drawn from work done in a most enjoyable collaboration with several people during these last years: G. Baym, F. Laloë, M. Holzmann, R. Mendez-Galain, D. Vautherin, N. Wschebor and J. Zinn-Justin. I would also like to express my gratitude to Achim Schwenk for insisting on having these lecture notes....and putting such a high cut-off on his patience as he waited for them.

Author information

Authors and Affiliations

  1. Institute of Theoretical Physics (IPhT), CEA Saclay, 91191, Gif-sur-Yvette Cedex, France

    Jean-Paul Blaizot

Authors
  1. Jean-Paul Blaizot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Paul Blaizot .

Editor information

Editors and Affiliations

  1. Institut für Kernphysik, und Extreme Matter Institute EMMI, Technische Universität Darmstadt, Schlossgarten 2, Darmstadt, 64289, Germany

    Achim Schwenk

  2. Lab. de Physique Therique, Université Louis Pasteur, rue de l'Université 3, Strasbourg, 67084, France

    Janos Polonyi

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Blaizot, JP. (2012). Nonperturbative Renormalization Group and Bose-Einstein Condensation. In: Schwenk, A., Polonyi, J. (eds) Renormalization Group and Effective Field Theory Approaches to Many-Body Systems. Lecture Notes in Physics, vol 852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27320-9_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27320-9_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27319-3

  • Online ISBN: 978-3-642-27320-9

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation