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Three-loop critical exponents, amplitude functions, and amplitude ratios from variational perturbation theory

H. Kleinert and B. Van den Bossche*

  • Institut für Theoretische Physik, Arnimallee 14 D-14195 Berlin, Germany

  • *On leave from Physique Nucléaire Théorique, B5, Université de Liège Sart-Tilman, 4000 Liège, Belgium.
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Phys. Rev. E 63, 056113 – Published 18 April, 2001

DOI: https://doi.org/10.1103/PhysRevE.63.056113

Abstract

We use variational perturbation theory to calculate various universal amplitude ratios above and below Tc in minimally subtracted φ4 theory with N components in three dimensions. In order to best exhibit the method as a powerful alternative to Borel resummation techniques, we consider only two- and three-loops expressions where our results are analytic expressions. For the critical exponents, we also extend existing analytic expressions for two loops to three loops.

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References (58)

  1. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific, Singapore, 1995). Variational perturbation theory is developed in Chaps. 5 and 17.
  2. H. Kleinert, Phys. Rev. D 57, 2264 (1998); ibid.58, 107702(Addendum) (1998).
  3. H. Kleinert, Phys. Rev. D 60, 085001 (1999).
  4. F. Jasch and H. Kleinert, cond-mat/9906246.
  5. H. Kleinert, Phys. Lett. B 434, 74 (1998); ibid.463, 69 (1999).
  6. H. Kleinert and V. Shulte-Frohlinde, cond-mat/9907214.
  7. K. Wilson, Phys. Rev. B 4, 3174 (1971).
  8. K. Wilson and M. Fisher, Phys. Rev. Lett. 28, 240 (1972); ibid.K. Wilson, 28, 548 (1972).
  9. E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, in Phase Transition and Critical Phenomana, edited by C. Domb and M. S. Green (Academic Press, New York, 1976), Vol. 6, p. 125.
  10. L. N. Lipatov, Leningrad Nuclear Physics Institute Report, 1976 (unpublished).
  11. E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Phys. Rev. D 15, 1544 (1977); ibid.15, 1558 (1977).
  12. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Science Publications, 3rd ed. (Oxford University Press, Oxford, 1996).
  13. H. Kleinert and V. Schulte-Frohlinde, Critical Properties ofφ4 Theories (World Scientific, Singapore, 2001), http://www.physik.fu-berlin.de/∼kleinert/b8.
  14. K. G. Chetyrkin, S. G. Gorishny, S. A. Larin, and F. V. Tkachov, Phys. Lett. 132B, 351 (1983).
  15. H. Kleinert, J. Neu, V. Schulte-Frohlinde, K. G. Chetkyrin, and S. A. Larin, Phys. Lett. 272B, 39 (1991); 319B, 545(E) (1993).
  16. D. I. Kazakov and D. V. Shirkov, Fortschr. Phys. 28, 465 (1980); J. Zinn-Justin, Phys. Rep. 70, 3 (1981); ibid.70, 109 (1981).
  17. J. C. Le Guillou and J. Zinn-Justin, J. Phys. (France) Lett. 46, L137 (1985); J. Phys. (Paris) 48, 19 (1987).
  18. R. Guida and J. Zinn-Justin, J. Phys. A 31, 8103 (1998); e-print cond-mat/9803240. See also J. Zinn-Justin, e-print hep-th/0002136.
  19. G. Parisi, J. Stat. Phys. 23, 49 (1980).
  20. G. A. Baker, B. G. Nickel, M. S. Green, and D. I. Meiron, Phys. Rev. Lett. 36, 1351 (1976); B. G. Nickel, D. I. Meiron and G. A. Backer, University of Guelph report, 1977 (unpublished); G. A. Baker, B. G. Nickel, and D. I. Meiron, Phys. Rev. B 17, 1365 (1978).
  21. R. Guida and J. Zinn-Justin, Nucl. Phys. B 489, 626 (1997).
  22. G. Münster and J. Heitger, Nucl. Phys. B 424, 582 (1994).
  23. C. Gutsfeld, J. Kuüster, and G. Münster, Nucl. Phys. B 479, 654 (1996).
  24. C. Bagnuls and C. Bervillier, Phys. Rev. B 32, 7209 (1985).
  25. C. Bagnuls, C. Bervillier, D. I. Meiron, and B. G. Nickel, Phys. Rev. B 35, 3585 (1987).
  26. P. C. Hohenberg, A. Aharony, B. I. Halperin, and E. D. Siggia, Phys. Rev. B 13, 2986 (1976).
  27. C. Bervillier, Phys. Rev. B 14, 4964 (1976).
  28. J. A. Lipa, D. R. Swanson, J. A. Nissen, T. C. P. Chui, and U. E. Israelsson, Phys. Rev. Lett. 76, 944 (1996); ibid.J. A. Lipa, D. R. Swanson, J. A. Nissen, Z. K. Geng, P. R. Williamson, D. A. Stricker, T. C. P. Chui, U. E. Israelsson, and M. Larson, 84, 4894 (2000). In the second paper, the value of α given in the first paper was corrected to α=−0.01056±0.0004.
  29. The presence and origin of confluent singular terms have been first understood from renormalization group analysis by F. J. Wegner, Phys. Rev. B 5, 4529 (1972). ibid.For a determination of the critical behavior of the first confluent singular term for the specific heat to the order ε2, see M.-C. Chang and A. Houghton, 21, 1881 (1980) (The ratio of the leading amplitude ratio for the heat capacity is incorrect in this later reference, as it is in [27]. ibid.For the corrected value for N=1, see [31], which also gives other various confluent terms and gives references to other wrong ratios in the literature. For the corrected value of A+/A− for all N, see C. Bervillier, 34, 8141 (1986), or Appendix C of Ref. [25].)
  30. C. Bagnuls and C. Bervillier, Phys. Rev. B 24, 1226 (1981).
  31. J. F. Nicoll and P. C. Albright, Phys. Rev. B 31, 4576 (1985).
  32. G. M. Avdeeva and A. A. Migdal, Pis’ma Zh. Éksp. Teor. Fiz. 16, 253 (1972) [JETP Lett. 16, 178 (1972)]; E. Brézin, D. J. Wallace, and K. G. Wilson, Phys. Rev. Lett. 29, 591 (1972); Phys. Rev. B 7, 232 (1972).
  33. D. J. Wallace and R. K. P. Zia, Phys. Lett. 46A, 261 (1973); J. Phys. C 7, 3480 (1974).
  34. C. Bervillier and M. Shpot, Phys. Rev. B 46, 955 (1992).
  35. A. R. Rajantie, Nucl. Phys. B 480, 729 (1996); ibid.513, 761(Addendum) (1998).
  36. V. Dohm, Z. Phys. B: Condens. Matter 60, 61 (1985).
  37. R. Schloms and V. Dohm, Nucl. Phys. B 328, 639 (1989).
  38. R. Schloms and V. Dohm, Phys. Rev. B 42, 6142 (1990); ibid.46, 5883(Addendum) (1992).
  39. See Chang and Houghton in Ref. [29].
  40. F. J. Halfkann and V. Dohm, Z. Phys. B: Condens. Matter 89, 79 (1992).
  41. S. A. Larin, M. Mönnigmann, M. Strösser, and V. Dohm, Phys. Rev. B 58, 3394 (1998).
  42. M. Strösser, L. A. Larin, and V. Dohm, Nucl. Phys. B 540, 654 (1999).
  43. B. Kastening, Phys. Rev. D 57, 3567 (1998).
  44. J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1985).
  45. H. KLeinert and B. Van den Bossche (unpublished).
  46. S. S. C. Burnett, M. Strösser and V. Dohm, Nucl. Phys. B 504, 665 (1997); ibid.509, 729(Addendum) (1998).
  47. J. S. Kang, Phys. Rev. D 13, 851 (1976).
  48. H. Kleinert and B. Van den Bossche (unpublished).
  49. H. Kleinert, Phys. Lett. B 463, 69 (1999).
  50. Note that the calculation we do in this paper are based on the normalization of the coupling constant of Dohm and co-workers [42][41][46] which differ from Kleinert’s [5][13]. The difference in conventions disappears at the level of the exponents, and the value of u* is only different by a factor 12, as mentioned after Eq. (11).
  51. B. Li, N. Madras, and A. D. Sokal, J. Stat. Phys. 80, 661 (1995).
  52. S. A. Antonenko and A. I. Sokolov, Phys. Rev. E 51, 1894 (1995).
  53. M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev. A 8, 1111 (1973).
  54. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic Press, London, 1965).
  55. H. J. Krause, R. Schloms, and V. Dohm, Z. Phys. B: Condens. Matter 79, 287 (1990); ibid.80, 313(Addendum) (1990).
  56. V. Privman, P. C. Hohenberg and A. Aharony, in Phase Transition and Critical Phenomana, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1991), Vol. 14, p. 1, and references therein.
  57. I. D. Lawrie, J. Phys. A 14, 2489 (1981).
  58. H. Kleinert and B. Van den Bossche (unpublished).

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Phys. Rev. E 63, 056113– Published 18 April, 2001
  • Received 21 November 2000

© 2001 American Physical Society

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