Advertisement

An Introduction to the Nonperturbative Renormalization Group

  • Bertrand DelamotteEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 852)

Abstract

We give in these notes a short presentation of both the main ideas underlying Wilson’s renormalization group (RG) and their concrete implementation under the form of what is now called the non-perturbative renormalization group (NPRG) or sometimes the functional renormalization group (which can be perturbative). Prior knowledge of perturbative field theory is not required for the understanding of the heart of the article.

Keywords

Renormalization Group Ising Model Critical Exponent Continuum Limit Renormalization Group Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

I want first to thank D. Mouhanna without whom the little group in Paris (with an antenna in Grenoble) working on the NPRG would have never existed. I owe my understanding of this subject to many discussions and common works with him. I also thank M. Tissier with whom I collaborated and from whom I have learned a lot. It is also a great pleasure to thank L. Canet and H. Chaté with whom I have collaborated and discussed many aspects of statistical mechanics and renormalization. I also thank I. Dornic and J. Vidal with whom I collaborated and G. Tarjus, J. Berges, C. Bervillier and C. Wetterich for many discussions about the NPRG. More recently, discussions with R. Mendez-Galain and N. Wschebor on their own version of the NPRG have greatly improved my understanding of this subject and I want to thank them for that. I also thank J-M. Caillol for several discussions. Finally, I thank Yu. Holovatch who encouraged me to write down my lecture notes, the students who helped me to improve them (P. Hosteins and G. Gurtner in particular) and A. Schwenk and J.-P. Blaizot who invited me in Trento to give a set of lectures on the NPRG.

References

  1. 1.
    Le Bellac, M.: Quantum and Statistical Field Theory. Clarendon Press, Oxford (1991)Google Scholar
  2. 2.
    Binney, J.J., Dowrick, N.J., Fisher, A.J., Newman, M.E.J.: The Theory of Critical Phenomena: An Introduction to the Renormalization Group. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  3. 3.
    Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley, Reading (1992)Google Scholar
  4. 4.
    Ryder, L.H.: Quantum Field Theory. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  5. 5.
    Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena, 3rd edn. Oxford University Press, New York (1989)Google Scholar
  6. 6.
    Delamotte, B.: Am. J. Phys. 72, 170 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    Kadanoff, L.P.: Physica 2, 263 (1966)Google Scholar
  8. 8.
    Wegner, F.J., Houghton, A.: Phys. Rev. A 8, 401 (1973)ADSCrossRefGoogle Scholar
  9. 9.
    Wilson, K.G., Kogut, J.: Phys. Rep., Phys. Lett. 12C, 75 (1974)Google Scholar
  10. 10.
    Wegner F.J.: The critical state, general aspects. In: Domb, C., Greene, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 6. Academic Press, New York (1976)Google Scholar
  11. 11.
    Weinberg, S.: Critical phenomena for field theorists. In: Zichichi, A. (ed.) Understanding the Fundamental Constituents of Matter. Plenum Press, New York (1978)Google Scholar
  12. 12.
    Polchinski, J.: Nucl. Phys. B 231, 269 (1984)ADSCrossRefGoogle Scholar
  13. 13.
    Shirkov, D.V., Kovalev, V.F.: Phys. Rep. 352, 219 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Fortuin, C.M., Kasteleyn, P.W.: Physica 57, 536 (1972)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Swendsen, R.H., Wang, J.S.: Phys. Rev. Lett. 58, 86 (1987)ADSCrossRefGoogle Scholar
  16. 16.
    Niemeijer, T., van Leuwen, J.: Renormalization: ising-like spin systems. In: Domb, C., Greene, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 6. Academic Press, New York (1975)Google Scholar
  17. 17.
    Golner, G.R., Riedel, E.K.: Phys. Rev. Lett. 34, 856 (1975)ADSCrossRefGoogle Scholar
  18. 18.
    Hasenfratz, A., Hasenfratz, P.: Nucl. Phys. B [FS] 270, 687 (1986)ADSCrossRefGoogle Scholar
  19. 19.
    Nicoll, J.F., Chang, T.S., Stanley, H.E.: Phys. Rev. Lett. 33, 540 (1974). err. ibid. 33, 1525 (1974)Google Scholar
  20. 20.
    Nicoll, J.F., Chang, T.S., Stanley, H.E.: Phys. Lett. A 57, 7 (1976)ADSCrossRefGoogle Scholar
  21. 21.
    Nicoll, J.F., Chang, T.S.: Phys. Lett. A 62, 287 (1977)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Ringwald, A., Wetterich, C.: Nucl. Phys. B 334, 506 (1990)ADSCrossRefGoogle Scholar
  23. 23.
    Wetterich, C.: Nucl. Phys. B 352, 529 (1991)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Tetradis, N., Wetterich, C.: Nucl. Phys. B 383, 197 (1992)ADSCrossRefGoogle Scholar
  25. 25.
    Wetterich, C.: Phys. Lett. B 301, 90 (1993)ADSCrossRefGoogle Scholar
  26. 26.
    Ellwanger, U.: Z. Phys. C 58, 619 (1993)ADSCrossRefGoogle Scholar
  27. 27.
    Ellwanger, U.: Z. Phys. C 62, 503 (1993)ADSCrossRefGoogle Scholar
  28. 28.
    Tetradis, N., Wetterich, C.: Nucl. Phys. B [FS] 422, 541 (1994)ADSCrossRefGoogle Scholar
  29. 29.
    Ellwanger, U.: Phys. Lett. B 335, 364 (1994)ADSCrossRefGoogle Scholar
  30. 30.
    Morris, T.R.: Int. J. Mod. Phys. A 9, 2411 (1994)ADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Morris, T.R.: Phys. Lett. B 329, 241 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Berges, J., Tetradis, N., Wetterich, C.: Phys. Rep. 363, 223 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    Delamotte, B., Mouhanna, D., Tissier, M.: Phys. Rev. B 69, 134413 (2004)ADSCrossRefGoogle Scholar
  34. 34.
    Newman, K.E., Riedel, E.K.: Phys. Rev. B 30, 6615 (1984)ADSCrossRefGoogle Scholar
  35. 35.
    Newman, K.E., Riedel, E.K., Muto, S.: Phys. Rev. B 29, 302 (1984)ADSCrossRefGoogle Scholar
  36. 36.
    Bervillier, C.: Phys. Lett. A 331, 110 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Kunz, H., Zumbach, G.: J. Phys. A 26, 3121 (1993)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Zumbach, G.: Phys. Rev. Lett. 71, 2421 (1993)ADSCrossRefGoogle Scholar
  39. 39.
    Zumbach, G.: Nucl. Phys. B 413, 754 (1994)ADSCrossRefGoogle Scholar
  40. 40.
    Zumbach, G.: Phys. Lett. A 190, 225 (1994)ADSCrossRefGoogle Scholar
  41. 41.
    Kleinert, H., Schulte-Frohlinde, V.: Critical Properties of Phi 4-Theories. World Scientific, Singapore (2001)zbMATHCrossRefGoogle Scholar
  42. 42.
    Liao, S., Polonyi, J., Strickland, M.: Nucl. Phys. B 567, 493 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Litim, D.: J. High Energy Phys. 0111, 059 (2001)MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    Litim, D.F.: Nucl. Phys. B 631, 128 (2002)ADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Canet, L., Delamotte, B., Mouhanna, D., Vidal, J.: Phys. Rev. D 67, 065004 (2003)ADSCrossRefGoogle Scholar
  46. 46.
    Canet, L., Delamotte, B., Mouhanna, D., Vidal J.: Phys. Rev. B 68, 064421 (2003)ADSCrossRefGoogle Scholar
  47. 47.
    Tissier, M., Delamotte, B., Mouhanna, D.: Phys. Rev. Lett. 84, 5208 (2000)ADSCrossRefGoogle Scholar
  48. 48.
    Bagnuls, C., Bervillier, C.: Phs.Rev. B 41, 402 (1990)ADSCrossRefGoogle Scholar
  49. 49.
    Bagnuls, C., Bervillier, C.: Phys. Rep. 348, 91 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  50. 50.
    Bervillier, C.: J. Phys. Condens. Matter 17, S1929 (2005)ADSCrossRefGoogle Scholar
  51. 51.
    Bagnuls, C., Bervillier, C.: Int. J. Mod. Phys. A 16, 1825 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  52. 52.
    Blaizot, J.-P., Galain, R.M., Wschebor, N.: Phys. Rev. E 74, 051116 (2006)MathSciNetADSCrossRefGoogle Scholar
  53. 53.
    Blaizot, J.-P., Galain, R.M., Wschebor, N.: Phys. Lett. B 632, 571 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  54. 54.
    Blaizot, J.-P., Galain, R.M., Wschebor, N.: Phys. Rev. E 051117 (2006)Google Scholar
  55. 55.
    Litim, D.: Mind the gap. Int. J. Mod. Phys. A 16, 2081 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  56. 56.
    Morris, T.R.: Phys. Lett. B 345, 139 (1995)ADSCrossRefGoogle Scholar
  57. 57.
    Gersdorff, G.V., Wetterich, C.: Phys. Rev. B 64, 054513 (2001)ADSCrossRefGoogle Scholar
  58. 58.
    Ballhausen, H., Berges, J., Wetterich, C.: Phys. Lett. B 582, 144 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  59. 59.
    Seide, S., Wetterich, C.: Nucl. Phys. B 562, 524 (1999)ADSCrossRefGoogle Scholar
  60. 60.
    Canet, L., Chaté, H., Delamotte, B.: Phys. Rev. Lett. 92, 255703 (2004)ADSCrossRefGoogle Scholar
  61. 61.
    Itzykson, C., Drouffe, J.M.: Statistical Field Theory. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar
  62. 62.
    Morris, T.R., Tighe, J.F.: J. High Energy Phys. 08, 007 (1999)ADSCrossRefGoogle Scholar
  63. 63.
    Gies, H.: hep-ph/0611146 (2006)Google Scholar
  64. 64.
    Pelissetto, A., Vicari, E.: Phys. Rep. 368, 549 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  65. 65.
    Calabrese, P., Parruccini, P., Pelissetto, A., Vicari, E.: Phys. Rev. B 70, 174439 (2004)ADSCrossRefGoogle Scholar
  66. 66.
    Canet, L., Delamotte, B., Deloubrière, O., Wschebor, N.: Phys. Rev. Lett. 92, 195703 (2004)ADSCrossRefGoogle Scholar
  67. 67.
    Canet, L., Chaté, H., Delamotte, B., Dornic, I., Munoz, M.A.: Phys. Rev. Lett. 95, 100601 (2005)ADSCrossRefGoogle Scholar
  68. 68.
    Tissier, M., Mouhanna, D., Vidal, J., Delamotte, B.: Phys. Rev. B 65, 140402 (2002)ADSCrossRefGoogle Scholar
  69. 69.
    Fisher, D.S.: Phys. Rev. Lett. 56, 1964 (1986)ADSCrossRefGoogle Scholar
  70. 70.
    Le Doussal, P., Wiese, K.J.: Phys. Rev. lett. 89, 125702 (2002)ADSCrossRefGoogle Scholar
  71. 71.
    Feldman, D.E.: Phys. Rev. Lett. 88, 177202 (2002)ADSCrossRefGoogle Scholar
  72. 72.
    Le Doussal, P., Wiese, K.J., Chauve, P.: Phys. Rev. E 69, 026112 (2004)ADSCrossRefGoogle Scholar
  73. 73.
    Tarjus, G., Tissier, M.: Phys. Rev. Lett. 93, 267008 (2004)ADSCrossRefGoogle Scholar
  74. 74.
    Tissier, M., Tarjus, G.: Phys. Rev. Lett. 96, 087202 (2006)ADSCrossRefGoogle Scholar
  75. 75.
    Strumia, A., Tetradis, N.: Nucl. Phys. B 542, 719 (1999)ADSCrossRefGoogle Scholar
  76. 76.
    Strumia, A., Tetradis, N.: JHEP, 9911, 023 (1999)ADSCrossRefGoogle Scholar
  77. 77.
    Diehl, S., Wetterich, C.: Phys. Rev. A 73, 033615 (2006)ADSCrossRefGoogle Scholar
  78. 78.
    Diehl, S., Gies, H., Pawlowski, J.M., Wetterich, C.: Phys. Rev. A 76, 021602 (2007)ADSCrossRefGoogle Scholar
  79. 79.
    Parola, A., Reatto, L.: Adv. Phys. 44, 211 (1995)ADSCrossRefGoogle Scholar
  80. 80.
    Caillol, J.-M.: Mol. Phys. 104, 1931 (2006)ADSCrossRefGoogle Scholar
  81. 81.
    Delamotte, B., Canet, L.: Condens. Matter Phys. 8, 163 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique de la Matière Condensée, CNRS-UMR 7600Université Pierre et Marie CurieParis Cedex 05France

Personalised recommendations