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Critical Phenomena pp 1–139Cite as

Scaling, universality and renormalization group theory

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Bibliography

  1. B. Widom in “Fundamental Problems in Statistical. Mechanics” vol. III, Ed. E. V. G. Cohen (North Holland, 1975) pp. 1–45.

    Google Scholar 

  2. P. Pfeuty and G. Toulouse, “Introduction to Renormalization Groups and Critical Phenomena” (Wiley, 1977).

    Google Scholar 

  3. M. E. Fisher, Rev. Mod. Phys. 46, 597–616 (1974).

    Article  ADS  Google Scholar 

  4. D. J. Wallace and R. K. P. Zia, Repts. Prog. Phys. 41, 1–85 (1978).

    Article  ADS  Google Scholar 

  5. A. Z. Patashinskii and V. L. Pokrovskii, “Fluctuation Theory of Phase Transitions” (Pergamon Press, 1979).

    Google Scholar 

  6. L. E. Reichl, “A Modern Course in Statistical Physics”, (Univ. of Texas Press, 1980), Chap. 4.

    Google Scholar 

  7. A. P. Young in “Ordering in Strongly Fluctuating Condensed Matter Systems”, Ed. T. Riste (Plenum Press, 1980) pp. 11–31, 271–284.

    Google Scholar 

  8. S.-K. Ma, “Modern Theory of Critical Phenomena” (Benjamin, 1976).

    Google Scholar 

  9. K. G. Wilson, Rev. Mod. Phys. 47, 773–840 (1975).

    Article  ADS  Google Scholar 

  10. K. G. Wilson and J. Kogut, Physics Reports 12 C (1974).

    Article  Google Scholar 

  11. B. I. Halperin in “Physics of Low-Dimensional Systems”, Eds. Y. Nagaoka and S. Hikami (Publ. Office, Prog. Theoret. Phys., Kyoto, 1979).

    Google Scholar 

  12. Articles by A. Aharony, by E. Brezin, J. C. Le Guillon and J. Zinn-Justin, by Th. Niemeijer and J. M. J. van Leeuwen, by F. J. Wegner and by others in “Phase Transitions and Critical Phenomena”, vol. 6, Eds. M. S. Green and C. Domb (Academic Press, 1976). Other volumes in this series provide a broad coverage of other aspects of critical phenomena.

    Google Scholar 

  13. D. J. Amit, “Field Theory, the Renormalization Group and Critical Phenomena” (McGraw-Hill, 1978).

    Google Scholar 

References

  1. See First Book of Kings, Chap. 7, verse 23.

    Google Scholar 

  2. For example, see the famous pictures reproduced in H. E. Stanley, “Introduction to Phase Transitions and Critical Phenomena” (Oxford Univ. Press, 1971) Fig.1.7.

    Google Scholar 

  3. D. Balzarini and K. Ohrn, Phys. Rev. Lett. 29, 840 (1972); D. Balzarini, Can. J. Phys. 52, 499 (1974).

    Article  ADS  Google Scholar 

  4. M. J. Sienko, J. Chem. Phys. 53, 566 (1970).

    Article  ADS  Google Scholar 

  5. M. I. Bagatskii, A. V. Voronel' and V. G. Gusak, Sov. Phys. JETP 16, 517 (1963).

    ADS  Google Scholar 

  6. G. Ahlers, Phys. Rev. Lett. 24, 1333 (1970).

    Article  ADS  Google Scholar 

  7. B. J. Lips, unpublished work at Stanford University (1981).

    Google Scholar 

  8. See, for example H. E. Stanley loc. cit. Figs. 11.4 and 11.5 or M. E. Fisher, Rep. Prog. Phys. 30, 615 (1967) Fig. 18, etc.

    Google Scholar 

  9. See, for example, H. B. Tarko and M. E. Fisher, Phys. Rev. B11, 1217 (1975) which lists data for various universal and nonuniversal parameters of two and three-dimensional Ising models.

    Article  ADS  Google Scholar 

  10. Based on the general scaling treatment presented in Proc. Nobel Symposium 24, Eds. B. Lundqvist and S. Lundqvist (Academic Press, 1974) pp. 16–37.

    Google Scholar 

  11. Frenkel's philosophy was expressed more fully in a 1946 review article devoted to the theory of metals. I came across his formulation two decades ago in I. E. Tamm's obituary of Yakov Il'ich Frenkel which appeared in Soviet Physics Uspekhi 5, No. 2, Sept./Oct. (1962).

    Google Scholar 

  12. See e.g. the review “Simple Ising models still thrive”: M. E. Fisher, Physics 106A, 28 (1981) and references therein.

    Google Scholar 

  13. C. Domb, Adv. Phys. 9, 149 (1960).

    Article  ADS  Google Scholar 

  14. See e.g. Domb.loc. cit..; C. Domb and M. F. Sykes, J. Math. Phys. 2, 63 (1961); M. E. Fisher, Rept. Prog. Phys. 30, 615 (1967) Figs. 13, 14; D. S Gaunt and A. J. Guttmann, in “Phase Transitions and Critical Phenomena”, Vol. 3, Eds. C. Domb and M. S. Green (Academic Press, London, 1974).

    Article  ADS  Google Scholar 

  15. See e.g. J. Zinn-Justin, J. Physique 42, 783 (1981). J.-H. Chen, M. E. Fisher and B. G. Nickel, Phys. Rev. Lett. 48, 630 (1982).

    Article  Google Scholar 

  16. See R. Balian and G. Toulouse, Phys. Rev. Lett. 30, 544 (1973); M. E. Fisher, Phys. Rev. Lett. 30, 679 (1973).

    Article  ADS  Google Scholar 

  17. We follow D. R. Nelson and M. E. Fisher, Ann. Phys. (N.Y.) 91, 226 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  18. Various concrete definitions of the correlation length and their inter-relations are discussed in an Ising model context by M. E. Fisher and R. J. Burford, Phys. Rev. 15b, 583 (1967) and M. B. Tarko and M. E. Fisher, Phys. Rev. B 11, 1217 (1975). In most simple critical phenomena all definitions agree up to a constant factor as the critical point is approached. In more complex situations, however, there may be different correlation lengths for different degrees of freedom even though (5.62) is maintained. Sometimes, as in dipolar interactions in Ising systems, two differently behaving correlation lengths are associated with orthogonal spatial directions in which case one may also need to generalize (5.62) to allow for different rescaling factors, say b‖ and b⊥, in different directions.

    Article  Google Scholar 

  19. See R. B. Griffiths, Physics 106A, 59 (1981); R. B. Griffiths and P. A. Pearce, J. Stat. Phys. 20, 499 (1979).

    ADS  Google Scholar 

  20. F. J. Wegner, J. Phys. C 7, 2098 (1974).

    Article  ADS  Google Scholar 

  21. L. P. Kadanoff, Physics 2, 263 (1966).

    Google Scholar 

  22. M. J. Buckingham in Critical Phenomena, Eds., M. S. Green and J. V. Sengers (National Bureau of Standards Misc. Publ. 273, Washington, D. C., 1966), p. 95.

    Google Scholar 

  23. See also P. Seglar and M. E. Fisher, J. Phys. C 13, 6613 (1980).

    Article  ADS  Google Scholar 

  24. V. L. Ginzburg, Sov. Phys. Solid St. 2, 1824 (1960).

    MathSciNet  Google Scholar 

  25. L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).

    Article  ADS  Google Scholar 

  26. P. C. Hohenberg in Fluctuations in Superconductors edited by W. S. Greee and F. Chilton (Stanford Research Inst., Menlo Park, 1968) p. 305.

    Google Scholar 

  27. J. Als-Nielsen and R. J. Birgeneau, Amer. J. Phys. 45, 554 (1977).

    Article  ADS  Google Scholar 

  28. Discovered originally by K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972) where the ε:-expansion was introduced.

    Article  ADS  Google Scholar 

  29. E. Brezin, J. C. LeGuillon, J. Zinn-Justin and B. G. Nickel, Phys. Lett. 44A, 227 (1973).

    Article  ADS  Google Scholar 

  30. A. A. Vladimirov, D. I. Kazakov, and O. Tarasov, Sov. Phys. JETP 50, 521 (1979).

    ADS  Google Scholar 

  31. M. E. Fisher, S.-K. Ma, and B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972).

    Article  ADS  Google Scholar 

  32. M. E. Fisher and D. R. Nelson, Phys. Rev. Lett. 32, 1352 (1974).

    Article  ADS  Google Scholar 

  33. M. E. Fisher and A. Aharony, Phys. Rev. Lett. 30, 559 (1973).

    Article  ADS  Google Scholar 

  34. A. I. Larkin and D. E. Khmel'nitskii, Sov. Phys. JETP 29, 1123 (1969) discovered some of these features before the general development of renormalization group theory.

    ADS  Google Scholar 

  35. See also A. Aharony, Phys. Lett. 44A, 313 (1973).

    Article  ADS  Google Scholar 

  36. E. K. Riedel and F. J. Wegner, Phys. Rev. Lett. 29, 349 (1972).

    Article  ADS  Google Scholar 

  37. R. Abe, Prog. Theoret. Phys. 48, 1414 (1972).

    Article  ADS  Google Scholar 

  38. A. M. Polyakov, Phys. Lett. 59B, 79 (1975); see also E. Brezin and J. ZinnJustin, Phys. Rev. Lett. 36, 691 (1976).

    Article  ADS  MathSciNet  Google Scholar 

  39. J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973).

    ADS  Google Scholar 

  40. J. M. Kosterlitz, J. Phys. C7, 1046 (1974).

    ADS  Google Scholar 

  41. J. V. Jose, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B16, 1217 (1977).

    Article  ADS  Google Scholar 

  42. B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978); E 41, 519 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  43. A. P. Young, Phys. Rev. B19, 1855 (1979).

    Article  ADS  Google Scholar 

  44. Th. Niemeijer and J. M. J. van Leeuwen, Physics (Utrecht) 71, 17 (1974).

    Article  ADS  Google Scholar 

  45. A. A. Migdal, Zh. Eksp.-Teor. Fiz. 69, 810, 1457 (1975).

    Google Scholar 

  46. L. P. Kadanoff, Ann. Phys. (N.Y.) 100, 359 (1976).

    Article  ADS  Google Scholar 

  47. See e.g., B. I. Halperin in “Collective Properties of Physical Systems” Nobel Symp. No. 24., Eds. B. Lundqvist and S. Lundqvist (Academic Press, 1974); P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977); J. Tobochnik, S. Sarker, and R. Cordery, Phys. Rev. Lett. 46, 1417 (1981).

    Google Scholar 

  48. D. R. Nelson, Phys. Rev. B11, 3504 (1975).

    Article  ADS  Google Scholar 

  49. J. Rudnick and D. R. Nelson, Phys. Rev. 13, 2208 (1976).

    Article  ADS  Google Scholar 

  50. K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).

    Article  ADS  Google Scholar 

  51. A review of graph theoretical terminology is given by J. W. Essam and M. E. Fisher, Rev. Mod. Phys. 42, 271 (1970).

    Article  ADS  MathSciNet  Google Scholar 

  52. K. G. Wilson, Phys. Rev. D7, 2911 (1973).

    ADS  Google Scholar 

  53. M. E. Fisher, in “Renormalization Group in Critical Phenomena and Ouantum Field Theory: Proceedings of a Conference”, Edited by J. D. Gunton and M. S. Green (Temple University, 1974) pp.65–68.

    Google Scholar 

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  1. Baker Laboratory, Cornell University Ithaca, 14853, New York, USA

    Michael E. Fisher

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F. J. W. Hahne

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Fisher, M.E. (1983). Scaling, universality and renormalization group theory. In: Hahne, F.J.W. (eds) Critical Phenomena. Lecture Notes in Physics, vol 186. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12675-9_11

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  • DOI: https://doi.org/10.1007/3-540-12675-9_11

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