Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
B. Widom in “Fundamental Problems in Statistical. Mechanics” vol. III, Ed. E. V. G. Cohen (North Holland, 1975) pp. 1–45.
P. Pfeuty and G. Toulouse, “Introduction to Renormalization Groups and Critical Phenomena” (Wiley, 1977).
M. E. Fisher, Rev. Mod. Phys. 46, 597–616 (1974).
D. J. Wallace and R. K. P. Zia, Repts. Prog. Phys. 41, 1–85 (1978).
A. Z. Patashinskii and V. L. Pokrovskii, “Fluctuation Theory of Phase Transitions” (Pergamon Press, 1979).
L. E. Reichl, “A Modern Course in Statistical Physics”, (Univ. of Texas Press, 1980), Chap. 4.
A. P. Young in “Ordering in Strongly Fluctuating Condensed Matter Systems”, Ed. T. Riste (Plenum Press, 1980) pp. 11–31, 271–284.
S.-K. Ma, “Modern Theory of Critical Phenomena” (Benjamin, 1976).
K. G. Wilson, Rev. Mod. Phys. 47, 773–840 (1975).
K. G. Wilson and J. Kogut, Physics Reports 12 C (1974).
B. I. Halperin in “Physics of Low-Dimensional Systems”, Eds. Y. Nagaoka and S. Hikami (Publ. Office, Prog. Theoret. Phys., Kyoto, 1979).
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.
D. J. Amit, “Field Theory, the Renormalization Group and Critical Phenomena” (McGraw-Hill, 1978).
References
See First Book of Kings, Chap. 7, verse 23.
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.
D. Balzarini and K. Ohrn, Phys. Rev. Lett. 29, 840 (1972); D. Balzarini, Can. J. Phys. 52, 499 (1974).
M. J. Sienko, J. Chem. Phys. 53, 566 (1970).
M. I. Bagatskii, A. V. Voronel' and V. G. Gusak, Sov. Phys. JETP 16, 517 (1963).
G. Ahlers, Phys. Rev. Lett. 24, 1333 (1970).
B. J. Lips, unpublished work at Stanford University (1981).
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.
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.
Based on the general scaling treatment presented in Proc. Nobel Symposium 24, Eds. B. Lundqvist and S. Lundqvist (Academic Press, 1974) pp. 16–37.
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).
See e.g. the review “Simple Ising models still thrive”: M. E. Fisher, Physics 106A, 28 (1981) and references therein.
C. Domb, Adv. Phys. 9, 149 (1960).
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).
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).
See R. Balian and G. Toulouse, Phys. Rev. Lett. 30, 544 (1973); M. E. Fisher, Phys. Rev. Lett. 30, 679 (1973).
We follow D. R. Nelson and M. E. Fisher, Ann. Phys. (N.Y.) 91, 226 (1975).
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.
See R. B. Griffiths, Physics 106A, 59 (1981); R. B. Griffiths and P. A. Pearce, J. Stat. Phys. 20, 499 (1979).
F. J. Wegner, J. Phys. C 7, 2098 (1974).
L. P. Kadanoff, Physics 2, 263 (1966).
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.
See also P. Seglar and M. E. Fisher, J. Phys. C 13, 6613 (1980).
V. L. Ginzburg, Sov. Phys. Solid St. 2, 1824 (1960).
L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).
P. C. Hohenberg in Fluctuations in Superconductors edited by W. S. Greee and F. Chilton (Stanford Research Inst., Menlo Park, 1968) p. 305.
J. Als-Nielsen and R. J. Birgeneau, Amer. J. Phys. 45, 554 (1977).
Discovered originally by K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972) where the ε:-expansion was introduced.
E. Brezin, J. C. LeGuillon, J. Zinn-Justin and B. G. Nickel, Phys. Lett. 44A, 227 (1973).
A. A. Vladimirov, D. I. Kazakov, and O. Tarasov, Sov. Phys. JETP 50, 521 (1979).
M. E. Fisher, S.-K. Ma, and B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972).
M. E. Fisher and D. R. Nelson, Phys. Rev. Lett. 32, 1352 (1974).
M. E. Fisher and A. Aharony, Phys. Rev. Lett. 30, 559 (1973).
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.
See also A. Aharony, Phys. Lett. 44A, 313 (1973).
E. K. Riedel and F. J. Wegner, Phys. Rev. Lett. 29, 349 (1972).
R. Abe, Prog. Theoret. Phys. 48, 1414 (1972).
A. M. Polyakov, Phys. Lett. 59B, 79 (1975); see also E. Brezin and J. ZinnJustin, Phys. Rev. Lett. 36, 691 (1976).
J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973).
J. M. Kosterlitz, J. Phys. C7, 1046 (1974).
J. V. Jose, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B16, 1217 (1977).
B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978); E 41, 519 (1978).
A. P. Young, Phys. Rev. B19, 1855 (1979).
Th. Niemeijer and J. M. J. van Leeuwen, Physics (Utrecht) 71, 17 (1974).
A. A. Migdal, Zh. Eksp.-Teor. Fiz. 69, 810, 1457 (1975).
L. P. Kadanoff, Ann. Phys. (N.Y.) 100, 359 (1976).
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).
D. R. Nelson, Phys. Rev. B11, 3504 (1975).
J. Rudnick and D. R. Nelson, Phys. Rev. 13, 2208 (1976).
K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
A review of graph theoretical terminology is given by J. W. Essam and M. E. Fisher, Rev. Mod. Phys. 42, 271 (1970).
K. G. Wilson, Phys. Rev. D7, 2911 (1973).
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.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/3-540-12675-9_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12675-1
Online ISBN: 978-3-540-38667-4
eBook Packages: Springer Book Archive