Critical Fluctuations in Normal-to-Superconducting Transition

  • R. Folk
  • Yu. Holovatch


Recent advances in our understanding of critical phenomena due to the application1 of the renormalization group (RG) approach2 are now well known (see, e.g., the textbooks3-5). The scale invariance at the critical point and the universality of certain features of critical phenomena can be explained by RG transformation, and lead to theory which provides a quantitative description of the critical behaviour of various thermodynamic quantities of interest.


Renormalization Group Critical Exponent Stable Fixed Point Pade Approximant Renormalization Group Transformation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. G. Wilson, Phys. Rev. B 4:3174 (1971); ibid 4:3184 (1971).Google Scholar
  2. 2.
    N. N. Bogoliubov, and D. V. Shirkov. “Introduction to the Theory of Quantized Fields,” Wiley & Sons, New York (1959).Google Scholar
  3. 3.
    D. J. Amit. “Field Theory, the Renormalization Group, and Critical Phenomena,” World Scientific, Singapore (1984).Google Scholar
  4. 4.
    M. Le Bellac. “Quantum and Statistical Field Theory,” Claredon Press, Oxford (1991).Google Scholar
  5. 5.
    J. Zinn-Justin. “Quantum Field Theory and Critical Phenomena,” Oxford University Press, Oxford (1996).Google Scholar
  6. 6.
    J. A. Lipa, D. R. Swanson, J. A. Nissen, T. C. P. Chui, and U. E. Israelsson, Phys. Rev. Lett. 76:944 (1996).ADSCrossRefGoogle Scholar
  7. 7.
    G. A. Baker Jr., B. G. Nickel, and D. I. Meiron, Phys. Rev. B 17:1365 (1978).ADSCrossRefGoogle Scholar
  8. 8.
    J. C. Le Guillou, and J. Zinn-Justin, Phys. Rev. B 21:3976 (1980).MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    C. Bagnuls, and C. Bervillier, Phys. Rev. B 32:7209 (1985).ADSCrossRefGoogle Scholar
  10. 10.
    C. Bagnuls, C. Bervillier, D. I. Meiron, and B. G. Nickel, Phys. Rev. B 35:3585 (1987).ADSCrossRefGoogle Scholar
  11. 11.
    R. Schloms, and V. Dohm, Europhys. Lett. 3:413 (1987).ADSCrossRefGoogle Scholar
  12. 12.
    R. Schloms, and V. Dohm, Nucl. Phys. B 328:639 (1989).MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    R. Schloms, and V. Dohm, Phys. Rev. B 42:6142 (1990).ADSCrossRefGoogle Scholar
  14. 14.
    I. A. Vakarchuk, Theor. Math. Phys. (Moscow) 36:122 (1978).Google Scholar
  15. 15.
    B. I. Halperin, T. C. Lubensky, and S. Ma, Phys. Rev. Lett. 32:292 (1974).ADSCrossRefGoogle Scholar
  16. 16.
    J. Lobb, Phys. Rev. B 36:3930 (1987).ADSGoogle Scholar
  17. 17.
    S. E. Inderhees, M. B. Salamon, N. Goldenfeld, J. P. Rice, B. G. Pazol, D. M. Ginsberg, J. Z. Liu, and G. W. Crabtree, Phys. Rev. Lett. 60:1178 (1988).ADSCrossRefGoogle Scholar
  18. 18.
    M. B. Salamon, S. E. Inderhees, J. P. Rice, B. G. Pazol, D. M. Ginsberg, and N. Goldenfeld, Phys. Rev. B 38:885 (1988).ADSCrossRefGoogle Scholar
  19. 19.
    S. E. Inderhees, M. B. Salamon, J. P. Rice, and D. M. Ginsberg, Phys. Rev. Lett. 66:232 (1991).ADSCrossRefGoogle Scholar
  20. 20.
    S. Regan, A. J. Lowe, and M. A. Howson, J. Phys.: Condens. Matter 3:9245 (1991).ADSCrossRefGoogle Scholar
  21. 21.
    G. Mozurkewich, and M. B. Salamon, Phys. Rev. B 46:11914 (1992).Google Scholar
  22. 22.
    M. B. Salamon, J. Shi, N. Overend, and M. A. Howson, Phys. Rev. B 47: 5520 (1993).ADSGoogle Scholar
  23. 23.
    A. Junod, E. Bonjour, R. Calemczuk, J. Y. Henry, J. Muller, G. Triscone, and J. C. Vallier, Physica C 211:304 (1993).ADSCrossRefGoogle Scholar
  24. 24.
    N. Overend, M. A. Howson, and I. D. Lawrie, Phys. Rev. Lett. 72:3238 (1994).ADSCrossRefGoogle Scholar
  25. 25.
    I. D. Lawrie, Phys. Rev. B 50:9456 (1994).ADSGoogle Scholar
  26. 26.
    J. H. Chen, T. C. Lubensky, and D. R. Nelson, Phys. Rev. B 17:4274 (1978).ADSCrossRefGoogle Scholar
  27. 27.
    J. C. Le Guillon, E. Brézin, and J. Zinn-Justin, Phys. Rev. D 15:1544 (1977).ADSCrossRefGoogle Scholar
  28. 28.
    L. N. Lipatov, Soy. Phys. JETP. 45:216 (1977).MathSciNetADSGoogle Scholar
  29. 29.
    E. Brezin, and G. Parisi, J. Stat. Phys. 19:269 (1978).MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    S. Kolnberger, and R. Folk, Phys. Rev. B 41:4083 (1990)ADSCrossRefGoogle Scholar
  31. 31.
    R. Folk, and Yu. Holovatch, J. Phys. A: Math. Gen. 29:3409 (1996).ADSCrossRefMATHGoogle Scholar
  32. 32.
    R. Folk, and Yu. Holovatch, J. Phys. Stud. (Lviv) 1:343 (1997).Google Scholar
  33. 33.
    H. Meyer-Ortmanns, Rev. Mod. Phys. 68:473 (1996)ADSCrossRefGoogle Scholar
  34. 34.
    Recall that for a given Landau-Ginsburg parameter (ratio of penetration depth to coherence length) k a superconductor with k < 1/i is called type-I and one with k > 1/v is called type-II.Google Scholar
  35. 35.
    K. G. Wilson, and M. E. Fisher, Phys. Rev. Lett. 28:240 (1972).ADSCrossRefGoogle Scholar
  36. 36.
    A. E. Filippov, A. V. Radievsky, and A. S. Zeltser, Phys. Lett. A 192:131 (1994); A. S. Zeltser, A. E. Filippov, J. Exp. Theor. Phys. 106:1117 (1994) (in Russian).Google Scholar
  37. 37.
    A. P. C. Malbouisson, F. S. Nogueira, and N. F. Svaiter, Europhys. Lett. 41:547 (1998).ADSCrossRefGoogle Scholar
  38. 38.
    S. Coleman, and E. Weinberg, Phys. Rev. D 7:1988 (1973).ADSGoogle Scholar
  39. 39.
    J. S. Kang, Phys. Rev. D 10:3455 (1974).ADSCrossRefGoogle Scholar
  40. 40.
    I. D. Lawrie, Nucl. Phys. B 200:1 (1982).ADSCrossRefGoogle Scholar
  41. 41.
    S. Hikami, Progr. Theor. Phys. 26:226 (1979).ADSCrossRefGoogle Scholar
  42. 42.
    S. W. Lovesey, Z. Physik B Condensed Matter 40:117 (1980)MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    C. Dasgupta, B. I. Halperin, Phys. Rev. Lett. 47:1556 (1981).ADSCrossRefGoogle Scholar
  44. 44.
    J. Bartholomew, Phys. Rev. B 28:5378 (1983).ADSCrossRefGoogle Scholar
  45. 45.
    N. C. Tonchev, and D. I. Uzunov, J. Phys. A 14:521 (1981).ADSCrossRefGoogle Scholar
  46. 46.
    D. Boyanovsky, and J. L. Cardy, Phys. Rev. B 25:7058 (1982).ADSCrossRefGoogle Scholar
  47. 47.
    D. I. Uzunov, E. R. Korutcheva, and Y. T. Millev, J. Phys. A 16:247 (1983).Google Scholar
  48. 48.
    C. Athorne, and I. D. Lawrie, Nucl. Phys. B 265:551 (1986)ADSCrossRefGoogle Scholar
  49. 49.
    G. Busiello, L. De Cesare, and D. I. Uzunov, Phys. Rev. B 34:4932 (1986).ADSCrossRefGoogle Scholar
  50. 50.
    E. J. Blagoeva, G. Busiello, L. De Cesare, Y. T. Millev, I. Rabuffo, and D. I. Uzunov, Phys. Rev. B 42:6124 (1990).ADSCrossRefGoogle Scholar
  51. 51.
    G. Busiello, L. De Cesare, Y. T. Millev, I. Rabuffo, and D. I. Uzunov, Phys. Rev. B 4:3:1150 (1991).ADSGoogle Scholar
  52. 52.
    The expression of (-function for the mass aggrees with C. Ford, I. Jack, and D. Jones, Nucl. Phys. B387:373 (1992).Google Scholar
  53. 53.
    I. D. Lawrie, and C. Athorne, J. Phys. A: Math. Gen. 16:L587 (1983).ADSCrossRefGoogle Scholar
  54. 54.
    B. Bergerhoff, F. Freire, D. F. Litim, S. Lola, and C. Wetterich, Phys. Rev. B 53:5734 (1996).ADSCrossRefGoogle Scholar
  55. 55.
    N. Tetradis, and C. Wetterich, Noel. Phys. B 422:541 (1994).ADSCrossRefGoogle Scholar
  56. 56.
    M. Gräter, and C. Wetterich, Phys. Rev. Lett. 75:378 (1995).ADSCrossRefGoogle Scholar
  57. 57.
    P. Arnold, and L. G. Yaffe, Phys. Rev. D 49:3003; P. Arnold, and L. G. Yaffe, Phys. Rev. D 55:1114 (1997) (erratum).Google Scholar
  58. 58.
    H. Kleinert, Lett. Nouvo Cimento 35:405 (1982) (we are thankful to Prof. H. Kleinert for attracting our attention to this reference).Google Scholar
  59. 59.
    M. Kiometzis, H. Kleinert, and A. M. J. Schakel, Phys. Rev. Lett. 73:1975 (1994). and Fortschr. Phys. 43:697 (1995).Google Scholar
  60. 60.
    A. J. Bray, Phys. Rev. Lett. 32:1413 (1974).ADSCrossRefGoogle Scholar
  61. 61.
    L. Radzihovsky, Europhys. Lett. 29:227 (1995).ADSCrossRefGoogle Scholar
  62. 62.
    I. F. Herbut, and Z. Tesanovie, Phys. Rev. Lett. 76:4588 (1996).ADSCrossRefGoogle Scholar
  63. 63.
    I. F. Herbut, J. Phys. A 30:423 (1997).MathSciNetADSCrossRefMATHGoogle Scholar
  64. 64.
    In fact an extensive two loop calculation have been already performed earlier by M. Machacek and M. Vaughn, Nucl. Phys. B222:83 (1983); B236:221 (1984); B249:70 (1985).Google Scholar
  65. 65.
    P. Olsson, and S. Teitel, Phys. Rev. Lett. 80:1964 (1998).ADSCrossRefGoogle Scholar
  66. 66.
    see e.g. X. Wen, and Y. Wu, Phys. Rev. Lett. 70:1501 (1993) and L. Pryadko, and S. Zhang, Phys. Rev. Lett. 73:3282 (1994).Google Scholar
  67. 67.
    K. K. Nanda, B. Kalta, Phys. Rev. B 57:123 (1998).ADSCrossRefGoogle Scholar
  68. 68.
    P. G. de Gennes, Solid State Commun. 10:753 (1972).ADSCrossRefGoogle Scholar
  69. 69.
    B. I. Halperin, and T. C. Lubensky, Solid State Commun. 14:997 (1974).ADSCrossRefGoogle Scholar
  70. 70.
    T. C. Lubensky, and J.-H. Chen, Phys. Rev. B 17:366 (1978).ADSCrossRefGoogle Scholar
  71. 71.
    G. B. Kasting, K. J. Lushington, and C. W. Garland, Phys. Rev. B 22:321 (1980).ADSCrossRefGoogle Scholar
  72. 72.
    M. A. Anisimov, P. E. Cladis, E. E. Gorodetskii, D. A. Huse, V. E. Podneks, V. G. Taratuta, W. van Saarloos, and V. P. Voronov, Phys. Rev. A 41:6749 (1990).ADSCrossRefGoogle Scholar
  73. 73.
    For the most updated comprehensive review of the experimental data on effective critical exponents governing nematic-smectic-A phase transitions see: C. W. Garland, and G. Nonnesis, Phys. Rev. E 49:2964 (1994).CrossRefGoogle Scholar
  74. 74.
    G. t’Hooft, and M. Veltman, Nucl. Phys. B 44:189 (1972).MathSciNetADSCrossRefGoogle Scholar
  75. 75.
    The first index is the number of rI fields, the second index is the number of A fields.Google Scholar
  76. 76.
    As an example for determination of critical exponents values for models with complicated symmetry by applying the resummation technique in different RG schemes see Refs.77–81; N. A. Shpot,. Phys.Lett. A 142:474 (1989); S. A. Antonenko, and A. I. Sokolov, Phys. Rev. B 49:15901 (1994); C. von Ferber, and Yu. Holovatch, Europhys. Lett. 39:31 (1997), Phys. Rev. E 56:6370 (1997); H. Kleinert, S. Thorns, and V. Schulte-Frohlinde, preprint (1997).Google Scholar
  77. 77.
    G. Jug, Phys. Rev. B 27:609 (1983).ADSCrossRefGoogle Scholar
  78. 78.
    I. O. Mayer, A. I. Sokolov, and B. N. Shalaev, Ferroelectrics 95:93 (1989).CrossRefGoogle Scholar
  79. 79.
    I. O. Mayer, J. Phys. A 22:2815 (1989).ADSCrossRefGoogle Scholar
  80. 80.
    . Yu. Holovatch, and M. Shpot, J. Stat. Phys. 66:867 (1989); Yu. Holovatch, and T. Yavors’kii, (1998) submitted to J. Stat. Phys.Google Scholar
  81. 81.
    H. K. Janssen, K. Oerding, and E. Sengespeick, J. Phys. A: Math. Gen. 28:6073 (1995).MathSciNetADSCrossRefMATHGoogle Scholar
  82. 82.
    G. Grinstein, and A. Luther, Phys. Rev. B 13:1329 (1976)ADSCrossRefGoogle Scholar
  83. 83.
    G.H. Hardy. “Divergent Series,” Oxford University, Oxford (1948).Google Scholar
  84. 84.
    Only when one of couplings is equal to zero one does obtain a series which is proven to be asymptotic.Google Scholar
  85. 85.
    P. W. Mitchell, R. A. Cowley, H. Yoshizawa, P. Böni, Y. J. Uemura, and R. J. Birgeneau. Phys. Rev. B 34:4719 (1986).ADSCrossRefGoogle Scholar
  86. 86.
    T. R. Thurston, C. J. Peters, R. J. Birgeneau, and P. M. Horn, Phys. Rev. B 37:9559 (1988).ADSCrossRefGoogle Scholar
  87. 87.
    J.-S. Wang, M. Wöhlert, H. Mühlenbein, and D. Chowdhury, Physica A 166:173 (1990).ADSCrossRefGoogle Scholar
  88. 88.
    J.-S. Wang, W. Selke, Vl. S. Dotsenko, and V. B. Andreichenko, Europhys. Lett. 11:301 (1994): A. L. Talapov, and L. N. Shchur, Europhys. Lett. 27:193 (1994).Google Scholar
  89. 89.
    T. Holey, and M. Fähnle, Phys. Rev. B 41:11709 (1990).ADSCrossRefGoogle Scholar
  90. 90.
    The results of resummation appear to be quite insensitive to the choice of p.Google Scholar
  91. 91.
    J. S. R. Chisholm, Math. Comp. 27:841 (1973).MathSciNetCrossRefMATHGoogle Scholar
  92. 92.
    P. J. S. Watson, J. Phys. A 7:L167 (1974).ADSCrossRefMATHGoogle Scholar
  93. 93.
    G. A. Baker Jr., and P. Graves-Morris. “Padé approximants,” Addison-Wesley Publ. Co., Reading, Mass. (1981).Google Scholar
  94. 94.
    G. Parisi, in: “Proceedings of the 1973 Cargrése Summer School,” unpublished. G. Parisi, J. Stat. Phys. 23:49 (1980).MathSciNetADSCrossRefGoogle Scholar
  95. 95.
    H. Kleinert, V. Schulte-Frohlinde, Phys. Lett. B 342:284 (1995).ADSCrossRefGoogle Scholar
  96. 96.
    For this model the system of fixed point equations is degenerate at one-loop level, resulting in particular in the /-expansion for critical exponents: D. E. Khmelnitskii, Zh. Eksp. Theor. Fiz. 68:1960 (1975); T. C. Lubensky, Phys. Rev. B 11:3573 (1975); Ref.82 Google Scholar
  97. 97.
    It corresponds to n = 0 fixed point of 0(n) symmetrical model, described by the de Gennes limit of self avoiding walk problem.Google Scholar
  98. 98.
    The results are given for the value of parameter p = 0 in the Borel-Leroy image.Google Scholar
  99. 99.
    Yu. Holovatch, Preprint, and C.E. Saclay, Service de Physique Theorique; S Ph T/92–123); Yu. Holovatch, Int. J. Mod. Phys. A 8:5329 (1993).Google Scholar
  100. 100.
    The last possibility has been chosen by G. Parisi94 in order to restore the presence of stable solution for the fixed point in the two-loop approximation.Google Scholar
  101. 101.
    The series in (37) appears to be alternating and this scheme can be applied without any difficulties.Google Scholar
  102. 102.
    M. Abramowitz, and A. I. Stegun, (editors) “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” National Bureau of Standards (1964).Google Scholar
  103. 103.
    In this case the principal value of integral (29) could be taken, but generally speaking it is preferable to avoid such situations (see Ref.7 as well).Google Scholar
  104. 104.
    We take t o=1.Google Scholar
  105. 105.
    . The value of y has been found by the scaling law: t1 = 2 - y/v.Google Scholar
  106. 106.
    V. Dohm, Phys. Rev. Lett. 53:1379 (1984).ADSCrossRefGoogle Scholar
  107. 107.
    V. Dohm, in: “Application of field theory to statistical mechanics,” L. Garrido Ed. p 263, Berlin, Heidelberg, New York, Tokyo, Springer (1985).Google Scholar
  108. 108.
    V. Dohm, Z. Physik Condensed Matter 60:61 (1985).ADSCrossRefGoogle Scholar
  109. 109.
    A. Singsaas and G. Ahlers, Phys. Rev. B 30:5103 (1984).ADSCrossRefGoogle Scholar
  110. 110.
    J. F. Annett, S. R. Renn, Phys. Rev. B 38:4660 (1988).ADSCrossRefGoogle Scholar
  111. 111.
    E. K. Riedel, and F. J. Wegner, Phys. Rev. B 9:294 (1974).ADSCrossRefGoogle Scholar
  112. 112.
    In fact we have solved flow equations (57) and (59) starting near unstable fixed points, for the initial value of flow parameter we have taken t = 1. The use of different initial values (u(1) and f(1) on the seperatrix) would amount to rescale the flow parameter.Google Scholar
  113. 113.
    I. D. Lawrie, Phys. Rev. Lett. 79:131 (1997); see also K. K. Nanda, and B. Kalta, Phys. Rev. B 57:123 (1998).Google Scholar
  114. 114.
    Because they are double series in two coupling constants we have represented them in a form of resolvent series which has enabled us then to pass to the Padé analysis.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • R. Folk
    • 1
  • Yu. Holovatch
    • 2
  1. 1.Institut für Theoretische PhysikJohannes Kepler Universität LinzLinzAustria
  2. 2.Institute for Condensed Matter PhysicsUkrainian Academy of SciencesLvivUkraine

Personalised recommendations