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Spin glasses, effective decrease of long-range interactions

  • Aernout C. D. van Enter
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 257)

Abstract

Long-range spin-glass models with random Hamiltonians
$$H = - \mathop \sum \limits_{i j} J(i,j)|i - j|^{ - \alpha } S_i S_j$$
where the J(i,j) are independent, identically distributed random variables with mean IE J(i,j) = 0, satisfying some moment conditions, in some respects behave like nonrandom models with Hamiltonians
$$H = - \sum |i - j|^{ - 2\alpha } S_i S_j .$$
This result is obtained through a study of the thermodynamic limit in any dimension, and the absence of long range order and the asymptotic correlation decay in dimension one and two. Hence the effective decrease of the interaction tends to be twice as fast as the original one.

Keywords

Thermodynamic Limit Spin Glass Relative Entropy Free Energy Density Free Boundary Condition 
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.

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References

  1. [1]
    S.F. Edwards and P.W. Anderson: J. Phys. F 5, 965 (1975).Google Scholar
  2. [2]
    Reviews on different aspects of spin glasses can be found in the proceedings of the Heidelberg Colloquium on Spin Glasses. Ed. J.L. van Hemmen and J. Morgenstern, Springer Lecture Notes in Physics 192 (1983).Google Scholar
  3. [2a]
    See also D. Chowdhury and A. Mookerjee: Phys. Rep. 114, 1 (1984).Google Scholar
  4. [3]
    J. Morgenstern and A. Ogielski: Phys. Rev. Lett. 54, 928 (1985) (transition d > 3).Google Scholar
  5. [4] a)
    R. Fisch and A.B. Harris: Phys. Rev. Lett. 38, 785 (1977).Google Scholar
  6. [4] b)
    J. Morgenstern and K. Binder: Phys. Rev. Lett. 43, 1615 (1979). (transition d > 4).Google Scholar
  7. [5]
    A.C.D. van Enter and R.B. Griffiths: Comm, Math. Phys. 90, 319 (1983).Google Scholar
  8. [6]
    R.B. Griffiths: Phys. Rev. Lett. 23, 17 (1969).Google Scholar
  9. [7]
    M. Cassandro, E. Olivieri and B. Tirozzi: Comm. Math. Phys. 87, 229 (1982).Google Scholar
  10. [8]
    A. Berretti: J. Stat. Phys. 38, 483 (1985).Google Scholar
  11. [9]
    J. Fröhlich and J. Imbrie: Comm. Math. Phys. 96, 148 (1985).Google Scholar
  12. [10]
    D. Sherrington and S. Kirkpatrick: Phys. Rev. Lett. 35, 1792 (1975).Google Scholar
  13. [11]
    G. Parisi: J. Phys. A 13, 1101, 1887 (1980).Google Scholar
  14. [12]
    G. Parisi: Phys. Rev. Lett. 50, 1946 (1983).Google Scholar
  15. [13]
    M. Cassandro, E. Olivieri and P. Picco: Rome preprint.Google Scholar
  16. [14]
    D. Fisher and H. Sompolinski: Phys. Rev. Lett. 54, 1063 (1985).Google Scholar
  17. [15] a)
    The thermodynamic limit for short-range random interactions has been treated in R.B. Griffths and L.J. Lebowitz: J. Math. Phys. 9, 1284 (1968).Google Scholar
  18. [15] b)
    F. Ledrappier: Comm. Math. Phys. 56, 297 (1977).Google Scholar
  19. [15] c)
    P. Vuillermot: J. Phys. A 10, 1319 (1977).Google Scholar
  20. [15] d)
    J.L. van Hemmen and R.G. Palmer: J. Phys. A 15, 3881 (1982).Google Scholar
  21. [16] a)
    The thermodynamic limit for long-range random interactions has been treated in K.M. Khanin and Ya.G. Sinai: J. Stat. Phys. 20, 573 (1979).Google Scholar
  22. [16] b)
    S. Goulart Rosa: J. Stat. Phys. A 15, L 51 (1982).Google Scholar
  23. [16] c)
    A.C.D. van Enter and J.L. van Hemmen: J. Stat. Phys. 32, 141 (1983).Google Scholar
  24. [17]
    K.M. Khanin: Theor. Math. Phys. 43, 445 (1980).Google Scholar
  25. [18]
    P. Picco: J. Stat. Phys. 32, 627 (1983.Google Scholar
  26. [19]
    P. Picco: J. Stat. Phys. 36, 489 (1984).Google Scholar
  27. [20]
    A.C.D. van Enter and J.L. van Hemmen: J. Stat. Phys. 39, 1 (1985).Google Scholar
  28. [21]
    A.C.D. van Enter and J. Fröhlich: Comm. Math. Phys. 98, 425 (1985).Google Scholar
  29. [22]
    A.C.D. van Enter: J. Stat. Phys. 41, 315 (1985).Google Scholar
  30. [23]
    L. Slegers, A. Vansevenant and A. Verbeure: Phys. Lett. 108 A, 267 (1985).Google Scholar
  31. [24] a)
    The existence of transitions in the corresponding ferromagnetic models is proven in F.J. Dyson: Comm. Math. Phys. 12, 212 (1969).Google Scholar
  32. [24] b)
    J. Fröhlich and T. Spencer: Comm. Math. Phys. 84, 87 (1982).Google Scholar
  33. [24] c)
    H. Kunz and C.E. Pfister: Comm. Math. Phys. 46, 245 (1976).Google Scholar
  34. [25]
    D.C. Mattis: Phys. Lett. 56 A, 421 (1976).Google Scholar
  35. [26]
    J. Luttinger: Phys. Rev. Lett. 37, 778 (1976).Google Scholar
  36. [27] a)
    J.L. van Hemmen: Phys. Rev. Lett. 49, 409 (1982).Google Scholar
  37. [27] b)
    J.L. van Hemmen, A.C.D. van Enter and J. Canisius: Z. Phys. B 50, 311 (1983).Google Scholar
  38. [27] c)
    J.L. van Hemmen, contribution to 2a).Google Scholar
  39. [28] a)
    J.P. Provost and G. Vallée: Phys. Rev. Lett. 50, 598 (1983).Google Scholar
  40. [28] b)
    F. Benamira, J.P. Provost and G. Vallée: J. de Phys. 46, 1269 (1985).Google Scholar
  41. [29]
    D. Amit, H. Gutfreund and H. Sompolinski: Jerusalem preprint.Google Scholar
  42. [30]
    P. Collet and J.P. Eckmann: Comm. Math. Phys. 93, 379 (1984).Google Scholar
  43. [31]
    P. Collet, J.P. Eckmann, V. Glaser and A. Martin: J. Stat. Phys. 36, 89 (1984).Google Scholar
  44. [32] a)
    A treatment of the thermodynamic limit can for example be found in D. Ruelle: Statistical Mechanics, Benjamin New York (1969).Google Scholar
  45. [32] b)
    R.B. Israel: Convexity in the theory of lattice gases, Princeton University Press, Princeton N.J. (1979).Google Scholar
  46. [33] a)
    Absence of phase transitions for one-dimensional models is proven (among others) in D. Ruelle: Comm. Math. Phys. 9, 367 (1968).Google Scholar
  47. [33] b)
    H. Araki: Comm. Math. Phys. 44, 1 (1975).Google Scholar
  48. [33] c)
    J. Bricmont, J.L. Lebowitz and C.E. Pfister: J. Stat. Phys. 21, 573 (1979).Google Scholar
  49. [33] d)
    M. Cassandro and E. Olivieri: Comm. Math. Phys. 80, 255 (1981).Google Scholar
  50. [34] a)
    Two-dimensional models where the rotation symmetry is not broken are treated in N.D. Mermin and H. Wagner: Phys. Rev. Lett. 17, 1133 (1966).Google Scholar
  51. [34] b)
    J. Fröhlich and C.E. Pfister: Comm. Math. Phys. 81, 277 (1981). and references mentioned there.Google Scholar
  52. [35] a)
    McBryan-Spencer estimates are given in O. McBryan and T. Spencer: Comm. Math. Phys. 53, 299 (1977).Google Scholar
  53. [35] b)
    J. Glimm and A. Jaffe: Quantum Physics §16.3, Springer-Verlag New York, Heidelberg, Berlin (1981).Google Scholar
  54. [35] c)
    A. Messager, S. Miracle-Sole and J. Ruiz: Ann. Inst. Henri Poincaré 40, 85 (1984).Google Scholar
  55. [35] d)
    K.R. Ito: J. Stat. Phys. 29, 747 (1982).Google Scholar
  56. [35] e)
    S.B. Shlosman: Theor. Math. Phys. 37, 1118 (1978).Google Scholar
  57. [36]
    G. Kotliar, P.W. Anderson and D.L. Stein: Phys. Rev. B 27, 602 (1983).Google Scholar
  58. [37] a)
    M.A. Akcoglu and U. Krengel: J. Reine Angew. Math. 323, 53 (1981).Google Scholar
  59. [37] b)
    U. Krengel: Ergodic theorems §6.2, de Gruyter, Berlin, New York (1985).Google Scholar
  60. [38]
    M. Campanino, A.C.D. van Enter and E. Olivieri: in preparation.Google Scholar
  61. [39]
    J. Fröhlich: private communication.Google Scholar
  62. [40]
    R.B. Griffiths: Phys. Rev. 176, 655 (1968).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Aernout C. D. van Enter
    • 1
  1. 1.SFB 123 Im Neuenheimer Feld 294Heidelberg, FRG

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