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Equilibrium theory of spin glasses: Mean-field theory and beyond

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Heidelberg Colloquium on Spin Glasses

Part of the book series: Lecture Notes in Physics ((LNP,volume 192))

Abstract

In this paper a new technique is analyzed to solve certain mean-field models, with particular emphasis on the spin-glass case. We also present an extended Gibbs formalism which is based on the observation that every equilibrium state can be decomposed uniquely into its ergodic components, and apply it to spin-glasses.

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References

  1. S.F. Edwards and P.W. Anderson, J.Phys.F 5 (1975) 965

    Google Scholar 

  2. J.A. Mydosh, in: Springer Lecture Notes in Physics 149 (1981) 87–106

    Google Scholar 

  3. A.C.D. van Enter and J.L. van Hemmen, Phys.Rev.A 29 (1984)

    Google Scholar 

  4. J.L. van Hemmen, Phys.Rev.Lett. 49 (1982) 409; J.L. van Hemmen, A.C.D. van Enter, and J. Canisius, Z.Phys.B 50 (1983) 311

    Google Scholar 

  5. A.P. Malozemoff and Y. Imry, Phys.Rev.B 24 (1981) 489

    Google Scholar 

  6. P. Monod and H. Bouchiat, J.Phys. (Paris) 43 (1982) L 45

    Google Scholar 

  7. R. Omari, J.J. Préjean, and J. Souletie, to be published; J. Souletie, these proceedings

    Google Scholar 

  8. N. Bontemps and J. Rajchenbach, to be published; H. Maletta, G. Aeppli, and S.M. Shapiro, J.Magn.Magn.Mater. 31–34 (1983) 1367

    Google Scholar 

  9. R.P. Feynman, Statistical Mechanics (Benjamin, Reading, 1972)p.1

    Google Scholar 

  10. L. Lundgren, P. Svedlindh, and O. Beckmann, Phys.Rev.B 26 (1982) 3990

    Google Scholar 

  11. H. Maletta, in: Excitations in disordered systems, edited-by M.F. Thorpe (Plenum, New York, 1982) pp.431–462

    Google Scholar 

  12. A.C.D. van Enter and J.L. van Hemmen, J.Stat.Phys. 32 (1983) 141 and work in preparation; K.M. Khanin and Ya.G. Sinai, J.Stat.Phys. 20 (1979) 573

    Google Scholar 

  13. P.M. Levy and A. Fert, Phys.Rev.B 23 (1981) 4667

    Google Scholar 

  14. P.W. Anderson, Rev.Mod.Phys. 50 (1978) 199

    Google Scholar 

  15. J.L. van Hemmen and R.G. Palmer, J.Phys.A: Math.Gen. 15 (1982) 3881

    Google Scholar 

  16. J.A. Mydosh, these proceedings

    Google Scholar 

  17. K. Binder and K. Schröder, Phys.Rev.B 14 (1976) 2142; in particular Fig.1

    Google Scholar 

  18. See also J.P. Provost and G. Vallée, Phys.Rev.Lett. 50 (1983) 598

    Google Scholar 

  19. By the strong law of large numbers applied to the ξ's and the η's. See L. Breiman, Probability (Addison-Wesley, Reading, 1968) Sec. 3.6

    Google Scholar 

  20. A.W. Roberts and D.E. Varberg, Convex functions (Academic Press, New York, 1973) pp. 30 and 110. This excellent book contains a wealth of information.

    Google Scholar 

  21. H. Cramér, Act.Sci.Ind., vol.736 (Herman, Paris, 1938) pp.5–23; H. Chernoff, Ann. Math.Stat. 23 (1952) 493. Textbook versions have been given by O.E. Lanford, Springer Lecture Notes in Physics 20 (1973) 35–49, and R.J. Serfling, Approximation theorems of mathematical statistics (Wiley, New York, 1980)pp. 326–328.

    Google Scholar 

  22. Ref.4 (Z.Phys.), Sec. VI

    Google Scholar 

  23. R.B. Griffiths, C.-Y. Weng, and J.S. Langer, Phys.Rev. 149 (1966) 301

    Google Scholar 

  24. D.C. Mattis, Phys.Lett. 56A (1976) 421

    Google Scholar 

  25. G. Toulouse, Commun.Phys. 2 (1977) 115

    Google Scholar 

  26. A.F.J. Morgownik and J.A. Mydosh, Phys.Rev.B 24 (1981) 5277

    Google Scholar 

  27. S. Nagata, P.H. Keesom, and H.R. Harrison, Phys.Rev.B 19 (1979) 1633

    Google Scholar 

  28. R.W. Knitter and J.S. Kouvel, J.Magn.Magn.Mater. 21 (1980) L316

    Google Scholar 

  29. P.W. Anderson, B.I. Halperin, C.M. Varma, Phil.Mag. 25 (1972) 1

    Google Scholar 

  30. L.E. Wenger, and P.H. Keesom, Phys.Rev.B 13 (1979) 4053

    Google Scholar 

  31. D.L. Martin, Phys.Rev.B 21 (1980) 1906

    Google Scholar 

  32. K. Binder, In: Fundamental problems in statistical mechanics, vol.V, edited by E.G.D. Cohen (North-Holland, Amsterdam, 1980) pp.21–51

    Google Scholar 

  33. E. Ising, Z.Phys. 31 (1925) 253, in particular pp.256–7

    Google Scholar 

  34. The idea reappears in the more recent literature; e.g. R.B. Griffiths, Phys.Rev. 152 (1966) 240, Fig. l. The formalism of section 4.2 nicely explains the main results of this paper. See also R.G. Palmer, Adv.Phys. 31 (1982) 669, Fig. 2

    Google Scholar 

  35. D.C. Mattis, The theory of magnetism I (Springer, New York, 1981) §6.6. Note that, if λ is large, the effective range of the interaction greatly exceeds that of I/R6; cf. section 1.2

    Google Scholar 

  36. K. Huang, Statistical Mechanics (Wiley, New York, 1963) chapters 8 and 9

    Google Scholar 

  37. G.E. Uhlenbeck and G.W. Ford, Lectures in Statistical Mechanics (American Mathematical Society, Providence, R.I., 1963) chapter I

    Google Scholar 

  38. C.N. Yang and T.D. Lee, Phys.Rev. 87 (1952) 404; Ref.36, §§15.1 and 15.2 39. R.B. Israel, Convexity in the theory of Zattice gases (Princeton University Press, Princeton, N.J., 1979)

    Google Scholar 

  39. In passing we note that a finite range of the interaction is not strictly necessary to derive the DLR equations.

    Google Scholar 

  40. A. Munster, Statistische Thermodynamik (Springer, Berlin, 1956) §5.12

    Google Scholar 

  41. R.L. Dobrushin and S.B. Shlosman, Sel.Math.Sov. 1 (1981) 317–338

    Google Scholar 

  42. J.M. Kosterlitz and D.J. Thouless, J.Phys.C 6 (1973) 1181; J. Fröhlich and T. Spencer, Commun.Math.Phys. 81 (1981) 527

    Google Scholar 

  43. L. Onsager, Phys.Rev. 65 (1944) 117; cf. Ref.36, chapter 17

    Google Scholar 

  44. G. Parisi, Phys.Rev.Lett 50 (1983) 1946

    Google Scholar 

  45. C. de Dominicis and A.P. Young, J.Phys.A: Math.Gen., 16 (1983) 2063

    Google Scholar 

  46. A. Houghton, S. Jain, and A.P.Young, J.Phys.C 16 (1983) L375

    Google Scholar 

  47. A.P. Young,these proceedings

    Google Scholar 

  48. Monte Carlo methods in statistical physics, edited by K. Binder (Springer Verlag, Berlin-Heidelberg-New York, 1979)

    Google Scholar 

  49. R.G. Palmer, Adv.Phys. 31 (1982) 669, and these proceedings

    Google Scholar 

  50. B. Simon and A. Sokal, J.Stat.Phys. 25 (1981) 679, sections 1 and 2

    Google Scholar 

  51. See, for instance, A.C.D. van Enter and R.B. Griffiths, Commun.Math.Phys. 90 (1983) 319

    Google Scholar 

  52. A.C.D. van Enter and J.L. van Hemmen, Ref.12, section 3

    Google Scholar 

  53. L. Landau, J.F. Perez, and W.F. Wreszinski, J.Stat.Phys. 26 (1981) 755; Ph.A. Martin, Nuovo Cimento 68B (1982) 302

    Google Scholar 

  54. A. Blandin, M. Gabay, and T. Garel, J.Phys.C 13 (1980) 403.

    Google Scholar 

  55. R. Kindermann and J.L. Snell, Markov random fields and their applications (American Mathematical Society, Providence, R.I. 1980). This is their Fig.18 on p.59. I thank the authors for their permission to reproduce it here.

    Google Scholar 

  56. All we need here is Kingman's subadditive ergodic theorem. See J.F.C. Kingman, in: Springer Lecture Notes in Mathematics 539 (1976) 168–223; also Y. Derriennic, C.R. Acad.Sci.Paris 281A (1975) 985–988

    Google Scholar 

  57. See Roberts and Varberg [20], section 51.

    Google Scholar 

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Authors and Affiliations

  1. Sonderforschungsbereich 123, Universität Heidelberg, Im Neuenheimer Feld 294, 6900, Heidelberg 1, Germany

    J. L. van Hemmen

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  1. J. L. van Hemmen
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J. L. van Hemmen I. Morgenstern

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van Hemmen, J.L. (1983). Equilibrium theory of spin glasses: Mean-field theory and beyond. In: van Hemmen, J.L., Morgenstern, I. (eds) Heidelberg Colloquium on Spin Glasses. Lecture Notes in Physics, vol 192. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12872-7_50

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  • DOI: https://doi.org/10.1007/3-540-12872-7_50

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