The Ising Model

  • Daniel C. Mattis
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 55)


In this prototype theory of ferromagnetism—and of many other physical phenomena as well—a spin Si =±1 is assigned to each of N sites on a fixed lattice. The spins, which live on the vertices of the lattice, interact with one another by means of bonds (the links of the lattice). These have strengths Jij in energy units. In addition, the spins can interact with external fields Bi of arbitrary strengths. The total energy is then given by:
$$H = - \sum {{J_{ij}}} {S_i}{S_j} - \sum {{B_i}} {S_i}$$
and can be directly evaluated in any of the 2N spin configurations. In the most familiar version of the Ising model, the interactions are limited to nearest-neighbors on the lattice and the magnetic field is homogeneous, Bi =constant.


Partition Function External Field Ising Model Transfer Matrix Critical Exponent 
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  1. 3.1
    S.G. Brush: The history of the Lenz-Ising 883 (1967)Google Scholar
  2. 3.2
    W. Lenz: Phys. Z. 21, 613 (1920)Google Scholar
  3. 3.3
    E. Ising: Z. Physik 31, 253 (1925)ADSCrossRefGoogle Scholar
  4. 3.4
    H. Bethe: Proc. Roy. Soc. (London) 9, 244 (1938). Also: F. CernuschiGoogle Scholar
  5. 3.5
    R. Fowler, E. Guggenheim: Statistical Thermodynamics (Cambridge Univ. Press, Cambridge 1939) Chap.13Google Scholar
  6. 3.6
    C.N. Yang, T.D. Lee: Phys. Rev. 87, 404 (1952)MATHADSMathSciNetCrossRefGoogle Scholar
  7. T.D. Lee, C.N. Yang: Phys. Rev. 87, 410 (1952) Recent work on the complex zeros includes:E. Marinari: Nucl. Phys. B235 (FS11), 123 (1984), 3D Ising modelGoogle Scholar
  8. K. De’Bell, M.L. Glasser: Phys. Lett. 104A, 255 (1984), Cayley tree W. Saarloos, D. Kurtze: J. Phys. A17, 1301 (1984), Ising modelGoogle Scholar
  9. A. Caliri, D. Mattis: Phys. Lett. 106A, 74 (1984), long-range model of (2.7.2) with J0 Google Scholar
  10. 3.7
    R. Peierls: Proc. Camb. Phil. Soc. 32, 477 (1936)MATHGoogle Scholar
  11. 3.8
    R. Griffiths: Phys. Rev. 136, A437 (1964). The reader will find it in- structive to determine where this proof fails for the XY model! See further corrections and extension inGoogle Scholar
  12. C.-Y. Weng, R. Griffiths, M. Fisher: Phys. Rev. 162, 475 (1967)ADSCrossRefGoogle Scholar
  13. 3.9
    L. Onsager: Phys. Rev. 65, 117 (1944), algebraic formulationGoogle Scholar
  14. B. Kaufman: Phys. Rev. 76, 1232 (1949), spinor reformulationGoogle Scholar
  15. L. Onsager: Nuovo Cimento (Suppl.) 6, 261 (1949), spontaneous magnetization;Google Scholar
  16. C.N. Yang: Phys. Rev. 85, 809 (1952), first derivation of Onsager’s formula for magnetization in the literatureGoogle Scholar
  17. 3.10
    H. Kramers, G. Wannier: Phys. Rev. 60, 252, 263 (1941)MATHADSMathSciNetCrossRefGoogle Scholar
  18. 3.11
    D.C. Mattis: The Theory of MagnetismI, Springer Ser. Solid-State Sci., Vol. 17 ( Springer, Berlin Heidelberg 1981 )Google Scholar
  19. 3.12
    C. Domb: On the Theory of Cooperative Phenomena in Crystals, Adv. Phys. 9, 149–361 (1960). The fit of Tc(d) on hypercubic lattices to 2 straight lines was performed by G. Cocho, G. Martinez-Mekler, R. Martinez-Enriquez: Phys. Rev. B26, 2666 (1982)Google Scholar
  20. 3.13
    M.E. Fisher: Phys. Rev. 162, 480 (1967)ADSCrossRefGoogle Scholar
  21. 3.14
    H.R. Ott et al.: Phys. Rev. B25, 477 (1982);Google Scholar
  22. Z. Chen, M. Kardar: Phys. Rev. B30, 4113 (1984)ADSGoogle Scholar
  23. 3.15
    M.E. Lines: Phys. Rpts. 55, 133(1979)Google Scholar
  24. 3.16
    E. Jahnke, F. Emde: Tables of Functions( Dover, New York 1945 )MATHGoogle Scholar
  25. 3.17
    T.A. Tjon: Phys. Rev. B2, 2411 (1970)ADSGoogle Scholar
  26. B. McCoy, J. Perk, R. Schrock: Nucl. Phys. B220, 35, 269 (1983) and references thereinGoogle Scholar
  27. 3.18
    E. Lieb, T. Schultz, D. Mattis: Ann. Phys. (NY) 16, 407 (1961)MATHADSMathSciNetCrossRefGoogle Scholar
  28. 3.19
    P. Pfeuty: Phys. Lett. 72A, 245 (1979)ADSMathSciNetGoogle Scholar
  29. 3.20
    T. Schultz, D. Mattis, E. Lieb: Rev. Mod. Phys. 36. 856 (1964)ADSMathSciNetCrossRefGoogle Scholar
  30. 3.21
    A review of Toeplitz matrices, and various improvements and applications thereof to statistical mechanics has been published by M. Fisher, R. Hartwig: Adv. Chem. Phys. 15, 333–354 (1968). The original application to the Ising model in the familiar literature seems to be E. Mon-troll, R. Potts, J. Ward: J. Math. Phys. 4, 308 (1963) in the Onsager anniversary issue of that Journal. But Montroll et al. disclaim first use, and credit Onsager #x00FD;ü this is one of the methods used by Onsager himself. Mark Kac alerted the authors to a limit formula for the calculation of large Toeplitz determinants which appear naturally in the theory of spin corre- lations in a two-dimensional Ising lattice. This formula was first“ discussed by Szeg? [Comm. Säminaire Math. Univ. Lund, tome suppl. (1952) déödie M. Riesz, p. 228]. Perusal of the Szeg? paper shows that the problem was proposed to Szeg? by the Yale mathematician S. Kakutani, who apparently heard it from Onsager…Google Scholar
  31. 3.22
    T. Oguchi: J. Phys. Soc. Jpn. 6, 31 (1951)ADSMathSciNetCrossRefGoogle Scholar
  32. 3.23
    M.F. Sykes: J. Math. Phys. 2, 52 (1961)ADSMathSciNetCrossRefGoogle Scholar
  33. 3.24
    G.A. Baker, Jr.: Phys. Rev. 124, 768 (1961)ADSCrossRefGoogle Scholar
  34. 3.25
    E. Barouch, B. McCoy, T.T. Wu: Phys. Rev. Lett. 31, 1409 (1973)ADSCrossRefGoogle Scholar
  35. 3.26
    M. Plischke, D. Mattis: Phys. Rev. B2, 2660 (1970)ADSGoogle Scholar
  36. 3.27
    E. Barouch: Physica 1D, 333 (1980) Generalizations of the Lee-Yang methods [3.6] to other models have recently appeared, notably: M. Bander, C. Itzykson: Phys, Rev. B30 6485 (1984) for 0(N) spin models D. Kurtze, M. Fisher: J. Stat. Phys. 19 205 (1978) for spherical modelsGoogle Scholar
  37. 3.28
    R. Baxter, I. Enting: J. Phys. All, 2463 (1978)Google Scholar
  38. 3.29
    G. Wannier: Phys. Rev. 79, 357 (1950)MATHADSMathSciNetCrossRefGoogle Scholar
  39. 3.30
    T. Utiyama: Progr. Theor. Phys. 6, 907 (1951)MATHADSCrossRefGoogle Scholar
  40. 3.31
    M. Sykes, M. Fisher: Physica 28, 919, 939 (1962)ADSGoogle Scholar
  41. 3.32
    E. Lieb, D. Ruelle: J. Math. Phys. 13, 781 (1972)ADSMathSciNetCrossRefGoogle Scholar
  42. 3.33
    E. Müller-Hartmann, J. Zittartz: Z. Physik B27, 261 (1977)ADSGoogle Scholar
  43. 3.34
    J. Zittartz: Z. Physik B40, 233 (1980)ADSMathSciNetGoogle Scholar
  44. 3.35
    K.Y. Lin, F.Y. Wu: Z. Physik B33, 181 (1979)ADSGoogle Scholar
  45. 3.36
    A. Bienenstock: J. Appl. Phys. 37, 1459 (1966)ADSCrossRefGoogle Scholar
  46. 3.37
    M. Plischke, D.C. Mattis: Phys. Rev. A3, 2092 (1971)ADSGoogle Scholar
  47. 3.38
    H. Blöte, W. Huiskamp: Phys. Lett. A29, 304 (1969)ADSGoogle Scholar
  48. 3.39
    L. de Jongh, A. Miedema: Experiments on Simple Magnetic Model Systems, Adv. Phys. 23, 1–260 (1974)Google Scholar
  49. 3.40
    M..Sykes, J. Essam, D. Gaunt: J. Math. Phys. 6, 283 (1965)ADSMathSciNetCrossRefGoogle Scholar
  50. M. Sykes, D. Gaunt, J. Essam, D. Hunter: J. Math. Phys. 14. 1060 (1973)ADSCrossRefGoogle Scholar
  51. M. Sykes, D. Gaunt, S. Mattingly, J. Essam, C. Elliott: J. Math. Phys. 14, 1066 (1973)ADSCrossRefGoogle Scholar
  52. M. Sykes, D. Gaunt, J. Martin, S. Mattingly, J. Essam: J. Math. Phys. 14, 1071 (1973)ADSCrossRefGoogle Scholar
  53. M. Sykes, D. Gaunt, J. Essam, B. Heap, C. Elliott, S. Mattingly: J. Phys. A6, 1498 (1973)ADSGoogle Scholar
  54. M. Sykes, D. Gaunt, J. Essam, C. Elliott: J. Phys. A6, 1506 (1973)ADSGoogle Scholar
  55. D. Gaunt, M. Sykes: J. Phys. A6, 1517 (1973)ADSGoogle Scholar
  56. 3.41
    M. Sykes, D. Gaunt, P. Roberts, J. Wyles: J. Phys. A5, 624, 640 (1972)ADSGoogle Scholar
  57. 3.42
    M. Sykes, D. Hunter, D. McKenzie, B. Heap: J. Phys. A5, 667 (1972)ADSGoogle Scholar
  58. D. Gaunt, J. Guttmann: Asymptotic Analysis of Coefficients, in Phase Transitions and Critical Phenomena, Vol.3, ed. by C. Domb and M. Green (Academic, New York 1974 )Google Scholar
  59. 3.43
    C. Domb, M. Green (eds.): Phase Transitions and Critical Phenomena, Vol. 3 ( Academic, New York 1974 )Google Scholar
  60. 3.44
    M. Sykes et al.: J. Phys. A5, 640 (1972) Appendix 3.45 See the recent analysis and references in S. Jensen, 0. Mouritsen: J. Phys. A15 2631 (1982) or [3.43]Google Scholar
  61. 3.45
    D. Gaunt, J. Guttmann: Asymptotic Analysis of Coefficients, in Phase Transitions and Critical Phenomena, Vol.3, ed. by C. Domb and M. Green (Academic, New York 1974 )Google Scholar
  62. 3.46
    R.B. Griffiths: J. Math. Phys. 10, 1559 (1969)ADSCrossRefGoogle Scholar
  63. 3.47
    R.B. Griffiths: J. Math. Phys. 8, 478, 484 (1967)ADSCrossRefGoogle Scholar
  64. 3.48
    M. Blume: Phys. Rev. 141, 517 (1966)ADSCrossRefGoogle Scholar
  65. H.W. Capel: Physica 37 423(1967) and references therein (Blume-Capel model)Google Scholar
  66. H. Chen, P.M. Levy: Phys. Rev. B7, 4267 (1973)ADSGoogle Scholar
  67. D. Furman, S. Dattagupta, R.B. Griffiths: Phys. Rev. B15, 441 (1977)ADSGoogle Scholar
  68. E.K. Riedel, F.J. Wegner: Phys. Rev. B9, 294 (1974)ADSGoogle Scholar
  69. G.B. Taggart: Phys. Rev. B20, 3886 (1979)ADSGoogle Scholar
  70. 3.49
    When Pythagoras established the theorem of the square upon the hypothenuse he sacrificed 1000 oxen to Apollo. Since then, whenever anyone has had a new idea, oxen everywhere have trembledGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Daniel C. Mattis
    • 1
  1. 1.Physics DepartmentUniversity of UtahSalt Lake CityUSA

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