Melting in Two Dimensions

  • J. M. Kosterlitz
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 77)


In these lectures I will try to review the present state of our understanding of the various phases which can occur in thin films which crystallise at low temperatures. Most experimental investigations are on systems of gases adsorbed on graphite in which the substrate plays an important role but often obscures the theoretically important effects. There is some recent work on freely suspended films of liquid crystal in which the problem of substrate is minimised but other complications arise. Perhaps the system closest to the theoretical ideal is electrons trapped on the surface of helium but the experimental probes of this are rather restricted.


Renormalisation Group Recursion Relation Triangular Lattice Reciprocal Lattice Vector Dislocation Theory 
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  1. 1.
    D.R. Nelson, Phys. Rev., B18 2318 (1978).ADSGoogle Scholar
  2. 2.
    D.R. Nelson and B.I. Halperin, Phys. Rev. B19 2457 (1979).ADSGoogle Scholar
  3. 3.
    D.R. Nelson and B.I. Halperin, Phys. Rev. B21 5312 (1980).ADSGoogle Scholar
  4. 4.
    S. Ostlund and B.I. Halperin, Phys. Rev. B23 335 (1981).MathSciNetADSGoogle Scholar
  5. 5.
    A. Zippelius, B.I. Halperin and D.R. Nelson, Phys. Rev. B22 2514 (1980).ADSGoogle Scholar
  6. 6.
    J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 1181 (1973);ADSGoogle Scholar
  7. 6a.
    J.M. Kosterlitz and D.J. Thouless, Prog. Low Temp. Phys. 7B 371 (1978).CrossRefGoogle Scholar
  8. 7. a)
    See for example: Phase Transitions in Surface Films (eds. J.G. Dash and J. Ruvalds) Plenum, New York) 1980.Google Scholar
  9. 7. b)
    Ordering in Strongly Fluctuating Condensed Matter Systems (ed. T. Riste) (Plenum, New York) 1980.Google Scholar
  10. 7. c)
    Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, Enschede, Netherlands -to be published.Google Scholar
  11. 7. d)
    Physics of Low Dimensional Systems, Proceedings of the Kyoto Summer Institute (eds. Y. Nagaoka and S. Hikami) Publication Office Progress Theoretical Physics, Kyoto) 1979.Google Scholar
  12. 8.
    F. Bloch, Z. Phys. 61 206 (1930);ADSMATHCrossRefGoogle Scholar
  13. 8a.
    R.E. Peierls, Ann. Inst. Henri Poincare 5 177 (1935):MathSciNetMATHGoogle Scholar
  14. 8b.
    L.D. Landau, Phys. Z. Sowjetunion II 26 (1937).Google Scholar
  15. 9.
    N.D. Mermin and M. Wagner, Phys. Rev. Lett. 17 1133 (1966);ADSCrossRefGoogle Scholar
  16. 9a.
    N.D. Mermin, J. Math Phys. 8 1061 (1967);ADSCrossRefGoogle Scholar
  17. 9b.
    N.D. Mermin, Phys. Rev. 176. 250 (1968);ADSCrossRefGoogle Scholar
  18. 9c.
    P.C. Hohenberg, Phys. Rev. 158 383 (1967).ADSCrossRefGoogle Scholar
  19. 10.
    H.E. Stanley and T.A. Kaplan, Phys. Rev. Lett. 17 913 (1966);MathSciNetADSCrossRefGoogle Scholar
  20. 10a.
    H.E. Stanley, Phys. Rev. 164 709 (1967);ADSCrossRefGoogle Scholar
  21. 10b.
    H.E. Stanley, Phys. Rev. Lett. 20 150 (1968).ADSCrossRefGoogle Scholar
  22. 11.
    V.L. Berezinskii, Sov. Phys. JETP 32 493 (1970);MathSciNetADSGoogle Scholar
  23. 11a.
    V.L. Berezinskii, Sov. Phys. JETP 34 610 (1972)ADSGoogle Scholar
  24. 12.
    F. Wegner, Z. Phys. 206 465 (1967);ADSCrossRefGoogle Scholar
  25. 12a.
    B. Jancoviei, Phys.Rev. Lett. 19 20 (1967);ADSCrossRefGoogle Scholar
  26. 12b.
    H.J. Mikeska and H. Schmidt, J. Low Temp. Phys. 2 371 (1970);ADSCrossRefGoogle Scholar
  27. 12c.
    Y. Imry and L. Gunther, Phys. Rev. B3 3939 (1971).ADSGoogle Scholar
  28. 13.
    L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Pergamon Press, N.Y.)1970.Google Scholar
  29. 14.
    F.R.N. Nabarro, Theory of Dislocations (Clarendon Press, Oxford) 1967.Google Scholar
  30. 15.
    See for example J.G. Dash, Films on Solid Surfaces (Academic, N.Y.) 1975.Google Scholar
  31. 16.
    For recent developments and further references: Proceedings of International Conference on Ordering in Two Dimensions 1980 (North Holland, N.Y.)Google Scholar
  32. 17.
    C.C. Grimes and G. Adams, Phys. Rev. Lett. 42 795 (1979).ADSCrossRefGoogle Scholar
  33. 18.
    D.S. Fisher, B.I. Halperin and P.M. Platzman, Phys. Rev. Lett. 42 798 (1979).ADSCrossRefGoogle Scholar
  34. 19.
    C.Y. Young, R. Pindak, N.A. Clark and R.B. Meyer, Phys. Rev. Lett. 40 773 (1978).ADSCrossRefGoogle Scholar
  35. 20.
    C. Rosenblatt, R. Pindak, N.A. Clark and R.B. Meyer, Phys. Rev. Lett. 42 1220 (1979).ADSCrossRefGoogle Scholar
  36. 21.
    D.E. Moncton and R. Pindak, Phys. Rev. Lett. 43 70 (1979).ADSCrossRefGoogle Scholar
  37. 22.
    R. Pindak, D.J. Bishop and W.O. Sprenger, Phys. Rev. Lett. 44 1461 (1980).ADSCrossRefGoogle Scholar
  38. 23.
    J.C. Tarczon and K. Miyano, Phys. Rev. Lett. 46 119 (1981).ADSCrossRefGoogle Scholar
  39. 24.
    D.J. Bishop and J. Reppy, Phys. Rev. Lett. 40 1727 (1978).ADSCrossRefGoogle Scholar
  40. 25.
    P.S. Pershan, J. de Physique Colloque 40 C3 (1979).Google Scholar
  41. 26.
    The sign of the angular term in eq. (2.4) is important and is given wrongly in many textbooks and papers.Google Scholar
  42. 27.
    J. Tobochnik and G.V. Chester, Cornell preprint (1980).Google Scholar
  43. 28.
    A.N. Berker and D.R. Nelson, Phys. Rev. B19 2488 (1979);ADSGoogle Scholar
  44. 28a.
    J. Cardy and D.J. Scalapino Phys.Rev. B19 1428 (1979);ADSGoogle Scholar
  45. 29.
    S. Ostlund, Harvard preprint 1980.Google Scholar
  46. 30.
    K.G. Wilson and J. Kogut, Phys. Rept. C12 75 (1974).ADSCrossRefGoogle Scholar
  47. 31.
    J.M. Kosterlitz, J. Phys. C7 1046 (1974).ADSGoogle Scholar
  48. 32.
    A.P. Young, Phys. Rev. B19 1855 (1979).ADSGoogle Scholar
  49. 33.
    D.J. Amit, Y.Y. Goldshmidt and G. Grinstein, J. Phys. A13 1507 (1980).Google Scholar
  50. 34.
    J.V. Jose, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Phys. Rev. B16 1217 (1977).ADSGoogle Scholar
  51. 35.
    Note that μ(T) + λ(T) is not equal to the bulk modulus B except at T = 0.Google Scholar
  52. 36.
    For a recent review see J. Villain in ref. 7B p.221 For recent developments see ref. 16.Google Scholar
  53. 37.
    S.N. Coppersmith, D.S. Fisher, B.I. Halperin, P.A. Lee and W.F. Brinkman, Phys. Rev. Lett. 46 (1981).Google Scholar
  54. 38.
    A.D. Novaco and J.P. McTague, Phys. Rev. Lett. 38 1286 (1977).ADSCrossRefGoogle Scholar
  55. 39.
    W.T. Reed and W. Shockley, Phys. Rev. 78 275 (1950).ADSCrossRefGoogle Scholar
  56. 40.
    D.S. Fisher, B.I. Halperin and R. Morf, Phys. Rev. B20 4692 (1979).ADSGoogle Scholar
  57. 41.
    S.T. Chui preprint 1981.Google Scholar
  58. 42.
    F.F. Abraham, Phys. Rev. Lett. 44 463 (1980).MathSciNetADSCrossRefGoogle Scholar
  59. 43.
    S. Toxvaerd, Phys. Rev. Lett. 44 1002 (1980).ADSCrossRefGoogle Scholar
  60. 44.
    D. Frenkel and J.P. McTague, Phys. Rev. Lett. 42 1632 (1979).ADSCrossRefGoogle Scholar
  61. 45.
    R.G. Gann, S. Chakravarty and G.V. Chester, Phys. Rev. B20 326 (1979).ADSGoogle Scholar
  62. 46.
    R. Morf, Phys. Rev. Lett. 43 931 (1979).ADSCrossRefGoogle Scholar
  63. 47.
    D.J. Thouless, J. Phys. C11, L 189 (1978).MathSciNetADSGoogle Scholar
  64. 47a.
    R.K. Kalia, P. Vashista and S.W. de Leeuw, Phys. Rev. B to be published.Google Scholar
  65. 48.
    J.P. McTague, D. Frenkel and M.P. Allen in ref. 16.Google Scholar
  66. 49.
    J.D. Weeks Private CommunicationGoogle Scholar
  67. 50.
    J. Toner and D.R. Nelson, Phys. Rev. B23 316 (1981).MathSciNetADSGoogle Scholar
  68. 51.
    R.A. Pelcovits and B.I. Halperin, Phys. Rev. B19 4614 (1979).ADSGoogle Scholar
  69. 52.
    D.R. Nelson and R.A. Pelcovits, Phys. Rev. B16 2191 (1977).ADSGoogle Scholar

Copyright information

© Plenum Press, New York 1982

Authors and Affiliations

  • J. M. Kosterlitz
    • 1
  1. 1.Department of Mathematical PhysicsBirmingham UniversityBirminghamEngland

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