Quick Jump: Vol./No: Page:

Progress of Theoretical Physics
Supplement No.94 (1988) pp. 1-41
URL : http://ptp.ipap.jp/link?PTPS/94/1/
DOI : 10.1143/PTPS.94.1

From Solitons to Knots and Links

Miki Wadati and Yasuhiro Akutsu*

Institute of Physics, College of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153
*Institute of Physics, Kanagawa University, Yokoahama 221

(Received March 26, 1988)

Development in the theory of solvable (integrable) models is reviewed. It covers from basic knowledge on completely integrable systems to recent work by the authors. First, soliton theory is briefly summarized. Through the inverse scattering method and its quantum extension, a central concept, commuting transfer matrices, and a key relation, the Yang-Baxter relation, are introduced. Second, it is shown that there exists at least ∞× ∞ number of solvable models in two-dimensional statistical mechanics. Third, quantum spin chains corresponding to solvable statistical mechanical models are discussed. In particular, finite temperature extension of Baxter's formula is given. Fouth, a new approach is presented to the classification problem of knots and links. It is shown that link polynomial, topological invariant for knots and links, can be associated with any solvable model in statistical mechanics. In the presentation, universality of the soliton picture in field theory, spin systems and statistical mechanics is observed.


[Full Text PDF(3176K)] [Buy This Article]
[Table of Contents] [Next Article] Citation:

References:

  1. N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15 (1965), 240[APS].
    There are many excellent review papers and books. See, for instance,
    A. C. Scott, F. Y. F. Chu and D. W. McLaughlin, Proc. IEEE 61 (1973), 1443.
    R. K. Bullough and P. J. Caudrey (ed.), Solitons (Speinger-Verlag, Berlin, Heidelberg, 1978).
    A. R. Bishop and T. Schneider (ed.), Solitons and Condensed Matter Physics (Springer-Verlag, Berlin, Heidelberg, 1978).
    R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, London, 1982).
    F. Calogero and A. Degasperis, Spectral Transform and Solitons (North-Holland, Amsterdam, 1982).
  2. C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 19 (1967), 1095[APS].
  3. M. Wadati and M. Toda, J. Phys. Soc. Jpn. 32 (1972), 1403[IPAP].
  4. V. E. Zakharov and A. B. Shabat, Sov. Phys. -JETP 34 (1972), 62.
  5. M. Wadati, J. Phys. Soc. Jpn. 32 (1972), 1681[IPAP]; ibid. 34 (1973), 1289[IPAP].
  6. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Phys. Rev. Lett. 31 (1973), 125[APS].
  7. V. E. Zakharov and L. D. Faddeev, Funct. Anal. Appl. 5 (1972), 280.
  8. H. Flaschka and A. C. Newell, Lecture Notes in Phys. 38 (Springer-Verlag, Berlin, Heidelberg, 1975), p. 355.
  9. L. D. Faddeev, Sov. Sci. Rev. Math. Phys. C 1 (1980), 107.
  10. H. B. Thacker, Rev. Mod. Phys. 53 (1981), 253[APS].
  11. P. P. Kulish and E. K. Sklyanin, Lecture Notes in Phys. 151 (Springer-Verlag, Berlin, Heidelberg, 1982), p. 61.
  12. M. Wadati, in Dynamical Problems in Soliton Systems, ed. S. Takeno (Springer-Verlag, Berlin, Heidelberg, 1985), p. 68.
    M. Wadati, T. Konishi an A. Kuniba, in Quantum Field Theory, ed. F. Mancini (North-Holland, Amsterdam, 1986), p. 305.
  13. K. Sogo and M. Wadati, Prog. Theor. Phys. 68 (1982), 85[IPAP].
  14. M. Wadati, E. Olmedilla and Y. Akutsu, J. Phys. Soc. Jpn. 56 (1987), 1340[IPAP].
    E. Olmedilla, M. Wadati and Y. Akutsu, J. Phys. Soc. Jpn. 56 (1987), 2298[IPAP].
    E. Olmedilla and M. Wadati, J. Phys. Soc. Jpn. 56 (1987), 4274[IPAP].
    E. Olmedilla and M. Wadati, Phys. Rev. Lett. 60 (1988), 1595[APS].
  15. C. N. Yang, Phys. Rev. Lett. 19 (1967), 1312[APS].
  16. M. Karowski, H. J. Thun, T. T. Truong and P. H. Weisz, Phys. Lett. B 67 (1977), 321[CrossRef].
  17. A. B. Zamolodchikov and A. B. Zamolodchikov, Ann. of Phys. 120 (1979), 253[CrossRef].
  18. K. Sogo, M. Uchinami, A. Nakamura and M. Wadati, Prog. Theor. Phys. 66 (1981), 1284[IPAP].
  19. K. Sogo, M. Uchinami, Y. Akutsu and M. Wadati, Prog. Theor. Phys. 68 (1982), 508[IPAP].
  20. A. B. Zamolodchikov, Commun. Math. Phys. 69 (1979), 165[CrossRef].
  21. R. J. Baxter, Ann. of Phys. 70 (1972), 193[CrossRef].
  22. R. J. Baxter, Ann. of Phys. 70 (1972), 323[CrossRef].
  23. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
  24. H. B. Thacker, Physica D 18 (1986), 348[CrossRef].
  25. K. Sogo and M. Wadati, Prog. Theor. Phys. 69 (1983), 431[IPAP].
  26. Yu. G. Stroganov, Phys. Lett. A 74 (1979), 116[CrossRef].
  27. R. J. Baxter, J. of Phys. A 13 (1980), L61[IoP STACKS].
  28. L. J. Rogers, Proc. London Math. Soc. 25 (1884), 318.
    S. Ramanujan, Proc. Cambridge Phillos. Soc. 19 (1919), 214.
  29. B. Gordon, Amer. J. Math. 83 (1961), 393.
  30. G. E. Andrews, The Theory of Partitions (Addison-Wesley, London, 1976).
  31. A. Kuniba, Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 1092[IPAP].
  32. R. J. Baxter and G. E. Andrews, J. Stat. Phys. 44 (1986), 249.
    G. E. Andrews and R. J. Baxter, J. Stat. Phys. 44 (1986), 731.
  33. A. Kuniba, Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 2166[IPAP]; Phys. Lett. A 116 (1986), 382[Elsevier].
  34. A. Kuniba, Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 3338[IPAP]; Phys. Lett. A 117 (1986), 358[Elsevier].
  35. G. E. Andrews, R. J. Baxter and P. J. Forrester, J. Stat. Phys. 35 (1984), 193.
  36. A. Kuniba, Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 2605[IPAP].
  37. Y. Akutsu, A. Kuniba and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 2907[IPAP].
  38. E. Date, M. Jimbo, T. Miwa and M. Okado, Lett. Math. Phys. 12 (1986), 209.
  39. E. Date, M. Jimbo, T. Miwa and M. Okado, Phys. Rev. B 35 (1987), 2105[APS].
  40. Y. Akutsu, A. Kuniba and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 1466[IPAP].
  41. Y. Akutsu, A. Kuniba and M. Wadati, J. Phys. Soc. Jpn. 55 (1986), 1880[IPAP].
  42. M. Wadati and Y. Akutsu, Exactly Solvable Models in Statistical Physics, Springer Series in Nonlinear Dynamics, ed. M. Lakshmanan (Springer-Verlag, Berlin Heidelberg, 1988), p. 282.
  43. Y. Akutsu, T. Deguchi and M. Wadati, J. Phys. Soc. Jpn. 57 (1988), 1173[IPAP].
  44. A. Kuniba and T. Yajima, J. Stat. Phys. (1988), to appear.
  45. H. N. V. Temperley and E. H. Lieb, Proc. Soc. London A 322 (1971), 251.
  46. V. Pasquier, J. of Phys. A 20 (1986), L217[IoP STACKS]; ibid. 20 (1986), L221[IoP STACKS].
  47. H. Au-Yang, B. M. McCoy, J. H. H. Perk, S. Tang and M. L. Yan, Phys. Lett. A 123 (1987), 219[CrossRef].
  48. V. V. Bazhanov and N. Yu. Reshetikhin, preprint.
  49. S. Katsura, Phys. Rev. 127 (1962), 1508[APS].
  50. P. Pfeuty, Ann. of Phys. 57 (1970), 79[CrossRef].
  51. M. Suzuki, Phys. Rev. B 31 (1985), 2957[APS].
  52. M. Suzuki and M. Inoue, Prog. Theor. Phys. 78 (1987), 787[IPAP].
  53. R. Z. Bariev, Theor. Math. Phys. 49 (1982), 1021.
  54. T. T. Troung and K. D. Schotte, Nucl. Phys. B 220 (1983), 77[CrossRef].
  55. D. B. Abraham, Phys. Rev. Lett. 51 (1983), 1279[APS].
  56. M. Barma and B. S. Shastry, Phys. Rev. B 18 (1978), 3351[APS].
  57. T. Koma, Prog. Theor. Phys. 78 (1987), 1213[IPAP].
  58. M. Suzuki, Prog. Theor. Phys. 56 (1976), 1454[IPAP]; J. Stat. Phys. 43 (1986), 883.
  59. M. Suzuki, Commun. Math. Phys. 51 (1976), 183[CrossRef].
  60. H. F. Trotter, Proc. Amer. Math. Soc. 10 (1959), 545.
  61. T. Asano, Phys. Rev. Lett. 24 (1970), 1409[APS]; J. Phys. Soc. Jpn. 29 (1970), 350[IPAP].
  62. R. B. Griffiths, J. Math. Phys. 5 (1964), 1215[AIP Scitation].
  63. E. H. Lieb, T. D. Schultz and D. C. Mattis, Ann. of Phys. 16 (1961), 407[CrossRef].
  64. B. McCoy and T. T. Wu, Nuovo Cim. B LVI (1968), 311.
    B. Sutherland, J. Math. Phys. 11 (1970), 3183[AIP Scitation].
    E. Barouch, Phase Transitions an Critical Phenomena, ed. C. Domb and M. S. Green (Academic Press, New York, 1972), vol. 1, p. 366-373.
  65. M. Inoue and M. Suzuki, Prog. Theor. Phys. 79 (1988), 645[IPAP].
  66. H. M. Babujan, Nucl. Phys. B 215 (1983), 317[CrossRef].
  67. B. S. Shastry, Phys. Rev. Lett. 56 (1986), 1529[APS]; ibid. 56 (1986), 2453[APS].
  68. H. J. de Vega and F. Woynarovitch, Nucl. Phys. B 251 (1985), 437[Elsevier].
  69. H. Tsuru and M. Wadati, Biopolymers 25 (1986), 2083.
    M. Wadati and H. Tsuru, Physica D 21 (1986), 213[CrossRef].
  70. J. S. Birman, Braids, Links, and Mapping Class Groups (Princeton University Press, Princeton, 1974).
  71. D. Rolfsen, Knots and Links (Publish or Perish, Inc., Berkeley, 1976).
  72. E. Artin, Ann. of Math. 48 (1947), 101.
  73. J. W. Alexander, Proc. Nat. Acad. 9 (1923), 93.
  74. A. A. Markov, Recueil Math. Moscow (1935), 73.
  75. Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 56 (1987), 839[IPAP].
  76. Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 56 (1987), 3039[IPAP].
  77. Y. Akutsu, T. Deguchi and M. Wadati, J. Phys. Soc. Jpn. 56 (1987), 3464[IPAP].
  78. T. Deguchi, Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 56 (1988), 757[IPAP].
  79. T. Deguchi, M. Wadati and Y. Akutsu, J. Phys. Soc. Jpn. 57 (1988), 1905[IPAP].
  80. Y. Akutsu and M. Wadati, Commun. Math. Phys. (in press).
  81. V. F. R. Jones, Bull. Am. Math. Soc. 12 (1985), 103.
  82. K. Sogo, Y. Akutsu and T. Abe, Prog. Theor. Phys. 70 (1983), 730[IPAP]; ibid. 70 (1983), 739[IPAP].
  83. A. B. Zamolodchikov and V. A. Fateev, Sov. J. Nucl. Phys. 32 (1980), 293.
  84. R. T. Powers, Ann. Math. 86 (1967), 138.
  85. M. Pimsner and S. Popa, unpublished.
  86. J. S. Birman, Invent. Math. 81 (1985), 287.
  87. N. Bourbaki, Groupes et algebras de Lie (Hermann, Paris, 1968), chap. 4.
  88. O. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millet and A. Ocneanu, Bull. Am. Math. Soc. 12 (1985), 239.
  89. L. H. Kauffman, Topology 26 (1987), 395.
  90. L. H. Kauffman, On Knot (Princeton University Press, Princeton, 1987).
  91. J. S. Birman and H. Wenzl, Preprint (1987).
  92. J. Murakami, Preprint (1987).
  93. A. A. Belavin, A. B. Zamolodchikov and A. M. Polyakov, J. Stat. Phys. 34 (1984), 763.
    A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241 (1984), 333[CrossRef].
  94. D. Friedan, Z. Qiu an S. Shenker, Phys. Rev. Lett. 52 (1984), 1575[APS].
  95. E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Nucl. Phys. B 290 (1987), 231[CrossRef].
  96. A. Kuniba and T. Yajima, J. of Phys. A 21 (1988), 519[IoP STACKS].

Citing Article(s) :

  1. Journal of the Physical Society of Japan 57 (1988) pp. 2921-2935 :
    Link Polynomials Constructed from Solvable Models in Statistical Mechanics
    Tetsuo Deguchi, Miki Wadati and Yasuhiro Akutsu
  2. Journal of the Physical Society of Japan 58 (1989) pp. 1153-1161 :
    Knot Theory and Conformal Field Theory: Reduction Relations for Braid Generators
    Miki Wadati, Yasuhiko Yamada and Tetsuo Deguchi
  3. Journal of the Physical Society of Japan 58 (1989) pp. 2219-2222 :
    Exact Landau Free-Energy of Solvable N-State Vertex Model
    Yasuhiro Akutsu
  4. Journal of the Physical Society of Japan 58 (1989) pp. 3111-3122 :
    Thermodynamic Quantities in the N-State Vertex (Sogo-Akutsu-Abe) Model
    Junji Suzuki
  5. Journal of the Physical Society of Japan 59 (1990) pp. 2667-2680 :
    A New Approach to Quantum Spin Chains at Finite Temperature
    Junji Suzuki, Yasuhiro Akutsu and Miki Wadati
  6. Journal of the Physical Society of Japan 63 (1994) pp. 3951-3961 :
    Correlation Length for the XXZ Spin Chain at Finite Temperature
    Hideyuki Mizuta, Taro Nagao and Miki Wadati
  7. Journal of the Physical Society of Japan 66 (1997) pp. 988-994 :
    Low–Temperature Expansions and the Euler–Maclaurin Formula
    Hideyuki Mizuta and Miki Wadati
  8. Journal of the Physical Society of Japan 66 (1997) pp. 3741-3750 :
    Low–Temperature Expansions of the XXZ Heisenberg Model
    Hideyuki Mizuta and Miki Wadati

Quick Search:

Author: Title/Abstract: Vol./No:
[PTP HOME] [PTP ONLINE] [SEARCH] [REGISTRATION]

Copyright (c) Progress of Theoretical Physics 1988 All rights reserved.
ptp@yukawa.kyoto-u.ac.jp