Baxterization, dynamical systems, and the symmetries of integrability

  • C. -M. Viallet
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 436)


We resolve the ‘baxterization’ problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, ...) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, beyond the narrow context of strict integrability. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the growth of the complexity of iterations.


Projective Space Coxeter Group Discrete Dynamical System Vertex Model Infinite Group 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M.P. Bellon, J-M. Maillard, and C-M. Viallet, Integrable Coxeter Groups. Physics Letters A 159 (1991), pp. 221–232.Google Scholar
  2. [2]
    M.P. Bellon, J-M. Maillard, and C-M. Viallet, Higher dimensional mappings. Physics Letters A 159 (1991), pp. 233–244.Google Scholar
  3. [3]
    M.P. Bellon, J-M. Maillard, and C-M. Viallet, Infinite Discrete Symmetry Group for the Yang-Baxter Equations: Spin models. Physics Letters A 157 (1991), pp. 343–353.Google Scholar
  4. [4]
    M.P. Bellon, J-M. Maillard, and C-M. Viallet, Infinite Discrete Symmetry Group for the Yang-Baxter Equations: Vertex Models. Phys. Lett. B 260 (1991), pp. 87–100.Google Scholar
  5. [5]
    M.P. Bellon, J-M. Maillard, and C-M. Viallet, Rational Mappings, Arborescent Iterations, and the Symmetries of Integrability. Physical Review Letters 67 (1991), pp. 1373–1376.Google Scholar
  6. [6]
    M.P. Bellon, J-M. Maillard, and C-M. Viallet, Quasi integrability of the sixteen-vertex model. Phys. Lett. B 281 (1992), pp. 315–319.Google Scholar
  7. [7]
    M.P. Bellon, S. Boukraa, J-M. Maillard, and C-M. Viallet, Towards threedimensional Bethe Ansatz. Phys. Lett. B 314 (1993), pp. 79–88.Google Scholar
  8. [8]
    M.P. Bellon, J-M. Maillard, G. Rollet, and C-M. Viallet, Deformations of dynamics associated to the chiral Potts model. Int. J. Mod. Phys. B 6 (1992), pp. 3575–3584.Google Scholar
  9. [9]
    G. Falqui and C.-M. Viallet, Singularity, complexity, and quasi-integrability of rational mappings. Comm. Math. Phys. 154 (1993), pp. 111–125.Google Scholar
  10. [10]
    L. Onsager, Crystal statistics: I. A two-dimensional model with an orderdisorder transition. Phys. Rev 65 (1944), pp. 117–149.Google Scholar
  11. [11]
    J.B. McGuire, Studies of exactly solvable one-dimensional N-body problems. J. Math. Phys. 5 (1964), pp. 622–636.Google Scholar
  12. [12]
    R.J. Baxter. Exactly solved models in statistical mechanics. London Acad. Press, (1981).Google Scholar
  13. [13]
    C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19(23) (1967), pp. 1312–1315.Google Scholar
  14. [14]
    L.A. Takhtajan and L.D. Faddeev, The quantum inverse problem and the XYZ Heisenberg model. Russian Math. Surveys 34(5) (1979), pp. 11–68.Google Scholar
  15. [15]
    L.D. Faddeev, E.K. Sklyanin, and L.A. Takhtajan, The quantum inverse problem method. Theor. Math. Phys 40 (1980), p. 688.Google Scholar
  16. [16]
    E.K. Sklyanin, Quantum version of the method of inverse scattering. transl. of Zap. Nauch. Sem. LOMI Steklov 95 (1980), pp. 55–128.Google Scholar
  17. [17]
    P.P. Kulish and E.K. Sklyanin. volume 151 of Lecture Notes in Physics, pages 61–119, (1982).Google Scholar
  18. [18]
    L.D. Faddeev. Integrable models in 1 + 1 dimensional quantum field theory. In Les Houches Lectures (1982), Amsterdam, (1984). Elsevier.Google Scholar
  19. [19]
    M. Gaudin. La fonction d'onde de Bethe. Collection du C.E.A. Série Scientifique. Masson, Paris, (1983).Google Scholar
  20. [20]
    E.K. Sklyanin. Quantum inverse scattering method. selected topics. In Quantum Groups and Quantum Integrable Systems, Singapore, (1992). World Scientific. Proceedings of the Nankai Lectures 1991 (and preprint hep-th-9211111).Google Scholar
  21. [21]
    H.S.M. Coxeter and W.O.J. Moser. Generators and relations for discrete groups. Springer Verlag, second edition, (1965).Google Scholar
  22. [22]
    V.F.R. Jones, Baxterization. Int. J. Mod. Phys. B 4 (1990), pp. 701–713. proc. of ‘Yang-Baxter equations, conformal invariance and integrability in statistical mechanics and field theory', Canberra, 1989.Google Scholar
  23. [23]
    R.J. Baxter, Eight-vertex model in lattice statistics. Phys. Rev. Lett. 26(14) (1971), pp. 832–833.Google Scholar
  24. [24]
    R.J. Baxter, One dimensional anisotropic Heisenberg chain. Phys. Rev. Lett. 26(14) (1971), p. 834.Google Scholar
  25. [25]
    A.B. Zamolodchikov, Tetrahedron equations and the relativistic S-matrix of straight-strings in 2+1 dimensions. Comm. Math. Phys. 79 (1981), p. 489.Google Scholar
  26. [26]
    M. T. Jaekel and J. M. Maillard, Symmetry relations in exactly soluble models. J. Phys. A15 (1982), pp. 1309–1325.Google Scholar
  27. [27]
    J.M. Maillet and F. Nijhoff, Integrability for multidimensional lattice models. Phys. Lett. B 224 (1989), pp. 389–396.Google Scholar
  28. [28]
    M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Scient. Ec. Norm. Sup. 4e série t.3 (1971), pp. 507–588.Google Scholar
  29. [29]
    H. Jung, Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184 (1942), pp. 161–172.Google Scholar
  30. [30]
    S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms. Ergod. Theory Dyn. Systems 9 (1989), pp. 67–99.Google Scholar
  31. [31]
    C. Fan and F.Y. Wu, General lattice statistical model of phase transition. Phys. Rev. B 2 (1970), pp. 723–733.Google Scholar
  32. [32]
    B.U. Felderhof, Diagonalization of the transfer matrix of the free-fermion model. II. Physica 66 (1973), pp. 279–297.Google Scholar
  33. [33]
    I.M. Krichever, Baxter's equations and algebraic geometry. Funct. Anal. and its Appl. 15 (1981), pp. 92–103.Google Scholar
  34. [34]
    H. Au-Yang, B.M. Mc Coy, J.H.H. Perk, S. Tang, and M.L. Yan, Commuting transfer matrices in the chiral Potts models: solutions of the star-triangle equations with genus ⩾ 1. Phys. Lett. A123 (1987), p. 219.Google Scholar
  35. [35]
    R.J. Baxter, J.H.H. Perk, and H. Au-Yang, New solutions of the startriangle relations for the chiral Potts model. Phys. Lett. A128 (1988), p. 138.Google Scholar
  36. [36]
    D. Hansel and J. M. Maillard, Symmetries of models with genus > 1. Phys. Lett. A 133 (1988), p. 11.Google Scholar
  37. [37]
    F. Jaeger, Strongly regular graphs and spin models for the Kaufman polynomial. Geometrige Ded. 44 (1992), pp. 23–52.Google Scholar
  38. [38]
    P. de la Harpe, Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model. Pacific J. Math. 162(1) (1994), pp. 57–96.Google Scholar
  39. [39]
    V.F.R Jones, On a certain value of the Kauffman polynomial. Comm. Math. Phys. 125 (1989), pp. 459–467.Google Scholar
  40. [40]
    R.J. Baxter and P.A. Pearce, Hard hexagons: interfacial tension and correlation length. J. Phys. A(15) (1982), pp. 897–910.Google Scholar
  41. [41]
    J. Avan, M. Talon, J.M. Maillard, and C.M. Viallet, New local relations for lattice models. Int. J. Mod. Phys. B 4 (1990), pp. 1895–1912.Google Scholar
  42. [42]
    H. Poincaré. Oeuvres. Tomes I-XI. Gauthier-Villars, Paris, (1952).Google Scholar
  43. [43]
    H. Poincaré. Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris, (1892).Google Scholar
  44. [44]
    G. D. Birkhoff, Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18 (1917), pp. 199–300.Google Scholar
  45. [45]
    A.S. Wightman, The mechanics of stochasticity in classical dynamical systems. Perspectives in Statistical Physics (1981), pp. 343–363. Reprinted in “Hamiltonian dynamical systems”, R.S. MacKay and J.D. Meiss editors, Adam Hilger (1987).Google Scholar
  46. [46]
    M. Henon, A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50 (1976), pp. 69–77.Google Scholar
  47. [47]
    C.L. Siegel, Iteration of analytic functions. Ann. Math. 43 (1942), pp. 807–812.Google Scholar
  48. [48]
    J.-C. Yoccoz, Conjugaison diférentiable des diféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sc. E.N.S. 4eme série t. 17 (1984), pp. 333–359.Google Scholar
  49. [49]
    G.R.W. Quispel, J.A.G. Roberts, and C.J. Thompson, Integrable Mappings and Soliton Equations. Phys. Lett. A 126 (1988), p. 419.Google Scholar
  50. [50]
    G.R.W. Quispel, J.A.G. Roberts, and C.J. Thompson, Integrable Mappings and Soliton Equations II. Physica D34 (1989), pp. 183–192.Google Scholar
  51. [51]
    J. Moser and A.P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Math. Phys. 139 (1991), pp. 217–243.Google Scholar
  52. [52]
    B. Grammaticos, A. Ramani, and V. Papageorgiou, Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67 (1991), pp. 1825–1827.Google Scholar
  53. [53]
    V.G. Papageorgiou, F.W. Nijhoff, and H.W. Capel, Integrable mappings and nonlinear integrable lattice equations. Phys. Lett. A147 (1990), pp. 106–114.Google Scholar
  54. [54]
    O. Ragnisco. Restricted flows of toda hierarchy as integrable maps. In Proceedings of the XIX LC.G.T.M.P, (1992).Google Scholar
  55. [55]
    Takuya Ikuta. Non-existence of spin models corresponding to non symmetric association schemes of class 2 on 4m + 2 vertices with m ≥ 1. preprint.Google Scholar
  56. [56]
    S. Boukraa, J-M. Maillard, and G. Rollet. Integrable mappings and polynomial growth. LPTHE preprint 93-26, to appear in Physica A.Google Scholar
  57. [57]
    M. Henon, Numerical study of quadratic area preserving maps. Q. J. Appl. Math. 27 (1969), pp. 291–312.Google Scholar
  58. [58]
    M.C. Gutzwiller. Chaos in Classical and Quantum Mechanics. Spinger Verlag, New-York, Berlin, Heidelberg, (1991). Interdisciplinary Applied Mathematics.Google Scholar
  59. [59]
    V.I. Arnold, Dynamics of complexity of intersections. Bol. Soc. Bras. Mat. 21 (1990), pp. 1–10.Google Scholar
  60. [60]
    A.P. Veselov, Growth and Integrability in the Dynamics of Mappings. Comm. Math. Phys. 145 (1992), pp. 181–193.Google Scholar
  61. [61]
    A. Neumaier, Duality in coherent configurations. Combinatorica 9(1) (1989), pp. 59–67.Google Scholar
  62. [62]
    M. Bellon. Addition on curves and complexity of quasi-integrable rational mappings. in preparation.Google Scholar
  63. [63]
    S. Boukraa, J-M. Maillard, and G. Rollet. Determinantal identities on integrable mappings. LPTHE preprint 93-25.Google Scholar
  64. [64]
    R.J. Baxter, The Inversion Relation Method for Some Two-dimensional Exactly Solved Models in Lattice Statistics. J. Stat. Phys. 28 (1982), pp. 1–41.Google Scholar
  65. [65]
    M. T. Jaekel and J. M. Maillard, Inverse functional relations on the Potts model. J. Phys. A15 (1982), pp. 2241–2257.Google Scholar
  66. [66]
    D. Hansel, J.M. Maillard, J. Oitmaa, and M.J. Velgakis, Analytical properties of the anisotropic cubic Ising model. J. Stat. Phys. 48 (1987), pp. 69–80.Google Scholar
  67. [67]
    R.J. Baxter, Solvable eight-vertex model on an arbitrary planar lattice. Phil. Trans. R. Soc. London 289 (1978), p. 315.Google Scholar
  68. [68]
    E.H. Lieb, Exact solution of the F model of an antiferroelectric. Phys. Rev. Lett. 18(24) (1967), pp. 1046–1048.Google Scholar
  69. [69]
    E.H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18(17) (1967), pp. 692–694.Google Scholar
  70. [70]
    V.E. Zakharov and L.D. Faddeev, The Korteweg-de Vries equation, a completely integrable hamiltonian system. Funct. Anal. and its Appl. 5 (1971), pp. 280–287.Google Scholar
  71. [71]
    L.D. Faddeev and L.A. Takhtajan. Hamiltonian methods in the theory of solitons. Springer Verlag, Heidelberg, (1986).Google Scholar
  72. [72]
    L.H. Kauffman, State models and the Jones polynomial. Topology 26 (1987), pp. 395–407.Google Scholar
  73. [73]
    V.G. Turaev, The Yang-Baxter equation and invariants of links. Invent. Math. 92 (1988), pp. 527–553.Google Scholar
  74. [74]
    V.F.R. Jones, On knots invariants related to some statistical mechanical models. Pacific J. Math. 137 (1989), pp. 311–334.Google Scholar
  75. [75]
    F.Y. Wu, Knot theory and statistical mechanics. Rev. Mod. Phys. 64(4) (1992), pp. 1099–1131.Google Scholar
  76. [76]
    M. Jimbo, A q-difference analogue of U(G) and the Yang-Baxter equation. Lett. Math. Phys. 10 (1985), p. 63.Google Scholar
  77. [77]
    V.G. Drinfel'd. Quantum groups. In Proceedings of the International Congress of Mathematicians. Berkeley, (1986).Google Scholar
  78. [78]
    S.L. Woronowicz, Twisted SU(2) Group. An Example of Noncommutative Differential Calculus. Publ. Res. Inst. Math. Sci. 23 (1987), pp. 117–181.Google Scholar
  79. [79]
    I.G. Korepanov. Vacuum Curves, Classical Integrable Systems in Discrete Space-Time and Statistical Physics. preprint hep-th 9312197, (1993).Google Scholar
  80. [80]
    H. Bethe, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71 (1931), pp. 205–226.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • C. -M. Viallet
    • 1
  1. 1.Laboratoire de Physique Théorique et des Hautes Energies, Centre National de la Recherche ScientifiqueUniversité de Paris 6- Paris 7Parisz Cedex 05France

Personalised recommendations