Quantum integrable models and discrete classical Hirota equations

Communications in Mathematical Physics

Volumn 188, Issue 2, 1997, Pages 267-304

  Krichever, I ^a,b   Lipan, O ^c   Wiegmann, P ^c   Zabrodin, A ^d,e  

^a Columbia University ^*  (United States)

^b LANDAU INSTITUTE FOR THEORETICAL PHYSICS   (Russian Federation)

^c UNIVERSITY OF CHICAGO   (United States)

^d SEMENOV INSTITUTE OF CHEMICAL PHYSICS   (Russian Federation)

^e INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041048843     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050165     Document Type: Article

References (62)

1. Airault, H., McKean, H., Moser, J.: Rational and elliptic solutions of the KdV equation and related many-body problem. Comm. Pure and Appl. Math. 30, 95-125 (1977)
2. Au-Yang, H., Perk, J.H.H.: Toda lattice equation and wronskians in the 2d lsing model. Physica 18D, 365-366 (1986)
3. Baxter, R.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193-228 (1972)
4. Baxter, R.: Exactly solved models in statistical mechanics. New York-London: Academic Press, 1982
5. Bazhanov, V., Reshetikhin, N.: Critical RSOS models and conformal field theory. Int. J. Mod. Phys. A4, 115-142 (1989)
6. Bazhanov, V., Reshetikhin, N.: Restricted solid on solid models connected with simply laced algebras and conformal field theory. J. Phys. A23, 1477-1492 (1990)
7. Bazhanov, V., Lukyanov, S., Zamolodchikov, A.: Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz. Preprint CLNS 94/1316, RU-94-98, hep-th/9412229
8. Bilal, A., Gervais, J.-L.: Extended c = ∞ conformal systems from classical Toda field theories. Nucl. Phys. B314, 646-686 (1989)
9. Bobenko, A., Kutz, N., Pinkall, V.: The discrete quantum pendulum. Phys. Lett. A177, 399-404 (1993)
10. Bogoliubov, N.M., Izergin, A.G., Korepin, V.E.: Quantum inverse scattering method and correlation functions. Cambridge: Cambridge University Press, 1993
11. Chudnovsky, D.V., Chudnovsky, G.V.: Pole expansions of non-linear partial differential equations. Nuovo Cimento 40B, 339-350 (1977)
12. Date, E., Jimbo, M., Miwa, T.: Method for generating discrete soliton equations I, II. J. Phys. Soc. Japan 4116-4131 (1982)
13. Faddeev, L.D., Takhtadjan, L.A.: Quantum inverse scattering method and the XY Z Heisenberg model. Uspekhi Mat. Nauk 34:5, 13-63 (1979)
14. Faddeev, L.D., Volkov, A.Yu.: Quantum inverse scattering method on a spacetime lattice. Teor. Mat. Fiz. 92, 207-214 (1992) (in Russian); Faddeev, L.D.: Current-like variables in massive and massless integrable models. Lectures at E. Fermi Summer School, Varenna 1994, hep-th/9406196
15. Faddeev, L.D., Volkov, A.Yu.: Quantum inverse scattering method on a spacetime lattice. Teor. Mat. Fiz. 92, 207-214 (1992) (in Russian); Faddeev, L.D.: Current-like variables in massive and massless integrable models. Lectures at E. Fermi Summer School, Varenna 1994, hep-th/9406196
16. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebras. Preprint q-alg/9505025 (1995)
17. Gaudin, M.: La fonction d'onde de Bethe. Paris: Masson, 1983
18. Gervais, J.-L., Neveu, A.: Novel triangle relation and absence of tachyons in Liouville string field theory. Nucl. Phys. B238, 125 (1984)
19. n-W-geometry. Commun. Math. Phys. 152, 317-368 (1993)
20. n-W-geometry. Commun. Math. Phys. 152, 317-368 (1993)
21. Griffiths, P., Harris, J.: Principles of algebraic geometry. A Wiley-Interscience Publication, New York: John Wiley & Sons, 1978
22. Hirota, R.: Nonlinear partial difference equations II; Discrete time Toda equations. J. Phys. Soc. Japan 43, 2074-2078 (1977) Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan 50, 3785-3791 (1981)
23. Hirota, R.: Nonlinear partial difference equations II; Discrete time Toda equations. J. Phys. Soc. Japan 43, 2074-2078 (1977) Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan 50, 3785-3791 (1981)
24. Hirota, R.: Nonlinear partial difference equations III; Discrete sine-Gordon equation. J. Phys. Soc. Japan 43, 2079-2086 (1977)
25. Hirota, R.: Discrete two-dimensional Toda molecule equation. J. Phys. Soc. Japan 56, 4285-4288 (1987)
26. Hirschfeld, J.W.P., Thas, J.A.: General Galois geometries. Oxford: Clarendon Press, 1991
27. Hodge, H.V.D., Pedoe, D.: Methods of algebraic geometry. Volume I, Cambridge: Cambridge University Press, 1947
28. Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS, Kyoto Univ. 19, 943-1001 (1983)
29. n-1 family of solvable lattice models. Mod. Phys. Lett. B1, 73-79 (1987)
30. Jorjadze, G., Pogrebkov, A., Polivanov, M., Talalov, S.: Liouville field theory: IST and Poisson bracket structure. J. Phys. A19, 121-139 (1986)
31. Kirillov, A.N.: Completeness of states in the generalized Heisenberg magnet. Zap. Nauchn. Sem. LOMI 134, 169-189 (1984) (in Russian)
32. Kirillov, A., Reshetikhin, N.: Exact solution of the integrable X X Z Heisenberg model with arbitrary spin, I, II. J. Phys. A20, 1565-1597 (1987)
33. Klümper, A., Pearce, P.: Conformal weights of RSOS lattice models and their fusion hierarchies. Physica A183, 304-350 (1992)
34. Krichever, I.M.: Elliptic solutions of Kadomtsev-Petviashvilii equation and integrable systems of particles. Funct. Anal. App. 14, n 4, 282-290 (1980)
35. Krichever, I., Babelon, O., Billey, E., Talon, M.: Spin generalization of the Calogero-Moser system and the Matrix KP equation. Preprint LPTHE 94/42
36. Krichever, I.M., Zabrodin, A.V: Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra. Uspekhi Mat. Nauk, 50:6, 3-56 (1995), hep-th/9505039
37. Kulish, P.P., Reshetikhin, N.Yu., Sklyanin, E.K.: Yang-Baxter equation and representation theory: I. Lett. Math. Phys. 5, 393-403 (1981)
38. 3-invariant solutions of the Yang-Baxter equation and associated quantum systems. Zap. Nauchn. Sem. LOMI 120, 92-121 (1982) (in Russian), Engl. transl.: J. Soviet Math. 34, 1948-1971 (1986)
39. 3-invariant solutions of the Yang-Baxter equation and associated quantum systems. Zap. Nauchn. Sem. LOMI 120, 92-121 (1982) (in Russian), Engl. transl.: J. Soviet Math. 34, 1948-1971 (1986)
40. Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method. Recent developments. Lecture Notes in Physics 151, Berlin-Heidelberg-New York: Springer, 1982, pp. 61-119
41. Kuniba, A., Nakanishi, T., Suzuki, J.: Functional relations in solvable lattice models, I: Functional relations and representation theory, II: Applications. Int. J. Mod. Phys. A9, 5215-5312 (1994)
42. Kuniba, A., Suzuki, J.: Analytic Bethe ansatz for fundamental representations of Yangians. hep-th/9406180
43. a. Physica 3D, 62-72 (1981); Group-theoretical methods for integration of nonlinear dynamical systems. Progress in Physics series 15, Basel: Birkhaüser-Verlag, 1992
44. a. Physica 3D, 62-72 (1981); Group-theoretical methods for integration of nonlinear dynamical systems. Progress in Physics series 15, Basel: Birkhaüser-Verlag, 1992
45. Lipan, O., Wiegmann, P., Zabrodin, A.: Fusion Rules for Quanatum Transfer Matrices as a Dynamical System on Grassmann Manifolds. Mod. Phys. Lett. A, 12 (19), 1369-1378 (1997)
46. McCoy, B., Wu, T.T.: Nonlinear partial difference equations for the two-dimensional Ising model. Phys. Rev. Lett. 45, 675-678 (1980); Nonlinear partial difference equations for the two-spin correlation function of the two-dimensional Ising model. Nucl. Phys. B180, 89-115 (1981); McCoy, B., Perk, J.H.H., Wu, T.: Ising field theory: quadratic difference equations for the n-point Green's functions on the lattice. Phys. Rev. Lett. 46, 757-760 (1981)
47. McCoy, B., Wu, T.T.: Nonlinear partial difference equations for the two-dimensional Ising model. Phys. Rev. Lett. 45, 675-678 (1980); Nonlinear partial difference equations for the two-spin correlation function of the two-dimensional Ising model. Nucl. Phys. B180, 89-115 (1981); McCoy, B., Perk, J.H.H., Wu, T.: Ising field theory: quadratic difference equations for the n-point Green's functions on the lattice. Phys. Rev. Lett. 46, 757-760 (1981)
48. McCoy, B., Wu, T.T.: Nonlinear partial difference equations for the two-dimensional Ising model. Phys. Rev. Lett. 45, 675-678 (1980); Nonlinear partial difference equations for the two-spin correlation function of the two-dimensional Ising model. Nucl. Phys. B180, 89-115 (1981); McCoy, B., Perk, J.H.H., Wu, T.: Ising field theory: quadratic difference equations for the n-point Green's functions on the lattice. Phys. Rev. Lett. 46, 757-760 (1981)
49. Miwa, T.: On Hirota's difference equations. Proc. Japan Acad. 58, 9-12 (1982)
50. Nijhof, F., Ragnisco, O., Kuznetsov, V.: Integrable time-discretization of the Ruijsenaars-Schneider model. Commun. Math. Phys. 176, 681-700 (1996)
51. Ohta, Y., Hirota, R., Tsujimoto, S., Imai, T.: Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy. J. Phys. Soc. Japan 62, 1872-1886 (1993)
52. Perk, J.H.H.: Quadratic identities for Ising model correlations. Phys. Lett. A79, 3-5 (1980)
53. Ravanini, F., Valleriani, A., Tateo, R.: Dynkin TBA's. Int. J. Mod. Phys. A8, 1707-1727 (1993)
54. Reshetikhin, N.Yu.: The functional equation method in the theory of exactly soluble quantum systems. Sov. Phys. JETP 57, 691-696 (1983)
55. Ruijsenaars, S.N.M., Schneider, H.: A new class of integrable systems and its relation to solitons. Ann. Phys. (NY) 170, 370-405 (1986)
56. Saito, S., Saitoh, N.: Linearization of bilinear difference equations. Phys. Lett. A120, 322-326 (1987); Gauge and dual symmetries and linearization of Hirota's bilinear equations. J. Math. Phys. 28, 1052-1055 (1987)
57. Saito, S., Saitoh, N.: Linearization of bilinear difference equations. Phys. Lett. A120, 322-326 (1987); Gauge and dual symmetries and linearization of Hirota's bilinear equations. J. Math. Phys. 28, 1052-1055 (1987)
58. Sato, M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. RIMS Kokyuroku 439, 30-46 (1981)
59. Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. IHES 61, 5-65 (1985)
60. Ueno, K., Takasaki, K.: Toda lattice hierarchy. Adv. Studies in Pure Math. 4, 1-95 (1984)
61. Zamolodchikov, A.B.: On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B253, 391-394 (1991)
62. n-1 fused lattice models. Preprint hep-th/9502067 (1995)