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Hirota equation and Bethe ansatz

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Abstract

Recent analyses of classical integrable structures in quantum integrable models solved by various versions of the Bethe ansatz are reviewed. Similarities between elements of quantum and classical theories of integrable systems are discussed. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton theory. Functional relations for quantum transfer matrices become the classical Hirota bilinear difference equation; solving this classical equation gives all the basic results for the spectral properties of quantum systems. Vice versa, typical Bethe ansatz formulas under certain boundary conditions yield solutions of this classical equation. The Baxter T-Q relation and its generalizations arise as auxiliary linear problems for the Hirota equation.

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References

  1. E. K. Sklyanin and L. D. Faddeev,Dokl. Akad. Nauk SSSR,243, 1430–1433 (1978); E. K Sklyanin,J. Sov. Math.,19, 1546–1596 (1982).

    Google Scholar 

  2. L. D. Faddeev and L. A. Takhtajan,The Hamiltonian Methods in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl., Springer, Heidelberg (1987).

    MATH  Google Scholar 

  3. T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch,Phys. Rev. B,13, 316–374 (1976).

    Article  ADS  Google Scholar 

  4. B. McCoy and T. T. Wu,Phys. Rev. Lett.,45, 675–678 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  5. J. H. H. Perk,Phys. Lett. A,79, 3–5 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  6. V. Korepin, A. Izergin, A. Its, and N. Slavnov,Int. J. Mod. Phys. B,4, 1003–1037 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin,Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ., Cambridge (1993).

    MATH  Google Scholar 

  8. Al. B. Zamolodchikov,Phys. Lett. B,253, 391–394 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  9. A. Klümper and P. Pearce,Physica A,183, 304–350 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  10. A. Kuniba, T. Nakanishi, and J. Suzuki,Int. J. Mod. Phys. A,9, 5215–5312 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. R. Hirota,J. Phys. Soc. Japan,50, 3785–3791 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Jimbo, T. Miwa, and M. Sato,Holonomic Quantum Fields [in Russian], Mir, Moscow (1983).

    Google Scholar 

  13. D. Bernard and A. LeClair,Phys. Lett. B,288, 113–120 (1992); “Differential equations for sine-Gordon correlation functions at the free fermion point,” Preprint CLNS 94/1276; SPhT-94-021; hep-th/9402144.

    Article  ADS  MathSciNet  Google Scholar 

  14. A. G. Izergin, D. A. Coker, and V. E. Korepin,J. Phys. A.,25, 4315–4334 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. I. G. Korepanov, “Vacuum curves, classical integrable systems in discrete space-time and statistical physics,” hep-th/9312197.

  16. R. M. Kashaev,Lett. Math. Phys.,35, 389–397 (1996).

    Article  MathSciNet  ADS  Google Scholar 

  17. I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin,Commun. Math. Phys.,188, 267–304 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. A. Zabrocin, “Bethe ansatz and classical Hirota equations,” in:Proceedings of the Second A. D. Sakharov International Conference. Moscow, Russia, 20–24 May 1996 (I. M. Dremin and A. M. Semikhatov, eds.), World Scientific, Singapore (1997), pp. 627–632.

    Google Scholar 

  19. P. Wiegmann, “Bethe ansatz and classical Hirota equation,” cond-mat/9610132.

  20. A. Zabrodin,Int. J. Mod. Phys. B,11, 3125–3158 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  21. O. Lipan, P. Wiegmann, and A. Zabrodin,Mod. Phys. Lett. A,12, 1369–1378 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. L. A. Takhtajan and L. D. Faddeev,Usp. Mat. Nauk,34, No. 5, 13–63 (1979).

    Google Scholar 

  23. P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin,Lett. Math. Phys.,5, 393–403 (1981).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  24. R. Hirota,J. Phys. Soc. Japan,43, 1424–1433 (1977).

    ADS  MathSciNet  Google Scholar 

  25. R. Hirota,J. Phys. Soc. Japan,43, 2074–2078 (1977).

    MathSciNet  ADS  Google Scholar 

  26. R. Hirota,J. Phys. Soc. Japan,43, 2079–2086 (1977).

    MathSciNet  ADS  Google Scholar 

  27. R. Hirota,J. Phys. Soc. Japan,45, 321–332 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  28. R. Hirota,J. Phys. Soc. Japan,46, 312–319 (1979).

    MathSciNet  ADS  Google Scholar 

  29. A. Zabrodin,Theor. Math. Phys.,113, 1347–1392 (1997).

    MathSciNet  Google Scholar 

  30. I. V. Cherednik,Funct. Anal. Appl.,20, 76–78 (1986);Dokl. Akad. Nauk SSSR,283, 1040–1046 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Jimbo, A. Kuniba, T. Miwa, and M. Okado,Commun. Math. Phys.,119, 691–696 (1988).

    Article  MathSciNet  Google Scholar 

  32. A. G. Izergin and V. E. Korepin,Dokl. Akad. Nauk SSSR,259, 76–79 (1981).

    MathSciNet  Google Scholar 

  33. K. Hasegawa,J. Phys. A. 26, 3211–3228 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. V. Bazhanov and N. Reshetikhin,J. Phys. A,23, 1477–1492 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. A. Kuniba, Y. Ohta, and J. Suzuki,J. Phys. A,28, 6211–6226 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. P. P. Kulish and N. Yu. Reshetikhin,Zap. Nauchn. Sem. LOMI,120, 92–121 (1982).

    MathSciNet  Google Scholar 

  37. N. Yu. Reshetikhin,JETP,57, 691–699 (1983).

    MathSciNet  Google Scholar 

  38. A. Belavin,Nucl. Phys. B,180, 189–200 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  39. M. Richey and C. Tracy,J. Stat. Phys.,42, 311–348 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  40. B. Hou and Y. Zhou,J. Phys. A,23, 1147–1154 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. I. V. Cherednik,Yad. Fiz.,36, 549–557 (1982).

    MathSciNet  Google Scholar 

  42. E. K. Sklyanin,Funct. Anal. Appl.,16, 263–270 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  43. Y.-H. Quano,Int. J. Mod. Phys. A,9, 2245–2281 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. R. Hirota,J. Phys. Soc. Japan,56, 4285–4288 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  45. A. Bilal and J.-L. Gervais,Nucl. Phys. B,314, 646–686 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  46. S. Kharchev, A. Marshakov, A. Mironov, A. Orlov, and A. Zabrodin,Nucl. Phys. B,366, 569–601 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  47. H. Airault, H. McKean, and J. Moser,Commun. Pure Appl. Math.,30, 95–125 (1977).

    ADS  MathSciNet  MATH  Google Scholar 

  48. I. M. Krichever,Funct. Anal. Appl.,14, 282–290 (1980).

    Article  Google Scholar 

  49. I. M. Krichever, P. B. Wiegmann, and A. V. Zabrodin, “Elliptic solutions to difference nonlinear equations and related many-body problems,” Preprint ITEP-TH-13/97, Inst. Theor. Exp. Phys., Moscow (1997); hep-th/9704090; to appear inCommun. Math. Phys.

    Google Scholar 

  50. S. Saito and N. Saitoh,J. Math. Phys.,28, 1052–1055 (1987).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. I. M. Krichever and A. V. Zabrodin,Russ. Math. Surv.,50, 1101–1150 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  52. I. Krichever, O. Babelon, E. Billey, and M. Talon, “Spin generalization of the Calogero-Moser system and the matrix KP equation,” Preprint LPTHE 94/42, Université Paris VI, Paris (1994).

    Google Scholar 

  53. A. N. Kirillov,Zap Nauchn. Sem. LOMI,134, 169–189 (1984).

    MathSciNet  MATH  Google Scholar 

  54. S. N. M. Ruijsenaars and H. Schneider,Ann. Phys.,170, 370–405 (1986).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. F. Nijhoff, O. Ragnisco, and V. Kuznetsov,Commun. Math. Phys.,176, 681–700 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. A. Leznov and M. Saveliev,Physica D,3, 62–72 (1981).

    Article  ADS  Google Scholar 

  57. A. N. Leznov and M. V. Saveliev,Group Theoretical Methods for Integration of Nonlinear Dynamical Systems [in Russian], Nauka, Moscow (1985); English transl., Birkhäuser, Basel (1992).

    Google Scholar 

  58. E. Frenkel and N. Reshetikhin,Commun. Math. Phys.,178, 237–264 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. A. Kuniba and J. Suzuki,Commun. Math. Phys.,173, 225–264 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. J.-J. Gervais and A. Neveu,Nucl. Phys. B,238, 125–141 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  61. G. P. Dzhordzhadze, A. K. Pogrebkov, and M. K. Polivanov,Theor. Math. Phys.,40, 706–714 (1979).

    Article  MathSciNet  Google Scholar 

  62. G. Jorjadze, A. Pogrebkov, M. Polivanov, and S. Talalov,J. Phys. A,19, 121–139 (1986).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. P. Kulish and E. Sklyanin, “Quantum spectral transform method. Recent developments,” in:Lect. Notes Phys. (J. Hietarinta and C. Montonen, eds.), Vol. 151,Integrable Quantum Field Theory, Springer, Heidelberg (1982), pp. 61–119.

    Google Scholar 

  64. R. Baxter,Ann. Phys.,70, 193–228 (1972).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  65. R. Baxter,Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).

    MATH  Google Scholar 

  66. E. K. Sklyanin, L. A. Takhtajan, and L. D. Faddeev,Theor. Math. Phys.,40, 688–705 (1979).

    Article  Google Scholar 

  67. A. Zabrodin,JETP Letters,66, 653–659 (1997); “Hidden quantumR-matrix in discrete time classical Heisenberg magnet”, Preprint ITEP-TH-54/97, Inst. Theor. Exp. Phys., Moscow (1997); solv-int/9710015.

    Article  ADS  MathSciNet  Google Scholar 

  68. P. Griffiths and J. Harris,Principles of Algebraic Geometry, Wiley, New York (1978).

    MATH  Google Scholar 

  69. M. Sato,RIMS Kokyuroku,439, 30–46 (1981).

    Google Scholar 

  70. M. Jimbo and T. Miwa,Publ. RIMS Kyoto Univ.,19, 943–1001 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  71. G. Segal and G. Wilson,Publ. IHES,61, 5–65 (1985).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Joint Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia

    A. V. Zabrodin

  2. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. V. Zabrodin

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 1, pp. 54–100, July, 1998.

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Zabrodin, A.V. Hirota equation and Bethe ansatz. Theor Math Phys 116, 782–819 (1998). https://doi.org/10.1007/BF02557123

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