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Off-Diagonal Bethe Ansatz for Exactly Solvable Models pp 23–65Cite as

The Algebraic Bethe Ansatz

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Abstract

The algebraic Bethe Ansatz method for quantum integrable models was proposed by the Leningrad Group [1–7] in the late 1970s, based on YBE. This method was then generalized to open boundary integrable systems by Sklyanin [8] in 1988, through developing an equation accounting for the integrable boundaries.

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References

  1. E.K. Sklyanin, L.D. Faddeev, Quantum mechanical approach to completely integrable field theory models. Sov. Phys. Dokl. 23, 902 (1978)

    ADS  Google Scholar 

  2. L.A. Takhtadzhan, L.D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model. Rush. Math. Surv. 34, 11 (1979)

    Google Scholar 

  3. E.K. Sklyanin, L.A. Takhtajan, L.D. Faddeev, Qunatum inverse problem method. Theor. Math. Phys. 40, 688 (1980)

    Article  MATH  Google Scholar 

  4. L.D. Faddeev, Quantum inverse scattering method. Sov. Sci. Rev. Math. C 1, 107 (1980)

    MATH  Google Scholar 

  5. E.K. Sklyanin, Quantum version of the method of inverse scattering problem. J. Sov. Math. 19, 1546 (1982)

    Article  MATH  Google Scholar 

  6. L.A. Takhtajan, Introduction to Bethe Ansatz. Lect. Notes Phys. 242, 175 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  7. P.P. Kulish, E.K. Sklyanin, Quantum spectral transform method: recent developments. Lect. Notes Phys. 151, 61 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  8. E.K. Sklyanin, Boundary conditions for integrable quantum systems. J. Phys. A 21, 2375 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. P.P. Kulish, Classical and quantum inverse problem method and generalized Bethe Ansatz. Physica 3D, 246 (1981)

    ADS  MathSciNet  Google Scholar 

  10. P.P. Kulish, N. Reshetikhin, Generalized Heisenberg ferromagnet and the Gross-Neveu model. Sov. Phys. JETP 53, 108 (1981)

    Google Scholar 

  11. H.J. de Vega, M. Karowski, Exact Bethe Ansatz solution of \(O(2N)\) symmetric theories. Nucl. Phys. B 280, 225 (1987)

    Article  ADS  Google Scholar 

  12. H.J. de Vega, E. Lopes, Exact solution of the Perk-Shultz model. Phys. Rev. Lett. 67, 489 (1991)

    Google Scholar 

  13. E. Lopes, Exact solution of the multi-component generalized six-vertex model. Nucl. Phys. B 370, 636 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  14. F.H.L. Essler, V.E. Korepin, Higher conservation laws and algebraic Bethe Ansatz for the supersymmetric t-J model. Phys. Rev. B 46, 9147 (1992)

    Article  ADS  Google Scholar 

  15. A. Foerster, M. Karowski, Algebraic properties of the Bethe Ansatz for an \(spl(2,1)\)-supersymmetric t-J model. Nucl. Phys. B 396, 611 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  16. A. Foerster, M. Karowski, The supersymmetric t-J model with quantum group invariance. Nucl. Phys. B 408, 512 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. H.J. de Vega, A. González-Ruiz, Exact solution of the \(SU_q(n)\)-invariant quantum spin chains. Nucl. Phys. B 417, 553 (1994)

    Google Scholar 

  18. H.J. de Vega, A. González-Ruiz, Exact Bethe Ansatz solution for \(A_{n-1}\) chains with non-\(SU_q(n)\) invariant open boundary conditions. Mod. Phys. Lett. A 9, 2207 (1994)

    Google Scholar 

  19. W. Galleas, M.J. Martins, Solution of the \(SU(N)\) vertex model with non-diagonal open boundaries. Phys. Lett. A 335, 167 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. M.J. Martins, The exact solution and the finite-size behaviour of the \(Osp(1|2)\)-invariant spin chain. Nucl. Phys. B 450, 768 (1995)

    Article  ADS  MATH  Google Scholar 

  21. M.J. Martins, P.B. Ramos, The algebraic Bethe Ansatz for rational braid-monoid lattice models. Nucl. Phys. B 500, 579 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. M.J. Martins, The thermodynamic limit and the finite-size behaviour of the fundamental \(Sp(2N)\) spin chain. Nucl. Phys. B 636, 583 (2002)

    Article  ADS  MATH  Google Scholar 

  23. R.-H. Yue, H. Fan, B.-Y. Hou, Exact diagonalization of the quantum supersymmetric \(SU_q(n|m)\) model. Nucl. Phys. B 462, 167 (1996)

    Google Scholar 

  24. G.-L. Li, R.-H. Yue, B.-Y. Hou, Nested Bethe Ansatz for Perk-Schultz model with open boundary conditions. Nucl. Phys. B 586, 711 (2000)

    Google Scholar 

  25. E.K. Sklyanin, The quantum Toda chain. Lect. Notes Phys. 226, 196 (1985)

    Google Scholar 

  26. E.K. Sklyanin, Separation of variables in the Gaudin model. J. Sov. Math. 47, 2473 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  27. E.K. Sklyanin, Separation of variables—new trends. Prog. Theor. Phys. Suppl. 118, 35 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  28. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  30. L.D. Faddeev, L.A. Takhtajan, What is the spin of a spin wave? Phys. Lett. A 85, 375 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  31. M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, Cambridge, 1999)

    Book  Google Scholar 

  32. N. Andrei, K. Furuya, J.H. Lowenstein, Solution of the Kondo problem. Rev. Mod. Phys. 55, 331 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  33. C.N. Yang, C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction. J. Math. Phys. 10, 1115 (1969)

    Article  ADS  MATH  Google Scholar 

  34. E.H. Lieb, W. Liniger, Exact analysis of an interacting Bose gas I. The general solution and the ground state. Phys. Rev. 130, 1605 (1963)

    Google Scholar 

  35. M. Gaudin, Thermodynamics of the Heisenberg-Ising ring for \(\Delta > 1\). Phys. Rev. Lett. 26, 1301 (1971)

    Google Scholar 

  36. M. Takahashi, One-dimensional Heisenberg model at finite temperature. Progress Theoret. Phys. 46, 401 (1971)

    Google Scholar 

  37. M. Takahashi, One-dimensional Hubbard model at finite temperature. Progress Theoret. Phys. 47, 69 (1972)

    Google Scholar 

  38. M. Takahashi, Low-temperature specific-heat of one-dimensional Hubbard model. Progress Theoret. Phys. 52, 103 (1974)

    Google Scholar 

  39. J.D. Johnson, B. McCoy, Low-temperature thermodynamics of the \(|\Delta | \ge 1\) Heisenberg-Ising ring. Phys. Rev. A 6, 1613 (1972)

    Google Scholar 

  40. Y. Wang, Fermi-liquid features of the one-dimensional Luttinger liquid. Int. J. Mod. Phys. B 12, 3465 (1998)

    Google Scholar 

  41. F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter, G.R.W. Quispel, Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models. J. Phys. A 20, 6397 (1987)

    Google Scholar 

  42. H.J. de Vega, A. González-Ruiz, Boundary K-matrices for the six vertex and the \(n(2n-1)\, A_{n-1}\) vertex models. J. Phys. A 26, L519 (1993)

    Google Scholar 

  43. H.J. de Vega, A. González-Ruiz, Boundary K-matrices for the XYZ, XXZ and XXX spin chains. J. Phys. A 27, 6129 (1994)

    Google Scholar 

  44. A. Lima-Santos, Reflection K-matrices for 19-vertex models. Nucl. Phys. B 558, 637 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. R. Malara, A. Lima-Santos, On \(A^{(1)}_{n-1}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n}, A^{(2)}_{2n-1}\) and \(D^{(2)}_{n+1}\) reflection K-matrices. J. Stat. Mech. P09013 (2006)

    Google Scholar 

  46. M. Gaudin, Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 386 (1971)

    Google Scholar 

  47. A.L. Owuarek, R.J. Baxter, Surface free energy of the critical six-vertex model with free boundaries. J. Phys. A 22, 1141 (1989)

    Article  ADS  Google Scholar 

  48. C.J. Hamer, G.R.W. Quispel, M.T. Batchelor, Conformal anomaly and surface energy for Potts and Ashkin-Teller quantum chains. J. Phys. A 20, 5677 (1987)

    Google Scholar 

  49. M.T. Batchelor, C.J. Hamer, Surface energy of integrable quantum spin chains. J. Phys. A 23, 761 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  50. P.A. Bares, G. Blatter, M. Ogata, Exact solution of the t-J model in one dimension at \(2t=\pm J\): ground state and excitation spectrum. Phys. Rev. B 44, 130 (1991)

    Article  ADS  Google Scholar 

  51. M.T. Grisaru, L. Mezincescu, R.I. Nepomechie, Direct calculation of the boundary S-matrix for the open Heisenberg chain. J. Phys. A 28, 1027 (1995)

    Google Scholar 

  52. F.H.L. Essler, The supersymmetric t-J model with a boundary. J. Phys. A 29, 6183 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. A. Kapustin, S. Skorik, Surface excitations and surface energy of the antiferromagnetic XXZ chain by the Bethe Ansatz approach. J. Phys. A 29, 1629 (1996)

    Article  ADS  MATH  Google Scholar 

  54. R. Murgan, R.I. Nepomechie, C. Shi, Boundary energy of the general open XXZ chain at roots of unity. J. High Energy Phys. 01, 038 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  55. A. Kitaev, Anyons in an exactly solved model and beyond. Ann. Phys. (NY) 321, 2 (2006)

    Google Scholar 

  56. B. Sutherland, Further results for the many-body problem in one dimension. Phys. Rev. Lett. 20, 98 (1968)

    Article  ADS  Google Scholar 

  57. C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312 (1967)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  58. B. Sutherland, Model for a multicomponent quantum system. Phys. Rev. B 12, 3795 (1975)

    Article  ADS  Google Scholar 

  59. A.M. Tsvelik, P.B. Wiegmann, Exact results in the theory of magnetic alloys. Adv. Phys. 32, 453 (1983)

    Article  ADS  Google Scholar 

  60. P. Schlottmann, Exact results for highly correlated electron systems in one dimension. Int. J. Mod. Phys. B 11, 415 (1997)

    Article  MathSciNet  Google Scholar 

  61. T. Deguchi, F.H.L. Essler, F. Göhmann, A. Klümper, V.E. Korepin, K. Kusakabe, Thermodynamics and excitations of the one-dimensional Hubbard model. Phys. Rep. 331, 197 (2000)

    Google Scholar 

  62. B. Sutherland, Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems (World Scientify Publishing, Singapore, 2004)

    Book  Google Scholar 

  63. P.P. Kulish, N.Y. Reshetikhin, E.K. Sklyanin, Yang-Baxter equation and representation theory: I. Lett. Math. Phys. 5, 393 (1981)

    Google Scholar 

  64. P.P. Kulish, N.Y. Reshetikhin, Quantum linear problem for the sine-Gordon equation and higher representation. J. Sov. Math. 23, 2435 (1983)

    Google Scholar 

  65. A.N. Kirillov, N.Y. Reshetikhin, Exact solution of the Heisenberg XXZ model of spin \(s\). J. Sov. Math. 35, 2627 (1986)

    Google Scholar 

  66. A.N. Kirillov, N.Y. Reshetikhin, Exact solution of the integrable XXZ Heisenberg model with arbitrary spin I. The ground state and the excitation spectrum. J. Phys. A 20, 1565 (1987)

    Google Scholar 

  67. N. Kitanine, J.M. Maillet, V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain. Nucl. Phys. B 554, 647 (1999)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Yupeng Wang & Junpeng Cao

  2. Institute of Modern Physics, Northwest University, Xi’an, China

    Wen-Li Yang & Kangjie Shi

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  1. Yupeng Wang
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Wang, Y., Yang, WL., Cao, J., Shi, K. (2015). The Algebraic Bethe Ansatz. In: Off-Diagonal Bethe Ansatz for Exactly Solvable Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46756-5_2

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