Introduction to algebraic Bethe ansatz

  • L. A. Takhtajan
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 242)


Heisenberg Model Monodromy Matrix Inverse Scattering Method Physical Vacuum Boltzmann Weight 
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Fundamental relation (1.16) was introduced by Baxter

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    R.J.Baxter. Partition-function of the eight-vertex lattice model.-Ann.Phys. (N.Y.), 1972, v.70, p.323–337, where the solution of the 8-vertex model was given. The YangBaxter relation is a far going generalization of the Onsager's famous star-triangle relation. Parametrization (1.22) of Boltzmann weights was extremely important for Baxter's approach. The role of equation (1.23) in statistical mechanics was manifested by BaxterCrossRefGoogle Scholar
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    A.B.Zamolodchikov. Factorized S-matrices and Lattice Statistical Systems. — Sov.Sci.Rev., 1980, Phys.Rev., v.2. Baxter's solution of the triangle equation — the R-matrix (1.24) was generalized to the higher matrix dimensions by BelavinGoogle Scholar
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The monodromy matrix was used in the full strength in the papers

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    E.K.Sklyanin, L.A.Takhtajan, L.D.Faddeev. Quantum inverse scattering method. I.-Teor. i matem. fisika, 1979, v.40, N 3, p.194–220 (in Russian).Google Scholar
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    L.A.Takhtajan, L.D.Faddeev. Quantum inverse scattering method and the XYZ-Heisenberg model.-Uspechi Mat. Nauk, 1979, v.34, N 5, p.13–63 (in Russian). In [5] the quantum inverse scattering method was proposed and exact solution of the Sine-Gordon model was given. In particular [5] contains the formulation of algebraic Bethe ansatz in Abc-form. In [6] the generalized algebraic Bethe ansatz for the 8-vertex model (and connected with it XYZ-Hamiltonian) was presented. In coordinate form it was given in the papersGoogle Scholar
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The coordinate Bethe ansatz method was formulated by Bethe

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    L.A.Takhtajan. Quantum inverse scattering method and algebraic matrix Bethe ansatz.-LOMI Proceedings, 1980, v. 101,p.158–183 (in Russian). Comment N 6 in lecture II belongs to S.V.Manakov (private communication). The requirement that { λλj} in Bethe states should be pairwise distinct is not really a restriction. This was proved in the paperGoogle Scholar
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    A.G. Isergin, V.E.Korepin. Pauli principle for one dimensional bosons and the algebraic Bethe ansatz.-LMP, 1982, v.6, N 4, p.283–288. Norms of Bethe states in the formalism of quantum inverse scattering method were calculated by KorepinCrossRefGoogle Scholar
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General scheme for obtaining local quantum Hamiltonians on a lattice can be found in

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    V.O.Tarasov, L.A.Takhtajan, L.D.Faddeev. Local Hamiltonians for quantum integrable models on the lattice.-Teoret. i matem. fisika, 1983, v.57, N 2, p.163–181. Integrable generalization of isotropic Heisenberg model of spins 1/2 to higher spins was given by Kulish, Reshetikhin and SklyaninGoogle Scholar
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    P.P.Kulish, N.Yu.Reshetikhin, E.K.Sklyanin. Yang-Baxter equation and representation theory. I.-LMP, 1981, v.5, N 5, p.393–403. Its solution and thermodynamical properties are given, correspondingly, in the papersGoogle Scholar
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    H.M.Babujian. Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spin 8.-Phys. Lett., 1982, v.90A, N 9, p.479–482. Properties of Bethe states for the XXX-Heisenberg model were established in the papersGoogle Scholar
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    L.D.Faddeev, L.A.Takhtajan. What is the spin of a spin wave?-Phys.Lett., 1981, v.85A, N 6-7, p.375–377.Google Scholar
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Complete solution of the nonlinear Schrödinger model by the quantum inverse scattering method was given by Sklyanin

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    E.K.Sklyanin. Quantum variant of the inverse scattering method.-LOMI Proceedings, 1980, v.95, p.55–128 (in Russian). The local L-operator (4.18) was introduced in the paperGoogle Scholar
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    A.G.Isergin, V.E.Korepin. Lattice model, connected with nonlinear Schrödinger equation.-Doklady Acad.Nauk USSR, 1981, v.259, N 1, p.76–79. About the connection between this model andXXX-Heisenberg model of arbitrary spins see [16]. Quantum sine-Gordon model was solved by the quantum inverse scattering method in [5]. Its lattice version was developed by Isergin and KorepinGoogle Scholar
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    A.G.Isergin, V.E.Korepin. The lattice quantum Sine-Gordon equation.-LMP, 1981, v.5, p.199–205 (see also their paper in Nucl.Phys.), where the local L-operator (4.30) was presentedCrossRefGoogle Scholar

Exposition in lecture V follows the paper [21], where an extensive list of references was given. Here we notice only that assumptions a) and b) were actually used by Bethe in [8]. The dynamical picture for the case of spins Swas presented in [18]. In this case the proof of an analog of completeness relation (5.21) requires more complicated combinatorics. This. proof was given by Kirillov

  1. 25.
    A.N.Kirillov. Combinatoric identities and the completeness of Bethe multiplets for the Heisenberg model.-LOMI Proceedings, 1983, v.131, p.88–105 (in Russian). The counting of Bethe states is very important when one has antiferromagnetic case. Thus in [20–21] it was shown that a spin wave-kink has spin 1/2 rather then 1 which was generally assumed to be true. It should be emphasized that the string hypothesis is literally true only whenN → ∞andis fixed. However in the anti ferromagnetic case we have l ≈ N/2, so the string hypothesis for such l needs verification. In the paperGoogle Scholar
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    C.Destry, J.H.Lowenstein. Analysis of the Bethe ansatz equations of the chiral-invariant Gross-Neveu model.-Nucl.Phys. v.B205, N 3, p.369–385 in was shown that in the absense of external magnetic field (H=0) and at zero temperature (T = 0) the string hypothesis literally is not true: complex solutions of the system (5.6) do not combine into a strings. By they do combine into a strings if H > 0 or T > 0 So in lecture V the following “regularization” is assumed: introduce a small magnetic fieldH ≪ 1, which permits to write down the system (5.12), and then set H → 0Google Scholar

It is impossible to give here all necessary references to such rapidly developing subject as quantum inverse scattering method. For mathematically oriented audience I should mention the good pedagogical course

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    L.D.Faddeev. Integrable models in 1 + 1 dimensions. — Les Houches Lectures 1982, Elsevier Sci.Publ., 1984 and the lectureGoogle Scholar
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    L.A.Takhtajan. Integrable models in classical and quantum field theory.-Proc.Inter.Congress of Mathem., Warsaw 1983.Google Scholar
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    For those who would like to follow the current papers on quantum inverse scattering method I shall point to the main journals where they are published: LOMI Proceedings, Teoret. i matem. fisika, JETP (these are Soviet journals) and Phys.Lett, Nucl.Phys., CMP, IMP, J.Phys. A and C, Physika D, Phys. Rev.D. The time interval starts from 1978–1979.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • L. A. Takhtajan
    • 1
  1. 1.Steklov Mathematical InstituteLeningrad BranchLeningradUSSR

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