# Introduction to algebraic Bethe ansatz

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## Keywords

Heisenberg Model Monodromy Matrix Inverse Scattering Method Physical Vacuum Boltzmann Weight
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## References

## Fundamental relation (1.16) was introduced by Baxter

- 1.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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## The monodromy matrix was used in the full strength in the papers

- 5.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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## The coordinate Bethe ansatz method was formulated by Bethe

- 8.H.Bethe. Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen Atomkette.-Zeit. Phys., 1931, B.71, N 3-4, p.205–226. It was generalized for the XYZ-Heisenberg model by Yang and YangCrossRefGoogle Scholar
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- 13.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 { λλ
_{j}} in Bethe states should be pairwise distinct is not really a restriction. This was proved in the paperGoogle Scholar - 14.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

- 16.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
- 17.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
- 18.L.A.Takhtajan. The picture of low-lying excitations in isotropic Heisenberg chain of arbitrary spin.-Phys.Lett., 1982, v.87A, N 9, p.479–482.Google Scholar
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## Complete solution of the nonlinear Schrödinger model by the quantum inverse scattering method was given by Sklyanin

- 22.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
- 23.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
- 24.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

- 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
- 26.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

- 27.L.D.Faddeev. Integrable models in 1 + 1 dimensions. — Les Houches Lectures 1982, Elsevier Sci.Publ., 1984 and the lectureGoogle Scholar
- 28.L.A.Takhtajan. Integrable models in classical and quantum field theory.-Proc.Inter.Congress of Mathem., Warsaw 1983.Google Scholar
- 29.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

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© Springer-Verlag 1985