How the Algebraic Bethe Ansatz Works for Integrable Models

“…For the first time, the spectrum of excitations was described correctly in the paper [10] by Faddeev and Takhtajan; it was shown that the spectrum contains magnons of spin 1/2. These authors used the algebraic Bethe Ansatz formulated by Faddeev, Sklyanin, and Takhtajan (see [9]) on the basis of R-matrices and the Yang-Baxter equation. The origin of these new techniques goes back to the work of Baxter [1].…”

Section: Consider the XXX Antiferromagnet Given By The Hamiltonianmentioning

confidence: 99%

A recursion formula for the correlation functions of an inhomogeneous XXX model

Boos

Jimbo

Miwa

et al. 2006

St. Petersburg Math. J.

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Abstract.A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It links the correlators involving n consecutive lattice sites to those with n − 1 and n − 2 sites. In a series of papers by V. Korepin and two of the present authors, it was discovered that the correlators have a certain specific structure as functions of the inhomogeneity parameters. The formula mentioned above makes it possible to prove this structure directly, as well as to obtain an exact description of the rational functions that were left undetermined in the earlier work. §1. Introduction Consider the XXX antiferromagnet given by the HamiltonianThis model was solved in the famous paper by Bethe [3] already in 1931, by using what is now called the coordinate Bethe Ansatz. Nevertheless, it took some time before the physical content of this model in the thermodynamic limit was clarified completely. For the first time, the spectrum of excitations was described correctly in the paper [10] by Faddeev and Takhtajan; it was shown that the spectrum contains magnons of spin 1/2. These authors used the algebraic Bethe Ansatz formulated by Faddeev, Sklyanin, and Takhtajan (see [9]) on the basis of R-matrices and the Yang-Baxter equation. The origin of these new techniques goes back to the work of Baxter [1]. Restricting ourselves to our example, we recall the role of R-matrices in solvable models. The XXX model is related to the rational R-matrix that acts on C 2 ⊗ C 2 (see (2.7) for the explicit formula). We employ the usual notation R 1,2 (λ), where 1, 2 label the corresponding spaces and λ is the spectral parameter. The relationship between the R-matrix and the XXX Hamiltonian is as follows. Consider the transfer matrix

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“…It is well known that solutions of the Yang-Baxter equation can be rather intricate [22,26]. None the less appealing to the Quantum Inverse Scattering Method [15,28] one can put forward a reasonable conjecture that they are composite objects having internal structure and that they are constructed out of elementary blocks. A more refined statement is that the R-operator admits factorization, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…In the case of restriction to two-dimensional space (spin s = 1 2 ) the formula (6) gives rise to the quantum L-operator [15]. In order to see it, let us choose the following basis in C…”

Section: Introductionmentioning

confidence: 99%

Matrix Factorization for Solutions of the Yang–Baxter Equation

Derkachov

Chicherin

2016

J Math Sci

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We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finitedimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sℓ 2 , the modular double (trigonometric deformation) and the Sklyanin algebra (elliptic deformation). The solutions are matrices with operator entries. The matrix elements are differential operators in the case of sℓ 2 , finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed out of Jacobi theta functions in the case of the Sklyanin algebra. We find a new factorized form of the rational, trigonometric, and elliptic solutions, which drastically simplifies them. We show that they are products of several simply organized matrices and obtain for them explicit formulae.

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How the Algebraic Bethe Ansatz Works for Integrable Models

Cited by 294 publications

References 22 publications

Classical integrability

Classical integrability

A recursion formula for the correlation functions of an inhomogeneous XXX model

Matrix Factorization for Solutions of the Yang–Baxter Equation

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