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arXiv:1012.1455 · Symmetry Integrability and Geometry Methods and Applications

Universal Bethe Ansatz and Scalar Products of Bethe Vectors

S. Belliard

Published Dec 14, 2010• 21 citations
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Abstract

An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra U q ( gl 3 ) is given.This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a "new" realization of the quantum affine algebra U q ( gl 3 ).

DOI
10.3842/sigma.2010.094

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An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra U q ( gl 3 ) is given.This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a "new" realization of the quantum affine algebra U q ( gl 3 ).
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Abstract We study the scalar products of Bethe vectors in integrable models solvable by the nested algebraic Bethe ansatz and possessing gl ( 2 ∣ 1 ) symmetry. Using explicit formulas of the monodromy matrix entries’ multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with gl ( 1 ∣ 1 ) symmetry.
Scalar products and norm of Bethe vectors for integrable models based on Uq(gl^n)U_q(\widehat{\mathfrak{gl}}_{n})Uq​(gl​n​)
We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra \uqgl{m} . We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients. In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant.