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arXiv:1605.09189

Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formula

Arthur Hutsalyuk, Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, E. Ragoucy, N. A. Slavnov

Published May 30, 2016• 2 citations
arXivHTMLPDF

Abstract

We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(2|1)$ 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 $\mathfrak{gl}(1|1)$ symmetry.

DOI
10.48550/arxiv.1605.09189

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We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing gl(2∣1)\mathfrak{gl}(2|1)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)\mathfrak{gl}(1|1)gl(1∣1) symmetry.
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.
We review the recent results we have obtained in the framework of algebraic Bethe ansatz based on algebras and superalgebras of rank greater than 1 or on their quantum deformation. We present different expressions (explicit, recursive or using the current realization of the algebra) for the Bethe vectors. Then, we provide a general expression (as sum over partitions) for their scalar products. For some particular cases (in the case of gl(3) g l ( 3 ) or its quantum deformation, or of gl(2|1) g l ( 2 | 1 ) ), we provide determinant expressions for the scalar products. We also compute the form factors of the monodromy matrix entries, and give some general methods to relate them. A coproduct formula for Bethe vectors allows to get the form factors of composite models.