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arXiv:1803.00103 · SciPost Physics Lecture Notes

Nested Algebraic Bethe Ansatz in integrable models: recent results

Stanislav Pakuliak, E. Ragoucy, Nikita A. Slavnov

Published Oct 29, 2018• 20 citations

Abstract

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.

DOI: 10.21468/scipostphyslectnotes.6

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Scalar products of Bethe vectors in models with

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symmetry 1. Super-analog of Reshetikhin formula

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.