The algebraic Bethe ansatz for scalar products inSU(3)-invariant integrable models

Abstract

We study SU (3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU (3)-invariant Heisenberg chain.

Why scalar products in the algebraic Bethe ansatz have determinant representation

“…In particular, the transfer matrix action in models with so(3)-symmetry contains similar contributions, however, a determinant representation for the scalar product of on-shell and off-shell vectors is known [34]. It is also worth mentioning that determinant formulas are known for some particular cases of the scalar products in models with gl(3) and gl(2|1) symmetry algebras [35][36][37]. This gives a hope that there is some generalization of our method to the models with higher symmetry rank.…”

Variations on Slavnov’s scalar product

“…No tractable expression, such as a determinant, is known for the corresponding objects in models based on higher rank algebras. For recent progress on this topic, see [16,17].…”

Scalar products and norm of Bethe vectors for integrable models based on $U_q(\widehat{\mathfrak{gl}}_{n})$

“…The calculation of scalar product and form factors have been addressed for some specific algebras. The case of the Y (gl 3 ) algebra has been studied in a series of works presenting some explicit forms of Bethe vectors [38], the calculation of their scalar product [39][40][41][42][43] and the expression of the form factors as determinants [44,45]. Results for models based on the deformed version U q ( gl 3 ) have been also obtained: explicit forms of Bethe vectors can be found in [46], their scalar products in [47][48][49] and a determinant expression for scalar products and form factors of diagonal elements was presented in [50].…”