On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

Abstract

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

New symmetries of $ {\mathfrak{gl}(N)}$ -invariant Bethe vectors

“…Our result allows us to obtain explicit formulas for the action of B g (u) onto the Bethe vectors using known action formulas of the operators T ij (u) [18]. This allowed us to prove the conjecture of [28] and show that it is valid only for special (symmetric) representations of the Yangian [27]. Concluding, we would like to mention that symmetries of the RT T -algebra, analogous to those considered in this paper, also exist for the RT T -relations associated to the U q ( gl n ) algebras and gl(m|n) superalgebras.…”

“…The new symmetry gives a description of the Bethe vector in terms of the entries of the monodromy matrix T ij (2.10). In paper [27], we have used already the symmetry of the Bethe vectors in the models with gl(3)-invariant R-matrix. In that paper the equivalence of the two representations was proved by the use of a recursion for the Bethe vectors.…”

Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame

“…Based on experimental evidence for small N and L, the authors of [7] conjectured the spectrum of B(u) and also conjectured that (1.11) are indeed transfer matrix eigenstates. The latter was proven in [10] for N = 3 and symmetric powers of the defining representation, however the proof is quite tedious and difficult for higher rank generalisations.…”

Introduction to the nested algebraic Bethe ansatz

“…Within this approach, the on-shell Bethe vectors have the form (1.8) provided the Bethe parameters satisfy Bethe equations. However, it was shown in [68] that if the Bethe parameters remain arbitrary complex numbers, then at least in the case of the R 3 algebra, the corresponding vectors coincide with the special case of off-shell Bethe vectors constructed within the NABA framework.…”