A note on $$ \mathfrak{g}{\mathfrak{l}}_2 $$-invariant Bethe vectors

Abstract

We consider $$ \mathfrak{g}{\mathfrak{l}}_2 $$ g l 2 -invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries on the twisted off-shell Bethe vectors.

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

“…A general twist K gen can be treated as further twisting of the matrix T (u). It is quite natural to expect that the effect of the general twist must be similar to what one has in the case of gl 2 based models [38]. Namely, we saw that for the minimal twist, the multiple action of B g was equivalent to the one semi-on-shell Bethe vector B a,b (ū,v).…”

“…However, as soon as we impose Bethe equations, only one term in this linear combination should survive. The proof of this property in the case of gl 2 -invariant models is very simple (see [38]). However, a generalization of this proof to the models with gl 3 -invariant R-matrix meats certain technical difficulties.…”

Scalar Products in Twisted XXX Spin Chain. Determinant Representation

“…First, the setū is divided into three subsets {ū 0 ,ū 2 ,ū 3 } ū, and then the subsetū 0 is divided once more as {ū 1 ,ū 4 } ū 0 . Obviously, the last sum over partitions of the subsetū 0 gives (1 + β −1 ) q 0 (see [7]), and we arrive at…”

Modified Algebraic Bethe Ansatz: Twisted XXX Case

“…The new actions of the modified operators on the weight vectors deserve to be studied in details. They also appear in the context of the separation of variable by introduction of the B good operators [18] (see also recent work of two of the authors [8] that considers the gl 2 case from the point of view of the ABA). These actions provide formulas for the development of the modified Bethe vector in terms of the original t 12 (u) creation operator.…”