Current presentation for the super-Yangian double $ DY(\mathfrak{gl}(m\vert n))$ and Bethe vectors

Abstract

We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians Y (gl(m|n) in terms of the current generators of the Yangian double DY (gl(m|n)). We use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double DY (gl(m|n)) we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.

Why scalar products in the algebraic Bethe ansatz have determinant representation

“…The most tempting direction is the study of scalar products in the models with symmetries of higher rank. In these models, the Bethe vectors depend on several sets of variables, therefore, equation (2.5) is replaced by a more complex formula [32,33]. In particular, the result of the transfer matrix action on the off-shell Bethe vector contains terms in which the argument of the transfer matrix replaces several Bethe parameters.…”

Bethe Vectors for Orthogonal Integrable Models

“…, corresponding to the algebras of the classical B, C, and D-series. This result immediately opens a possibility to address the algebraic Bethe ansatz method to the models with so 2n+1 , sp 2n+2 , or so 2n+2 symmetries using the current approach [13].…”

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

“…However, in the general gl(N ) case, the form of the Bethe vectors is much more involved (see e.g. [19]). In the gl(N ) based models, an off-shell Bethe vector B(t) depends on N − 1 sets of complex numberst = {t 1 ,t 2 , .…”

“…Section 3 is devoted to the description of the properties of the Bethe vectors. The main results of our paper are given in section 4, where we use an identification of the Bethe vectors with certain combination of the generators of the Yangian double [19] to prove the claimed symmetry of the Bethe vectors. In section 5 we study symmetry properties of the scalar products of the Bethe vectors.…”