Bethe Vectors for Orthogonal Integrable Models

Abstract

We consider quantum integrable models associated with so 3 algebra. We describe Bethe vectors of these models in terms of the current generators of the DY (so 3 ) algebra. To implement this approach we use isomorphism between R-matrix and Drinfeld current realizations of the Yangians and their doubles for classical types B, C, and D series algebras. Using these results we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We show that these action formulas lead to recursions for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The action formulas can also be used for calculating the scalar products in the models associated with so 3 algebra. 1 a.

Why scalar products in the algebraic Bethe ansatz have determinant representation

“…A series of results suggests that it can be solved. 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].…”

Spinorial R operator and Algebraic Bethe Ansatz

“…In general by this relation the spinorial RT T algebra, the generators of which are contained in the matrix T , is mapped to the ordinary RT T algebra, the generators of which are matrix elements of T. Moreover, it provides a way to solve the spectral problem for the trace of the ordinary monodromy matrix by solving the spectral problem for a trace involving the spinorial monodromy matrix. The entries of the inverse-transposeT ij (u) = T −1 N +1−j,N +1−i (u) to the monodromy matrix T defined over the fundamental (vector) auxiliary space used in [10] is given by the quantum minors divided by the quantum determinant. In contrast, due to the inversion relation for L (4.4) the inverse of spinor monodromy matrix T (u) is given by the same matrix with the shifted spectral parameter.…”

“…The fusion relation (4.8) results in expressions of the 9 elements of the vector monodromy matrix T in terms of the 4 spinorial RT T generators in T (u) for all representations admitting this relation. The study of [10] resulted in relations among the matrix elements of T, in particular the three elements in the upper triagle are expressed in terms of one of them. This confirms that (4.8) establishes the equivalence of the spinorial so(3) type RT T algebra and the ordinary Yangian of sℓ(2) type.…”

“…In last years the Yangian algebras of the B, C, D types and the corresponding Algebraic Bethe Ansatz (ABA) attracted increasing interest, e.g. [5,6,7,8,9,10,11].…”

New compact construction of eigenstates for supersymmetric spin chains

“…When finishing the draft we learned about a work in progress [52] where related results were obtained independently.…”