Scalar products in generalized models with SU(3)-symmetry

TL;DR

This paper derives explicit determinant formulas for scalar products in SU(3)-invariant models, extending known SU(2) results and providing new tools for analyzing eigenvectors in integrable systems.

Contribution

It evaluates a partition function related to scalar products in SU(3) models and derives a new determinant expression for off-shell scalar products when one set of variables tends to infinity.

Findings

Explicit determinant formulas for scalar products in SU(3) models.

Reduction of partition functions to domain wall partition functions.

Extension of SU(2) scalar product results to SU(3) case.

Abstract

We consider a generalized model with SU(3)-invariant R-matrix, and review the nested Bethe Ansatz for constructing eigenvectors of the transfer matrix. A sum formula for the scalar product between generic Bethe vectors, originally obtained by Reshetikhin [11], is discussed. This formula depends on a certain partition function Z(\{\lambda\},\{\mu\}|\{w\},\{v\}), which we evaluate explicitly. In the limit when the variables \{\mu\} or \{v\} approach infinity, this object reduces to the domain wall partition function of the six-vertex model Z(\{\lambda\}|\{w\}). Using this fact, we obtain a new expression for the off-shell scalar product (between a generic Bethe vector and a Bethe eigenvector), in the case when one set of Bethe variables tends to infinity. The expression obtained is a product of determinants, one of which is the Slavnov determinant from SU(2) theory. It extends a result of Caetano [13].