Scalar products and norm of Bethe vectors for integrable models based on $U_q(\widehat{\mathfrak{gl}}_{n})$

TL;DR

This paper derives recursion formulas and a sum formula for scalar products of Bethe vectors in models based on quantum affine algebra, revealing their norms as Gaudin determinants when on-shell.

Contribution

It introduces new recursion formulas and a sum formula for Bethe vector scalar products in $U_q( ext{gl}_n)$ models, including explicit expressions for their norms.

Findings

Scalar product expressed as a sum over partitions of Bethe parameters.

Recursion formulas for highest coefficients in scalar products.

On-shell Bethe vectors have norms given by Gaudin determinants.

Abstract

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients. In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant.