A new integral representation for the scalar products of Bethe states for the XXX spin chain

TL;DR

This paper introduces a novel integral representation for scalar products of Bethe states in the SU(2) XXX spin chain, utilizing separation of variables and contour integrals, which may aid in analyzing semi-classical limits.

Contribution

It develops a new integral formula for Bethe state scalar products in the XXX spin chain using separation of variables and boundary twists, enhancing analytical tools for integrable models.

Findings

Derived the integration measure and spectrum of separated variables.

Expressed the scalar product as a multiple contour integral involving Baxter functions.

Formulated an integral reminiscent of matrix model eigenvalue integrals.

Abstract

Based on the method of separation of variables due to Sklyanin, we construct a new integral representation for the scalar products of the Bethe states for the SU(2) XXX spin 1/2 chain obeying the periodic boundary condition. Due to the compactness of the symmetry group, a twist matrix must be introduced at the boundary in order to extract the separated variables properly. Then by deriving the integration measure and the spectrum of the separated variables, we express the inner product of an on-shell and an off-shell Bethe states in terms of a multiple contour integral involving a product of Baxter wave functions. Its form is reminiscent of the integral over the eigenvalues of a matrix model and is expected to be useful in studying the semi-classical limit of the product.