# Inner products in integrable Richardson-Gaudin models

## TL;DR

This paper derives determinant formulas for inner products of eigenstates in Richardson-Gaudin models, linking Bethe rapidities and conserved charge eigenvalues, and connects different integrable model approaches.

## Contribution

It introduces two classes of Gaudin-like determinant expressions for inner products, bridging Bethe rapidities and conserved charge eigenvalues in integrable models.

## Key findings

- Derived determinant formulas for inner products.
- Connected Bethe rapidities with conserved charge eigenvalues.
- Expressed inner products as domain wall boundary partition functions.

## Abstract

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. This framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.