# Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

## TL;DR

This paper introduces a unified algebraic framework and quadratic Bethe equations for solving rational Gaudin magnets in arbitrary magnetic fields, simplifying the eigenstate construction without rotating the spin basis.

## Contribution

It presents a novel implementation of the quantum inverse scattering method that uses a common unrotated reference state for any magnetic field orientation, enabling straightforward determinant expressions.

## Key findings

- Unified framework for Gaudin magnets in arbitrary fields
- Simplified eigenstate construction via algebraic Bethe ansatz
- Determinant formulas for scalar products in the unrotated basis

## Abstract

In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach.