New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains

TL;DR

This paper introduces a novel method for constructing eigenstates and separating variables in SU(N) quantum spin chains, providing a compact alternative to the nested algebraic Bethe ansatz and generalizing Sklyanin's approach.

Contribution

The authors propose a new construction of eigenstates using a single operator Bgood(u), simplifying the algebraic Bethe ansatz for SU(N) chains and extending the separation of variables framework.

Findings

Conjecture of a new eigenstate construction verified in several cases

Identification of roots of Bgood(u) as separated variables

Evidence supporting the method's validity for rational spin chains

Abstract

We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator Bgood(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin's approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.