Bethe Vectors in Quantum Integrable Models with Classical Symmetries

TL;DR

This work develops a unified, RTT-based definition of off-shell Bethe vectors for classical Yangians associated with $rak{gl}_n$, $rak{o}_{2n+1}$, $rak{sp}_{2n}$, and $rak{o}_{2n}$, proving they become on-shell under the Bethe equations. The authors show that core BV properties—action formulas of monodromy entries, rectangular recurrence relations, and coproduct structures—arise directly from the definition, without invoking Drinfeld’s current realization. They provide explicit eigenvalue formulas for on-shell BV and extend the BV construction to all four classical series via two discrete parameters that encode algebraic differences. The resulting framework is universal and model-independent (up to highest-weight representations), enabling systematic derivations of BV, their recurrences, and coproducts, with potential applications to form factors, scalar products, and higher-rank integrable systems across both finite and (suggested) affine settings.

Abstract

The first goal of this paper is to give a precise and simple definition for off-shell Bethe vectors in a generic $g$-invariant integrable model for $g=gl_n$, $o_{2n+1}$, $sp_{2n}$ and $o_{2n}$. We prove from our definition that the off-shell Bethe vectors indeed become on-shell when the Bethe equations are obeyed. Then, we show that some properties for these off-shell Bethe vectors, such as the action formulas of monodromy entries on these vectors, their rectangular recurrence relations and their coproduct formula, are a consequence of our definition.

Theorems & Definitions (25)

• Definition 2.1
• Remark 2.1
• Proposition 2.1
• Proposition 2.2
• Corollary 2.1
• Proposition 2.3
• Proposition 2.4
• Definition 4.1
• Proposition 4.1
• Proposition 4.2