[Paper Review] Algebraic Bethe ansatz for $\mathfrak{o}_{2n+1}$-invariant integrable models

A. Liashyk, S. Pakuliak|arXiv (Cornell University)|Aug 9, 2020

Algebraic structures and combinatorial models27 references7 citations

TL;DR

This paper develops the algebraic Bethe ansatz for $\mathfrak{o}_{2n+1}$-invariant quantum integrable models using the Drinfeld currents of the double Yangian $DY(\mathfrak{o}_{2n+1})$. It constructs off-shell Bethe vectors via current projections and derives explicit action formulas for monodromy matrix entries on these vectors, enabling scalar product analysis. The key result is a closed-form expression for the action of $T^{+}_{-n,n}(z)$ on off-shell Bethe vectors in terms of shifted Bethe parameters and structure functions.

ABSTRACT

A class of $\mathfrak{o}_{2n+1}$-invariant quantum integrable models is investigated in the framework of algebraic Bethe ansatz method. A construction of the $\mathfrak{o}_{2n+1}$-invariant Bethe vector is proposed in terms of the Drinfeld currents for the double of Yangian $\mathcal{D}Y(\mathfrak{o}_{2n + 1})$. Action of the monodromy matrix entries onto off-shell Bethe vectors for these models is calculated. Recursion relations for these vectors were obtained. The action formulas can be used to investigate structure of the scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant models.

Motivation & Objective

• To extend the algebraic Bethe ansatz to $\mathfrak{o}_{2n+1}$-invariant integrable models beyond the $\mathfrak{sl}_3$ case.
• To construct off-shell Bethe vectors in terms of Drinfeld currents of the double Yangian $DY(\mathfrak{o}_{2n+1})$.
• To derive explicit action formulas for monodromy matrix entries on these off-shell Bethe vectors.
• To provide a foundation for studying scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant models.

Proposed method

• Uses the Drinfeld current realization of the double Yangian $DY(\mathfrak{o}_{2n+1})$ to define generating series for algebra generators.
• Constructs off-shell Bethe vectors as projections of ordered products of currents using Gaussian coordinates.
• Applies the projection method to compute matrix element actions, leveraging the Gauss decomposition of the monodromy matrix.
• Derives action formulas by reordering current products using commutation relations, particularly for $T^{+}_{-n,n}(z)$.
• Introduces shifted spectral parameters $\tilde{z}_i = z_i - c(i - 1/2)$ to handle non-trivial current commutation relations.
• Uses the unitarity and Yang-Baxter relations of the $\mathfrak{o}_{2n+1}$-invariant R-matrix to verify consistency of derived formulas.

Experimental results

Research questions

• RQ1 How can off-shell Bethe vectors be systematically constructed for $\mathfrak{o}_{2n+1}$-invariant integrable models using the current realization of $DY(\mathfrak{o}_{2n+1})$?
• RQ2 What is the explicit action of monodromy matrix entries, particularly $T^{+}_{-n,n}(z)$, on these off-shell Bethe vectors?
• RQ3 How do the action formulas for $\mathfrak{o}_{2n+1}$-invariant models reduce to known results in simpler cases like $\mathfrak{o}_3$?
• RQ4 Can the derived action formulas be used to study scalar products of Bethe vectors in $\mathfrak{o}_{2n+1}$-invariant models?

Key findings

• The action of $T^{+}_{-n,n}(z)$ on an off-shell Bethe vector $B(\bar{t})$ is given by $T^{+}_{-n,n}(z) \cdot B(\bar{t}) = -\kappa \, g(z_1, \bar{t}_0) \, h(z, \bar{t}_{n-1}) / (h(z, \bar{t}_0) \, g(z_n, \bar{t}_{n-1})) \cdot \lambda_n(z) \cdot B(\bar{w})$, where $\bar{w} = \{\bar{t}_0, z, \tilde{z}_0, \dots, \bar{t}_{n-1}, \tilde{z}_{n-1}\}$.
• The formula reduces correctly to the known $\mathfrak{o}_3$ case when $n=1$, confirming consistency with prior work.
• The derivation relies on a non-trivial reordering of current products using the commutation relations $ (t - t' - c) F_i(t) F_{i+1}(t') = (t - t') F_{i+1}(t') F_i(t) $ for $0 \leq i \leq n-2$.
• The action is non-singular upon setting $z_\ell = z_{\ell'} = z$, despite apparent poles, due to cancellation in structure functions.
• The result is expressed in terms of rational functions $f(u,v)$, $g(u,v)$, and $h(u,v)$, which encode the R-matrix structure and spectral parameter dependence.
• The method establishes a framework for computing scalar products in $\mathfrak{o}_{2n+1}$-invariant models by providing the action of lowering operators on Bethe vectors.