[Paper Review] Actions of the monodromy matrix elements onto $\mathfrak{gl}(m|n)$-invariant Bethe vectors

Arthur Hutsalyuk, A. Liashyk+3 more|arXiv (Cornell University)|May 19, 2020

Algebraic structures and combinatorial models19 references4 citations

TL;DR

This paper develops a novel zero modes method to compute multiple actions of monodromy matrix elements onto off-shell $χ\mathfrak{gl}(m|n)$-invariant Bethe vectors in quantum integrable models. By leveraging commutation relations involving $T_{i+1,i}[0]$ zero modes and the $T_{1,N+1}(z)$ matrix element, the authors derive explicit formulas for the action of lower-triangular elements, enabling recursion relations for scalar product highest coefficients. The method is rigorously proven for the non-supersymmetric $\mathfrak{gl}(m)$ case and extended to the supersymmetric $\mathfrak{gl}(m|n)$ case with minimal proof.

ABSTRACT

Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the $\mathfrak{gl}(m|n)$-invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the $\mathfrak{gl}(m)$ case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.

Motivation & Objective

• To develop an alternative, more efficient method for computing actions of monodromy matrix elements—especially lower-triangular ones—onto off-shell Bethe vectors in $χ\mathfrak{gl}(m|n)$-invariant quantum integrable models.
• To provide explicit recursion formulas for the highest coefficients in the scalar product of Bethe vectors, which are essential for computing correlation functions.
• To generalize the zero modes method, previously used for upper-triangular and diagonal elements, to the full monodromy matrix using only fundamental commutation relations and zero mode operators.
• To establish a systematic framework that bypasses the cumbersome projection method, particularly for lower-triangular element actions.

Proposed method

• The method relies on the commutation relation $[T_{i,j}(z), T_{π+1,π}[0]] = \delta_{i,π}\kappa_i T_{i+1,j}(z) - \delta_{π,j-1}\kappa_j T_{i,j-1}(z)$, derived from the RTT algebra, to recursively compute matrix element actions.
• It uses the action of $T_{1,N+1}(z)$ and zero mode operators $T_{i+1,i}[0]$ as initial data, obtained via the projection method, to build the full action formulas.
• The approach avoids direct computation via the projection method by focusing on algebraic recursion through commutation relations and normalization conditions.
• For the $χ\mathfrak{gl}(m)$ case, detailed proofs are provided using induction over the cardinality of the Bethe parameter sets.
• The method is extended to the $χ\mathfrak{gl}(m|n)$ case by analogy, with results stated without full proof, relying on structural similarity of the RTT algebra.
• The framework is validated by showing that the transfer matrix acts diagonally on Bethe vectors when the Bethe equations are satisfied, confirming their eigenvector property.

Experimental results

Research questions

• RQ1 How can the action of lower-triangular monodromy matrix elements on off-shell Bethe vectors be computed efficiently without relying on the full projection method?
• RQ2 What algebraic structure underlies the recursion relations for the highest coefficients in the scalar product of Bethe vectors?
• RQ3 Can the zero modes method be generalized to include all monodromy matrix elements, including lower-triangular ones, in $χ\mathfrak{gl}(m|n)$-invariant models?
• RQ4 How do the commutation relations between $T_{i,j}(z)$ and zero mode operators $T_{k+1,k}[0]$ facilitate the derivation of multiple action formulas?
• RQ5 What conditions ensure that unwanted terms in the transfer matrix action vanish, confirming the Bethe vector eigenvector property?

Key findings

• The zero modes method successfully computes multiple actions of monodromy matrix elements onto off-shell Bethe vectors using only the action of $T_{1,N+1}(z)$ and $T_{i+1,i}[0]$ zero modes, along with their commutation relations.
• For the $χ\mathfrak{gl}(m)$ case, the method yields explicit recursion formulas for the highest coefficients in the scalar product, proven via induction on the cardinality of the Bethe parameter sets.
• The action of the transfer matrix $\mathfrak{t}(z)$ on off-shell Bethe vectors produces a sum over partitions of the Bethe parameters, with the correct eigenvalue emerging only when the Bethe equations are satisfied.
• Unwanted terms in the transfer matrix action cancel out precisely when the Bethe equations hold, confirming that Bethe vectors become eigenvectors of the transfer matrix under these conditions.
• The method generalizes to the $χ\mathfrak{gl}(m|n)$ case, where similar recursion formulas for scalar product coefficients are expected, though proofs are omitted.
• The framework provides a systematic, algebraic alternative to the projection method for computing monodromy actions, particularly effective for lower-triangular elements.