Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

TL;DR

This paper develops an algebraic Bethe ansatz approach for the open XXZ spin chain with non-diagonal boundary terms, overcoming symmetry-breaking challenges by leveraging $U_q(sl_2)$ symmetry and Schur-Weyl duality.

Contribution

It introduces a novel algebraic construction using $U_q(sl_2)$-invariant spin chains with Verma modules to derive Bethe equations for non-diagonal boundary conditions.

Findings

Derived Bethe equations for the open XXZ chain with non-diagonal boundaries.

Established Schur-Weyl duality between $U_q(sl_2)$ and the two-boundary Temperley-Lieb algebra.

Provided algebraic interpretation of Nepomechie boundary condition via quantum group fusion rules.

Abstract

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.