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

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.

References in corpus (6)

• Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
• Conformal boundary loop models
• The two-boundary Temperley-Lieb algebra
• Exact solution of the anisotropic special transition in the O(n) model in 2D
• Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain