Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 046, 47 pages      arXiv:2212.09696

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

Dmitry Chernyak ab, Azat M. Gainutdinov c, Jesper Lykke Jacobsen abd and Hubert Saleur be
a) Laboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France
b) Institut de Physique Théorique, Paris Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France
c) Institut Denis Poisson, CNRS, Université de Tours, Parc de Grandmont, 37200 Tours, France
d) Sorbonne Université, École Normale Supérieure, CNRS, Laboratoire de Physique (LPENS), 75005 Paris, France
e) USC Physics and Astronomy Department, Los Angeles Ca 90089, USA

Received January 26, 2023, in final form July 04, 2023; Published online July 16, 2023

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.

Key words: quantum integrable models; non-diagonal K-matrices; Verma modules; Temperley-Lieb algebras.

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