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Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry: Algebraic Bethe ansatz for the open XXZ spin chain with non-diagonal boundary terms via \(U_{\mathfrak{q}}\mathfrak{sl}_2\) symmetry

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Název: Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry: Algebraic Bethe ansatz for the open XXZ spin chain with non-diagonal boundary terms via \(U_{\mathfrak{q}}\mathfrak{sl}_2\) symmetry
Autoři: Chernyak, Dmitry, Gainutdinov, Azat M., Jacobsen, Jesper Lykke, Saleur, Hubert
Přispěvatelé: HEP, INSPIRE
Zdroj: Symmetry, Integrability and Geometry: Methods and Applications.
Publication Status: Preprint
Informace o vydavateli: SIGMA (Symmetry, Integrability and Geometry: Methods and Application), 2023.
Rok vydání: 2023
Témata: High Energy Physics - Theory, algebra: Temperley-Lieb, Yang-Baxter equations, non-diagonal K-matrices, quantum group: SL(2), FOS: Physical sciences, Groups and algebras in quantum theory and relations with integrable systems, Fusion categories, modular tensor categories, modular functors, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, 01 natural sciences, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Exactly and quasi-solvable systems arising in quantum theory, Generalized knots (virtual knots, welded knots, quandles, etc.), Temperley-Lieb algebras, Mathematics - Quantum Algebra, 0103 physical sciences, FOS: Mathematics, Quantum Algebra (math.QA), XXZ model, 0101 mathematics, algebra: lattice, Quantum groups and related algebraic methods applied to problems in quantum theory, Exactly solvable models, Bethe ansatz, Condensed Matter - Statistical Mechanics, Mathematical Physics, quantum integrable models, Statistical Mechanics (cond-mat.stat-mech), Symmetry breaking in quantum theory, Verma modules, Verma module, symmetry: U(1), Mathematical Physics (math-ph), [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], boundary condition, High Energy Physics - Theory (hep-th), [PHYS.HTHE] Physics [physics]/High Energy Physics - Theory [hep-th], [PHYS.COND] Physics [physics]/Condensed Matter [cond-mat]
Popis: 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.
Druh dokumentu: Article
Popis souboru: application/xml; application/pdf
ISSN: 1815-0659
DOI: 10.3842/sigma.2023.046
DOI: 10.48550/arxiv.2212.09696
Přístupová URL adresa: http://arxiv.org/abs/2212.09696
Rights: CC BY SA
Přístupové číslo: edsair.doi.dedup.....4d86b3adce1da92ef632eb15a2626732
Databáze: OpenAIRE
Popis
Abstrakt: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.
ISSN:18150659
DOI:10.3842/sigma.2023.046