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Symmetry Integrability and Geometry Methods and Applications

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

Dmitry Chernyak, Azat M. Gainutdinov, Jesper Lykke Jacobsen, Hubert Saleur, Institut de Physique Theorique, France

Published Jul 16, 2023• 2 citations
Publisher

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.

DOI
10.3842/sigma.2023.046

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Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via Uqsl2U_{\mathfrak{q}}\mathfrak{sl}_2Uq​sl2​ Symmetry
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 U(1)\mathsf{U}(1)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 Uqsl2U_{\mathfrak{q}}\mathfrak{sl}_2Uq​sl2​-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 Uqsl2U_{\mathfrak{q}}\mathfrak{sl}_2Uq​sl2​ 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.
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Uqsl2U_q\mathfrak{sl}_2Uq​sl2​-invariant non-compact boundary conditions for the XXZ spin chain
We introduce new Uqsl2U_q\mathfrak{sl}_2Uq​sl2​-invariant boundary conditions for the open XXZ spin chain. For generic values of qqq we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for q=eiπpq=e^{\frac{iπ}{p}}q=epiπ​ a 2p2p2p-th root of unity $ - $ to its ppp-dimensional analogue. Both cases are parametrised by a continuous "spin" α∈Cα\in\mathbb{C}α∈C. To motivate our construction, we first specialise to q=iq=iq=i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the (η,ξ)(η,ξ)(η,ξ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic qqq we investigate the algebraic properties of the underlying lattice algebras. We show that if qα∉±qZq^α\notin\pm q^{\mathbb{Z}}qα∈/±qZ, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to Uqsl2U_q\mathfrak{sl}_2Uq​sl2​. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of qqq and ααα. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.
Uqsl2 {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 Uq​sl2​-invariant non-compact boundary conditions for the XXZ spin chain
A bstract We introduce new Uqsl2 {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 Uq​sl2​ U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of q \mathfrak{q} q q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for q=eiπp \mathfrak{q}={e}^{\frac{i\pi}{p}} q=epiπ​ q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to q \mathfrak{q} q q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if q \mathfrak{q} q q α ∉ ± q \mathfrak{q} q q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to Uqsl2 {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 Uq​sl2​ U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of q \mathfrak{q} q q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.
We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin- s representation of quantum-deformed s l ( 2 ) . We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.