Functional relations and analytic Bethe ansatz for twisted quantum affine algebras

Atsuo Kuniba, Junji Suzuki

Published Aug 25, 1994• 29 citations

Abstract

Functional relations are proposed for transfer matrices of solvable vertex models associated with the twisted quantum affine algebras U_q(X^{(\\kappa)}_n) where X^{(\\kappa)}_n = A^{(2)}_n, D^{(2)}_n, E^{(2)}_6 and D^{(3)}_4. Their solutions are obtained for A^{(2)}_n and conjectured for D^{(3)}_4 in the dressed vacuum form in the analytic Bethe ansatz.

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Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted $$ {D}_3^{(2)} $$ D 3 2 algebra (or the $$ {D}_3^{(2)} $$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the $$ {D}_{n+1}^{(2)} $$ D n + 1 2 model and other high rank integrable models.

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Functional relations and analytic Bethe ansatz for twisted quantum affine algebras

Atsuo Kuniba, Junji Suzuki

Published Aug 25, 1994• 29 citations

OpenAlex

Abstract

Functional relations are proposed for transfer matrices of solvable vertex models associated with the twisted quantum affine algebras U_q(X^{(\kappa)}_n) where X^{(\kappa)}_n = A^{(2)}_n, D^{(2)}_n, E^{(2)}_6 and D^{(3)}_4. Their solutions are obtained for A^{(2)}_n and conjectured for D^{(3)}_4 in the dressed vacuum form in the analytic Bethe ansatz.

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Zengo Tsuboi

Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2], [3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2], [3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).

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G. K. Sampa, A. Lima-Santos

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Guang-Liang Li, Xiaotian Xu, Kun Hao +5 more

Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted ${D}_3^{(2)}$ D 3 2 algebra (or the ${D}_3^{(2)}$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the ${D}_{n+1}^{(2)}$ D n + 1 2 model and other high rank integrable models.Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted $$ {D}_3^{(2)} $$ D 3 2 algebra (or the $$ {D}_3^{(2)} $$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the $$ {D}_{n+1}^{(2)} $$ D n + 1 2 model and other high rank integrable models.

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Analytic Bethe ansatz and functional relations related to tensor-like representations of type-II Lie superalgebras B ( r | s ) and D ( r | s )

Zengo Tsuboi

An analytic Bethe ansatz is carried out related to tensor-like representations of the type-II Lie superalgebras B ( r | s ) = osp (2 r +1|2 s ) ( r 0 , s 1 ) and D ( r|s ) = osp (2 r|2s ) ( r 2 , s 1 ). We present eigenvalue formulae of transfer matrices in dressed vacuum forms (DVFs) labelled by Young (super) diagrams. A class of transfer matrix functional relations ( T -system) is discussed. In particular for the B (0| s ) = osp (1|2 s ) ( s 1 ) case, a complete set of functional relations is proposed by using duality among DVFs.An analytic Bethe ansatz is carried out related to tensor-like representations of the type-II Lie superalgebras B ( r | s ) = osp (2 r +1|2 s ) ( r 0 , s 1 ) and D ( r|s ) = osp (2 r|2s ) ( r 2 , s 1 ). We present eigenvalue formulae of transfer matrices in dressed vacuum forms (DVFs) labelled by Young (super) diagrams. A class of transfer matrix functional relations ( T -system) is discussed. In particular for the B (0| s ) = osp (1|2 s ) ( s 1 ) case, a complete set of functional relations is proposed by using duality among DVFs.

arXiv:0911.5393Journal of Physics A Mathematical and GeneralSep 28, 199920 citations

B_{n},

D_{n}

and

C_{n}

Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.

On Bethe Ansatz for a Supersymmetric Vertex Model with

\mathcal{U}_{\rm q}[{\rm osp}(2|2)^{(2)}]

The Algebraic Bethe ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra

\mathcal{U}_{q}[\mathrm{osp}(2|2)^{(2)}]

is presented in detail. The eigenvalues and eigenvectors of the row-to-row transfer matrix are calculated and the corresponding Bethe Ansatz equations are obtained and analyzed numerically.

Exact solution of the quantum integrable model associated with the twisted

{\mathrm{D}}_3^{(2)}

algebra

Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted

{D}_3^{(2)}

D 3 2 algebra (or the

{D}_3^{(2)}

D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the

{D}_{n+1}^{(2)}

D n + 1 2 model and other high rank integrable models.