© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2309.16660 · Nuclear Physics B

Folding QQ-relations and transfer matrix eigenvalues: Towards a unified approach to Bethe ansatz for super spin chains

Zengo Tsuboi

Published Jun 28, 2024• 5 citations

Abstract

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)).

DOI: 10.1016/j.nuclphysb.2024.116607

Related papers

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.

arXiv:0911.5393•Journal of Physics A Mathematical and General•Sep 28, 1999•20 citations

Generalized Perk–Schultz models: solutions of the Yang–Baxter equation associated with quantized orthosymplectic superalgebras

Maithili Mehta, K. A. Dancer, M. D. Gould +1 more

The Perk–Schultz model may be expressed in terms of the solution of the Yang–Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra U q [ gl ( m | n )], with a multiparametric coproduct action as given by Reshetikhin. Here, we present analogous explicit expressions for solutions of the Yang–Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras U q [ osp ( m | n )]. In this manner, we obtain generalizations of the Perk–Schultz model.

arXiv:nlin/0509019•Journal of Physics A Mathematical and General•Dec 7, 2005•2 citations

On the Exact Solution of Models based on Non-Standard Representations

J. Gruneberg

(revised version) The algebraic Bethe ansatz is a powerful method to diagonalize transfermatrices of statistical models derived from solutions of (graded) Yang Baxter equations, connected to fundamental representations of Lie (super-)algebras and their quantum deformations respectively. It is, however, very difficult to apply it to models based on higher dimensional representations of these algebras in auxiliary space, which are not of fusion type. A systematic approach to this problem is presented here. It is illustrated by the diagonalization of a transfer-matrix of a model based on the product of two different fourdimensional representations of Uq ( ̂ gl ′ (2,1; C)). I.

•Jan 1, 1999•4 citations

QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains

Gwenaël Ferrando, Rouven Frassek, Vladimir Kazakov

Abstract We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the Dr Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through r + 1 basic Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length.

arXiv:2008.04336•Journal of High Energy Physics•Feb 23, 2021•24 citations

Functional relations and analytic Bethe ansatz for twisted quantum affine algebras

Atsuo Kuniba, Junji Suzuki

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.

•Aug 25, 1994•29 citations

Boundary algebraic Bethe Ansatz for a nineteen vertex model with U q [osp(2|2) (2) ] symmetry

R S Vieira, A Lima-Santos

Abstract The boundary algebraic Bethe Ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra U q [ o s p ( 2 | 2 ) ( 2 ) ] ≃ U q [ C ( 2 ) ( 2 ) ] is presented. The eigenvalues and eigenvectors of Sklyanin’s transfer matrix, with diagonal reflection K -matrices, are calculated and the corresponding Bethe Ansatz equations are obtained. A numerical analysis for a small-length chain is also presented.

arXiv:1705.08953•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2017•1 citation

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)).

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.

The Perk–Schultz model may be expressed in terms of the solution of the Yang–Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra U q [ gl ( m | n )], with a multiparametric coproduct action as given by Reshetikhin. Here, we present analogous explicit expressions for solutions of the Yang–Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras U q [ osp ( m | n )]. In this manner, we obtain generalizations of the Perk–Schultz model.

(revised version) The algebraic Bethe ansatz is a powerful method to diagonalize transfermatrices of statistical models derived from solutions of (graded) Yang Baxter equations, connected to fundamental representations of Lie (super-)algebras and their quantum deformations respectively. It is, however, very difficult to apply it to models based on higher dimensional representations of these algebras in auxiliary space, which are not of fusion type. A systematic approach to this problem is presented here. It is illustrated by the diagonalization of a transfer-matrix of a model based on the product of two different fourdimensional representations of Uq ( ̂ gl ′ (2,1; C)). I.