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arXiv:hep-th/0106139 · Letters in Mathematical Physics

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

D. Arnaudon, A. Sedrakyan, Tigran Sedrakyan, P. Sorba

Published Dec 1, 2001• 9 citations
arXivHTMLPDF
DOI
10.1023/a:1014504526934

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Generalization of the U\mathcal{U} U q (gl(N)) Algebra and Staggered Models
Abstract We consider quantum integrable models based on the Lie algebra gl ( n ) and non-skew-symmetric classical r -matrices associated with Z 2 -gradings of gl ( n ) of the following type: gl ( n ) = gl ( n ) 0 ¯ + gl ( n ) 1 ¯ , where gl ( n ) 0 ¯ = gl ( n 1 ) ⊕ gl ( n − n 1 ) . Among the considered models are Gaudin-type models with an external magnetic field, used in nuclear physics to produce proton–neutron Bardeen–Cooper–Schrieer-type models, n -level many-mode Jaynes–Cummings–Dicke-type models of quantum optics, matrix generalization of Bose–Hubbard dimers, etc. We diagonalize the constructed models by means of the ‘generalized’ nested Bethe ansatz.
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.