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arXiv:hep-th/9303052 · Nuclear Physics B

Yang-Baxter symmetry in integrable models: new light from the Bethe ansatz solution

C. Destri, Hector J. de Vega

Published Oct 1, 1993• 14 citations
arXivHTMLPDF
DOI
10.1016/0550-3213(93)90002-7

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Solutions of the classical Yang–Baxter equation provide a systematic method to construct integrable quantum systems in an algebraic manner. A Lie algebra can be associated with any solution of the classical Yang–Baxter equation, from which commuting transfer matrices may be constructed. This procedure is reviewed, specifically for solutions without skew-symmetry. A particular solution with an exotic symmetry is identified, which is not obtained as a limiting expansion of the usual Yang–Baxter equation. This solution facilitates the construction of commuting transfer matrices which will be used to establish the integrability of a multi-species boson tunnelling model. The model generalises the well-known two-site Bose–Hubbard model, to which it reduces in the one-species limit. Due to the lack of an apparent reference state, application of the algebraic Bethe Ansatz to solve the model is prohibitive. Instead, the Bethe Ansatz solution is obtained by the use of operator identities and tensor product decompositions.
Recently, researchers have proposed the Asymmetric Bethe ansatz method - a theoretical tool that extends the scope of Bethe ansatz-solvable models by "breaking" partial mirror symmetry via the introduction of a fully reflecting boundary. Within this framework, the integrability conditions which were originally put forward by Gaudin have been further generalized. In this work, building on Gaudin's generalized kaleidoscope model, we present a detailed investigation of the relationship between DN symmetry and its integrability. We demonstrate that the mathematical essence of integrability in this class of models is characterized by a newly proposed Kaleidoscope Yang-Baxter Equation. Furthermore, we show that the solvability of a model via the coordinate Bethe ansatz depends not only on the consistency relations satisfied by scattering matrices, but also on the model's boundary conditions and the symmetry of the subspace where solutions are sought. Through finite element method based numerical studies, we further confirm that Bethe ansatz integrability arises in a specific symmetry sector. Finally, by analyzing the algebraic structure of the Kaleidoscope Yang-Baxter Equation, we derive a series of novel quantum algebraic identities within the framework of quantum torus algebra.
Recently, researchers have proposed the Asymmetric Bethe Ansatz method, a theoretical tool that extends the scope of Bethe Ansatz-solvable models by “breaking” partial mirror symmetry via the introduction of a fully reflecting boundary. Within this framework, the integrability conditions which were originally put forward by Gaudin have been further generalized. In this work, building on Gaudin’s generalized kaleidoscope model, we present a detailed investigation of the relationship between dihedral symmetry and its Bethe-type solvability. We identify a scattering-consistency relation, which we refer to as the Kaleidoscope Yang-Baxter Equation (KYBE), and use it to organize the Bethe-type construction in the present two-dimensional setting. Furthermore, we show that the solvability of a model via the coordinate Bethe Ansatz depends not only on the consistency relations satisfied by scattering matrices, but also on the model’s boundary conditions and the symmetry of the subspace where solutions are sought. Through finite-element-based numerical studies, we further confirm that Bethe Ansatz integrability arises in a specific symmetry sector. Finally, by analyzing the algebraic structure of the Kaleidoscope Yang-Baxter Equation, we derive a family of algebraic identities within the framework of quantum torus algebra.